THE 
 
 MECHANICAL PRINCIPLES 
 
 ENGINES KING 
 
 ARCHITECTURE. 
 
 HENRY MOSELET, I. A. F.R.S. 
 
 , 
 
 CHAPLAIN IN ORDINARY TO THE QUEEN, CANON OF BRISTOL, VICAR OF OLVESTON ; 
 
 CORRESPONDING MEMBER OF TUG INSTITUTE OF FRANCE, AND FORMERLY PROFESSOB 
 OF NATURAL PHILOSOPHY AND ASTRONOMY IN KING'S COLLEGE, LONDON. 
 
 Second American from Second London Edition 
 WITH ADDITIONS BY 
 
 D. H. M A H A N , LL.D. 
 
 U. S. MILITARY ACADEMY. 
 
 V.MTIJ ri, LUSTRATIONS ON WOOD. 
 
 NEW YORK : 
 
 JOHN WILEY & SON, 535 BROADWAY 
 1 
 
ENTERFO according *o Act of Congress, <jt\ the year 1856, Ij 1 
 WILEY & HA-LSTRD, 
 
 fc the Clerk's Office of the District Court of the United States, for the Southu a District 
 
 of New York. 
 
EDITOR'S PREFACE. 
 
 THE high place that Professor Moseley occupies in the 
 scientific world, as an original investigator, and the clear- 
 ness and elegance of the methods he has employed in this 
 work have made it a standard text book on the subjects it 
 treats of. In undertaking its revision for the press, at the 
 request of the publishers of this edition, it has been deemed 
 advisable, in view of the class of students into whose hands 
 it may fall, to make some slight addition to the original. 
 This has been done in the way of Notes thrown into an 
 Appendix, the matter of which has been gathered from 
 various authorities ; but chiefly from notes taken by the 
 editor, whilst a pupil at the French military school at Metz, 
 of lectures delivered by General Poncelet, at that time, 1829, 
 professor in that school. It is a source of great pleasure to 
 the editor to have this opportunity of publicly acknowledg- 
 ing his obligations to the teachings of this eminent savan, 
 who is distinguished not more for his high scientific attain- 
 ment, and the advancement he has given to mechanical 
 science, than for having brought these to minister to the 
 wants of the industrial classes, the intelligent success of 
 whose operations depends so much upon mechanical science, 
 by presenting it in a form to render it attainable by the most 
 ordinary capacities. 
 
 Hi 
 
iv EDITOR'S PREFACE. 
 
 The editor would remark that lie has carefully refrained 
 from making any alterations in the text revised, except cor- 
 rections of typographical errors, and in one instance where, 
 from a repetition of apparently one of these, he apprehended 
 some difficulty might be offered to the student if allowed 
 to remain exactly as printed in the original. 
 
 UNITED STATES MILITARY ACADEMY, 
 
 Went Point March 8, 1866. 
 
PKEFACE TO THE SECOND EDITION. 
 
 I HAVE added in this Edition articles : first, " On the 
 Dynamical Stability of Floating Bodies ;" secondly, " On 
 the Kolling of a Cylinder ;" thirdly, " On the descent of a 
 body upon an inclined plane, when subjected to variations of 
 temperature, which would otherwise rest upon it ;" fourthly, 
 u On the state bordering upon motion of a body moveable 
 about a cylindrical axis of finite dimensions, when acted 
 upon by any number of pressures." 
 
 The conditions of the dynamical stability of floating 
 bodies include those of the rolling and pitching motion of 
 ships. The discussion of the rolling motion of a cylinder 
 includes that of the rocking motion to which a locomotive 
 engine is subject, when its driving wheels are falsely 
 balanced, and that of the slip of the wheel due to the same 
 cause. The descent of a body upon an inclined plane 
 when subjected to variations in temperature, which other- 
 wise would rest upon it, appears to explain satisfactorily the 
 descent of glaciers. 
 
 The numerous corrections made in the text, I owe chiefly 
 to my old pupils at King's College, to whom the lectures 
 of which it contains the substance, were addressed. For 
 
VI PREFACE TO THE SECOND EDITION. 
 
 several important ones I am, however, indebted to Mr 
 Eobinson, Master of the School for Shipwrights' Apprentices, 
 in Her Majesty's Dockyard, Portsea ; to whom I have also to 
 express my warm acknowledgments for the care with 
 which he has corrected the proof sheets whilst going through 
 the press. 
 
 May, 1855 
 
PREFACE. 
 
 IN the following work, I have proposed to myself to apply: 
 the principles of mechanics to the discussion of the most 
 important and obvious of those questions which present 
 themselves in the practice of the engineer and the architect ; 
 and I have sought to include in that discussion all the 
 circumstances on which the practical solution of such ques- 
 tions may be assumed to depend. It includes the substance 
 of a course of lectures delivered, to > the students of King's 
 College in the department of engineering and architecture, 
 during the years 1840, 1841, 1842.* 
 
 In the first part I have treated of those portions of the 
 science of STATICS, which have their application in the theory 
 of machines and the theory of construction. 
 
 In the second, of the science of DYNAMICS, and, under this 
 head, particularly of that union of a continued pressure with 
 a continued motion which has received from English writers 
 the various names of "dynamical effect," "efficiency," "work 
 done," "labouring force," "work," &c. ; and "moment 
 d'activite"," "quantite d' action," "puissance mecanique," 
 " travail," from French writers. 
 
 Among the latter this variety of terms has at length given 
 place to the most intelligible and the simplest of them,, 
 
 * The first 170 pages of the work were printed for the use of my pupils in the- 
 year 1840. Copies of them were about the same time in the possession of 
 several of my friends in the Universities. 
 
Vlll PREFACE. 
 
 " travail." The English word " work " is the obvious trans- 
 lation of " travail," and the use of it appears to be recom- 
 mended by the same considerations. The work of overcoming 
 a pressure of one pound through a space of one foot has, in 
 this country, been taken as the unit, in terms of which any 
 other amount of work is estimated ; and in France, the work 
 of overcoming a pressure of one kilogramme through a space 
 of one metre. M. Dupiii has proposed the application of the 
 term dyname to this unit. 
 
 I have gladly sheltered myself from the charge of having 
 contributed to increase the vocabulary of scientific words, 
 by assuming the obvious term " unit of work " to represent 
 concisely and conveniently enough the idea which is attached 
 to it. 
 
 The work of any pressure operating through any space is 
 evidently measured in terms of such units, oy multiplying 
 the number of pounds in the pressure by the number of feet 
 in the space, if the direction of the pressure be continually 
 that in which the space is described. If not, it follows, by 
 a simple geometrical deduction, that it is measured by the 
 product of the number of pounds in the pressure, by the 
 number of feet in the projection of the space described,* 
 upon the direction of the pressure ; that is, by the product 
 of the pressure by its virtual velocity. Thus, then, we 
 conclude at once, by the principle of virtual velocities, that 
 if a machine work under a constant equilibrium of the 
 pressures applied to. it, or if it work uniformly, then is the 
 aggregate work of those pressures which tend to accelerate 
 its motion equal to the aggregate work of those which tend 
 to retard it ; and, by the principle of vis viva, that if the 
 machine do not work under an equilibrium of the forces 
 impressed upon it, then is the aggregate work of those which 
 tend to accelerate the motion of the machine greater or less 
 
 * If the direction of the pressure renfain always parallel to itself, the space 
 described may be any finite space ; if it do not, the space is understood to be 
 so small, that the direction of the pressure may be supposed to remain parallel 
 to itself whilst that space is described. 
 
PREFACE. IX 
 
 than the aggregate work of those which tend to retard its 
 motion by one half the aggregate of the vires vivce acquired 
 or lost by the moving parts of the system, whilst the work is 
 being done upon it. In no respect have the labours of the 
 illustrious president of the Academy of Sciences more con- 
 tributed to the development of the theory of machines than 
 in the application which he has so successfully made to it of 
 this principle of vis viva.* In the elementary discussion of 
 this principle, which is given by M. Poncelet, in the intro- 
 duction to his Mecanique Industrielle, he has revived the 
 term vis inertia (vis inertias, vis insita, Newton), and, 
 associating with it the definitive idea of a force of resistance 
 opposed to the acceleration or the retardation of a body's 
 motion, he has shown (Arts. 66. and 122.) the work expended 
 in overcoming this resistance through any space, to be 
 measured by one half the vis viva accumulated through the 
 space ; so that throwing into the consideration of the forces 
 under which a machine works, the vires inerticB of its moving 
 elements, and observing that one half of their aggregate vis 
 viva is equal to the aggregate work of their vires inertice, it 
 follows, by the principle of virtual velocities, that the differ- 
 ence between the aggregate work of those forces impressed 
 upon a machine, which tend to accelerate its motion, and 
 the aggregate work of those which tend to retard the motion, 
 is equal to the aggregate work of the vires inerticB of the 
 moving parts of the machine : under which form the prin- 
 ciple of vis viva resolves itself into the principle of virtual 
 velocities. So many difficulties, however, oppose themselves 
 to the introduction of the term vis inertice, associated with 
 the definitive idea of a force, into the discussion of questions 
 of mechanics, and especially of practical and elementary 
 mechanics, that I have thought it desirable to avoid it. It 
 is with this view that I have given a new interpretation to 
 that function of the velocity of a moving body which is 
 known as its vis viva. One half that function I have inter- 
 preted to represent the number of units of work accumulated 
 
 * See Poncelet, Mecanique Industrielle, troisieme partie. 
 
PREFACE. 
 
 in the body so long as its motion is continued. This number 
 of units of work it is capable of reproducing upon any resist- 
 ance opposed to its motion. A very simple investigation 
 (Art. 66.) establishes the truth of this interpretation, and 
 gives to the principle of vis viva the following more simple 
 enunciation : " The difference between the aggregate work 
 done upon the machine, during any time, by those forces 
 which tend to accelerate the motion, and the aggregate 
 work, during the same time, of those which tend to retard 
 the motion, is equal to the aggregate number of units of 
 work accumulated in the moving parts of the machine 
 during that time if the former aggregate exceed the latter, 
 and lost from them during that time if the former aggregate 
 fall short of the latter." Tims, then, if the aggregate work 
 of the forces which tend to accelerate the motion of a 
 machine exceeds that of the forces which tend to retard it, 
 then is the surplus work (that done upon the driving points, 
 above that expended upon the prejudicial resistances and 
 upon the working points) continually accumulated in the 
 moving elements of the machine, and their motion is thereby 
 continually accelerated. And if the former aggregate be 
 less than the latter, then is the deficiency supplied from the 
 work already accumulated in the moving elements, so that 
 their motion is in this case continually retarded. 
 
 The moving power divides itself whilst it operates in a 
 machine, first, into that which overcomes the prejudicial 
 resistances of the machine, or those which are opposed by 
 friction and other causes, uselessly absorbing the work in its 
 transmission. Secondly, into that which accelerates the 
 motion of the various moving parts of the machine, and which 
 accumulates in them so long as the work done by the moving 
 power upon it exceeds that expended upon the various 
 resistances opposed to the motion of the machine. Thirdly, 
 into that which overcomes the useful resistances, or those 
 which are opposed to the motion of the machine at the 
 working point, or points, by the useful work which is done 
 by it. 
 
PREFACE. XI 
 
 Between these three elements there obtains in every 
 machine a mathematical relation, which I have called its 
 MODULUS. The general form of this modulus I have discussed 
 in a memoir on the " Theory of Machines " published in the 
 Philosophical Transactions for the year 1841. The deter- 
 mination of the particular moduli of those elements of 
 machinery which are most commonly in use, is the subject 
 of the third part of the following work. From a combination 
 of the moduli of any such elements there results at once the 
 modulus of the machine compounded of them." 
 
 "When a machine has acquired a state of uniform motion, 
 work ceases to accumulate in its moving elements, and its 
 modulus assumes the form of a direct relation between the 
 work done by the motive power upon its driving point and 
 that yielded at its working points. I have determined by a 
 general method' 35 ' the modulus in this case, from that statical 
 relation between the driving and working pressures upon 
 the machine which obtains in the sfate bordering upon its 
 motion, and which may be deduced from the known condi- 
 tions of equilibrium and the established laws of friction. In 
 making this deduction I have, in every case, availed myself 
 of the following principle, first published in my paper on the 
 theory of the arch, read before the Cambridge Philosophical 
 Society in Dec. 1833, and printed in their Transactions of 
 the following year: "In the state bordering upon motion 
 of one body upon the surface of another, the resultant 
 pressure upon their common surface of contact is inclined 
 to the normal, at an angle whose tangent is equal to the 
 coefficient of friction." 
 
 This angle I have called the limiting angle of resistance. 
 Its values calculated, in respect to a great variety of surfaces 
 of contact, are given in a table at the conclusion of the 
 second part, from the admirable experiments of M. Morin,f 
 into the mechanical details of which precautions have been 
 introduced hitherto unknown to experiments of this class, 
 
 * Art. 152. See Phil. Trans., 1841, p. 290. 
 
 f Nouvelles Experiences sur le Frottement, Paris, 1833. 
 
Xll PEEFACE. 
 
 and which have given to our knowledge of the laws of 
 friction a precision and a certainty hitherto unhoped for. 
 
 Of the various elements of machinery those which rotate 
 about cylindrical axes are of the most frequent occurrence 
 and the most useful application; I have, therefore, in the 
 first place sought to establish the general relation of the 
 state bordering upon motion between the driving and the 
 working pressures upon such a machine, reference being 
 had to the weight of the machine.* This relation points out 
 the existence 'of a particular direction in which the driving 
 pressure should be applied to any such machine, that the 
 amount of work expended upon the friction of the axis may 
 be the least possible. This direction of the driving pressure 
 always presents itself on the same side of the axis with that 
 of the working pressure, and when the latter is vertical it 
 becomes parallel to it ; a principle of the economy of power 
 in machinery which has received its application in the 
 parallel motion of the marine engines known as the Gorgon 
 Engines. 
 
 I have devoted a considerable space in this portion of my 
 work to the determination of the modulus of a system of 
 toothed wheels ; this determination I have, moreover, 
 extended to bevil wheels, and have included in it, with the 
 influence of the friction of the teeth of the wheels, that of 
 their axes and their weights. An approximate form of this 
 modulus applies to any shape of the teeth under which they 
 may be made to work correctly ; and when in this approxi- 
 mate form of the modulus the terms which represent the 
 influence of the friction of the axis and the weight of the 
 wheel are neglected, it resolves itself into a well known 
 theorem of M. Poncelet, reproduced by M. ISTavier and the 
 Rev. Dr. Whewell.f In respect to wheels having epicy- 
 
 * In my memoir on the " Theory of Machines " (Phil. Trans. 1841), I have 
 extended this relation to the case in which the number of the pressures and 
 their directions are any whatever. The theorem which expresses it is given in 
 the Appendix of this work. 
 
 f In the discussion of the friction of the teeth of wheels, the direction of the 
 mutual pressures of the teeth is determined by a method first applied by me to 
 
PREFACE. xiij 
 
 cloidal and involute teeth, the modulus assumes a character 
 of mathematical exactitude and precision, and at once 
 establishes the conclusion (so often disputed) that the loss of 
 power is greater before the teeth pass the line of centres 
 than at corresponding points afterwards ; that the contact 
 should, nevertheless, in all cases take place partly before 
 and partly after the line of centres has been passed. In the 
 case of involute teeth, the proportion in which the arc of 
 contact should thus be divided by the line of centres is 
 determined by a simple formula ; as also are the best 
 dimensions of the base of the involute, with a view to the 
 most perfect economy of power in the working of the 
 wheels. 
 
 The greater portion of the discussions in the third part of 
 my work I believe to be new to science. In the fourth part 
 I have treated of " the theory of the stability of structures," 
 referring its conditions, so far as they are dependent upon 
 the rotation of the parts of a structure upon one another, to 
 the properties of a certain line which may be conceived to 
 traverse every structure, passing through those points in it 
 where its surfaces of contact are intersected by the resultant 
 pressures upon them. To this line, whose properties I first 
 discussed in a memoir upon " the Stability of a System of 
 Bodies in Contact," printed in the sixth volume of the Carrib. 
 Phil. Trans., I have given the name of the line of resist- 
 ance ; it differs essentially in its properties from a line 
 referred to by preceding writers under the name of the 
 curve of equilibrium or the line of pressure. 
 
 The distance of the line of resistance from the extrados of 
 a structure, at the point where it most nearly approaches it, 
 I have taken as a measure of the stability of a structure,* and 
 
 that purpose in a popular treatise, entitled Mechanics applied to the Arts, 
 published in 1834. 
 
 * This idea was suggested to me by a rule for the stability of revetement 
 walls attributed to Vauban, to the effect, that the resultant pressure should 
 intersect the base of such a wall at a point whose distance from its extrados is 
 iths the distance between the extrados at the base and the vertical through 
 the centre of gravity. 
 
X1T PREFACE. 
 
 have called it the modulus of stability; conceiving thia 
 measure of the stability to be of more obvious and easier 
 application than the coefficient of stability used by the 
 French writers. 
 
 That structure in respect to every independent element 
 of which the modulus of stability is the same, is evidently 
 the structure of the greatest stability having a given quantity 
 of material employed in its construction ; or of the greatest 
 economy of material having a given stability. 
 
 The application of these principles of construction to the 
 theory of piers, walls supported by counterforts and shores, 
 buttresses, walls supporting the thrust of roofs, and the 
 weights of the floors of dwellings, and Gothic structures, 
 has suggested to me a class of problems never, I believe, 
 before treated mathematically. 
 
 I have applied the well known principle of Coulomb to 
 the determination of the pressure of earth upon revetement 
 walls, and a modification of that principle, suggested by M. 
 Poncelet, to the determination of the resistance opposed to 
 the overthrow of a wall backed by earth. This determina- 
 tion has an obvious application to the theory of foundations. 
 
 In the application of the principle of Coulomb I have 
 availed myself, with great advantage, of the properties of 
 the limiting angle of resistance. All my results have thus 
 received a new and a simplified form. 
 
 The theory of the arch I have discussed upon principles 
 first laid down in my memoir on " the Theory of the Stability 
 of a System of Bodies in Contact," before referred to, and 
 subsequently in a memoir printed in the "Treatise on 
 Bridges" by Professor Hosking and Mr. Hann.* They 
 differ essentially from those on which the theory of Coulomb 
 is founded ;f when, nevertheless, applied to the case treated 
 
 * I have made extensive use of the memoir above referred to in the following 
 work, by the obliging permission of the publisher, Mr. Weale. 
 
 f The theory of Coulomb was unknown to me at the time of the publication 
 of my memoirs printed in the Camb. Phil. Trans. For a comparison of the 
 two methods see Mr. Hann's treatise. 
 
PKEFACE. XT 
 
 by the French mathematicians, they lead, to identical results, 
 I have inserted at the conclusion of my work the tables of 
 the thrust of circular arches, calculated by M. Garidel from 
 formulae founded on the theory of Coulomb. 
 
 The fifth part of the work treats of the "strength of 
 materials," and applies a new method to the determination 
 of the deflexion of a beam under given pressures. 
 
 In the case of a beam loaded uniformly over its whole 
 length, and supported at four different points, I have deter^ 
 mined the several pressures upon the points of support by a 
 method applied by M. Navier to a similar determination in 
 respect to a beam loaded at given points.* 
 
 In treating of rupture by elongation I have been led to a 
 discussion of the theory of the suspension bridge. This 
 question, so complicated when reference is had to the weight 
 of the roadway and the weights of the suspending rods, and : 
 when the suspending chains are assumed to tte of uniform 
 thickness, becomes comparatively easy when the section of 
 the chain is assumed so to vary its dimensions as to be every 
 where of the same strength. A suspension bridge thus 
 constructed is obviously that which, being of a given 
 strength, can be constructed with the least quantity of 
 materials ; or, which is of the greatest strength having a 
 given quantity of materials used in its construction.! 
 
 The theory of rupture by transverse strain has suggested 
 a new class of problems, having reference to the forms of 
 girders having wide flanges connected by slender ribs or by 
 open frame work : the consideration of their strongest forms 
 leads to results of practical importance. 
 
 In discussing the conditions of the strength of breast- 
 summers, my attention has been directed to the best positions 
 of the columns destined to support them, and to a comparison 
 
 * As in fig. p. 487. of the following work. 
 
 f That particular case of this problem, in which the weights of the suspending 
 rods are neglected, has been treated by Mr. Hodgkinson in the fourth vol. of 
 Manchester Transactions, with his usual ability. He has not, however, suc- 
 ceeded in effecting its complete solution. 
 
XVI PREFACE. 
 
 of the strength of a beam carrying a uniform load and sup- 
 ported freely at its extremities, with that of a beam similarly 
 loaded but having its extremities firmly imbedded in 
 masonry. 
 
 In treating of the strength of columns I have gladly 
 replaced the mathematical speculations upon this subject, 
 which are so obviously founded upon false data, by the 
 invaluable experimental results of Mr. E. Hodgkinson, 
 detailed in his well known paper in the Philosophical 
 Transactions for 1840. 
 
 The sixth and last part of my work treats on " impact ;" 
 and the Appendix includes, together with tables of the 
 mechanical properties of the materials of construction, the 
 angles of rupture and the thrusts of arches, and complete 
 elliptic functions, a demonstration of the admirable theorem 
 of M. Poncelet for determining an approximate value of the 
 square root of the sum or difference of two squares. 
 
 In respect to the following articles of my work I have tc 
 acknowledge my obligations to the work of M. Poncelet, 
 entitled Mecanique Industrielle. The mode of demonstration 
 is in some, perhaps, so far varied as that their origin might 
 with difficulty be traced ; the principle, however, of each 
 demonstration all that constitutes its novelty or its value 
 belongs to that distinguished author. 
 
 30,* 38, 40, 45, 46, 47, 52, 58, 62, 75, 108,f 123, 202, 
 267,t 268, 269, 270, 349, 354, 365. 
 
 * The enunciation only of this theorem is given in the Mec. Ind., 2me partie, 
 Art. 38. 
 
 f Some important elements of the demonstration of this theorem are taken 
 from the Mec. Ind., Art. 79. 2me partie. The principle of the demonstration 
 is not, however, the same as in that work. 
 
 \ In this and the three following articles I have developed the theory of the 
 9y-wheel, under a different form from that adopted by M. Poncelet (Mec. Ind., 
 Art. 56. 3me partie). The principle of the whole calculation is, however, 
 taken from his work. It probably constitutes one of the most valuable of hia 
 contributions to practical science. 
 
 The idea of determining the work necessary to produce a given deflection 
 of a beam from that expended the compression and the elongation of its com- 
 ponent fibres was suggested by an observation in the Mec. Ind., Art. 75. 3me 
 partie. 
 
CONTENTS. 
 
 STATICS. 
 
 1*1 
 
 The Parallelogram of Pressures ........ g 
 
 The Principle of the Equality of Moments ...... 6 
 
 The Polygon of Pressures ......... 10 
 
 The Parallelopipedon of Pressures ... ..... 14 
 
 Of Parallel Pressures .......... 16 
 
 The Centre of Gravity .......... 20 
 
 The Properties of Guldinus ......... 3$ 
 
 PART II. 
 
 DYNAMICS. 
 
 Motion .............. 47 
 
 Velocity ............. 48 
 
 WORK ............. 48 
 
 Work of Pressures applied in different Directions to a Body moveable 
 
 about a fixed Axis .......... 57 
 
 Accumulation of Work .......... 58 
 
 Angular Velocity ........... 65 
 
 The Moment of Inertia .......... 70 
 
 THE ACCELERATION OF MOTION BY GIVEN MOVING FORCES . . .79 
 
 The Descent of a Body upon a Curve ....... 83 
 
 The Simple Pendulum .......... 85 
 
 Impulsive Force ........... 86 
 
 The Parallelogram of Motion ......... 86 
 
 The Polygon of Motion .......... 88 
 
 The Principle of D'Alembert ......... 89 
 
 Motion of Translation .......... 90 
 
 Motion of Rotation about a fixed Axis ....... 91 
 
 The Centre of Percussion ....... . .96 
 
 The Centre of Oscillation .......... 96 
 
 Projectiles ............ 99 
 
 Centrifugal Force ........... 106 
 
 ft 
 
XV111 CONTENTS. 
 
 Page 
 
 The Principle of virtual Velocities 112 
 
 The Principle of Vis Viva 115 
 
 Dynamical Stability * 121 
 
 FRICTION 124 
 
 Summary of the Laws of Friction 130 
 
 The limiting Angle of Resistance 131 
 
 The Cone of Resistance 133 
 
 The two States bordering upon Motion 133 
 
 THE RIGIDITY OF CORDS . . . . 142 
 
 PART III. 
 
 THE THEORY OP MACHINES. 
 
 The Transmission of Work by Machines . 146 
 
 The Modulus of a Machine moving with a uniform or periodical Motion . 148 
 The Modulus of a Machine moving with an accelerated or a retarded 
 
 Motion 150 
 
 The Velocity of a Machine moving with a variable Motion . . . 151 
 To determine the Co-efficients of the Modulus of a Machine . . .153 
 General Condition of the State bordering upon Motion in a Body acted 
 upon by Pressures in the same Plane, and moveable about a cylindrical 
 
 Axis 154 
 
 The Wheel and Axle 155 
 
 The Pulley 160 
 
 System of one fixed and one moveable Pulley . . . . . .161 
 
 A System of one fixed and any Number of moveable Pulleys . . .163 
 
 A Tackle of any Number of Sheaves 166 
 
 The Modulus of a compound Machine 169 
 
 The Capstan 194 
 
 The Chinese Capstan 199 
 
 The Horse Capstan, or the Whim Gin 202 
 
 The Friction of Cords 207 
 
 The Friction Break 213 
 
 The Band 215 
 
 The modulus of the Band 217 
 
 The Teeth of Wheels 227 
 
 Involute Teeth 234 
 
 Epicycloidal and Hypocycloidal Teeth 236 
 
 To set out the Teeth of Wheels 239 
 
 A Train of Wheels 241 
 
 The Strength of Teeth 243 
 
 To describe Epicycloidal Teeth 245 
 
 To describe involute Teeth 251 
 
 The Teeth of a Rack and Pinion . . 253 
 
CONTENTS. XLX 
 
 r 
 
 Page 
 The Teefh of a Wheel working with a Lantern or Trundle . . . 25 r < 
 
 The driving and working Pressures on Spur Wheels 259 
 
 The Modulus of a System of two Spur Wheels . . . . . . 268 
 
 The Modulus of a Rack and Pinion 282 
 
 Conical or Bevil Wheels 284 
 
 The Modulus of a System of two Bevil Wheels 288 
 
 The Modulus of a Train of Wheels 301 
 
 The Train of least Resistance 310 
 
 The Inclined Plane 312 
 
 The Wedge driven by Pressure . 321 
 
 The Wedge driven by Impact 823 
 
 The mean Pressure of Impact 325 
 
 The Screw 326 
 
 Applications of the Screw 329 
 
 The Differential Screw 331 
 
 Hunter's Screw 332 
 
 The Theory of the Screw with a Square Thread in reference to the vari- 
 able Inclination of the Thread at different Distances from the Axis . 333 
 
 The Beam of the Steam Engine 337 
 
 The Crank 341 
 
 The Dead Points in the Crank 845 
 
 The Double Crank 346 
 
 The Crank Guide 351 
 
 The Fly-wheel 353 
 
 The Friction of the Fly-wheel 362 
 
 The Modulus of the Crank and Fly-wheel 363 
 
 The Governor . 364 
 
 The Carriage-wheel 368 
 
 On the State of the accelerated or retarded Motion of a Machine . .373 
 
 PART IV. 
 
 THE THEORY OF THE STABILITY OF STRUCTURES. 
 
 General Conditions of the Stability of a Structure of Uncemented Stones 87 7 
 
 The Line of Resistance . 371 
 
 The Line of Pressure 37;) 
 
 The Stability of a Solid Body 38 > 
 
 The Stability of a Structure 381 
 
 The Wall or Pier . . . 382 
 
 The Line of Resistance in a Pier 383 
 
 The Stability of a Wall supported by Shores 387 
 
 The Gothic Buttress 396 
 
 The Stability of Walls sustaining Roofs ....... 397 
 
 The Plate Band 402 
 
 The sloping Buttress 40J 
 
XX CONTENTS. 
 
 Page 
 The Stability of a Wall sustaining the Pressure of a Fluid . . .408 
 
 Earth Works 412 
 
 Revetement Walls 416 
 
 The Arch 429 
 
 The Angle of Rupture 437 
 
 The Line of Resistance in a circular arch whose Voussoirs are equal, and 
 whose Load is distributed over different Points of its Extrados . . 440 
 
 A segmental Arch whose Extrados is horizontal 441 
 
 A Gothic Arch, the Extrados of each Semi- Arch being a straight Line 
 inclined at any given Angle to the Horizon, and the Material of the 
 
 Loading different from that of the Arch 442 
 
 A circular Arch having equal Voussoirs and sustaining the Pressure of 
 
 Water 444 
 
 The Equilibrium of an Arch, the Contact of whose Voussoirs is geometri- 
 cally accurate 446 
 
 Applications of the Theory of the Arch 448 
 
 Tables of the Thrust of Arches .... .454 
 
 PART V. 
 
 THE STRENGTH OP MATERIALS. 
 
 Elasticity 458 
 
 Elongation $ 459 
 
 The Moduli of Resilience and Fragility 452 
 
 Deflection . 467 
 
 The Deflexion of Beams loaded uniformly .... .481 
 
 The Deflexion of Breast Summers . 486 
 
 Rupture 502 
 
 Tenacity 502 
 
 The Suspension Bridge 505 
 
 The Catenary 50g 
 
 The Suspension Bridge of greatest Strength 510 
 
 - Rupture by Compression . . . * 618 
 
 The Section of Rupture in a Beam ..... 520 
 
 General Conditions of the Rupture of a Beam 521 
 
 The Beam of greatest Strength t 527 
 
 The Strength of Breast Summers < 540 
 
 The best Positions of their Points of Support .... 542 
 
 Formulas representing the absolute Strength of a Cylindrical Column to 
 
 sustain a Pressure in the Direction of its Length 545 
 
 Torsion 
 
CONTENTS. 
 
 XXI 
 
 PART VI. 
 
 IMPACT. 
 
 Page 
 The Impact of two Bodies whose centres of Gravity move in the same 
 
 right Line 553 
 
 Greatest Compression of the Surface of the Bodies ... . 555 
 
 Velocity of two elastic Bodies after Impact ... . 556 
 
 The Pile Driver 534 
 
 ADDITIONS BY THE AMERICAN EDITOR .... .671 
 
 APPENDIX. 
 
 Note A 631 
 
 Note B. Poncelet's Theorems 632 
 
 Note C. On the Rolling of Ships 637 
 
 Note D 653 
 
 Note E. On the Rolling Motion of a Cylinder 655 
 
 Note F. On the Descent upon an Inclined Plane of a Body subject to 
 Variations of Temperature, and on the Motion of Glaciers . . . 675 
 
 Note G. The best Dimensions of a Buttress 683 
 
 Note H. Dimensions of the Teeth of Wheels 684 
 
 Note I. Experiments of M. Morin on the Traction of Carriages . . 685 
 
 N"ote K. On the Strength of Columns , 686 
 
 Table I. The Numerical Values of complete Elliptic Functions of the 
 first and second Orders for Values of the Modulus Jc corresponding to 
 
 each Degree of the Angle $in- l k 687 
 
 Table II. Showing the Angle of Rupture * of an Arch whose Loading 
 is of the same Material with its Voussoirs, and whose Extrados is 
 
 inclined at a given Angle to the Horizon 688 
 
 Table III. Showing the Horizontal Thrust of an Arch, the Radius of 
 whose Intrados is Unity, and the Weight of each Cubic Foot of its 
 
 Material and that of its Loading, Unity 691 
 
 Table IV. Mechanical Properties of the Materials of Construction . . 694 
 Table V. Useful Numbers . 698 
 
THE 
 
 MECHANICAL PRINCIPLES 
 
 or 
 
 CIVIL ENGINEERING. 
 
 PA.RT I. 
 
 STATICS, 
 
 1. FORCE is that which, tends to cause or to destroy 
 motion, or which actually causes or destroys- it. 
 
 The direction of a force is that straight line in which it 
 tends to cause motion in the point to which it is applied, or 
 in which it tends to destroy the motion in it.* 
 
 When more forces than one are applied to a body, and 
 their respective tendencies to communicate motion to it 
 counteract one another, so that the body remains at rest, 
 these forces are said to be in EQUILIBRIUM, and are called 
 PRESSURES. 
 
 It is found by experiment f that the effect of a pressure, 
 when applied to a solid body, is the same at whatever point 
 in the line of its direction it is applied ; so that the condi- 
 tions of the equilibrium of that pressure, in respect to other 
 pressures applied to the same body, are not altered, if, with 
 out altering the direction of the pressure, we remove its. 
 point of application, provided only the point to which we 
 remove it be in the straight line in the direction of which it 
 acts. 
 
 The science of STATICS is that which treats of the equili- 
 brium of pressures. When two pressures only are applied to 
 
 * Note (a) Ed. Appendix. f Note (6) Ed. Appendix. 
 
THE UNIT OF PKESSURE. 
 
 a body, and hold it at rest, it is found by experiment that 
 these pressures act in opposite directions, and have their 
 directions always in the same straight line. Two such pres- 
 sures are said to be equal. 
 
 If, instead of applying two pressures which are thus equal 
 in opposite directions, we apply them both in the same 
 direction, the single pressure which must be applied in a 
 direction opposite to the two to sustain them, is said to be 
 double of either of them. If we take a third pressure, equal 
 to either of the two first, and apply the three in the same 
 direction, the single pressure, which must be applied in a 
 direction opposite to the three to sustain them, is said to be 
 triple of either of them : and so of any number of pressures. 
 Thus, fixing upon any one pressure, and ascertaining how 
 many pressures equal to this are necessary, when applied in 
 an opposite direction, to sustain any other greater pressure, 
 we arrive at a true conception of the amount of that greater 
 pressure in terms of the first. 
 
 That single pressure, in terma of which the amount of any 
 other greater pressure is thus ascertained, is called an UNIT 
 of pressure. 
 
 Pressures, the amount of which are determined in terms 
 of some known unit of pressure, are said to be measured. 
 
 Different pressures, the amounts of which can be deter- 
 mined in terms of the same unit, are said to be commensur- 
 able. 
 
 The units of pressure which it is found most convenient to 
 use, are the weights of certain portions of matter, or the 
 pressures with which they tend towards the centre of the 
 earth. The units of pressure are different in different coun- 
 tries. With us, the unit of pressure from which all the rest 
 are derived is the weight of 22-S15* cubic inches of distilled 
 water. This w r eiglit is one pound troy ; being divided into 
 5760 equal parts, the weight of each is a grain troy, and 
 TOGO such grains constitute the pound avoirdupois. 
 
 If straight lines be taken in the directions of any number 
 of pressures, and have their lengths proportional to the 
 numbers of units in those pressures respectively, then these 
 lines having to one another the same proportion in length 
 that the pressures have in magnitude, and being moreover 
 draw^n in the directions in which those pressures respectively 
 act, are said to represent them in magnitude and direction. 
 
 * This standard was fixed by Act of Parliament, in 1824. The temperature 
 of the water is supposed to be 62 Fahrenheit, the weight to be taken in air, 
 and the barometer to stand at 30 inches. 
 
THE PARALLELOGRAM OF PRESSURES. 3 
 
 A system of pressures being in equilibrium, let any num- 
 ber of them be imagined to be taken away and replaced by 
 a single pressure, and let this single pressure be such that 
 the equilibrium which before existed may remain, then this 
 single pressure, producing the same effect in respect to the 
 equilibrium that the pressures which it replaces produced, is 
 said to be the RESULTANT. 
 
 The pressures which it replaces are said to be the COMPO- 
 NENTS of this single pressure ; and the act of replacing them 
 by such a single pressure, is called the COMPOSITION of 
 pressures. 
 
 If, a single pressure being removed from a system in equi- 
 librium, it be replaced by any number of other pressures, 
 such, that whatever effect was produced by that which they 
 replace singly, the same effect (in respect to the conditions of 
 the equilibrium) may be produced by those pressures con- 
 jointly, then is that single pressure said to have been RE- 
 SOLVED into these, and the act of making this substitution 
 of two or more pressures for one, is called the RESOLUTION 
 of pressures. 
 
 THE PARALLELOGRAM OF PRESSURES. 
 
 2. The resultant of any two pressures applied to a point, 
 is represented in direction by the diagonal of a paral- 
 lelogram, whose adjacent sides represent those pressures in 
 magnitude and direction* 
 
 (Duchayla's Method.f) 
 
 To the demonstration of this* proposition, after the excel- 
 lent method of Duchayla, it is necessary in the first place 
 to show, that if there be any two pressures P 2 and P 3 whose 
 directions are in the same straight line, and a third pressure 
 P x in any other direction, and if the proposition be true in 
 respect to Pj and P 2 , and also in respect to P 1 and P 3 , then 
 it will be true in respect to Pj and P 2 -f-P 3 
 
 Let P 15 P 2 , and P 3 , form part of any system of pressures in 
 
 ? ? equilibrium, and let them be applied to the point 
 
 ^;*C % ;Tr\ A; take AB and AC to represent, in magnitude 
 
 \ x \;V^v, and direction, the pressures Y l and P Q , and CD 
 
 >-."* fo Q p ressTire P 3? an d complete the parallelograms 
 
 CB and DF. Suppose the proposition to be true with regard 
 
 * This proposition constitutes the foundation of the entire science of Statics. 
 f Note (c) Ed. App. 
 
4: THE PAKALLELOGRAM 
 
 to P, and P 2 , tlie resultant of 'P l and P 2 will then be in the 
 direction of the diagonal AF of the parallelogram BO, whose 
 adjacent sides AC and AB represent P, and P 2 in magnitude 
 and direction. Let P, and P ? be replaced by this resultant. 
 It matters not to the equilibrium where in the line AF it is 
 applied ; let it then be applied at F. But thus applied at 
 F it may, without affecting the conditions of the equilibrium, 
 be in its turn replaced by (or resolved into) two other pressures 
 acting in OF and BF, and these will manifestly be equal to 
 P, and P 2 , of which P, may be transferred without altering 
 the conditions to 0, and P 2 to E. Let this be done, and let 
 P 3 be transferred from A to C, we shall then have Pj and 
 !P 3 acting in the directions CF and CD at C and P 2 , in the 
 direction FE at E, and the conditions of the equilibrium will 
 not have been affected by the transfer of them to these 
 points. .Now suppose that the proposition is also true in 
 respect to P x and P 3 as well as P x and P 2 . Then since CF 
 and CD represent t t l and P 3 in magnitude and direction, 
 therefore their resultant is in the direction of the diagonal 
 CE. Let them be replaced by this resultant, and let it be 
 transferred to E, and let it then be resolved into two other 
 pressures acting in the directions DE and FE ; these will 
 evidently be P a and P 8 . We have now then transferred all 
 the three pressures P 1? P 2 , P 3 , from A to E, an.d they act at E 
 in directions parallel to the directions in which they acted at 
 A, and this has been done without affecting the conditions of 
 the equilibrium ; or, in other words, it has been shown that 
 the pressures P 1? P 2 , P 3 , produce the same effect as it re- 
 spects the conditions of the equilibrium, whether they be 
 applied at A or E. The residtant of P 1? T 2 , P 3 , must there- 
 fore produce the same effect as it regards the conditions of 
 the equilibrium, whether it be applied at A or E. But in 
 order that this resultant may thus produce the same effect 
 when acting at A or E, it must act in the straight line AE, 
 because a pressure produces the same effect when applied at 
 two different points only when both those points are in the 
 line of its direction. On the supposition made, therefore, 
 the resultant of P,, P 2 , and P 3 , or of P, and P 2 + P 3 
 acts in the direction of the diagonal AE of the parallel- 
 ogram BD, whose adjacent sides AD and AB represent 
 P a + P 3 and P> in magnitude and direction ; and it has been 
 shown, that if the proposition be true in respect to P t and 
 P 2 , and also in respect to P, and P 3 , then it is true in respect 
 to Pj and P a + P 3 . Now this being the case for all values 
 f P P a , P 35 it is the case when P 1? P 2 , and P 8 , are equal 
 
OF PRESSURES* 5 
 
 to one another. But if P x be equal to P a their resultant 
 will manifestly have its direction as much towards one of 
 these pressures as the other ; that is, it will have its direc- 
 tion midway between them, and it will bisect the angle BAG : 
 but the diagonal AF in. this case also bisects the angle BAG, 
 since P, being equal to P 2 , AC is equal to AB ; so that in 
 this particular case the direction of the resultant is the 
 direction of the diagonal, and the proposition is true, and 
 similarly it is true of P t and P 3 , since these pressures are 
 equal. Since then it is true of P l and P 2 when they are 
 equal, and also of P x and P 3 , therefore it is true in this case 
 of P, and P 2 + P s , that is of P 1 and 2 P r And since it is 
 true of Pj and P 2 , and also of P 1 and 2 P n therefore it is true 
 of _P 1 and P 2 + 2 P 15 that is of P, and 3 P, ; and so of P, and 
 m P 15 if m be any w^hole number ; and similarly since it is 
 true of m P x and P 1? therefore it is true of m P a and 2 P,, &c., 
 and of m P t and n P 1 where n is any whole number. There- 
 fore the proposition is true of any two pressures raP x and 
 n P 1 which are commensurable. 
 
 It is moreover true when the pressures are in- 
 j,^.........^ cot)imensura ii e ^ y or i e t AC anc i AB represent 
 
 |:;V:~\y/r:-::i? anv two such pressures P! and P 2 in magnitude 
 and direction, and complete the parallelogram 
 ABDC, then will the direction of the resultant of P, and 
 P 2 be in AD ; for if not, let its direction be AE, and draw 
 EG parallel to CD. Divide AB into equal parts, each less 
 than GO, and set oif on AC parts equal to those from A 
 towards C. One of the divisions of these will manifestly 
 fall in GC. Let it be H, and complete the parallelogram 
 AHFB. Then the pressure P 2 being conceived to be 
 divided into as many equal units of pressure as there are 
 equal parts in the line AB, AH may be taken to represent a 
 pressure P 3 containing as many ot these units of pressure 
 as there are equal parts in AH, and these pressures P 2 and 
 P 3 will be commensurable, being measured in terms of the 
 same unit. Their resultant is therefore in the direction AF, 
 and this resultant of P 3 and P 2 has its direction nearer to 
 AC than the resultant AE of P, and P 2 has ; which is 
 absurd, since P a is greater than P 3 . 
 
 Therefore AE is not in the direction of the resultant ot 
 P and P a ; and in the same manner it may be shown that no 
 other than AD is in that direction. Therefore, &c. 
 
THE PRINCIPLES OF THE 
 
 3. The resultant of two pressures applied in any directions 
 to a point, is represented in magnitude as well as in direc- 
 twnoy the diagonal of the parallelogram whose adjacent 
 sides represent those pressures in magnitude and in direc- 
 tion. 
 
 Let BA and CA represent, in magnitude and 
 %., direction, any two pressures applied to the point 
 
 " A. Complete the parallelogram BC. Then by 
 
 the last proposition AD will represent the result- 
 ant of these pressures in direction. It will also 
 represent it in magnitude ; for, produce DA to G, and con- 
 ceive a pressure to be applied in GA equal to the resultant 
 of BA and CA, and opposite to it, and let this pressure be 
 represented in magnitude by the line GA. Then will the 
 pressures represented by the lines BA, CA, and GA, mani- 
 festly be pressures in equilibrium. Complete the parallelo- 
 gram BG r then is the resultant of GA and BA in the 
 direction FA; also since GA and BA are in equilibrium 
 w T ith CA, therefore this resultant is in equilibrium, with CA, 
 but when two pressures are in equilibrium, their directions 
 are in the same straight line ; therefore FAC is a straight 
 line. But AC is parallel to BD, therefore FA is parallel to 
 BD, and FB is, by construction, parallel to GD, therefore 
 AFBD is a parallelogram, and AD is equal to FB and 
 therefore to AG. But AG represents the resultant of CA 
 and BA in magnitude, AD therefore represents it in magni- 
 tude. Therefore, &c* 
 
 THE PRINCIPLE OF THE EQUALITY OF MOMENTS. 
 
 4. DEFINITION. If any number of pressures act in the 
 same plane, and any point be taken in that plane, and per- 
 pendiculars be drawn from it upon the directions of all these 
 pressures, produced if necessary, and if the number of units 
 in each pressure be then multiplied by the number of units 
 in the corresponding perpendicular, then this product is 
 called the moment of that pressure about the point from 
 which the perpendiculars are drawn, and these moments are 
 said to be measured from that point. 
 
 * Note (d) Ed. App. 
 
EQUALITY OF MOMENTS. 
 
 5. If three pressures be in equilibrium, and their moments 
 ~be taken about any point in the plane in which they act, 
 then the sum of the moments of those two pressures which 
 tend to turn the plane in one direction about the point 
 from which the moments are measured, is equal to the 
 moment of that pressure which tends to turn it in the 
 opposite direction. 
 
 . ..... *,c P 15 P a , P 3 , acting in the directions 
 
 P A. P A p 3 0, be any three pressures in 
 ~D,....-^>|--iB equilibrium. Take any point A in the plane 
 
 '*-"' in which they act, and measure their moments 
 
 from A, then will the sum of the moments of P 2 and P 8 , 
 which tend to turn the plane in one direction about A, equal 
 the moment of P 1? which tends to turn it in the opposite 
 direction. 
 
 Through A draw DAB parallel to OP 15 and produce OP, 
 to meet it in D. Take OD to represent P 3 , and take DB 
 such a length that OD may have the same proportion to 
 DB that P 2 has to P,. Complete the parallelogram ODBC, 
 then will OD and OC represent P 2 and P 1 in magnitude and 
 direction. Therefore OB will represent P 3 in magnitude 
 and direction. 
 
 Draw AM, AN, AL, perpendiculars on OC, OD, OB, 
 and join AO, AC. Now the triangle OBC is equal to the 
 triangle OAC, since these triangles are upon the same base 
 and between the same parallels. 
 
 Also, A ODA+ AOAB = AOBD = AOBC, 
 .-.A PDA -f AO AB= A OAC, 
 
 AN + P 3 x AL=Px AM. 
 
 Now Pj x AM, P 2 x AN, P 3 x AL, are the moments of P,, 
 P 2 , P 3 , about A (Art. 4.) 
 
 ..m t P 9 + m t P > = m t P 1 ...... (1). 
 
 Therefore, &c. &c. 
 
 6. If E be the resultant of P 2 and P^then since E is 
 equal to P 1 and acts in the same straight line, rr^E = mtPj, 
 
 The sum of the moments therefore, about any point, of 
 two pressures, P a and P 3 in the same plane, which tend to 
 
O THE PRINCIPLE OF THE 
 
 turn it in the same direction about that point, is equal to 
 the moment of their resultant about that point. 
 
 If they had tended to turn it in opposite directions, then 
 the difference of their moments would have equalled the 
 moment of their resultant. For let R be the resultant of 
 P t and P 3 , which tend to turn the plane in opposite direc- 
 tions about A, &c. Then is R equal to P 2 , and in the same 
 straight line with it, therefore moment R is equal to 
 moment P 2 . But by equation (1) m. t P 1 m'P,, = m^ ; 
 .-.mT, m t P 3 = m t R. 
 
 Generally, therefore, m* P, 4- m* P 2 = m 1 R (2), 
 
 the moment, therefore, of the resultant of any two pressures 
 in the same plane is equal to the sum or difference of the 
 moments of its components, according as they act to turn the 
 plane in the same direction about the point from which the 
 moments are measured, or in opposite directions.* 
 
 7. If any number of pressures in the same plane be in equi- 
 librium* and any point be taken, in that plane, from 
 which their moments are measured, then the sum of the 
 moments of those pressures which tend to turn the plane 
 in one direction about that point is equal to the sum of the 
 moments of those which tend to turn it in the opposite 
 direction. 
 
 Let P 15 P a , P 3 P, be any number of pressures in 
 
 the same plane which are in equi- 
 librium, and A any point in the 
 plane from which their moments 
 are measured, then will the .sum of 
 the moments of those pressures 
 which tend to turn the plane in one direction about A equal 
 the sum of the moments of those which tend to turn it in 
 the opposite direction. 
 
 Let R, be the resultant of P 1 and P 2 , 
 
 R 2 R, and P 3 , 
 
 R 3 R a and P 4 , 
 
 &c &c. 
 
 Therefore, by the last proposition, it being understood 
 that the moments of those of the pressures r 1? P 2 , which 
 ) to the left of A 
 
 * Note (c) Ed. App. 
 
 tend to turn the plane to the left of A, are to be taken nega- 
 tively, we have 
 
EQUALITY OF MOMENTS. 
 
 m* K, = m 4 P t + m* P 2 . 
 
 m* E 2 = m* E! + m* P,, 
 
 m 4 E 3 = m* K 2 + m' P 4 , 
 
 &c. = &c. &c. 
 
 m* E n _ = m* 
 
 Adding these equations together, and striking out the 
 terms common to both sides, we have 
 
 m* P, + in* P 2 4- m 1 P 3 -f ..... .+ m* P n 
 
 (3), where Rn_i is the resultant of all the pressures P 1? 
 
 But these pressures are in equilibrium ; they have, there- 
 fore, no resultant. 
 
 .-.Kn-i = .-. m'En-! = 0, 
 .-.m 4 P, + m* P 2 + m 1 P 3 , + ..... m* P n = . . . . (4). 
 
 Now, in this equation the moments of those pressures which 
 tend to turn the system to the left hand are to be taken 
 negatively. Moreover, the sum of the negative terms must 
 equal the sum of the positive terms, otherwise the whole 
 sum could not equal zero. It follows, therefore, that the 
 sum of the moments of those pressures which tend to turn 
 the system to the right must equal the sum of the moments 
 of those which tend to turn it to the left. Therefore, &c. &c. 
 
 8. If any number of pressures acting in the same plane fie in 
 equilibrium, and they be imagined to be moved parallel^ to 
 their existing directions, and all applied to the samepoint^ 
 so as all to act upon that point ^n directions parallel to 
 those in which they before acted upon different points, then 
 will they he in equilibrium about that point. 
 
 For (see the preceding figure) the pressure E, at whatever 
 point in its direction it be conceived to be applied, may be 
 resolved at that point into two pressures parallel and equal 
 to Pj and P 2 : similarly, E 2 may be resolved, at any point in 
 its direction, into two pressures parallel and equal to Ej and 
 P 3 , of which E x may be resolved into two, parallel and equal 
 to P, and P 2 , so that E 2 may be resolved at any point of its 
 direction into three pressures parallel and equal to P n P 2 , P 8 : 
 and, in like manner, E 3 may be resolved into two pressures 
 parallel and equal to E 2 and P 4 , and therefore into four pres- 
 sures parallel and equal to P 1? P 2 , P 3 , P 4 , and so of the rest. 
 
10 THE POLYGON 
 
 Therefore K_i may, at any point of its direction be resolved 
 
 into n pressures parallel and equal to P n P 2 , P 3 , P n ; 
 
 if, therefore, n such pressures were applied to that point, 
 they would just be held in equilibrium by a pressure equal 
 and opposite to R n _ i. But R_i = 0; these n pressures 
 would, therefore, be in equilibrium with one another if 
 applied to this point. 
 
 Now it is evident, that if, being thus applied to this point, 
 they would be in equilibrium, they would be in equilibrium 
 if similarly applied to any other point. Therefore, &c. 
 
 THE POLYGON OF PRESSURES. 
 
 9. The conditwns of the equilibrium of any number of pres- 
 sures applied to a point. 
 
 Let OP 15 OP 2 , OP 3 , &c., represent in mag- 
 nitude and direction pressures P x , P 2 , &c., 
 applied to the same point O. Complete the 
 parallelogram OPj AP 2 , and draw its diago- 
 nal OA ; then will OA represent in magni- 
 tude and direction the resultant of Pj and 
 P 2 . Complete the parallelogram OABP 3 , then will OB 
 represent in magnitude and direction the resultant of OA 
 and P 3 ; but OA is the resultant of P x and P 2 , therefore OB 
 is the resultant of P 1? P 2 , P 3 ; similarly, if the parallelogram 
 OBCP 4 be completed, its diagonal OC represents the result- 
 ant of OB and P 4 , that is, of P,, P 2 , P 3 , P 4 , and in like 
 manner OD, the diagonal of the parallelogram OCDP 5 , 
 represents the resultant of P,, P 2 , P 3 , P 4 , P B . 
 
 ISTow let it be observed, that AP X is equal and parallel to 
 OP 2 , AB to OP 3 , BO to OP 4 , CD to OP., so that P,A, AB, 
 BC, CD, represent P 2 , P 3 , P 4 , P 6 , respectively in magnitude, 
 and are parallel to their directions. Moreover, OP X is in the 
 direction of Pj and represents it in magnitude, so that the 
 sides OP,, P,A, AB, BC, CD, of the polygon OP 1? ABCDO 
 represent the pressures P a , P 2 , P 8 , P 4 , P 5 , respectively in 
 magnitude, and are parallel to their directions ; whilst the 
 side OD, which completes that polygon, represents the 
 resultant of those pressures in magnitude and direction. 
 
 If, therefore, the pressures P P 2 , P 8 , P 4 , P 5 , be in equili- 
 brium, so that they have no resultant, then the side OD of 
 the polygon must vanish, and the point D coincide with O. 
 Thus, then, if any number of pressures be applied to a point 
 
OF PRESSURES. H 
 
 and lines be drawn parallel to the directions of those pres- 
 sures, and representing them in magnitude, so as to form 
 sides of a polygon (care being taken to draw each line from 
 the point where it unites with the preceding, towards the 
 direction in which the corresponding pressure acts), then the 
 line thus drawn parallel to the last pressure, and representing 
 it in magnitude, will pass through the point from which the 
 polygon commenced, and will just complete it if the pres- 
 sures be in equilibrium ; and if they be not in equilibrium, 
 then this last line will not complete the polygon, and if a 
 line be drawn completing it, that line will represent the 
 resultant of all the pressures in magnitude and direction. 
 
 This principle is that of the POLYGON OF PRESSURES ; it 
 obtains in respect to pressures applied to the same point, 
 whether they be in the same plane or not. 
 
 10. If any number of pressures in the same plane "be in equi- 
 librium, and each be resolved in directions parallel to any 
 two rectangular axes, then the sum of all those resolved 
 pressures, whose tendency is to communicate motion in one 
 direction along either axis, is equal to the sum of those 
 whose tendency is in the opposite direction. 
 
 Let the polygon of pressures be formed in respect to any 
 number of pressures, P n P Q , P 3 , P 4 , in the same plane and in 
 equilibrium (Arts. 8, 9), and let the sides of 
 ^" s Ptyg n be projected on any straight line 
 Ax in the same plane. ]STow it is evident, 
 that the sum of the projections of those sides 
 & of the polygon which form that side of the 
 figure which is nearest to Ax, is equal to the sum of the pro- 
 jections of those sides which form the opposite side of the 
 polygon : moreover, that the former are those sides of the 
 polygon which represent pressures tending to communicate 
 motion from A towards x, or from left to right in respect to 
 the line Ax ; and the latter, those which tend to communi- 
 cate motion in the opposite direction. Now each projection 
 is equal to the corresponding side of the polygon, multiplied 
 by the cosine of its inclination to Ax. The sum of all those 
 sides of the polygon which represent pressures tending to 
 communicate motion from A towards x, multiplied each by 
 the cosine of its inclination to Ax, is equal, therefore, to the 
 sum of all the sides representing pressures whose tendency 
 is in the opposite direction, each being similarly multiplied 
 by the cosine of its inclination to Ax. Now the sides of the 
 
12 THE RESOLUTION 
 
 polygon represent the pressures in magnitude, and are 
 inclined at the same angles to Ax. Therefore, each pressure 
 being multiplied by the cosine of its inclination to Aa?, the 
 sum of all these products, in respect to those which tend to 
 communicate motion in one direction, equals the sum simi- 
 larly taken in respect to those which tend to communicate 
 motion in the opposite direction ; or, if in taking this sum it 
 be understood that each term into which there enters a pres- 
 sure, whose tendency is from A towards a?, is to be taken 
 positively, whilst each into which there enters a pressure 
 which tends from x towards A is to be taken negatively, 
 then the sum of all these terms will equal zero ; that is, 
 calling the inclinations of the directions of P 15 P 2 , P 3 . . . P 4 
 to Aa?, 1? 2 , 3 . . . . a n respectively, 
 
 P, cos. > -f P 2 cos. 2 + P 3 cos. 3 + + P n cos. a n =0 ... (5), 
 
 in which expression all those terms are to be taken nega- 
 tively which include pressures, whose tendency is from x 
 towards A. 
 
 This proposition being true in respect to any axis, Aa? is 
 true in respect to another axis, to which the inclinations of 
 the directions of the pressures are represented by /3 15 j3 2 , /3 3 , 
 P n , so that, 
 
 P, cos. ft + P 2 cos. ft + . . . . + P^cos. p n =0. 
 Let this second axis be at right angles to the first : 
 
 then ft = a x /. cos. ft = sin. o 1? j3 a = 2 , .% cos. ft 
 
 = sin. a a , &c. = &c. 
 
 /. P, sin. a x + P 2 sin. 2 + ....+ P w sin. a n = . . . . (6) ; 
 
 those terms in this equation, involving pressures which tend 
 to communicate motion in one direction, in respect to the 
 axis Ay being taken with the positive sign, and those which 
 tend in the opposite direction with the negative sign. 
 
 If the pressures P 15 P 2 , &c. be each of them resolved 
 into two others, one of which is parallel to the axis Aa?, and 
 the other to the axis Ay, it is evident that the pressures 
 thus resolved parallel to Aa?, will be represented by P, cos. a l5 
 P 2 cos. a,, &c., and those resolved parallel to Ay, by 
 P! sin. a, P 2 sin. a a , &c. Thus then it follows, that if 
 any system of pressures in equilibrium be thus resolved 
 parallel to two rectangular axes, the sum of those resolved 
 pressures, whose tendency is in one direction along either 
 
OF PRESSURES. 13 
 
 axis, is equal to the sum of those whose tendency is in the 
 opposite direction.* 
 
 This condition, and that pf the equality of moments, are 
 necessary to the equilibrium of any number of pressures in 
 the same plane, and they are together sufficient to that equi- 
 librium. 
 
 11. To determine the resultant of any number of pressures 
 in the same plane. 
 
 If the pressures Pj P a .... P w be not in 
 equilibrium, and have a resultant, then one 
 side is wanting to complete the polygon of 
 pressures, and that side represents the res- 
 ultant of all the pressures in magnitude, 
 and is parallel to its direction (Art. 9). 
 Moreover it is evident, that in this case the sum of the pro- 
 jections on Ax (Art. 10) of those lines which form one 
 side of the polygon, will be deficient of the sum of those of 
 the lines which form the other side of the polygon, by the 
 projection of this last deficient side ; and therefore, that the 
 sum of the resolved pressures acting in one direction along 
 the line A#, will be less than the sum of the resolved pres- 
 sures in the opposite direction, by the resolved part of the 
 resultant along this line. Now if R represent this resultant, 
 and 6 its inclination to AOJ, then R cos. is the resolved part 
 of R in the direction of A.X. Therefore the signs of the terms 
 being understood as before, we have 
 
 R cos. 0=P a cos. OJ + PS cos. +.... +P w cos. an . . (7). 
 And reasoning similarly in respect to the axis Ay, we have 
 R sin. 0=P 1 sin. a. + P, sin. a 2 + .... +P n sin. a n . . . (8). 
 
 Squaring these equations and adding them, and observing 
 that R 2 sin. 2 0-f R 2 cos. 2 R a (sin. 2 0+cos. a 0) =R 2 , we have 
 
 R 2 =(2P sin. ) 2 + (2P cos. a) 2 ........ (9), 
 
 where 2P sin. a is taken to represent the sum P, sin. ^ -r 
 P 2 sin. a a + P 3 sin. a s + &c., and 2p cos. a to represent the 
 sum P x cos. o^+P,, cos. 2 + P 3 cos. a 3 + &c. 
 Dividing equation (8) by equation (7), 
 
 r, 2P sin. a / im 
 
 tan. 0=-_ - ....... (10). 
 
 SP cos. a 
 
 Thus then by equation (9) the magnitude of the resultant 
 Note(/)Ed. App. 
 
14: TIIE PARALLELOPIPEDON 
 
 K is known, and by equation (10) its inclination 6 to the axis 
 Ax is known. In order completely to determine it, we have 
 yet to find the perpendicular distance at which it acts from 
 the given point A. For this w r e must have recourse to the 
 condition of the equality of moments (Art. Y). 
 
 If the sum of the moments of those of the pressures, P 1? 
 P a . . . . P, , which tend to turn the system in one direc- 
 tion about A, do not equal the sum of the moments of those 
 which tend to turn it the other way, then a pressure being 
 applied to the system, equal and opposite to the resultant R, 
 will bring about the equality of these two sums, so that the 
 moment of R must be equal to the difference of these sums. 
 Let then p equal the perpendicular distance of the direction 
 of E from A. Therefore 
 
 a-f .... +m t P w . . . (11), 
 
 in the second member of which equation the moments of 
 those pressures are to be taken negatively, which tend to 
 communicate motion round A towards the left. 
 Dividing both sides by E- we have 
 
 
 Thus then by equations (9), (10), (12), the magnitude of 
 the resultant R, its inclination to the given axis Aa?, and the 
 perpendicular distance of its direction from the point A, are 
 known; and thus the resultant pressure is completely deter- 
 mined in magnitude and direction. 
 
 THE PARALLELOPIPEDON OF PRESSURES. 
 
 12. Three pressures, P l? P Q , P 3 , being applied to the same 
 point A, in directions a?A, ^/A, 0A, which are not in the 
 same plane, it is required to determine their resultant. 
 
 Take the lines P, A, P 2 A, P 3 A, to represent the pressures 
 Pj, P 2 , P 3 , in magnitude and direction. 
 Complete the parallelopipedoii RP a P 8 P l5 
 of which APj, AP 2 , AP 3 , are adjacent edges, 
 and draw its diagonal RA ; then will RA 
 represent the resultant of P,, P 2 , P 3 , in 
 direction and magnitude. For since 
 PjSPjjA is a parallelogram, whose adjacent 
 sides Pj A, P 2 A, represent the presurea 
 P a and P a in magnitude and direction, therefore its diagonal 
 
OF THREE PRESSURES. 15 
 
 SA represents the resultant of these two pressures. And 
 similarly KA, the diagonal of the parallelogram KSAP 3 , re- 
 presents in magnitude and direction the resultant of SA and 
 P 8 , that is, of F 15 P 2 and P 3 , since SA is the resultant of 
 P, and P 9 . 
 
 It is evident that the fourth pressure necessary to produce 
 an equilibrium with P n P 9 , P 8 , being equal and opposite to 
 their resultant, is represented in magnitude and direction 
 by AR. 
 
 13. Three pressures, P,, P 2 , P 8 , "being in equilibrium, it is 
 required to determine the third P 3 in terms of the other 
 two, and their inclination to one another. 
 
 Let APj and AP a represent the pressures P l and P 2 in 
 magnitude and direction, and let the inclination 
 P, AP, of JP 1 to P 2 be represented by A- Com- 
 plete the parallelogram AP X RP 2 , and draw its 
 diagonal AR. Then does AR represent the 
 resultant of P, and P 2 in magnitude and direc- 
 tion. But this resultant is in equilibrium with P 3 , since P, 
 and P 2 are in equilibrium with P 3 . It acts, therefore, in the 
 same straight line with P 3 , but in an opposite direction, and 
 is equal to it. Since then AR represents this resultant in 
 magnitude and direction, therefore RA represents P 3 in mag- 
 nitude and direction. 
 
 Now, A]?=AP> 2AF X . PJK . cos. 
 also, AP.R^Tr ^AP^Tr A, P 1 R=AP,, and AP 15 AP 2) 
 AR, represent P,, P 2 , P 3 , in magnitude. 
 
 /. ?/ = ?,- 2?^. COB. (7T A) +P,-). 
 NOW COS. (7T A) = COS. A, .'. P.'zrP^ + SP,?, COS. A 
 
 (13). 
 
 14. If three pressures, P 1? P 2 , P 3 , he in equilibrium, any two 
 of them are to one, another inversely as the sines of their 
 inclinations to the third. 
 
 Let the inclination of Pj to P 3 be represented by A> an( i 
 that of P a to P, by A- 
 
 rrrTT P.AP.rrzTA, " SlU. P.ARrzrsin. & ' 
 
 P 1 RA=P,AR=^ P 2 AP 3 =rr A, /. sin. P 1 KA= S in. A- 
 
16 OF PARALLEL PRESSURES. 
 
 AF, AP, sin. P,KA 
 
 Also, 
 
 AP 2 ~~ P,R ~ sin. 
 
 sn. 
 
 That is, P, is to P 2 inversely, as the sine of the inclina- 
 tion of PJ to P 3 is to the sine of the inclination of P 2 to P 3 . 
 Therefore, &c. &c. [Q. E. D.] 
 
 OF PARALLEL PRESSURES. 
 
 15. The principle of the equality of moments obtains in 
 respect to pressures in the same plane whatever may be 
 their inclinations to one another, and therefore if their 
 inclinations be infinitely small, or if they he parallel. 
 
 In this case of parallel pressures, the same line AB, which 
 3 is drawn from a given point A, perpendicular 
 to one of these pressures, is also perpendicular 
 to all the rest, so that the perpendiculars are 
 here the parts of this line AM 1? AM 2 , &c. 
 intercepted between the point A and the direc- 
 tions of the pressures respectively. The principle is not how- 
 ever in this case true only in respect to the intercepted parts 
 of this perpendicular line AB, but in respect to the inter- 
 cepted parts of any line AC, drawn through the point A 
 across the directions of the pressures, since the intercepted 
 parts Am x , Am 2 , Am 3 , &c. of this second line are proportional 
 to those, AM 15 AM 2 , &c. of the first. 
 
 Thus taking the case represented in the figure, since by 
 the principle of the equality of moments we have, 
 
 AM, . P, + AM 4 . P 4 =AM 2 . 
 dividing both sides by AM 6 , 
 
 . - , 
 
 AM. ' r ' + AM 5 ' r *~~ AM 5 ' r '+ AM 5 ' 
 
 AM, Am, AM 2 Am 2 
 Butbysimilartriangles, = 
 
 ' Am. ' Am, ' 4 ~~Am B ' Am ' 
 Therefore multiplying by Am 6 , 
 
 Am, . P.+ Am t . P 4 =Am^ . P 3 + Am, . P 8 + Am~ . P 6 . 
 Therefore, &c. [Q.B.D.] 
 
OF PARALLEL PRESSURES. 17 
 
 16. To find the resultant of any number of parallel pressures 
 in the same plane. 
 
 It is evident that if a pressure equal and opposite to the 
 resultant were added to the system, the whole would be in 
 equilibrium. And being in equilibrium it has been shown 
 (Art. 8.), that if the pressures were all moved from their 
 present points of application, so as to remain parallel to their 
 existing directions, and applied to the same point, they are 
 such as would be in equilibrium about that point. ' But 
 being thus moved, these parallel pressures would all have 
 their directions in the same straight line. Acting therefore all 
 in the same straight line, and being in equilibrium, the sum 
 of those pressures whose tendency is in one direction along 
 that line must equal the sum of those whose tendency is in 
 the opposite direction. Now one of these sums incluaes the 
 resultant R. It is evident then that before R was introduced 
 the two sums must have been unequal, and that R equals, the 
 excess of the greater sum over the less ; and generally that if 
 2P represent the sum of any number of parallel pressures, . 
 those whose tendency is in one direction being taken with: 
 the positive sign, and those whose tendency is in the opposite 
 direction, with the negative sign ; then 
 
 R = 2P (15). 
 
 the sign of R indicating whether it act in the direction of 
 those pressures which are taken positively, or those which are 
 taken negatively. 
 
 Moreover since these pressures, including R, are in equi- 
 librium, therefore the sum of the moments about any point, 
 of those whose tendency is to communicate motion in one 
 direction, must equal the sum of the moments of the rest 
 these moments being measured on any line, as AC ; but one 
 of these sums includes the moment of R ; these 
 two sums must therefore, before the introduc- 
 tion of R, have been unequal, and the moment 
 of R must be equal to the excess of the greater 
 sum over the less, so that, representing the 
 sum of the moments of the pressures (R not being included) 
 by 2 m* P, those whose tendency is to communicate motion 
 in one direction, having the positive sign, and the rest the 
 negative ; and representing by x the ^ distance from A, mea- 
 sured along the line AC, at which R intersects that line, we 
 have, since xR is the moment of R, xR = 2 m l P, where the 
 
18 OF PARALLEL PKESSURES. 
 
 sign of a?R indicates the direction in which R tends to turn 
 the system about A, but E- = 2P, 
 
 . (16). 
 
 2P 
 
 Equations (15) and (16) determine completely the magni- 
 tude and the direction of the resultant of a system of parallel 
 pressures in the same plane. 
 
 IT. To determine tlw resultant of any number of parallel 
 pressures not in the same plane. 
 
 Let Pj 'and P 2 be the points of application of any two of 
 these pressures, and let the pressures themselves 
 be represented by P 1 and P 2 . Also let their 
 resultant Rj intersect the line joining the points 
 PI and P 2 in the point Rj ; produce the line 
 P 1? P a , to intersect any plane given in position, 
 in the point L. Through the points P 15 P 2 , and R 15 draw 
 P.Mj, P a M 2 , and Rj]^ perpendicularly to this plane: these 
 lines will be in the same plane with one another and with 
 Pj L ; let the intersection of this last mentioned plane with 
 the first be LM,, then will PjMj, P 2 M 2 , and R^ be per- 
 pendiculars to LM : ; moreover by the last proposition, 
 
 But by similar triangles, 
 
 LP_PM l LP 2 _P 2 1VI 2 
 
 Let now the resultant, R 2 , of R : and P s 
 intersect the line joining the points R, and 
 P 3 in the point R 2 , and similarly let the 
 resultant, R 3 , of R 2 and P 4 intersect the 
 me j mm g the points R 2 and P 4 in the 
 point Rg, and so on : then by the last equa- 
 tion. 
 
OF PARALLEL PRESSURES. 
 
 Similarly, E, . R + P 3 . P.M. = E 2 
 
 &c. + &c. = &c. 
 E 
 
 7i _ 2 
 
 Adding these equations, and striking out terms common to 
 both sides, 
 
 P, . P + P 2 P + . . . + P. ."P3L =R- 1 . 
 
 P 2 M 2 + ..... +P W . P n M n ; 
 
 . p N- -' > 
 
 in which expression those of the parallel pressures P 15 P 2 , 
 &c. which tend in one direction, are to be taken positively, 
 whilst those which tend in the opposite direction are to be 
 taken negatively. 
 
 The line E n _i N_i represents the perpendicular distance 
 from the given plane of a point through which the resultant 
 of all the pressures P 1? P 2 . . . . P n , passes. In the same 
 manner may be determined the distance of this point from 
 any other plane. Let this distance be thus determined in 
 respect to three given planes at right angles to one another. 
 Its actual position in space will then be known. Thus then 
 we shall know a point through which the resultant of all the 
 pressures passes, also the direction of that resultant, for it is 
 parallel to the common direction of all the pressures, and we 
 shall know its amount, for it is equal to the sum of all the 
 pressures with their proper signs. Thus then the resultant 
 pressure will be completely known. The point E^i is called 
 the CENTRE OF PARALLEL PRESSURES. 
 
 18. The product of any pressure by its perpendicular dis- 
 tance from a plane (or rather the product of the number of 
 units in the pressure by the number of units in the perpen- 
 dicular), is called the moment of the pressure, in respect to 
 that plane. Whence it follows from equation (17) that the 
 sum of the moments of any number of parallel pressures in 
 
20 THE CENTRE OF GRAVITY. 
 
 respect to a given plane is equal to the moment of their 
 resultant in respect to that plane. 
 
 19. It is evident, from equation (18), that the distance 
 ~N n i of the centre of pressure of any number of 
 parallel pressures from a given plane, is independent of the 
 directions of these parallel pressures, and is dependent 
 wholly upon their amounts and the perpendicular distances 
 PjM^ P 2 M 2 , &c. of their points of application from the 
 given plane. 
 
 So that if the directions of the pressures were changed, 
 provided that their amounts and points of application 
 remained the same, their centre of pressure, determined as 
 above, would remain unchanged; that is, the resultant, 
 although it would alter its direction with the directions of 
 the component pressures, would, nevertheless, always pass 
 through the same point. 
 
 The weights of any number of different bodies or different 
 parts of the same body, constitute a system of parallel pres- 
 sures ; the direction, therefore, through this system of the 
 resultant weight may be determined by the preceding pro- 
 position ; their centre of pressure is their centre of gravity. 
 
 THE CENTRE OF GRAVITY. 
 
 20. The resultant of the weights of any number of bodies 
 or parts of the same body unitea into a system of inva- 
 riable form passes through the same point in it, into what- 
 ever position it may be turned. 
 
 For the effect of turning it into different positions is to 
 cause the directions of the weights of its parts to traverse 
 the heavy body or system in different directions, at one time 
 lengthwise for instance, at another across, at another 
 obliquely and the effect upon the direction of the resultant 
 weight through the body, produced by thus turning it into 
 different positions, and thereby changing the directions in 
 which the weights of its component parts traverse its mass, 
 is manifestly the same as would be produced, if without alter- 
 ing the position of the body, the direction of gravity could 
 be changed so as, for instance, to make it at one time tra- 
 verse that body longitudinally, at another obliquely, at a 
 third transversely. But by Article 19, this last mentioned 
 change, altering the common direction of the parallel pres- 
 
THE CENTRE OF GRAVITY. 21 
 
 sures through the body without altering their amounts or 
 their points of application, would not alter the position of 
 their centre of pressure in the body ; therefore, neither would 
 the first mentioned change. Whence it follows that the 
 centre of pressure of the weights of the parts of a heavy 
 body, or of a system of invariable form, does not alter its 
 position in the body, whatever may be the position into 
 which the body is turned; or in other words, that the 
 resultant of the weights of its parts passes always through 
 the same point in the body or system in whatever position 
 it may be placed. 
 
 This point, through which the resultant of the weights of 
 the parts of a body, or system of bodies of invariable form, 
 passes, in whatever position it is placed ; or, if it be a body 
 or system of variable form, through which the resultant 
 would pass, in whatever position it were placed, if it became 
 rigid or invariable in its form, is called the CENTRE OF 
 GRAVITY. 
 
 21. Since the weights of the parts of a body act in 
 parallel directions, and all tend in the same direction, there- 
 fore their resultant is equal to their sum. Now, the result- 
 ant of the weights of the parts of the body would produce, 
 singly, the same effect as it regards the conditions of the 
 equilibrium of the body, that the weights of its parts 
 actually do collectively, and this weight is equal to the sum 
 of the weights of the parts, that is, to the whole weight of 
 the body, and in every position it acts vertically downwards 
 through the same point in the body, viz. the centre of 
 gravity. Thus then it follows, that in every position of the 
 body and under every circumstance, the weights of its parts 
 produce the same effect in respect to the conditions of its 
 equilibrium, as though they were all collected in and acted 
 through that one point of it its centre of gravity* 
 
 * That the resultant of the weights of all the parts of a rigid body passes 
 in all the positions of that body through the same point in it is a property of 
 many and most important uses in the mechanism of the universe, as well as in 
 the practice of the arts ; another proof of it is therefore subjoined, which 
 may be more satisfactory to some readers than that given in the text. The 
 system being rigid, the distance PI, P 2 , of the points of 
 application of any two of the pressures remains the 
 same, into whatever position the body may be turned : 
 the only difference produced in the circumstance under 
 which they are applied is an alteration in the inclina- 
 tions of these pressures to the line PI, P 2 : now being 
 weights, the directions of these pressures always remain 
 parallel to one another, whatever may be their inclina- 
 tion ; thus then it follows by the principle of the equa- 
 
22 THE CENTRE OF GRAVITY. 
 
 22. To determine the position jf the centre o gravity of 
 two weights ', P x and reforming part of a rigid system. 
 
 Let it be represented by G. Then since the resultant of 
 
 T@ ^ Pj and P Q passes through G, we have by equa- 
 
 P * tion (16), taking P t as the point from which the 
 moments are measured, 
 
 P 4-P P (1 P P P 
 x^j-f jr t . r a vr 1 3 . JT jiT.,, 
 
 P P P 
 
 p r\ _ 2j_^_i^_ 2 . 
 
 ilU -TVfPT 
 
 whence the position of G is known. 
 
 23. It is required to determine the centre of gravity of three 
 weights P 15 P 2 , P 3 , not in the same straight line, and form- 
 ing part of a rigid system. 
 
 Find the centre of gravity G 15 of P x and P 2 , as in the last 
 
 proposition. Suppose the weights P 1 and P 2 to 
 
 jft be collected in G 15 and find as before the com- 
 
 ^^Jc, mon centre of gravity G 2 of this weight Pj + P,, 
 
 r.*-- Vj so collected in G a , and the third weight P 3 . It 
 
 Lg is evident that this point G 2 is the centre of 
 
 gravity required. Since G 2 is the centre of 
 
 gravity of P 3 and P^ P 2 collected in G 15 we have by the 
 
 last proposition 
 
 G?, . P., 
 
 G.P..F. 
 
 lity of moments (Art. 15), that Pi4-P 2 PiRi P 2 . PiP 2 , so that for every 
 such inclination of the pressures to PI P 2 , the line PI HI is of the same length, 
 and the point Rj therefore the same point ; therefore, the line P 3 Ri is always 
 the same line in the body; and RI which equals P!-f-P 2 , is always the same 
 pressure, as also is P 8 , and these pressures always remain parallel, therefore, 
 for the same reason as before, R 2 is always the same point in the body in 
 
 whatever position it may be turned, and so of R 3 , R4 and R.-I. That 
 
 is, in every position of the body, the resultant of the weights of its parts 
 passes through the same point R-i in it. Since the resultant of the weights 
 of the parts of a body always passes through its centre of gravity, it is 
 evident, 'that a single force applied at that point equal and opposite to this 
 resultant, that is, equal in amount to the whole weight of the body, and in a 
 direction vertically upwards, would in every position of the body sustain it. 
 This property of the centre of gravity, viz. that it is a point in the body where 
 a single force would support it is sometimes taken as the' definition of it. 
 
OF A TRIANGLE. 23 
 
 If P 15 P 2 , P 3 , be all equal, then 
 
 %<M= 
 
 Moreover in this case, 
 
 24. To find the centre of gravity of four weights not in thf> 
 
 same plane. 
 
 Let P 1? P 2 , P 3 , P 4 , represent these weights; find the 
 
 centre of gravity G 2 of the weights P,, P 3 , 
 
 y5_ P 3 , as in the last proposition ; suppose these 
 
 /j\ three weights to he collected in G 2 , and then 
 
 //I \ find the centre of gravity G 3 of the weight 
 
 /iift... \ tnus collected in G- 2 and t\. G 3 will he the 
 
 jgjSfc^ 3 ^ centre of gravity required, and since G 3 is 
 
 the centre of gravity of P 4 acting at the 
 
 point P 4 , and of P^Pa+P, collected at G 2 , 
 
 If all these weights be equal, then by the above equation, 
 
 also, _G 1 G ? =i_G 1 P 3 , 
 and &?!=$ P.P.- 
 
 25. THE CENTRE OF GRAVITY OF A TRIANGLE. 
 
 Let the sides AB and EC of the triangular lamina ABC 
 be bisected in E and D, and the lines CE and 
 AD drawn to the opposite angles, then is the 
 intersection G of these lines the centre of gravity 
 of the triangle : for the triangle may be supposed 
 to be made up of exceedingly narrow rectangular 
 strips or bands, parallel to JBC, each of which will 
 be bisected by the line AD; for by similar triangles 
 PK : DB :: AE : AD :: KQ : DC, therefore, alternando, 
 PK : KQ::DB : DC; but DB=DC; therefore PR=PvQ. 
 
 Therefore, each of the elementary bands, or rectangles 
 parallel to BC, which compose the triangle ABC, would 
 separately balance on the line AD ; therefore, all of them 
 
24 THE CENTRE OF GRAVITY 
 
 joined together would balance on the line AD, therefore the 
 centre of gravity of the triangle is in AD. 
 
 In the same manner it may be shown that the centre of 
 gravity of the triangle is in the line CE ; therefore, the cen- 
 tre of gravity is at the intersection G of these lines. 
 
 Now DG=J DA : for imagine the triangle to be without 
 weight, and three equal weights to be placed at the angles 
 A, 6, and C, then it is evident that these three weights will 
 balance upon AD ; for AD being supported, the weight A 
 will be supported, since it is in that line ; moreover, B and 
 C will be supported since they are equidistant from that 
 line. 
 
 Since, then, all three of the weights will balance upon 
 AD, their centre of gravity is in AD. In like manner it 
 may be shown that the centre of gravity of all three weights 
 is in CE ; therefore it is in G, and coincides with the centre 
 of gravity of the triangle. 
 
 ]N ow, suppose the weights B and C to be collected in their 
 centre of gravity D, and suppose each weight to be repre- 
 sented in amount by A, a weight equal to 2A will then be 
 collected in D, and a weight equal to A at A, and the centre 
 of gravity of these is in G ; therefore DA x A = DG x 
 (2A + A), 
 
 . . D A = 3 DG, or DG = DA.* [Q.E.D.] 
 
 26. THE CENTKE OF GRAVITY OF THE PYRAMID. 
 
 Let ABC be a pyramid, and suppose it to be 
 made up of elementary laminae l)cd, parallel to 
 the base BCD. Take G, the centre of gravity 
 of the base BCD, and join AG; then AG will 
 pass through the centre of gravity g of the 
 lamina Icd^ therefore each of the laminae will separately 
 balance on the straight line AG ; therefore the laminae when 
 combined will balance upon this line ; therefore the whole 
 figure will balance on AG, and the centre of gravity of 
 the whole is in AG. In like manner if the centre of gravity 
 H of the face ABD be taken, and CH be joined, then it may 
 be shown that the centre of gravity of the whole is in Cli ; 
 
 
 
 * Note (g) Ed. App. 
 
 f For produce the plane ABG- to intersect the plane ADC in AM, then by 
 similar triangles DM : MC : : dm : me, but DM = MC ; therefore dm = me. Also 
 by similar triangles GM : BM::#m : bm, but GM = i BM; therefore gm = $ 
 bm. Since then dm = J dc and gm = $ bm, therefore g is the centre of gravity 
 of the triangle bdc. 
 
OF A PYRAMID. 25 
 
 therefore the lines AGr and CII intersect, and the centre of 
 gravity is at their intersection K. 
 
 Now GK is one-fourth of GA ; for suppose equal weights 
 to be placed at the angles A, B, C, and D of the pyramid 
 (the pyramid itself being imagined without weight), then 
 will these four ^ weights balance upon the line AG, for one 
 of them, A, is in that line, and the line passes through the 
 centre of gravity G of the other three. 
 
 Since, then, the equal weights A, B, C, and D balance 
 upon the line AG, their centre of gravity is in AG ; in the 
 same manner it may be shown that the centre of gravity of 
 the four weights is in CH, therefore it is in K, and coincides 
 with the centre of gravity of the pyramid. 
 
 Now let the number of units in each weight be repre- 
 sented by A, and let the three weights B, C, and D be 
 supposed to be collected in their centre of gravity G ; the 
 four weights will then be reduced to two, viz. 3A at G, and 
 A at A, whose common centre of gravity is K, 
 
 /. GKx3A+A = GAxA, 
 
 /. 4GK = GA or GK = J GA.* [Q.E.D.] 
 
 27. The centre of gravity of a pyramid with a polygonal lase 
 is situated at a vertical height from the base, equal to one 
 fourth the whole height of the pyramid. 
 
 For any such pyramid ABCDEF may be supposed to 
 be made up of triangular pyramids ABOF, 
 A CDF, and ADEF, whose centres of gravity 
 G, H, and K, are situated in lines AL, AM, 
 and AN, drawn to the centres of gravity L, M, 
 and N of their bases' ; LG being one-fourth of 
 LA, Mil one-fourth of MA, and NK one-fourth 
 of NA. The points G, H, and K, are therefore in a plane 
 parallel to the base of the pyramid, and whose vertical dis- 
 tance from the base equals one-fourth the vertical height of 
 the pyramid. 
 
 Since then the centres of gravity G, H, and K of the ele- 
 mentary triangular pyramids which compose the whole poly- 
 gonal pyramid are in this plane, therefore the centre of gravity 
 of the whole is in this plane, i. e. the centre of gravity of the 
 whole polygonal pyramid is situated at a vertical height from 
 the base, equal to one fourth the vertical height of the whole 
 
 * Note (h) Ed. App. 
 
26 THE CENTEE OF GRAVITY 
 
 pyramid, or at a vertical depth from the vertex, equal to three 
 fourths of the whole. JSTow the above proportion is true, 
 whatever be the number of the sides of the polygonal base, 
 and therefore if they be infinite in number ; and therefore it 
 is true of the cone, which may be considered a pyramid hav- 
 ing a polygonal base, of an infinite number of sides ; and it 
 is true whether the cone or pyramid be an oblique or a right 
 cone or pyramid. 
 
 28. If a body be of a prismatic form, and symmetrical 
 about a certain plane, then its whole weight may be sup- 
 posed to be collected in the surface of that plane, and uni- 
 formly distributed through it. For let 
 ACBEFD represent such a prismatic 
 body, and dbc a plane about which it is 
 symmetrical : take m, an element of uni- 
 form thickness whose sides are parallel to 
 B the sides of the prism, and which is 
 terminated by the faces ACB and DFE of the prism ; 
 it is evident that this element m will be bisected by the 
 plane abc, and that its centre of gravity will therefore 
 lie in that plane, so that its whole weight nlay be sup- 
 posed collected in that plane ; and this being true of 
 every other similar element, and all these elements be- 
 ing equal, it follows that the whole weight of the body 
 may be supposed to be collected in and uniformly dis- 
 tributed through that plane. It is in this sense only that we 
 can speak with accuracy of the weight and the centre of gra- 
 vity of a plane, whereas a plane being a surface only, and 
 having no thickness, can have no weight, and therefore no 
 centre of gravity. In like manner when we speak of the 
 centre of gravity of a curved surface, we mean the centre of 
 gravity of a body, the weights of all whose parts may be sup- 
 posed to be collected and uniformly distributed throughout 
 that curved surface. It is evident that this condition is 
 approached to whenever the body being hollow, its material 
 is exceedingly thin. Its whole weight may then be conceived 
 to be collected in a surface equidistant from its two external 
 surfaces. In like manner an exceedingly thin uniform curved 
 rod may be imagined to have its weight collected uniformly 
 in a line passing along the centre of its thickness, and in this 
 sense we may speak of the centre of gravity of a line, 
 although a line having no breadth or thickness can have no 
 weight, and therefore no centre of gravity. 
 
OF ANY QUADRILATERAL FIGURE. 27 
 
 29. THE CENTRE OF GRAVITY OF A TRAPEZOID. 
 
 Let AD and BO be the parallel sides of the trapezoid, of 
 B which AD is the less. Let AD be represented 
 by 0, BC by 5, and the perpendicular distance 
 
 ~ ' 
 
 tlie tw sides b y ^* Draw DE parallel 
 to AB. Let G x be the intersection of 
 t ] le ^i a g 0na ls o f the parallelogram ABED, 
 then will 4 G t be the centre of gravity of that parallelo- 
 gram. Bisect OE in L, join DL, and take DG 2 =f DL, 
 then will G 3 be the centre of gravity of the triangle DEC. 
 Draw GjM, and G 2 M 2 perpendiculars to AD ; then since 
 AG,=i AE, therefore G.M^i FE=4 h. And since 
 DG 2 = f DL, therefore G 2 M 2 = | NL = f h. Suppose the 
 whole parallelogram to be collected in its centre of gravity 
 G 1? and the whole triangle in its centre of gravity G a . Let 
 G be the centre of gravity of the whole trapezoid, and draw 
 GM perpendicular to AD. Then would the whole be sup- 
 ported by a single force equal to the weight of the trapezoid 
 acting upwards at G. Therefore (Art. 17), 
 
 MG . ABCD=~G^M 1 . ABED + G~M 2 . "CED 
 
 Now, ABCD = i h (a + 5), ABED = ha, 
 
 CED = i A (7,_a), G.M, = J A, G 2 M 2 = f A, 
 .-. MG' . -J- h (a+~b} = % h . ha+% h . \ li(ba\ 
 .-. MG (a+6) = Aa+| h (I a) = -J h 
 
 (19). 
 
 30. THE CENTRE OF GRAVITY OF ANY QUADRILATERAL FIGURE. 
 
 Draw the diagonals AC and BD of any quadrilateral figure 
 ABCD, and let them intersect in E, 
 and from the greater of the two parts, 
 BE and DE, of either diagonal BD set 
 off a part BF equal to the less part. 
 Bisect the other diagonal AC in H, join 
 HF and take HG equal to one third of 
 
 HF ; then w^ill G be the centre of gravity of the whole 
 
 figure. 
 
 For if not, let g be the centre of gravity, join HB and HD 
 
 and take HG X \ HB and HG 2 = i HD, then will G x and 
 
 G 2 be the centres of gravity of the triangles ABC and ADC 
 
28 THE CENTRE OF GRAVITY. 
 
 respectively (Art. 25). Suppose these triangles to be col- 
 lected in their centres of gravity G 15 G 2 ; it is evident that 
 the centre of gravity ^, of the whole figure, will be in the 
 straight line joining the points G x G 2 : let this line intersect 
 AC in K ; then since a pressure equal to the weight of the 
 whole figure acting upwards at </, will be in equilibrium with 
 the weights of the triangles collected in Gj and G 2 , we have, 
 by the principle of the equality of moments (Art. 15), 
 
 E^ . ABCD^KG, . ABC KG 2 . ADC. 
 
 Now since HG, = \ HB, and HG 2 = \ HD, therefore G, G 2 
 is parallel to DB, therefore KG^i BE, and KG a = J DE. 
 Now let the angle AED = BEC i. Therefore the perpen- 
 dicular from B upon AC BE sin. , and that from 33 = DE 
 
 sin. , therefore area of triangle ABC % AC . BE sin. , 
 and area of triangle ADC = J AC . DE sin. i, therefore area 
 of quadrilateral ABGD = j AC . BE~ sin. i+J AC . DE 
 
 sin. = J (BE + DE) AC sin. *. Substituting these values in 
 the preceding equation, 
 
 !Lg . J (BE+DE)J[C sin^iJBE . J AC . BE sin. i 
 DE . AC . DE sin. i 
 
 (BE+DE) = -i- (BF DE 5 ), 
 
 But sin-ce HG = i HF, .\KG = ^ FE, .-.E^ = KG; that 
 is, the true centre of gravity g coincides with the point G. 
 Therefore, &c. [Q.E.D.] 
 
 *31. In the examples hitherto given, the centre of pressure 
 of a system of weights, or their centre of gravity, lias been 
 determined by methods which are indirect as compared with 
 the direct and general method indicated in Article IT. That 
 method supposes, however, a determination of the sum of the 
 moments of the weights of all the various elements of the 
 body in respect to three given planes. Now in a continuous 
 body these elements are infinite in number, each being infi- 
 nitely small ; this determination supposes, therefore, the sum- 
 mation of an infinite number of infinitely small quantities, 
 and requires an application of the principles of the intregal 
 calculus. 
 
 Let AM be taken to represent any small element of the 
 
THE CENTRE OP GRAVITY. 
 
 29 
 
 volume M of a continuous body, and x its perpendicular 
 distance from a given plane. Then will OJJA AM represent 
 the moment of the weight of this element about that plane, 
 fx representing the weight of each unit of the volume M. 
 Let \&x AM represent the sum of all such moments, taken in 
 respect to all the small elements, such as AM, which make 
 up the volume of the body. Then if G^ represent the dis- 
 tance of the centre of gravity of the body from the given 
 plane ; since i^2a?AM represents the sum of the moments of a 
 system of parallel pressures about that plane, f*M the sum of 
 those pressures, and G x the distance of their centre of pres- 
 sure from the plane (Art. 19), it follows by equation (18) that 
 
 4^3^ (20) _ 
 
 Now it is proved in the theory of the integral calculus,* 
 that a sum, such as is represented by the above expression 
 2a?AM, whose terms are infinite in number, and each the pro- 
 duct of a finite quantity a?, and an infinitely small quantity 
 AM, and in which M is, as in this case, a function of x (and 
 therefore x a function of M), is equal to the definite integral 
 
 / a 
 
 Therefore, generally, 
 
 fx 
 
 M 
 
 Similarly, 
 
 J'ydM. 
 
 (21). 
 
 M 
 
 In the two last of which equations y and s are taken to repre- 
 sent, respectively, the distances of the element AM of the 
 
 * Poisson, Journal de 1'Ecole Polytechnique, 18me cahier, p. 320, or Art. 2, 
 in the Treatise on Definite Integrals in the Encyclopaedia Metropolitana by the 
 author of this work. See Appendix, note A. 
 
30 THE CENTRE OF GRAVITY. 
 
 body from two other planes, as x represents its distance from 
 the first plane ; and G t and G 2 to represent the distances of 
 its centre of gravity from those planes. The distances G,, 
 G 2 , G 3 , of the centre of gravity from three different planes 
 being thus known, its actual position in space is fully deter- 
 mined. These three planes are usually taken at right angles 
 to one another, and are then called rectangular co-ordinate 
 planes, and their common intersections rectangular co-ordi- 
 nate axes. 
 
 If the centre of gravity of the body be known to lie in a 
 certain plane, and one of the co-ordinate planes spoken of 
 above, as for instance that from which G 3 is measured, be 
 taken to coincide with this plane in which the centre of gra- 
 vity is known to lie, then G 3 == 0, and the position of the cen- 
 tre of gravity is determined by the two first only of the above 
 three equations. This case occurs when the body, whose 
 centre of gravity is to be determined, is symmetrical about a 
 certain plane, since then its centre of gravity evidently lies 
 in its plane of symmetry. If the centre of gravity of the 
 body be known to lie in a certain line, and two of the co-or- 
 dinate planes, those for instance from which G 2 and G 3 are 
 measured, be taken so as to intersect one another in that line, 
 then the centre of gravity will be in both those planes ; there- 
 fore G 2 = and G 3 = 0, and its position is determined by the 
 first of the preceding equations alone. This case occurs 
 when^ the body is symmetrical about a given line ; its centre 
 of gravity is then manifestly in that line. 
 
 "*32. THE CENTRE OF GRAVITY OF A CURVED LINE WHICH LIES 
 WHOLLY IN THE SAME PLANE. 
 
 Taking M to represent the length S of such a line, we 
 have, by equations (21), 
 
 .- , .. 
 
 EXAMPLE. Let it ~be required to determine the centre of 
 gravity of a circular arc EF. 
 
 The centre of gravity of such an arc is evidently in the 
 radius CA, which bisects it; since the arc 
 is symmetrical about that radius. Take a 
 plane Cy perpendicular to this radius, and 
 passing through the centre, to measure the 
 moments from. Let x represent the dis- 
 tance PM of any point P in this arc from 
 
OF A CURVED LINE. 31 
 
 this plane ; also let s represent the arc P A, and S the aro 
 E AF, a the radius C A, and C the chord EF. 
 
 .-. x = PM = OF cos. CPM = CP cos. AGP = a cos. - 
 #8 IS 
 
 .'. f xdS= a I cos. ds=a* /.cos. d\ ) 2& 2 sin.( )> 
 / / a v a \a/ \af 
 
 the integral being taken between the limits -JS and JS, 
 because these are the values of s which correspond to the 
 extreme points F and E of the arc. 
 
 Now 2a sin. J ( ) = chord of EAF = C, /.Juarffl = 00, 
 
 "Gi = lT ...... (23). 
 
 The distance of the centre of gravity of a circular arc from 
 the centre of the circle is therefore a fourth proportional to 
 the length of the arc, the length of the chord, and the radius 
 of the arc. 
 
 *33. THE CENTRE OF GRAVITY OF A CURVILINEAR AREA 
 WHICH LIES WHOLLY IN THE SAME PLANE. 
 
 'Let BAG represent such an area. If x and y represent 
 the perpendicular distances PN and PM of any 
 point P in the- curve AB from planes AC and 
 AD, perpendicular to the plane of the given area 
 and to one another, and M represent the area 
 PAM, then, considering this area to be made up 
 of rectangles parallel to PM, the width of each 
 of which is represented by the exceedingly small quantity 
 Aa?, the area AM of each such rectangle will be -represented 
 by yAaj, and its moment about AD by v-xy&x. 
 
 j 
 
 xydx 
 
 Therefore by equation (20), G, = = -g . . (24). 
 
 A similar expression determines the value o G 2 ; butane 
 more convenient for calculation is obtained, if we consider 
 the weight of each of the rectangles, whose length is y, to 
 be collected in its centre of gravity, whose distance from AC 
 
32 THE CENTRE OF GRAVITY. 
 
 is \y. The moment of the weight of each rectangle about 
 AC will then be represented ^y*&x ; whence it follows that 
 
 i 
 tfdx 
 
 ..... ( 25 ). 
 
 - 2 M 
 
 EXAMPLE. Suppose the curve APB to be a parabola, whose 
 axis is AC. 
 
 Dl B By the equation to the -parabola y 1 = 4##, if a 
 
 ] be the distance of the focus from the vertex. 
 Moreover, the limits between which the integral 
 is to be taken are and aj t and and y 1? since at 
 A, x 0, y = 0, and at C, x = a? 1? y y^ 
 
 therefore Cxydx 2 \/ a /*x%dx~ \f ax 1 2 ; also, M = C yd 
 
 C 2 
 
 051 4 3 
 
 = 2 ya Cx\dx g i/tf^f, therefore Gj^ ^^. 
 
 o 
 
 1 1 ^,4 ,1 _ . t 
 
 Also, jy^dx 4^ /xdx= %ax? =^ 
 J J b< 
 
 o 
 
 Q 
 
 therefore G 2 Q^. 
 
 o 
 
 If, then, G be the centre of gravity of the parabolic area 
 
 ACB, then AH = ? AC, HG = - CB. 
 
 5 -8 
 
 * 34. THE CENTRE OF GRAVITY OF A SURFACE OF REVOLUTION. 
 
 Any surface of revolution BAC is evidently symmetrical 
 about its axis of revolution AD, its centre of 
 gravity is therefore in that axis. Let the mo- 
 ments be measured from a plane passing through 
 A and perpendicular to the axis AD, and let x 
 and y be co-ordinates of any point P in the 
 generating curve APB of the surface, and s the 
 length of the curve AP. Then M being taken to represent 
 the area of the surface, and being supposed to be made up 
 of bands parallel to PQ, the area AM of each such band is 
 represented (see Art. 40.)* by %ny&s, and its moment by 
 
 * Church's Diff. Calculus, Art. 91. 
 
OF A SUKFACE. 
 
 33 
 
 s, 
 
 27T,/ 
 
 
 (26> . 
 
 EXAMPLE. To determine the centre of gravity of the 
 face of any zone or segment of a sphere. 
 
 B //\ 7 j^ j^ACj represent the surface of a sphere v 
 :, D whose centre is D, and whose radius DP is repre- 
 sented by #, and the arc AP by s. Then x = DM 
 
 = DP cos. 
 
 sn., 
 
 =# sn. -, 
 a 
 
 a 
 
 = %a? sin. - cos. - = a* sin. . 
 a a a 
 
 S, S, 
 
 ./* /* 2^ 
 
 .*. STT / xyds = TTO? I sin. ds 
 
 s, s, 
 
 = 4 j cos. ^i_ cos. ^1 
 
 = TO- icos. 3 ?? - cos. 3 il ..... (27). 
 { J 
 
 < 
 
 where S 4 and S 2 are the values of s at the points 
 03 B, and B a , where the zone is supposed to ter- 
 
 minate. 
 
 Also, since = 27ry, /. M = tor 
 
 = 27ra /*sin. 1 <fo = 2^ 9 i cos. ^- a - cos. ?2 1 
 J a ( a a > 
 
34: THE CENTRE OF GRAVITY 
 
 ... .(28), 
 
 if E be the bisection of EJE,. 
 
 If S 2 = 0, or the zone commence from A, then 
 
 G, = -a \ 1 + cos. 4 = cos. 2 -?i. . . . (29). 
 2 ( a ) 2a 
 
 *35. THE CENTRE OF GRAVITY OF A SOLID OF REVOLUTION. 
 
 Any solid of revolution BAG is evidently symmetrical 
 about its axis of revolution AD, its centre of 
 gravity is therefore in that line ; and taking a 
 plane passing through A and perpendicular to 
 that axis as the plane from which the moments 
 are measured, we have only to determine the 
 distance AG of ^the centre of gravity, from 
 that plane. 
 
 Now, if x and y represent the co-ordinates of any point P 
 in the generating curve, and M the volume of the portion 
 PAQ of this solid, then, conceiving it to be made up of 
 cylindrical laminse parallel to PQ, the thickness of each of 
 which is A#, the volume of each is represented by iry*&x, and 
 its moment by *pxy*&x. 
 
 i 
 
 *l xy*dx 
 ^xy^x_^ ...... 
 
 M M 
 
 EXAMPLE. To determine the centre of gravity of any solid 
 segment of a sphere. 
 
 Let BjACj represent any such segment of a 
 sphere whose centre is D and its radius a. Let x 
 and y represent the co-ordinates AM and MP of 
 any point P, x being measured from A ; then by 
 the equation to the circle y*=2axx*, 
 
 :. if fxy*dx= fx (2axx*) dx 
 
 a? 2 o 
 
 x, x, 
 
 Also, M.=fydx = * f(2axx*) dx= 
 
OF THK SEGMENT OF AN AKCH. 35 
 
 (31). 
 
 a x 
 
 v JL A 
 
 If the segment become a hemisphere, x t =a 9 /.G l =-|a. 
 36. The centre of gravity of the sector of a circle. 
 
 Let CAB represent such a sector ; conceive the arc ADB 
 to be a polygon of an infinite number of sides 
 and lines, to be drawn from all the angles of the 
 polygon to the centre C of the circle, these will 
 divide the sector into as many triangles. Now 
 /B the centre of gravity of each triangle will be at 
 a distance from C equal to f- the line drawn from the vertex 
 C of that triangle to the bisection of its base, that is equal 
 to f the radius of the circle, so that the centres of gravity of 
 all the triangles will lie in a circular arc FE, whose centre is 
 C and its radius CF equal to fCA, and the weights of the 
 triangles may be supposed to be collected in this arc FE, 
 and to be uniformly distributed through it, so that the cen- 
 tre of gravity G of the whole sector CAB is the centre of 
 gravity of the circular arc FE. Therefore by equation (23), 
 if S 1 , C 1 , and a\ represent the arc FE, its chord FE, and its 
 radius CF, and S, C, &, the similar arc, chord, and radius of 
 
 ADB, then CG ^ ; but since the arcs AB and FE are 
 
 similar, and that a 1 = \a, :. C 1 = f C and S 1 = f S. Substi- 
 tuting these values in the last equation, we have 
 
 C*C* 2. ^ /QO\ 
 
 =| -s- (32) " 
 
 37. The centre of gravity of any portion of a circular ring 
 or of an arch of equal voussoirs. 
 
 2 represent any such portion of a circular ring 
 whose centre is A. Let a l represent the 
 radius, and C t the chord of the arc BA> and 
 Sj its length, and let a C 2 similarly represent 
 the radius and chord of the arc B 2 C 2 , and S a 
 the length of that arc. 
 
 Also let G, represent the centre of gravity of the sector 
 ~ 15 G 2 that of the sector AB 2 C 2 , and Q- the centre of 
 gravity of the ring. Then 
 
 AG 2 x sect, AB 2 C 2 + AG x ring B 1 C 1 B 2 C 2 = AG; x sect. ABA 
 
 ISTow (by equation 32), AG 1= -|i, AG 2 =f 
 
 8 
 
36 THE PROPERTIES 
 
 also sector AB 1 C I = ^8^, sector AB 2 C 2 = ^S 2 # 2 , 
 .-. ring B 1 O l 1 B 1 =sect. AB.C, sect. AB 2 C 2 = JSA 
 
 .-. AG . (SA ^,0=1 (OA* CA*), 
 
 .-. AG = | ^ "" fy* a (33). 
 
 38. THE PROPERTIES OF GULDENUS. 
 
 If NL represent any plane area, and AB be any axis, in the 
 same plane, about which the area is made to 
 revolve, so thai NL is by this revolution made to 
 generate a solid of revolution, then is the volume 
 of this solid equal to that of a prism whose base 
 ^s NL, and whose height is equal to the length 
 of the path which the centre of gravity G of the 
 area NL is made to describe. 
 
 For take any rectangular area PRSQ in NL, whose sides 
 are respectively parallel and perpendicular to AB, and let 
 MT be the mean distance of the points P and Q, or R and 
 S, from AB. Now it is evident that in the revolution of 
 NL about AB, PQ will describe a superficial ring. 
 
 Suppose this to be represented by QFPK, let M be the 
 centre of the ring, and let the arc subtended by 
 the angle QMF at distance unity from M be repre- 
 sented by d, then the area FQPK equals the sector 
 FQM the sector ~ 
 
 Now the solid ring generated by PRSQ is evidently equal 
 to the superficial ring generated by PQ, multiplied by the 
 distance PR. This solid ring equals therefore 6 (MT x PQ 
 xPR) or dxMTxPRSQ. Now suppose the area PRSQ 
 to be exceedingly small, and the whole area NL to be made 
 up of such exceedingly small areas, and let them be repre- 
 sented by # 15 a a 3 , &c. and their mean distances MT by x : , 
 a? 2 , a? 3 , &c. then the solid annuli generated by these areas 
 respectively will (as we have shown), be represented by 
 &x z a z , &c. &c. ; and the sum of these annuli, 
 
OF GTJLDINTJS. 37 
 
 or the whole solid, will be represented by .^^ , . 
 to,a, + &c., or by 6 (x^a, + x,a, + x,a 3 + &c.). Now if p repre- 
 sent the weight of any superficial el'ement of the plane NX, 
 x l a 1 ^=thQ moment of the weight of a, about the axis AB* 
 35,0^= that of the area <z 2 about the same axis AB, and so 
 on, therefore the sum (fl? 1 <^,-|-fiP,aj-hQ9 t a f -4 i &o.)f*=;the moment 
 of the whole area NL about AB ; but if G be the centre of 
 gravity of NL, and GI its distance from AB, then the 
 moment of NL about AB=GI> 
 therefore the whole solid =6 . Gl 
 
 but 6 . GI equals the length of the circu- 
 lar path described by G ; therefore the 
 volume of the solid equals NL multi- 
 plied by the length of the path de- 
 scribed by G, i. e. it equals &^sm NM, 
 whose base is NL, and whose height GH 
 is the length of the path described by 
 G ; which is the first property of GUL- 
 DDTCJS. 
 
 39. The above proposition is applicable to finding the 
 solid contents of the thread of a screw of variable diame- 
 ter, or of the material in a spiral staircase: for it is 
 evident that the thread of a screw may be supposed to be 
 made up of an infinite number of small solids of revolution, 
 arranged one above another like the steps of a staircase; all 
 of which (contained in one turn of the thread) might be 
 made to slide along the axis, so that their surfaces should all 
 lie in the same plane ; in which case they would manifestly 
 form one solid of revolution, such as that whose volume has 
 been investigated. The principle is moreover applicable to 
 determine the volume of any solid (however irregular may 
 be its form otherwise), provided only that it may be con- 
 ceived to be generated by the motion of a given plane area, 
 perpendicular to a given curved line, which always passes 
 through the same point in the plane. For it 
 is evident that whatever point in this curved 
 line the plane may at any instant be traver- 
 sing, it may at that instant be conceived to be revolving 
 about a certain fixed axis, passing through the centre of 
 curvature of the curve at that point; and thus revolving 
 about a fixed axis, it is generating for an instant a solid of 
 revolution about that axis, the volume of which elementary 
 solid of revolution is equal to the area of the plane miilti- 
 
38 THE PEOPEETIES 
 
 plied by the length of the path described by its centre of 
 gravity ; and this being true of all such elementary solids. 
 each being equal to the' product of the plane by the corres- 
 ponding elementary path of the centre of gravity, it follows 
 that the whole volume of the solid is equal to the product 
 of the area by the whole length of the path. 
 
 40. If AB represent any curved line made to revolve about 
 the axis AD so as to generate the sur- 
 face of revolution BAG, and G l)e the 
 centre of gravity of this curved line, 
 then is the area of this surface equal 
 to the product of the length of the 
 curved line AB, by the length of the 
 path described lyy thepoint G, during 
 the revolution of the curve about AD. This is the second 
 property of Guldinus. 
 
 Let PQ be any small element of the generating curve, 
 and PQFK a zone of the surface generated by this element, 
 this zone may be considered as a portion of the surface of a 
 cone whose apex is M, where the tangents to the curve at T 
 and V, which are the middle points of PQ and FK, meet 
 when produced. Let this band PQFK of the cone QMF be 
 developed*, and let PQFK represent its develop- 
 ment ; this figure PQFK will evidently be a circu- 
 lar ring, whose centre is M ; since the develop- 
 ment of the whole cone is evidently a circular 
 sector MQF whose centre M corresponds to the 
 apex of the cone, and its radius MQ to the side MQ of the 
 cone. 
 
 Now, as was shown in the last proposition, the area of 
 this circular ring when thus developed, and therefore of the 
 conical band before it was developed, is represented by 
 
 6 . MT . PQ, where represents the arc subtended by QMF 
 at distance unity. Now the arc whose radius is MT is 
 represented by & . MT ; but this arc, before it w r as developed 
 from the cone, formed a complete circle whose radius was 
 NT, and therefore its circumference 2^NT ; since then the 
 circle has not altered its length by its development, we 
 have 
 
 * If the cone be supposed covered with a flexible sheet, and a band such 
 as PQFK be imagined to be cut upon it, and then unwrapped from the cone 
 and laid upon a plane, it is called the development of the band. 
 
OF GULDINUS. ^ J^lOTttT^^ 
 
 <*OaUfr*ri*l 
 
 Substituting this value of dMT in the expression for the area 
 of the band we have 
 
 area of zone PQFK=2* . NT . PQ. 
 
 Let the surface be conceived to be divided into an infinite 
 number of such elementary bands, and let the lengths of 
 the corresponding elements of the curve AB be represented 
 by s l9 $, $ 3 , &c. and the corresponding values of NT by y l5 
 2/2? 2/3? & c - Then will the areas of the corresponding zones 
 b 
 
 e represented by 2tf?/ 1 s 1 , 2tfy 2 s 2 , 2#y 3 s 3 , &c. and the area of 
 the whole surface BAG by %*y 1 s l + 2tfy 2 s 2 + 2*y, 8 + .... or 
 by 2-7r(y 1 s 1 + y 2 s 2 + y 3 $ 3 -f ....). But since G is the centre of 
 gravity of the curved line AB, therefore AB . GHjx repre- 
 sents the moment of the weight of a uniform thread or wire 
 of the form of that line about AD, j* being the weight of 
 each unit in the length of the line : moreover, this moment 
 equals the sum of the moments of the weights s^, s^, $,, 
 &c. of the elements of the line. 
 
 /.AB . GH=y 1 1 
 
 __ 
 Therefore area of surface BAC=2*AB . GH=AB 
 
 But 2-rrGH equals the length of the circular path described 
 by G in its revolution about AD. Therefore, &c. 
 
 This proposition, like the last, is true not only in respect to 
 a surface of revolution, but of any surface generated by a 
 plane curve, which traverses perpendicularly another curve 
 of any form whatever, and is always intersected by it in the 
 same point. It is evident, indeed, that the same demonstra- 
 tion applies to both propositions. It must, however, be ob- 
 served, that neither proposition applies unless the motion of 
 the generating plane or curve be such, that no two of its con- 
 secutive positions intersect or cross one another. 
 
 41. The volume of any truncated prismatic or cylindrical 
 lody ABCD, of which one extremity CD is perpendicular 
 to the sides of the prism, and the other AB inclined to 
 them, is equal to that of an upright jwism ABEF, having 
 for its lose the plane AB, and for its height the perpen- 
 dicular height GN of the centre of gravity G of the plane 
 DC, above the plane of AB. 
 
 For let i represent the inclination of the plane DC to AB ; 
 
40 THE PROPERTIES OF GULDINUS. 
 
 take m, any small element of the plane 
 CD, and let mr be a prism whose base is m 
 and whose sides are parallel to AD and 
 BC ; of elementary prisms similar to which 
 the whole solid ABCD may be supposed 
 to be made up. Now the volume of this prism, whose base 
 is m and its height mr, equals mr xm = sec. i x (mr . cos. i) 
 xm = sec. x (mr . sin. mm) m = sec. * x mn x m. 
 
 Therefore the whole solid equals the sum of all such pro- 
 ducts as mn x m, each such product being multiplied by the 
 constant quantity sec. i, or it is equal to the sum just spoken 
 of, that sum being divided by cos. i. Let this sum be repre- 
 sented by 2mn x m, therefore the volume of the solid is re- 
 
 , 'Zmn xm AT ' ^ T . , 
 
 presented by JNow suppose CD to represent a 
 
 thin lamina of uniform thickness, the weight of each square 
 unit of which is f*, then will the weight of the element m be 
 represented by M* X m, and its moment about the plane ABIsT 
 by M- x mn x m, and ^mn x m will represent the sum of the 
 moments of all the elements of the lamina similar to m about 
 that plane. Now by Art. 15. this sum equals the moment of 
 the whole weight of the lamina f* x CD supposed to be col- 
 lected in G, about that plane. Therefore 
 I* x CD x ~NG=^mn x m, 
 :. T)D x NG Zmn x m. 
 Substituting this value of 2mn x n, we have 
 
 volume of solid = sec. * x CD x NGL 
 
 But the plane CD is the projection of AB, therefore CD 
 = AB cos. i, .'. CD x sec. i = AB ; 
 
 .-. vol. of solid ABCD = AB x SX5 = vol. of prism ABEF. 
 Therefore, &c. 
 
 [Q. E. D .] 
 
MOTION. 41 
 
 T II. 
 
 DYNAMICS. 
 
 42. MOTION is change of place. 
 
 The science of DYNAMICS is that which treats of the laws 
 which govern the motions of material bodies, and of their 
 relation to the forces whence those motions result. 
 
 The SPACES described by a moving body are the distances 
 between the positions which it occupies at different succes- 
 sive periods of time. 
 
 UNIFORM MOTION is that in which equal spaces are de- 
 scribed in equal successive intervals of time. 
 
 The VELOCITY of uniform motion is the space which a 
 body moving uniformly describes in each second of time. 
 Thus if a body move uniformly with a velocity represented 
 by Y, and during a time represented in seconds by T, then 
 the space S described by it in those T seconds is represented 
 
 by TY, or S=TY. Whence it follows that Y = |-and T=L 5 
 
 so that if a body move uniformly, the space described by it 
 is equal to the velocity multiplied by the time in seconds, 
 the velocity is equal to the space divided by the time, and 
 the time is equal to the space divided by the velocity. 
 
 43. It is a law of motion, established from constant obser- 
 vation upon the motions of the planets, and by experiment 
 upon the motions of the bodies around us, that when once 
 communicated to a body, it remains in that body, unaffected 
 by the lapse of time, carrying it forward for ever with the 
 same velocity and in the same direction in which it first be- 
 
 n to move, unless some force act afterwards in a contrary 
 irection to destroy it.* 
 
 * This is the first LAW OP MOTION. For numerous illustrations of this fun- 
 damental law of motion, the reader is referred to the author's work, entitled, 
 ILLUSTRATIONS OF MECHANICS, Art. 193. 
 
42 VELOCITY. 
 
 The velocity, at any instant, of a body moving with a 
 VARIABLE MOTION, is the space which it would describe in 
 one second of time if its motion were from that instant to 
 become UNIFORM. 
 
 An ACCELERATING FORCE is that which acting continually 
 upon a body in the direction of its motion, produces in it a 
 continually increasing velocity of motion. 
 
 A RETARDING FORCE is that which acting upon a body in 
 a direction opposite to that of its motion produces in it a 
 continually diminishing velocity. 
 
 An IMPULSIVE FORCE is that which having communicated 
 motion to a body, ceases to act upon it after an exceedingly 
 small time from the commencement of the motion. 
 
 44. A UNIFORMLY accelerating or retarding force is that 
 which produces equal increments or decrements of velocity 
 in equal successive intervals of time. If f represent the 
 additional velocity communicated to a body by a uniformly 
 accelerating force in each successive second of time, and T 
 the number of seconds during which it moves, then since by 
 the first law of motion it retains all these increments of velo- 
 city (if its motion be unopposed), it follows that after T 
 seconds, an additional velocity represented by f T, will have 
 been communicated to it ; and if at the commencement of 
 this T seconds its velocity in the same direction was Y, then 
 this initial velocity having been retained (by the first law of 
 motion), its whole velocity will have become Y+/T. 
 
 If, on the contrary, f represents the velocity continually 
 taken away from a body in each successive second of time, 
 by a uniformly retarding force, and Y the velocity with 
 which it began to move in a direction opposite to that in 
 which this retarding force acts, then will its remaining velo- 
 city after T seconds be represented by Y /T; so that gene- 
 rally the velocity Y of a body acted upon by a uniformly 
 accelerating or retarding force is represented, after T seconds, 
 by the formula 
 
 (34). 
 
 The force of gravity is, in respect to the descent of bodies 
 near the earth's surface, a constantly accelerating force, 
 increasing the velocity of their descent by 32 j feet in each 
 successive second, and if they be projected upwards it is a 
 constantly retarding force, diminishing their velocity by that 
 quantity in each second. The symbol g is commonly used to 
 
VELOCITY. 43 
 
 represent this number 32| ; so that in respect to gravity the 
 above formula becomes v=V 0T, the sign being taken 
 according as the body is projected downwards or upwards. 
 
 A VARIABLE accelerating force is that which communicates 
 unequal increments of velocity in equal successive intervals 
 of time ; and a variable retarding force that which takes 
 away unequal decrements of velocity.* 
 
 45. To DETERMINE THE RELATION BETWEEN THE VELOCITY AND 
 THE SPACE, AND THE SPACE AND TIME OF A BODY'S MOTION. 
 
 Let AM 1? M^, M 2 M 8 , &c. represent the exceeding small 
 successive periods of a body's motion, and 
 AP the velocity with which it began to 
 move, MjPj the velocity at the expiration 
 of the first interval of time, M 2 P 2 that at 
 the expiration of the second, M 3 P 3 of the 
 third interval of time, and so on; and 
 instead of the body varying the velocity of its motion con- 
 tinually throughout the period AM 1? suppose it to move 
 through that interval with a velocity which is a mean 
 between the velocity AP at A, and that M^ at M 15 or with 
 a velocity equal to ^(AP + M.P,). 
 
 Since on this supposition it moves with a uniform motion, 
 the space it describes during the period AM 1 equals the 
 product of that velocity by that period of time, or it equals 
 KAP-fMjP^AMj. Now this product represents the area 
 of the trapezoid AM^P. The space described then in the 
 interval AM n on the supposition that the body moves during 
 that interval with a velocity which is the mean between 
 those actually acquired at the commencement and termi- 
 nation of the interval, is represented by the trapezoidal 
 areaAM^P. 
 
 Similarly the areas P^,, P 2 M 3 , &c. represent the spaces 
 the body is made to describe in the successive intervals 
 MjM 2 , M 2 M 3 , &c. ; and therefore the whole polygonal area 
 APCB represents the whole space the body is made to 
 describe in the whole time AB, on the supposition that it 
 moves in each successively exceeding small interval of time 
 with the mean velocity of that interval. Now the less the 
 intervals are, the more nearly does this mean velocity of each 
 interval approach the actual velocity of that interval ; and 
 if they be infinitely small, and therefore infinitely great in 
 
 * Kote (i) Ed. App. 
 
Me MOTION UNIFORMLY 
 
 number, then the mean velocity coincides with the actual 
 velocity of each interval, and in this case the polygonal area 
 passes into the curvilinear area APCB. 
 
 Generally, therefore, if we represent by the abscissa of a 
 curve the times through which a body has moved, and by 
 the corresponding ordinates of that curve the velocities which 
 it has acquired after those times, then the area of that curve 
 will represent the space through which the body has moved ; 
 or in other words, if a curve PC be taken such that the num- 
 ber of equal parts in any one of its abscissae AM 9 being taken 
 to represent the number of seconds during which a body has 
 moved, the number of those equal parts in the corresponding 
 ordinate M 3 P 3 will represent the number of feet in the velo- 
 city then acquired; then the space which the body has 
 described will be represented by the number of these equal 
 parts squared which are contained in the area of that curve. 
 
 46. To DETERMINE THE SPACE DESCRIBED IN A GIVEN TIME BY 
 A BODY WHICH IS PROJECTED WITH A GIVEN VELOCITY, AND 
 WHOSE MOTION IS UND7ORMLY ACCELERATED, OR UNIFORMLY 
 RETARDED. 
 
 Take any straight line AB to represent the whole time T, 
 in seconds, of the body's motion, and draw AD 
 perpendicular to it, representing on the same 
 scale its velocity at the commencement of its 
 motion. Draw DE parallel to AB, and accord- 
 ing as the motion is accelerated or retarded 
 draw DC or DF inclined to DE, at an angle w T hose tangent 
 equals/", the constant increment or decrement of the body's 
 velocity. Then if any abscissa AM be taken to represent a 
 number of seconds t during which the body has moved, the 
 corresponding ordinate MP or MQ will represent the velocity 
 then acquired by it, according as its motion is accelerated or 
 retarded. For PR = EQ = DK tan. PDE= AM tan. PDE ; 
 but AM = , and tan. PDE=/: therefore PR = KQ=/j{. 
 Also BM=AD=V, therefore JdP==BM+PB==V-f/^, and 
 MQ=EM RQ=V- -ft\ therefore by equation (34), MP or 
 MQ represents the velocity after the time AM according as 
 the motion is accelerated or retarded. The same being true 
 of every other time, it follows, by the last proposition, that 
 the whole space described in the time T or AB is represented 
 by the area ABCD if the motion be accelerated, and by the 
 area ABFD if it be retarded. 
 
ACCELEEATED OK RETARDED. 45 
 
 ISTow area ABCD=iAB(AD+BO), but AB=T, AD=Y 
 BC=Y+/T, 
 
 /. area ABOD=iT(V+V+/T)=YT+i/T'. 
 
 Also area ABFD=JAB (AD + BF), where AB and AD 
 have the same values as before, and BF=Y /T, 
 
 /. area ABFD=iT(Y+Y-/T)^YT-i/T 2 . 
 
 Therefore generally, if S represent the space described after 
 T seconds, 
 
 S-YTi/T a (35); 
 
 in which formula the sign is to be taken according as the 
 motion is accelerated or retarded. 
 
 47. To DETERMINE A RELATION BETWEEN THE SPACE DESCRD3ED 
 AND THE VELOCITY ACQUIRED BY A BODY WHICH IS PROJECTED 
 WITH A GIVEN VELOCITY, AND WHOSE MOTION IS UNIFORMLY 
 ACCELERATED OR RETARDED. 
 
 Let v be the velocity acquired after T seconds, then by 
 equation (34), v = Y /T, .-. T = ^ 
 
 p J*\ c ]Sr w area ABCI) = * AB ( AD + BC )> where 
 
 E AB=T= ^' 
 p f 
 
 (v V} 
 :. area ABCD= i^ 1 " ("V+ ^) = 
 
 f 
 area ABFD = JAB (AD+BF), where AB=T = - 
 
 Therefore generally, if S represent the space through which 
 the velocity v is acquired, then S=iJ- 3 4 
 
 (36); 
 
 in which formula the sign is to be taken according as 
 the motion is accelerated or retarded. 
 
 If the body's motion be retarded, its velocity v will eventu- 
 ally be destroyed. Let Sj be the space which will have been 
 
4:6 THE UNIT OF WORK. 
 
 described when v thus vanishes, then by the last equation 
 0-Y 2 = - 2/S,. 
 
 A Y 2 -2/S 1 (37), 
 
 where Y is the velocity with which the body is projected 
 in a direction opposite to the force, and S t the whole space 
 which by this velocity of projection it can be made to 
 describe. 
 
 If the body's motion be accelerated, and it fall from rest, 
 or have no velocity of projection, then ir 2 - = +2/*S, 
 
 ..tf=2/S (38). 
 
 Let S 2 be the space through which it must in this case 
 move to acquire a velocity V equal to that with which it 
 was projected in the last case, therefore Y 2 = 2/*S a . Whence 
 it follows that S 1 =S a , or that the whole space S t through 
 which a body will move when projected with a given velo- 
 city Y, and uniformly retarded by any force, is equal to the 
 space S a , through which it must move to acquire that velo- 
 city when uniformly accelerated by the same force. 
 
 In the case of bodies moving freely, and acted upon by 
 gravity, f equals 32| feet and is represented by g ; and the 
 space S a , through which any given velocity Y is acquired, is 
 then said to be that due to that velocity. 
 
 WOBK. 
 
 48. WORK is the union of a continued pressure with a 
 continued motion. And a mechanical agent is thus said to 
 WORK when a pressure is continually overcome, and a point 
 (to which that pressure is applied) continually moved by it. 
 Neither pressure nor motion alone is sufficient to constitute 
 work / so that a man who merely supports a load upon his 
 shoulders, without moving it, no more works, in the sense in 
 which that term is here used, than does a column which sus- 
 tains a heavy weight upon its summit ; and a stone, as it falls 
 freely in vacuo, no more works than do the planets as they 
 wheel unresisted through space.* 
 
 49. THE UNIT OF WORK. The unit of work used in this 
 country, in terms of which to estimate every other amount 
 
 * Note (,; ) Ed. App. 
 
VARIABLE WORK. 47 
 
 of work, is the work necessary to overcome a pressure of one 
 pound through a distance of one foot, in a direction opposite 
 to that in which a pressure acts. Thus, for instance, if a 
 pound weight be raised through a vertical height of one foot, 
 one unit of work is done ; for a pressure of one pound is 
 overcome through a distance of one foot, in a direction oppo- 
 site to that in which the pressure acts. 
 
 50. The number of units of work necessary to overcome a 
 pressure of M pounds through a distance of 1ST feet, is 
 equal to the product MN. 
 
 For since, to overcome a pressure of one pound through 
 one foot requires one unit of work, it is evident that to over- 
 come a pressure of M pounds through tho same distance of 
 one foot, will require M units. Since, then, M units of work 
 are required to overcome this pressure through one foot, it 
 it evident that N" times as many units (i. e. KM) are required 
 to overcome it through IN" feet. Thus, if we take U to repre- 
 sent the number of units of work done in overcoming a con- 
 stant pressure of M pounds through N feet, we have 
 
 (39).* 
 
 51. To ESTIMATE THE WOEK DONE UNDER A VARIABLE 
 PRESSURE. 
 
 Let PC be a curved line and AB its axis, such that any 
 one of its abscissae AM 3 , containing as many 
 equal parts as there are units in the space 
 through which any portion of the work has 
 M M been done, the corresponding ordinate M S P 3 
 
 may contain as many of those equal parts, 
 as there are in the pressure under which it is then being 
 done. Divide AB into exceedingly small equal parts, AM 19 
 MjM 2 , &c., and draw the ordinates MjPj, M 2 P 2 , &c. ; then if 
 we conceive the work done through the space AMj (which 
 is in reality done under pressures varying from AP to M,?,), 
 to be done uniformly under a pressure, which is the arith- 
 metic mean between AP and M^, it is evident that the 
 number of units in the work done through that small space 
 will equal the number of square units in the trapezoid 
 APPjM, (see Art. 45.), and similarly with the other trape- 
 
 * Note (fc) Ed. App. 
 
48 THE RESOLUTION 
 
 zoids ; so that the number of units in the whole work done 
 through the space AB will equal the number of square units 
 in the whole polygonal area APPff^ &c., CB. 
 
 But since AM,, M.M^ &c., are exceedingly small, 'this 
 polygonal area passes into the curvilinear area APCB ; the 
 whole work done is therefore represented by the number of 
 square equal parts in this area. 
 
 Now, generally, the area of any curve is represented by 
 
 the integral I ydx, where y represents the ordinate, and x 
 
 the corresponding abscissa. But in this case the variable 
 pressure P is represented by the ordinate, and the space S 
 described under this variable pressure by the abscissa. If 
 therefore U represent the work done between the values S a 
 and S a of S, we have. 
 
 s 
 
 (40). 
 
 8, 
 
 Mean pressure is that under which the same work would 
 be done over the same space, provided that pressure, instead 
 of varying throughout that space, remained 
 the same : thus, the mean pressure in re- 
 spect to an amount of work represented by 
 the curvilinear area AEFC, is that under 
 which an amount of work would be done 
 represented by the rectilineal area ABDC, the area ABDC 
 being equal to the curvilinear area AEFC ; the mean pres- 
 sure in this case is represented by AB. Thus, to determine 
 the mean pressure in any case of variable pressure, we have 
 only to find a curvilinear area representing the work done 
 under that variable pressure, and then to describe a rectan- 
 .gular parallelogram on the same base AC, which shall have 
 an area equal to the curvilinear area. 
 
 If S represent the space described under a variable pres- 
 sure, U the work done, and p the mean pressure, then 
 
 pS = U, therefore p = -^ .* 
 
 52. To estimate the work of a pressure, whose direction is not 
 that in which its point of application is made to move. 
 
 Hitherto the work of a force has been estimated only on 
 
 * Note (I) Ed. App. 
 
OF WOBK. 
 
 the supposition that the point of applica- 
 - tion of that force is moved in the direction 
 
 in which the force operates, or in the oppo- 
 
 site direction. Let PQ be tfye direction of 
 a pressure, whose point of application Q 
 is made to move in the direction of the 
 straight line AB. Suppose the pressure P to remain con- 
 stant, and its direction to continue parallel to itself. It is 
 required to estimate the work done, whilst the point of 
 application has been moved from A to Q. 
 
 Eesolve P into R and S, of which R is parallel and S per- 
 pendicular to AB. Then since no motion takes place in the : 
 direction of SQ, the pressure S does no work, and the whole.- 
 work is done by R ; therefore the work = R . AQ. 
 
 ]STow R=P . cos. PQR, therefore the work=:P . AQcos;. 
 PQR. From the point A draw AM perpendicular to PQ> 
 then AQ cos. PQR^QM ; therefore work^P . QM. There- 
 fore the work of any pressure as above, not acting ia the 
 direction of the motion of the point of application of that 
 pressure, is the same as it would have been if the point of 
 application had been made to move in the direction of the 
 pressure, provided that the space through which it was so 
 moved had been the projection of the space through which 
 
 it actually moves. The product P . QM may be called the 
 work of r resolved in the direction of P. 
 
 The above proposition which has been proved, whatever 
 may be the distance through which the point of application 
 is moved, in that particular case only in which the pressure 
 remains the same in amount and always parallel to itself, is 
 evidently true for exceedingly small spaces of motion, even 
 if the pressure be variable both in amount and direction ; 
 since for such exceedingly small variations in the positions 
 of the points of application, the variations of the pressures 
 themselves, both in amount and direction, arising from these 
 variations of position, must be exceedingly small, and there- 
 fore the resulting variations in the work exceedingly small 
 as compared with the whole work.* 
 
 * Note (m) Ed. App. 
 
50 THE WOKK OF 
 
 53. If any number of pressures P,, P 2 , P 3 , le applied to the 
 same point A, and remain constant and parallel to them- 
 selves, whilst the point A is made to move through the 
 straight lirfe AB, then the whole work done is equal to the 
 sum o the works of the different pressures resolved in the 
 directions of those pressures, each being taken negatively 
 whose point of application is made to move in an opposite 
 to the pressure upon it. 
 
 Let a i? a 25 a g? &c. represent the inclinations of the pres- 
 sures P,, P 2 , &c. to the line AB, then will 
 the resolved parts of these pressures in the 
 direction of that line be Y l cos. a l? P 2 cos. 
 2 , P 3 cos. 3 , &c. and they will be equiva- 
 lent to a single pressure in the direction 
 of that line represented by P, cos. t -f P 2 
 cos. a 2 + P 3 cos. a 35 &c. in which sum all 
 those terms are to be taken negatively which involve pres- 
 sures whose direction is from B towards A (since the single 
 pressure from A towards B is manifestly equal to the differ- 
 ence between the sum of those resolved pressures which act 
 in that direction, and those in the opposite direction). There- 
 fore the whole work is equal to jPj cos. ^ + P 2 cos. 2 -f- P 3 
 cosv ..... }. AB = P, . AB cos, a, + P 2 . AB cos. a 2 
 
 + P 3 ABcos. a 3 + ... =P 1 
 
 ; in which expression the successive terms are the 
 
 works of the different pressures resolved in the several 
 directions of those pressures, each being taken positively or 
 negatively, according as the direction of the corresponding 
 pressure is towards the direction of the motion or opposite 
 to it. 
 
 Thus if U represent the whole work and Uj and U 2 the 
 sums of those done in opposite directions, then 
 U=U,-U, (41). 
 
 54. If any number of pressures applied to a point he in equi- 
 librium, and their point of application he moved, the whole 
 work done by these pressures in the direction of the motion 
 will equal the whole work done in the opposite direction. 
 
 For if the pressures P,, P 2 , P 3 , &c, (Art. 53) be in equi- 
 librium, then the sums of their resolved pressures in opposite 
 
 * Note (n) Ed. App. 
 
CENTRAL FORCES. 51 
 
 directions along AB will be equal (Art. 10) ; therefore the 
 whole work U along AB, which by the last proposition is 
 equal to the work of a pressure represented by the difference 
 of these sums, will equal nothing, therefore = tJ, U 2 , 
 therefore IT, U 2 , that is, the whole work done in one direc- 
 tion along AB, by the pressures P,, P 2 , &c. is equal to the 
 whole work done in the opposite direction. 
 
 55. If a body be acted upon by a force whose direction is 
 always towards a certain point S, called a centre of force, 
 and be made to describe any given curve PA in a direction 
 opposed to the action of that force, and Sp be measured on 
 SA equal to SP, then will the work done in moving the 
 body through the curve PA be equal to that which would 
 be necessary to move it in a straight line from p to A. 
 
 For suppose the curve PA to be a portion of a polygon of 
 an infinite number of sides, PP,, P,P 2 , &c. 
 Through the points P, P,, P 2 , &c. describe circu- 
 lar arcs with the radii SP, SP,, SP 2 , &c. and let 
 them intersect S A in p, p^ p^ &c. Then since 
 PP, is exceedingly small, the force may be consi- 
 dered to act throughout this space always in a 
 direction parallel to SP, ; therefore the work done 
 through PP, is equal to the work which must be 
 done to move the body through the distance raP, (Art. 52.), 
 since mP, is the projection of PP, upon the direction SP, of 
 the force. But mP l =pp l ; therefore the work done through 
 PP, is equal to that which would be required to move the 
 body along the line S A through the distance pp^ ; and simi- 
 larly the work done through P,P 2 is equal to that which 
 must be done to move the body through p^p so that 
 the work through PP 2 is equal to that through pp^ and so 
 of all other points in the curve. Therefore the work through 
 PA is equal to that through pA.* Therefore, &c. [Q.E.D.] 
 
 * Of course it is in this proposition supposed that the force, if it be not 
 constant, is dependant for its amount only on the distance of the point at 
 which it acts from the centre of force S ; so that the distances of p and P 
 from S being the same, the force at p is equal to that at P ; similarly the 
 force at pi is equal to that at P 1} the force at p* equal to that at P 2 , &c. 
 
52 THE WOBK OF 
 
 56. If S ~be at an exceedingly great distance as compared 
 with AJ?, then all the lines drawn from S to AP may ~be con- 
 sidered parallel. This is the case with the force of gravity 
 at the surface of the earth, which tends towards a point, the 
 earth's centre, situated at an exceedingly great distance, as 
 compared with any of the distances through which the work 
 of mechanical agents is usually estimated. 
 
 Thus then it follows that the work necessary to move a 
 heavy body up any curve PA, or inclined plane, is the same 
 as would be necessary to raise it in a vertical line pA. to the 
 same height. 
 
 The dimensions of the body are here supposed to be ex- 
 ceeding small. If it be of considerable dimensions, then 
 whatever be the height through which its centre of gravity 
 is raised along the curve, the work expended is the same 
 (Art 60.) as though the centre of gravity were raised verti- 
 cally to that height.* 
 
 57. In the preceding^ propositions the work has been esti- 
 mated on the supposition that the body is made to move so 
 as to increase its distance from the centre S, or in a direction 
 opposed to that of the force impelling it towards S. It is 
 evident, nevertheless that the work would have been precisely 
 the same, if instead of the body moving from P to A it had 
 moved from A to P, provided only that in this last case 
 there were applied to it at every point such a force as would 
 prevent its motion from being accelerated by the force con- 
 tinually impelling it towards S ; for it is evident that to pre- 
 vent this acceleration, there must continually be applied to 
 the body a force in a direction from S equal to that by which 
 it is attracted towards it ; and the work of such a force is 
 manifestly the same, provided the path be the same, whether 
 the body move in one direction or the other along that path, 
 being in the two cases the work of the same force over the 
 same space, but in opposite directions. 
 
 * The only force acting upon the body is in this proposition supposed to be 
 that acting towards S. No account is taken of friction or any other forces 
 "which oppose themselves to its motion. 
 
PARALLEL FORCES. 53 
 
 58. If there ~be any number of parallel pressures, P n P aJ P 3 , 
 &c. whose points of application are transferred, each 
 through any given distance from one position to another, 
 then is the work which would he necessary to transfer their 
 resultant through a space equal to that ~by which their 
 centre of pressure is displaced in this change of position, 
 equal to the difference between the aggregate work of those 
 pressures whose points of application have been moved in 
 the directions in which the pressures applied to them act, 
 and those whose points of application have been moved in 
 the opposite directions to their pressures. 
 
 For (Art. IT.), if y 15 y a , ?/ 3 , &c. represent the distances of 
 the points of application of these pressures from any given 
 plane in their first position, and h the distance of their centre 
 of pressure from that plane, and if Y t , Y 2 , Y 8 &c. and H re- 
 present the corresponding distances in the second position, 
 and if P 15 P 2 , P 3 , &c. be taken positively or negatively ac- 
 cording as their directions are from or towards the given 
 plane, h { 
 
 , (Y 3 -y 2 ) 
 
 '+P 3 (Y 3 -y 3 ) + ..... (42); 
 
 in the second member of which equation the several terms 
 are evidently positive or negative, according as the pressure 
 P corresponding to each, arid the difference Y y of its dis- 
 tances from the plane in its two positions, have the same or 
 contrary signs. Now by supposition P is positive or negative 
 according as it acts from or towards the plane ; also Y y is 
 evidently positive or negative according as the point of appli- 
 cation of P is moved from or towards the plane ; each term 
 is therefore positive or negative, according as the correspond- 
 ing point of application is transferred in a direction towards 
 that in which its applied pressure acts, or in the opposite 
 direction. 
 
 Now the plane from which the distances of the points of 
 application are measured may be any plane whatever. Let 
 it be a plane perpendicular to the directions of the pressures. 
 
54 THE WORK OF 
 
 Let A.xy represent this plane, and let P 
 P' represent the two positions of the point 
 of application of the pressure P (the path 
 described by it between these two positions 
 having been any whatever). Let MP and 
 M'P' represent the perpendicular dis- 
 tances of the points P and P' from the 
 plane, and draw Pm from P perpendicular 
 
 to M'P'. Then P (Y y)=P(M'P'-MP)^P . mP'; but by 
 Art. 55., P . mP' equals the work of P as its point of applica- 
 tion is transferred from P to P'. Thus each term of the second 
 member of equation (42) represents the work of the corre- 
 sponding pressure, so that if 2-z^, represent the aggregate 
 work of those pressures whose points of application are trans- 
 ferred towards the directions in which the pressures act, and 
 2^ 2 the work of those whose points of application are moved 
 opposite to the directions in which they severally act, then 
 the second member of the equation is represented by 2^ 
 2-z/, a . Moreover the first member of the equation is evidently 
 the work necessary to transfer the resultant pressure P 2 + 
 P 2 -f P 3 &c. through the distance H A, which is that by 
 which the centre of pressure is removed from or towards the 
 given plane, so that if U represent the quantity of work 
 necessary to make this transfer of the centre of pressure, 
 
 11=2^-2^ (43). 
 
 t 
 
 59. If the sum of those parallel pressures whose tendency 
 is in one direction equal the sum of those whose tendency 
 
 is in the opposite direction, then Pj + Pg + Pg-j- =0. 
 
 In this case, therefore, 11=0, therefore 2^2^0, there- 
 fore 2^= 2^ 2 ; so that when in any system of parallel pres- 
 sures the sum of those whose tendency is in one direction 
 equals the sum of those whose tendency is in the opposite direc- 
 tion, then the aggregate work of those whose points of appli- 
 cation are moved in the directions of the pressure severally 
 applied to them is equal to the aggregate work of those whose 
 points of application are moved in the opposite directions. 
 
 This case manifestly obtains when the parallel pressures 
 are in EQUILIBRIUM, the sum of those whose tendency is in 
 one direction then equalling the sum of those whose tendency 
 is in the opposite direction, since otherwise, when applied to 
 a point, these pressures could not be in equilibrium about 
 that point (Art. 8.). 
 
PAEALLEL FORCES. 55 
 
 60. The preceding proposition is manifestly true in respect 
 to a system of weights, these being pressures whose directions 
 are always parallel, wherever their points of application may 
 be moved. 'Now the centre of pressure of a system of 
 weights is its centre of gravity (Art. 19). Thus then it fol- 
 lows, that if the weights composing such a system be sepa- 
 rately moved in any directions whatever, and through any 
 distances whatever, then the difference between the aggre- 
 gate work done upwards in making this change of relative 
 position and that done downwards is equal to the work 
 necessary to raise the sum of all the weights through a height 
 equal to that through which their centre of gravity is raised 
 or depressed.* Moreover that if .such a system of weights 
 be supported in equilibrium by the resistance of any fixed 
 point or points, and be put in motion, ttVn (since the work 
 of the resistance of each such point is nothing) the aggregate 
 
 * This proposition has numerous applications. If, for instance, it be required 
 to determine the aggregate expenditure of work in raising the different ele- 
 ments of a structure, its stone, cement, &c., to the different positions they 
 occupy in it, we make this calculation by determining the work requisite to 
 raise the whole weight of material at once^to the height of the centre of gra- 
 vity of the structure. If these materials have been carried up by labourers, and 
 we are desirous to include the whole of their labour in the calculation, we 
 ascertain the probable amount of each load, and conceive the weight of a la- 
 bourer to be added to each load, and then all these at once to be raised to the 
 height of the centre of gravity. 
 
 Ag:iin, if it be required to determine the expenditure of work made in rais- 
 ing the material excavated from a well, or in pumping the water out of it, we 
 know that (neglecting the effect of friction, and the weight and rigidity of the 
 cord) this expenditure of work is the same as though the whole material had 
 been raised at one lift from the centre of gravity of the shaft to the surface. 
 Let us take another application of this principle which offers so many practical 
 results. The material of a railway excavation of considerable length is to be 
 removed so as to form an embankment across a valley at some distance, and it 
 is required to determine the expenditure of work made in this transfer of th3 
 material. Here each load of material is made to traverse a different distance, 
 a resistance from the friction, &c., of the road being continually opposed to its 
 motion. These resistances on the different loads constitute a system of paral- 
 lel pressures, each of whose points of application is separately transferred fro n 
 one given point to another given point, tb.e directions of transfer being als > 
 parallel. Now by the preceding proposition, the expenditure of work in all 
 these separate transfers is the same as it would have been had a pressure equil 
 to the sum of all these pressures been at once transferred from the centre of 
 resistance of the excavation to the centre of resistance of the embankment. 
 Now the resistances of the parts of the mass moved are the frictions of its ele- 
 ments upon the road, and these frictions are proportional to the weights of the 
 elements ; their centre of resistance coincides therefore with the centre of gra- 
 vity of the mass, and it follows that the expenditure of work is the same as 
 though all the material had been moved at once from the centre of gravity of 
 the excavation to that of the embankment. To allow for the weight of the 
 carriages, as many times the weight of a carriage must be added to the weight 
 of the material as there are journeys made. 
 
56 STABILITY OF THE CENTRE. 
 
 work of those weights which are made to descend, is equal 
 to that of those which are made to ascend. 
 
 61. If a plane ~be taken perpendicular to the directions of any 
 number of parallel pressures and there he two different po- 
 sitions ojr the points of application of certain of these pi^es- 
 sures in which they are at different distances from the 
 plane-, whilst the points of application of the rest of tJwse 
 pressures remain at the same distance from that plane, 
 and if in both positions the system be in equilibrium, then 
 the centre of pressure of the first mentioned pressures will 
 be at the same distance from the plane in both positions. 
 
 For since in both positions the system is in equilibrium, 
 therefore in both positions Pj + P 2 + P 8 + ... =0, 
 
 Now let P w be any one of the pressures whose points ol appli- 
 cation is at the same distance from the given plane in both 
 positions, 
 
 Y n =y n , and Y.-y. = 0, 
 ..(Y 1 - 2 / 1 )P 1 + (Y 2 - 
 .-. Y^+YJP, +. . . 
 . Y 1 P 1 +Y 2 P 2 + . . . 
 
 where H w _, represents the distance of the centre of pressure 
 of P 15 P 2 . . . P n _,, from the given plane in the first position, 
 and A n _ 1 its distance in the second position. Its distance in 
 the first position is therefore the same as in the second, 
 Therefore, &c. 
 
 From this proposition, it follows that if a system of weights 
 be supported by the resistances of one or more fixed points, 
 and if there be any two positions whatever of the weights in 
 both of which they are in equilibrium with the resistances 
 of those points, then the height of the common centre of 
 gravity of the weights is the same in both positions. And 
 that if there be a series of positions in all of which the 
 weights are in equilibrium about such a resisting point or 
 points, then the centre of gravity remains continually at the 
 same height as the system passes through this series of posi- 
 tions. 
 
 If all these positions of equilibrium be infinitely near to 
 
WORK OF PRESSURES. 57 
 
 one another, then it is only during an infinitely small motion 
 of the points of application that the centre of gravity ceases 
 to ascend or descend ; and, conversely, if for an infinitely 
 small motion of the points of application the centre of 
 gravity ceases to ascend or descend, then in two or more 
 positions of the points of application of the system, infi- 
 nitely near to one another, it is in equilibrium. 
 
 WORK OF PRESSURES APPLIED IN DIFFERENT DIRECTIONS TO 
 A BODY MOVEABLE ABOUT A FIXED AxiS. 
 
 62. The work of a pressure applied to a body moveable about 
 a fixed axis is the same at whatever point in its proper 
 direction that pressure may be applied. 
 
 For let AB represent the direction of a pressure applied 
 to a body moveable about a fixed axis 
 O ; the work done by this pressure 
 will be the same whether it be ap- 
 plied at A or B. For conceive the 
 body to revolve about O, through an 
 exceedingly small angle AOC, or 
 BOD, so that the points A and D may describe circular arcs 
 AC and BD. Draw Cm, Dn, and OE, perpendiculars to 
 AB, then if P represent the pressure applied to AB, P . Am 
 will represent the work done by P when applied at A (Art. 
 52.), and P . l&n will represent the work done by P when 
 applied at B ; therefore the work done by P at A is the same 
 as that done by P at B, if Am is equal to B^. 
 
 Now AC and BD being exceedingly small, they may be 
 conceived to be straight lines. Since BD and BE are 
 respectively perpendicular to OB and OE, therefore /DBE 
 / BOE ;'* and because AC and AE are perpendicular to 
 O A and OE, therefore / CAE = /_ AOE. Now Am = 
 
 CA 
 CA . cos. CAE = CA . cos. AOE = ^ . OA . cos. AOE 
 
 PA 
 
 = 7 . - x OE. Similarly B^ = DB cos. DBE = DB . cos. 
 
 BOE = 55 OB cos. BOE = ^ x OE, i.e. Am = OE . 
 
 * It is a well-known principle of Geometry, that if two lines be inclined at 
 any angle, and any two others be drawn perpendicular to these, then the incli- 
 nation of the last two to one another shall equal that of the first two. 
 
58 THE ACCUMULATION OF WORK. 
 
 CA n BD CA BD 
 
 PT-T-, and tin= OE -^5. But 7 = 7 ^^ since the /_ AOC 
 
 V/JTX \)JL> W-OL \JJ 
 
 / BOD, therefore Am = ~Bn* 
 
 63. If any number of pressures oe ^n equilibrium about a 
 fixed axis, then the whole work of those which tend to move 
 the system in one direction about that axis is equal to the 
 whole work of those which tend to move it in the opposite 
 direction about the same axis. For let P be any one of such 
 a system of pressures, and O a fixed axis, and OM perpen- 
 dicular to the direction of P, then whatever may be the 
 point of application of P, the work of that pressure is the 
 same as though it were applied at M. Suppose the whole 
 system to be moved through an exceeding small 
 angle d about the point O, and let OM be repre- 
 sented by Pi then will pQ represent the space 
 described by the point M, which will be actually 
 in the direction of the force P, therefore the work 
 of P=P . p . 6. Now let P,, P 2 , P,, &c. represent those 
 pressures which act in the direction of the motion, and P' x , 
 P' 2 , &c. those which act in the opposite direction, and let 
 Pupvpv &c. be the perpendiculars on the first, and j/ l5 p' n 
 p ' &c. be the perpendiculars on the second ; therefore by 
 the principle of the equality of moments Pj^ + P^ + ^^P* 
 f &c. = P'^ + P' 2 y 2 + P' 8 y 3 + &c. ; therefore multiplying 
 both sides by 0, P j>^ + Pj^ + P 3 ^ = P',^ + P'^pV -f 
 P'^V + &c. ; but Pjj?^, P',^>'^, &c. are the works of the 
 forces P 15 P' 15 &c. ; therefore the aggregate work of those 
 which tend to move the system in one direction is equal to 
 the aggregate of those which tend to move it in the opposite 
 direction. 
 
 64. THE ACCUMULATION OF WOEK IN A MOVING BODY. 
 
 In every moving bod^ there is accumulated, by the action 
 of the forces whence its motion has resulted, a certain 
 amount of power which it reproduces upon any resistance 
 opposed to its motion, and which is measured by the work 
 done by it upon that obstacle. Not to multiply terms, we 
 shall speak of this accumulated power of working, thus 
 measured by the work it is capable of producing, as ACCU- 
 MULATED WOEK. It is in this sense that in a ball fired from 
 
 * Note (o) Ed. App. 
 
THE ACCUMULATION OF WORK. 59 
 
 a cannon there is understood to be accumulated the work it 
 reproduces upon the obstacles which it encounters in its 
 flight ; that in the water which flows through the channel 
 of a mill is accumulated the work which it yields up to the 
 wheel ; * and that in the carriage which is allowed rapidly 
 to descend a hill is accumulated the work which carries it a 
 considerable distance up the next hill. It is when the pres- 
 sure under which any work is done, exceeds the resistance 
 opposed to it, that the work is thus accumulated in a moviftg 
 body ; and it will subsequently be shown (Art. 69.) that in 
 every case the work accumulated is precisely equal to the 
 work done upon the body beyond that necessary to over- 
 come the resistances opposed to its motion, a principle 
 which might almost indeed be assumed as in itself evident. 
 
 65. The amount of work thus accumulated in a body 
 moving with a given velocity, is evidently the same, what- 
 ever may have been the circumstances under which its 
 velocity has been acquired. Whether the velocity of a ball 
 has been communicated by projection from a steam gun, or 
 explosion from a cannon, or by being allowed to fall freely 
 from a sufficient height, it matters not to the result ; pro- 
 vided the same velocity be communicated to it in all three 
 cases, and it be of the same weight, the work accumulated 
 in it, estimated by the effect it is capable of producing, is 
 evidently the same. 
 
 In like manner, the whole amount of work which it is 
 capable of yielding to overcome any resistance is the same, 
 whatever may be the nature of that resistance. 
 
 66. To ESTIMATE THE NUMBER OF UNITS OF WORK ACCUMU- 
 LATED IN A BODY MOVING WITH A GIVEN VELOCITY. 
 
 Let w be the weight of the body in pounds, and v its 
 velocity in feet. 
 
 Now suppose the body to be projected with the velocity v 
 in a direction opposite to gravity, it will ascend to the height 
 h from which it must have fallen, to acquire that same velo- 
 city v (Art. 47.); there must then at the instant of projection 
 have been accumulated in it an amount of work sufficient to 
 raise it to this height h ; but the number of units of work 
 
 * This remark applies more particularly to the under-shot wheel, which is 
 carried round by the rush of the water. 
 
60 THE ACCUMULATION OF WORK. 
 
 requisite to raise a weight w to a height A, is represented by 
 wh ; this then is the number of units of work accumulated 
 in the body at the instant of projection. But since h is the 
 height through which the body must fall to acquire the velo- 
 
 city v, therefore u 2 %gh (Art. 47.) ; therefore h=% ; whence 
 
 it follows that if U represent the number of units of work 
 accumulated, 
 
 nn 
 
 Moreover it appears by the last article that this expression 
 represents the work accumulated in a body weighing w 
 pounds, and moving with a velocity of v feet, whatever may 
 have been the circumstances under which that velocity was 
 accumulated. 
 
 The product j'W 2 is called the vis VIVA of the body, so 
 that the accumulated work is represented by half the vis 
 viva, the quotient ( J is called the MASS of the body.* 
 
 67. To estimate the work accumulated in a body, or lost by 
 it, as it passes from one velocity to another. 
 
 In a body whose weight is w, and which moves with a 
 velocity v there is accumulated a number of units of work 
 
 w 
 represented (Art. 66.) by the formula v\ After it has 
 
 passed from this velocity to another V, there will be accumu- 
 
 w 
 lated in it a number of units of work, represented by ^ V 2 , 
 
 so that if its last velocity be greater than the first, there 
 will have been added to the work accumulated in it a num- 
 
 ber of units represented by J Y 2 J v 2 ; or if the second 
 
 velocity be less than the first, there will have been taken 
 from the work accumulated in it a number of units repre- 
 
 sented by i> 2 J V 2 . So that generally if U represent 
 
 the work accumulated or lost by the body, in passing from 
 the velocity v to the velocity Y, then 
 
 * Note (p), Ed. App. 
 
THE ACCUMULATION OF WOKK. 61 
 
 U=i-{y-t^ ---- (45), 
 
 where the sign is to be taken according as the motion is 
 accelerated or retarded. 
 
 68. The work accumulated in a body, whose motion is accele- 
 rated through any given space by given forces is equal to 
 the work which it would he necessary to do upon the body 
 to cause it to move hack again through the same space 
 when acted upon by the same forces. 
 
 For it is evident that if with the velocity which a body 
 has acquired through any space AB by the 
 action of any forces whose direction is from A 
 towards B, it be projected back again from B 
 towards A, then as it returns through each 
 successive small part or element of its path, it 
 will be retarded by precisely the same forces as those by 
 which it was accelerated when it before passed through it ; 
 so that it will, in returning through each such element, lose 
 the same portion of its velocity as before it gained there ; 
 and when at length it has traversed the whole distance BA, 
 and reached the point A, it will have lost between B and A 
 a velocity, and therefore an amount of work (Art. 67.), 
 precisely equal to that which before it gained between A 
 and B. Now the work lost between B and A is the work 
 necessary to overcome the resistances opposed to the motion 
 through 'B A. The work accumulated from A to B is there- 
 fore equal to the work which would be necessary to over- 
 come the resistances between B and A, or which would be 
 necessary to move the body from a state of rest, and with a 
 uniform motion, in opposition to these resistances, through 
 BA. Let this work be represented by U ; also let v be the 
 velocity with which the body started from A, and Y that 
 
 which it has acquired at B. Then will J (Y 3 v*) repre- 
 sent the work accumulated between A and B, 
 
 If the body, instead of being accelerated, had been 
 retarded, then the work lost being that expended in over- 
 coming the retarding forces, is evidently that necessary to 
 
62 THE ACCUMULATION OF WORK. 
 
 move the body uniformly in opposition to these retarding 
 forces through AB ; so that if this force be represented by 
 
 w 
 
 TJ, then, since -| (V 2 V 2 ) is in this case the work lost, we 
 
 t/ 
 
 shall have v* Y 2 =^-. Therefore, generally, 
 
 (46), 
 
 where the sign is to be taken according as the motion is 
 accelerated or retarded. 
 
 69. The work accumulated in a body which has moved 
 through any space acted upon by any force, is equal to the 
 excess of the work which has been done upon it by those 
 forces which tend to accelerate its motion above that which 
 has been done upon it by those which tend to retard its 
 motion. 
 
 For let R be the single force which would at any point P 
 (see last fig.) be necessary to move the body back again 
 through an exceeding small element of the same path (the 
 other forces impressed upon it remaining as before) ; then it 
 follows by Art. 54:. that the work of R over this element of 
 the path is equal to the excess of the work over that 
 element of the forces which are impressed upon the body in 
 the direction of its motion above the work of those 
 impressed in the opposite direction. Now this is true at 
 every point of the path ; therefore the whole work of the 
 force II necessary to move the body back again from B to A 
 is equal to the excess of the work done upon it, by the 
 impressed forces in the direction of its motion, #bove the 
 work done upon it by them in a direction opposed to its 
 motion ; whence also it follows, by the last proposition, that 
 the accumulated work is equal to this excess. There- 
 fore, &c. 
 
 *70. If P represent the force in the direction of the 
 motion which at a given distance S, measured along the 
 path, acts to accelerate the motion of the body, this force 
 being understood not to be counteracted by any other, or to 
 be the surplus force in the direction of the motion over and 
 
THE ACCUMULATION OF WORK. (>3 
 
 above any resistance opposed to it, then will f PdS be the 
 
 o 
 
 work which must be done in an opposite direction to over- 
 
 s 
 come this force through the space S, or U f 
 
 o 
 
 by equation (46), Y 2 -V:= - ...... (47). 
 
 71. If the force P tends at first towards the direction in 
 which the body moves, so as to accelerate the motion, and 
 if after a certain space has been described it changes its 
 direction so as to retard the motion, and U 1 represent the 
 value of U in respect to the former motion, and Y x the 
 velocity acquired when that motion has terminated, whilst 
 U 2 is the value of U in respect to the second or retarded 
 motion, and if v be the initial and Y the ultimate or actual 
 velocity, then 
 
 V-V- 
 
 " " 
 
 W ' 
 
 . ; . . . . (48). 
 
 As U 2 increases, the actual velocity Y of the body con- 
 tinually diminishes ; and when at length U^Uj, that is 
 when the whole work done (above the resistances) in a 
 direction opposite to the motion, comes to equal that done, 
 before, in the direction of the motion, then Y=v, or the 
 velocity of the body returns again to that which it had 
 when the force P begjan to act upon it. This is that gene- 
 ral case of reciprocating motion which is so frequently pre- 
 sented in the combinations of machinery, and of which the 
 crank motion is a remarkable example. 
 
 If the force which accelerates the body's motion act 
 always towards the same centre S, and SJ be taken equal to 
 
64: THE ACCUMULATION OF WOKK. 
 
 SB, it has been shown (Art 55.) that the work 
 necessary to move the body along the curve from 
 B to A, 'is equal to that which would be necessary 
 to move it through the straight line &A. The 
 accumulated work is therefore equal to that neces- 
 sary to move the body through the difference 5A 
 of the two distances SA and SB (Art. 68.). If these 
 distances be represented by 3^ and R a , and P 
 represent the pressure with which the body's motion along 
 JA would be resisted at any distance R from the point S, 
 
 RI 
 then / ~PdH will represent this work. Moreover the work 
 
 R<2 
 
 accumulated in the body between A and B is represented 
 
 by -J (Y 2 v*), if Y represent the velocity at B and v that 
 
 at A, 
 
 73. The work accumulated in the body while it descends 
 the curve AB, is the same as that which it would acquire in 
 falling directly towards S through the distance A5, for both 
 of these are equal to the work which w^ould be necessary to 
 raise the body from b to A. Since then the work accumu- 
 lated by the body through AB is equal to that which it would 
 accumulate if it fell through A5, it follows that velocity 
 acquired by it in falling, from rest, through AB is equal to 
 that which it would acquire in falling through AZ>. For if 
 Y represent the velocity acquired in the one case, and Y t 
 that in the other, then the accumulated work in the first case 
 
 W W 
 
 is represented by $ Y a , and that in the second case by -J- Y! a , 
 
 W W 
 
 therefore i Y a = J Y, 8 , therefore Y= V,. 
 
 From this it follows, that if a body descend, being pro- 
 jected obliquely into free space, or sliding from rest upon 
 any curved surface or inclined plane, and be acted upon only 
 by the force of gravity (that is, subject to no friction or 
 resistance of the air or other retarding cause), then the velo- 
 
TIIE ACCUMULATION OF WORK. 65 
 
 city acquired by it in its descent is precisely the same as 
 though it had fallen vertically through the same height. 
 
 74. DEFINITION. The ANGULAR VELOCITY of a body which 
 rotates about a fixed axis is the arc which every particle of 
 the body situated at a distance unity from the axis describes 
 in a second of time, if the body revolves uniformly ; or, if 
 the body moves with a variable motion, it is the arc which it 
 would describe in a second of time if (from the instant when 
 its angular velocity is measured) its revolution were to 
 become uniform. 
 
 75. THE ACCUMULATION OF WORK IN A BODY WHICH, 
 ROTATES ABOUT A FIXED AXIS. 
 
 Propositions 68 and 69 apply to every case of the motion 
 of a heavy body. In every such case the work accumulated 
 or lost by the action of any moving force or pressure, whilst 
 the body passes from any one position to another, is equal 
 to the work which must be done in an opposite direction, to 
 cause it to pass back from the second position into the first. 
 Let us suppose U to represent this work in respect to a body 
 of any given dimensions, which has rotated about a fixed 
 axis from one given position into another, by the action of 
 given forces. 
 
 Let , be taken to represent the ANGULAR VELOCITY of the 
 body after it has passed from one of these positions into 
 another. Then since a is the actual velocity of a particle at 
 distance unity from the axis, therefore the velocity of a par- 
 ticle at any other distance p t from the axis is ap,. Let j* 
 represent the weight of each unit of the volume of the body, 
 and m t the volume of any particle whose distance from the 
 axis is pj, then will the weight of that particle be pm l ; also 
 its velocity has been shown to be ap 15 therefore the amount 
 of work accumulated in that particle is represented by 
 
 -^, or by ia 2 
 
 Similarly the different amounts of work accumulated in 
 the other particles or elements of the body whose distances 
 from the axis are represented by p 2 , p 3 , . . . and their 
 
 i* 
 
 volumes by m 2 , m a , ra< . . . ., are represented by i a2 -^ a p a a , 
 
 y 
 
 a 3 - 77^p 3 3 , &c. ; so that the whole work accumulated is repre- 
 y 
 
66 ANGULAR VELOCITY. 
 
 U Ui ' ttt 
 
 eented by the sum ia 2 -m 1 p 1 2 + -Ja 2 -m 2 p 2 2 +-|a 2 -m 3 p 8 s + . 
 
 y . y ff 
 
 , or by a 2 - {m 1 p 1 2 +m 2 p 2 2 +m 3 p 3 2 + }. 
 
 The sum m 1 ?? + m 2 p 2 2 + m g p 3 2 + . . . ., or 2mp 2 taken in 
 respect to all the particles or elements which compose the 
 body, is called its MOMENT OF INERTIA in respect to the 
 particular axis about which the rotation takes place. Let it 
 
 be represented by I ; then will -Ja 2 . I J . I, represent the 
 
 whole amount of work accumulated in the body whilst it has 
 been made to acquire the angular velocity a from rest. If 
 therefore U represent the work which must be done in an 
 opposite direction to cause the body to pass back from its 
 last position into its first, 
 
 =# )? 
 
 If instead of the body's first position being one of rest, it 
 had in its first position been moving with an angular velocity 
 x which had passed, in its second position, into a velocity 
 a ; and if U represent, as before, the work which must be 
 done in an opposite direction, to bring this body back from 
 
 its second into its first position, then is -Ja 2 ( -j I %a* ( -j I, 
 or J ( - ) (a 2 a x s ) I, the work accumulated between the first 
 
 9 
 
 and second positions ; therefore 
 
 ('-<)!= TI, 
 
 where the sign is to be taken according as the motion is 
 accelerated or retarded between the first and second posi- 
 tions, since in the one case the angular velocity increases 
 during the motion, so that a 2 is greater than a^, whilst in the 
 latter case it diminishes, so that a 2 is less than a^. 
 
 76. If during one part of the motion, the work of the 
 
ANGULAR VELOCITY. 67 
 
 impressed forces tends to accelerate, and during another to 
 retard it, and the work in the former case be represented by 
 Uj, and in the latter by U 9 , then 
 
 From this equation it follows that when 112=11,, or when 
 the work U 2 done by the forces which tend to resist the 
 motion at length, equals that done by the forces which tend 
 to accelerate the motion, then a a n or the revolving body 
 then returns again to the angular velocity from which it set 
 out. "Whilst, if UV * never becomes equal to U, in the course 
 of a revolution, then the angular velocity a does not return 
 to its original value, but is increased at each revolution ; 
 and on the other hand, if U 2 becomes at each revolution 
 greater than U^, then the angular velocity is at each revolu- 
 tion diminished. 
 
 The greater the moment of inertia I of the revolving 
 mass, and the greater the weight f* of its unit of volume 
 (that is, the heavier the material of which it is formed), the 
 less is the variation produced in the angular velocity a by 
 any given variation of II or U 1 U 2 at different periods of 
 the same revolution, or from revolution to revolution ; that 
 is, the more steady is the motion produced by any variable 
 action of the impelling force. It is on this principle that 
 the fly-wheel is used to equalize the motion of machinery 
 under a variable operation of the moving power, or of the 
 resistance. It is simply a contrivance for increasing the 
 moment of inertia of the revolving mass, and thereby 
 giving steadiness to its revolution, under the operation of 
 variable impelling forces, 011 the principles stated above. 
 This great moment of inertia is given to the fly-wheel, by 
 collecting the greater part of its material on the rim, or 
 about the circumference of the wheel, so that the distance 
 p of each particle which composes it, from the axis about 
 which it revolves, may be the greatest possible, and thus 
 the sum 2mp 2 , or I, may be the greatest possible. ^ At the 
 same time the greatest value is given to the quantity f*, by 
 constructing the wheel of the heaviest material applicable 
 to the purpose. 
 
 What has here been said will best be understood in its 
 application to the CRANK. 
 
68 ANGULAH VELOCITY. 
 
 77. If we conceive a constant pressure Q to act upon the 
 B arm CB of the crank 
 
 ... "X^^^^P i n tne direction AB of 
 
 / VVCi > ^ :: ^3jL ^6 cran k ro( ^? an ^ a 
 
 I c @r j t '^ ^-^r~ constant resistance R 
 
 /y* to be opposed to the 
 
 revolution of the axis 
 
 C always at the same perpendicular distance from that axis, 
 it is evident that since the perpendicular distance at which 
 Q acts from the axis is continually varying (being at one 
 time nothing, and at another equal to the whole length CB 
 of the arm of the crank), the effective pressure upon the 
 arm CB must at certain periods of each revolution exceed the 
 constant resistance opposed to the motion of that arm, and 
 at other periods fall short of it ; so that the resultant of 
 this pressure and this resistance, or the unbalanced pressure 
 P upon the arm, must at one period of each revolution have 
 its direction in the direction of the motion, and at another 
 time opposite to it. Representing the work done upon the 
 arm in the one case by tF,, and in the other by U 2 , it follows 
 that if U^TJ, the arm will return in the course of each 
 revolution, from the velocity which it had when the work 
 Uj began to be done, to that velocity again when the work 
 U 2 is completed. If on the contrary \J l exceed U a , then the 
 velocity will increase at each revolution ; and if Uj be less 
 than u 2 , it will diminish. It is evident from equation (52), 
 that the greater the moment of inertia I of the body put 
 in motion, and the greater the weight M- of its unit of 
 volume, the less is the variation in the value of a, pro'duced 
 by any given variation in the value of Uj U 2 ; the less 
 therefore is the variation in the rotation of the arm of the 
 crank, and of the machine to which it gives motion, pro- 
 duced by the varying action of the forces impressed upon it. 
 Now the fly-wheel being fixed upon the same axis with the 
 crank arm, and revolving with it, adds its own moment of 
 inertia to that of the rest of the revolving mass, thereby 
 increasing greatly the value of I, and therefore, on the prin- 
 ciples stated above, equalizing the motion, whilst it does not 
 otherwise increase the resistance to be overcome, than by 
 the friction of its axis, and the resistance which the air 
 opposes to its revolution.* 
 
 * We shall hereafter treat fully of the crank and fly-wheel 
 
ANGULAR VELOCITY. 69 
 
 78. The rotation of a T)ody about a fixed axis when acted 
 upon ly no other moving force than its weight. 
 
 Let U represent the work necessary to raise it from its 
 second position into the first if it be descending, or from its 
 first into its second position if it be ascending, and let x be 
 its angular velocity in the first position, and a in the second ; 
 
 fT>m-> ~V\-tr Ck/rna-f-i r\ri ('\~\\ 
 
 then by equation (51), 
 
 "Now it has been shown (Art. 60.), that the work necessary 
 to raise the body from its second position into the first if it 
 be descending, or from its first into its second if it be 
 ascending (its weight being the only force to be overcome), 
 is the same as would be necessary to raise its whole weight 
 collected in its centre of gravity from the one position into 
 the other position of its centre of gravity. Let CA repre- 
 sent the one, and CA 1 the other position of 
 the body, and G and G^ the two correspond- 
 ing positions of the centre of gravity, then 
 will the work necessary to raise the body 
 from its position CA to its position CA 1? be 
 equal to that which is necessary to raise its 
 whole weight "W, supposed collected in G, 
 from that point to Gj ; which by Article 56, is the same as 
 that necessary to raise it through the vertical height GM. 
 
 Let now CG=CG,=:A, let CD be a vertical line through 
 C, let G.CD ^ and GCD=d, in the case in which the 
 body descends, and conversely when it ascends; therefore 
 GM=KN" 1 =CK CN^h cos. 4 h cos. ^ when the body 
 descends, or =h cos. X h cos. 6 when it ascends from the 
 position AC to AC 1? since in this last case GCD=^ and 
 Q 1 GD=6. Therefore GM=A (cos. A cos. 6\ the sign 
 being taken according as the body ascends or descends. 
 U=W . GM=WA cos. 4 cos. 
 
 .'. by equation (51) a*=ai,* + { 3\ (cos. 6 cos. ^). 
 If M represent the volume of the revolving body Mf*= W, 
 (cos. d-cos. ^) ..... (53). 
 
 When the body has descended into the vertical position, 
 
TO MOMENT OF INEKTIA. 
 
 0=0, so that (cos. 6 cos. ^)=1 cos. ^=2 sin. 2 -^. When 
 it has ascended into that position d=tf, so that (cos. 6 cos. 
 *,)= (1 + cos. ^)= 2 cos. 2 ^. 
 In the first case, therefore, 
 
 In the second case, 
 
 e.'tt ..... (55). 
 
 When the body has descended or ascended into the hori- 
 
 if 
 zontal position d=~, so that ( cos - ^ cos - ^) cos - ^i- But 
 
 it is to be observed, that if the body have descended into 
 
 the horizontal position, 0, must have been greater than ^, 
 
 2 
 
 and therefore cos. 6 l must be negative and equal to cos. 
 BCG, ; so that if we suppose 6 t to be measured from CB or 
 CD, according as the body descends or ascends, then (cos. 
 6 cos. fl,)=cos. d 1? and we have for this case of descent 
 or ascent to a horizontal position 
 
 (56.) 
 
 If the body descend from a state of rest, a l =0. 
 .-. by equation (53) <*?=-$ (cos. d cos. d,) . . . (57). 
 
 Thus the angular velocity acquired from rest is less as the 
 moment of inertia I is greater as compared with the volume 
 M, or as the mass of the body is collected farther from its 
 axis. 
 
 THE MOMENT OF INERTIA. 
 
 79. Having given the moment of inertia of a ~body, or system 
 of bodies, about an axis passing through its centre of 
 gravity, to find its moment of inertia about an axis, par- 
 allel to the first, passing through any other point in the 
 body or system. 
 
 Let m, be any element of the body or system, 7 
 
MOMENT OF INERTIA. 71 
 
 plane perpendicular to the axis, about 
 which the moments are to be measured, A 
 \ the point where this plane is intersected 
 by that axis, and G the point where it is 
 intersected by the parallel axis passing 
 through the centre of gravity of the body. Join AG, 
 A.m^ Gra,, and draw m 1 M 1 perpendicular to AG. Let 
 
 JSTow (Euclid, 212.), Am' = AG^+G^'+aAG . GM^, 
 
 if therefore the volume of the element be represented by 
 m a and both sides of the above equation be multiplied by it, 
 pj'mj = ffm^ + r 1 2 m 1 -f 2 Aa^m^ 
 
 And if m 2 , m^ m 4 , &c. represent the volumes of any other 
 elements, and p s , r 2 , a? 2 ; Pa , r 3 , a? 3 , &c. be similarly taken in 
 respect to those elements, then, 
 
 Adding these equations we have, p 1 9 m 1 + p 2 2 m 2 + p*m a + 
 ,+ m 3 + 
 
 or 
 
 ~Now 2xm is the sum of the moments of all the elements 
 of the body about a plane perpendicular to AG, and passing 
 through the centre of gravity G of the body. Therefore 
 (Art. IT.) 
 
 Also 2pV& is the moment of inertia of the body about the 
 given axis passing through A, and 2r 2 m is the moment of 
 inertia about an axis parallel to this, passing through the 
 centre of gravity of the body. Let the former moment be 
 represented by X ; and the latter by I ; and let the volume 
 of the body ^m be represented by M, 
 
 .-. ];=A f M+I ..... (58). 
 
 From which relation the moment of inertia (I,) about any 
 axis may be found, that (I) about an axis parallel to it, and 
 passing through the centre of gravity of the body being 
 known. 
 
 80. THE RADIUS OF GYRATION. If we suppose ^ to be the 
 distance from the axis passing through A, at which distance, 
 
72 MOMENT OF INERTIA. 
 
 if the whole mass of the body were collected, the moment of 
 inertia would remain the same, so that & t *M=I,, then \ is 
 called the RADIUS OF GYRATION, in respect to that axis. 
 
 If k be the radius of gyration, similarly taken in respect 
 to the axis passing through G 1? so that & a M=I, then, substi 
 tuting in the preceding equation, and dividing by M, 
 
 The following are examples of the determination of the 
 moments of inertia of bodies of some of the more common 
 geometrical forms, about the axes passing through their cen- 
 tres of gravity : they may thence be found about any other 
 axes parallel to these, by equation (58). 
 
 1. The moment of inertia of a thin uniform rod about an 
 axis perpendicular to its length and passing through its 
 middle point. 
 
 Let m represent an element of the rod contained between 
 two plane sections perpendicular to its 
 faces, the area of each of which is , and 
 whose distance from one another is Ap, 
 I and let K and Ap be so small -that every 
 
 point in this element may be considered to be at the same 
 distance p from the axis A, about which the rod revolves. 
 Then is the volume of the element represented by Ap ? and 
 its moment of inertia about A by p 2 Ap. So that the whole 
 moment of inertia I of the bar is represented by 2/cp a Ap, or, 
 since K is the same throughout (the bar being uniform), by 
 ;2p a Ap; or since Ap is infinitely small, it is represented by 
 
 the definite integral KJ p 2 <#p, where I is the whole length 
 
 \i 
 of the bar, 
 
 or I= T V^ 3 ..... (60). 
 
 *82. The moment of inertia of a thin rectangular lamina 
 about an axis, passing through its centre of gravity r , and 
 parallel to one of its sides. 
 
 It is evident that such a lamina may be conceived to be 
 
MOMENT OF INERTIA. 
 
 made up of an infinite number of slendei 
 rectangular rods of equal length, each of 
 which will be bisected by the axis AB, 
 and that the moment of inertia of the 
 whole lamina is equal to the sum of the 
 moments of inertia of these rods. Now if K be the section 
 of any rod, and I the length of the lamina, then the moment 
 of inertia of that rod is, by the last proposition, represented 
 by T V/^ 3 ; so that if the section of each rod be the same, and 
 they be n in number, then the whole moment of inertia of 
 the lamina is -j?nicF. Now UK, is the area of the transverse 
 section of the lamina, which may be represented by K, so 
 that the moment of inertia of the lamina about the axis AB 
 is represented by the formula 
 
 (61). 
 
 *83. The moment of inertia of a rectangular parallelopipe- 
 don about an axis, passing through its centre of gravity, 
 and parallel to either of its edges. 
 
 Let CD be a rectangular parallelopipedon, and AB an 
 axis passing through its centre of gravity and 
 parallel to either of its edges ; also let ab be 
 an axis parallel to the first, passing through 
 the centre of gravity of a lamina contained 
 by planes parallel to either of the faces of the 
 parallelepiped. Let a, 5, c, represent the 
 three edges ED, EF, EG, of the parallele- 
 piped, then will the moment of inertia of the lamina about 
 the axis ab be represented by T yK& 8 , where K is the trans- 
 verse section of the lamina (equation 61). Now let the 
 perpendicular distance between the two axes AB and ab be 
 represented by x. Then (by equation 58) the moment of 
 inertia of the lamina about the axis AB is represented by 
 the formula o^M+y^KJ 3 , where M represents the volume of 
 the lamina. Let the thickness of the lamina be represented 
 by AX ; .*. M ab^x, K a&x ; .*. m* in a of lam a = dbx*&x + 
 jijfl&'Aaj ; .-. whole m* in a of parallelepiped = ab'Zx'Ax + 
 jJyd^'SAoj ; or taking &x infinitely small, and representing the 
 moment of inertia of the parallelepiped by I. 
 
 / + j-c /* +|o 
 
 ! abj x*dx + j-sCtb'J dx ; 
 
 ~\c |o 
 
MOMENT OF INEKTIA. 
 
 or 1= 
 
 (62). 
 
 moment of inertia of an upright triangular 2^1 
 about a vertical axis passing through its centre of gravity. 
 
 Let AB be a vertical axis passing through the centre of 
 gravity of a prism, whose horizontal section is 
 an isosceles triangle having the equal sides ED 
 and EF. 
 
 Let two planes be drawn parallel to the face 
 DF of the prism, and containing between them 
 a thin lamina pq of its volume. Let Cm, the 
 perpendicular distance of an axis passing through 
 the centre of gravity of this lamina from the 
 axis AB, be represented by a? ; also let A# represent the 
 thickness of the lamina. 
 
 Let DF= #, DG = 5, and let the perpendicular from the 
 vertex E to the base DF of the triangle DEF be represented 
 
 :.pq = - (fc a?) ; also transverse section K of lamina = 
 c 
 
 .'.volume M of lamina = - (ftf oc)&oc. Therefore by equa- 
 
 c 
 
 tions (58) and (61), 
 
 m* in a of lam a about AB=^(fc x)x'Ax+ 7 ^ 3 (%c-x)*Ax-, 
 
 c c 
 
 :. m* in a of prism about 
 ob 
 
 C -*c 
 
 Performing the integrations here indicated, and represent- 
 ing the inertia of the prism about AB by I, we have 
 
 ...... (63). 
 
MOMENT OF LNEETIA. 
 
 *85. The moment of inertia of a solid cylinder about its 
 axis of symmetry. 
 
 Let AB be the axis of such a cylinder, whose radius AC 
 is represented by #, and its height by b. Con- 
 ceive the cylinder to be made up of cylindrical 
 rings having the same axis ; let AP p be the 
 internal radius of one of these, and let its thick- 
 ness PQ be represented by Ap, so that p+ Apis 
 D the exteral radius AQ of the ring. Then will 
 the volume of the ring be represented by 
 tf(p-|-Ap) 2 <rrZ>p 2 , or by tf#[2pAp-|-(Ap) 2 ] ; or if Ap 
 be taken exceedingly small,, so that (Ap) 2 may vanish as com- 
 pared with 2pAp, then is the volume of the ring represented 
 by 2 < n$pAp. 
 
 Now this being the case, the ring may be considered as an 
 element AM of the volume of the solid, every part of which 
 element is at the same distance p from the axis AB, so that 
 the whole moment of inertia 2p 2 AM of the cylinder = 
 
 *86. The moment of inertia of a hollow cylinder about 
 axis of symmetry. 
 
 be the external radius AC, and # 2 the internal 
 radius AP, and b the height of the cylinder ; 
 then by the last proposition the moment of in- 
 ertia of the cylinder CD, if it were solid, would 
 be ^~ba* ; also the moment of inertia of the 
 cylinder PK, which is taken from this solid to 
 form the hollow cylinder, would be %*la*. Now 
 let I represent the moment of inertia of the hol- 
 low cylinder CP, therefore 
 
 Let the thickness a, a, of the hollow cylinder be repre- 
 
 sented by c, and its mean radius 
 
 therefore 
 
76 MOMENT OF INERTIA. , 
 
 Substituting these values in the preceding equation, we ob 
 tain 
 
 (65). 
 
 \1. The moment of inertia of a cylinder about an axis 
 passing through ^ts centre of gravity, and perpendicular 
 to its axis of symmetry. 
 
 Let AB be such an axis, and let PQ represent a lamina 
 contained between planes perpendicular to 
 this axis, and exceedingly near to each other. 
 Let CD, the axis of the cylinder, be repre- 
 , sented by 5, its radius by tz, and let CM=a?. 
 Take &x to represent the thickness of the 
 lamina, and let MP=y. Now this lamina 
 may be considered a rectangular parallelo 
 piped traversed through its centre of gravity by the axis AB ; 
 therefore by equation (62) its moment of inertia about that axis 
 is represented by i-VC^O^O^) ft* + (^2/) 2 } =y& \b*y + ty*\ &x. 
 Now the whole moment of inertia I of the cylinder about 
 AB is evidently equal to the sum of the moments of inertia 
 of all such laminae ; 
 
 .'.1= 
 
 Also, since x and y are the co-ordinates of a point in a 
 circle from its centre, therefore y ($ 2 a? 2 )*. Substituting 
 this value of y, and integrating according to the well known 
 rules of the integral calculus,* we have 
 
 *88. The moment of inertia of a cone about its axis of 
 symmetry. 
 
 The cone may be supposed to be made up of laminae, such 
 as PQ, contained by planes perpendicular to 
 the axis of symmetry AB, and each having its 
 centre of gravity in that axis. Let BP #?, and 
 let Aa? represent the thickness of the lamina, 
 and y its radius PR. Then, since it may be 
 considered a cylinder of very email height, its 
 moment of inertia about AB (equation 64) is 
 represented by fyty^x. Now the moment of 
 
 Church's Diff. and Intcg. Calculus, Arts. 148, 149. 
 
MOMENT OF INEKTIA. 77 
 
 inertia I of the whole cone is equal to the sum of the mo- 
 ments of all such elements, 
 
 Let the radius of the base of the cone be represented by 
 
 and its height by 5 ; therefore- =-, therefore Aa?= -Ay ; 
 
 y a a 
 
 (6T). 
 
 89. The moment of inertia of a sphere about one of its 
 diameters. 
 
 Let C be the centre of the sphere and AB the diameter 
 about which its moment is to be determined. 
 
 eLet PQ be any lamina contained by planes 
 perpendicular to AB ; let CM=#, and let &x 
 represent the thickness of the lamina, and y its 
 radius ; also let CA=a ; then since this lamina, 
 being exceedingly thin, may be considered a 
 cylinder, its moment of inertia about the axis AB is (equa- 
 tion 64) %*y*&x ; and the moment of inertia I of the whole 
 sphere is the sum of the moments of all such laminae, 
 
 Now by the equation to the circle y*=a* a? 2 , therefore 
 2/*=& 4 2V+# 4 . If this value be substituted for y 4 , and 
 the integration be completed according to the common 
 methods, we shall obtain the equation, 
 
 (68). 
 
 90. The moment of inertia of a cone about an axis gassing 
 through its centre of gravity and perpendicular to its axis 
 of symmetry. 
 
 Let CD be an axis passing through the centre of gravity 
 
78 MOMENT OF INERTIA. 
 
 G of the cone, and perpendicular to its axis of 
 symmetry, and let GP the distance of the lamina 
 from G, measured along the axis, be represented 
 by x ; also let the thickness of the lamina be re- 
 presented by &x. Now this lamina may be con- 
 sidered a cylinder of exceedingly small thick- 
 ness. If its radius be represented by y, its mo- 
 ment of inertia about an axis parallel to CD passing through 
 its centre, is therefore (equation 66) represented by 
 J^y 2 jy Q +-J(Aa?) 2 }Aa? ? or if ACC be assumed exceedingly email, 
 it is represented by fay*&x. Now this being the moment of 
 the lamina about an axis parallel to CD, passing through its 
 centre of gravity, and the distance of this axis from CD be- 
 ing a?, and also the volume of the lamina being flry'Aaj, it fol- 
 lows (equation 58), that the moment of the lamina about CD 
 is represented by tfyVAoj+J^Aa? * jyV-f Jy 4 J A#. 
 
 Now the moment I of the whole cone about CD equals 
 the sum of the moments of all such elements, 
 
 Now if a be the radius of the base of the cone and ~b its 
 height, then since BG=f 5, 
 
 j x 1} T> 
 
 91. The moment of inertia of a segment of a sphere about 
 a diameter parallel to the plane of section. 
 
 Let ADBE represent any such portion of a sphere, and 
 
 T> AB a diameter parallel to the plane of section. 
 
 /^JlX Let CD=tf, CE=5, and let PQ be any lamina 
 
 p ^:::::::--'-^ contained by planes parallel to the plane of 
 
 \ Z'-S- -"""/*" section : let the distance of the lamina from 
 
 C=#, and let its thickness be &x and its radius 
 
 , Then considering it a cylinder of exceeding small thick- 
 
MOVING FORCES. 79 
 
 ness, its moment of inertia about an axis passing through its 
 centre of gravity and parallel to AB, is represented (equa- 
 tion 66) by i^y 2 fy 2 + i( A a?) 2 ! AOJ, or (neglecting powers of A# 
 above the first by %*y*&x. Hence, therefore, the moment of 
 this lamina about the axis AB is represented (equation 
 58) by 7r?/ 2 (A2>),' 2 _|_ J-7ry 4 A# ) or by <7r 'ji?/V 2 4-J?/ 4 J A#?; now the 
 whole moment I of inertia of ADBE about AB is evidently 
 equal to the sum of the moments of all such laminae, 
 
 .; = y 
 
 -b 
 
 *=a?x\ therefore ^V-f Jy 4 =Jj2&V 3^ 4 + ^ 4 j. 
 Substituting this value in the integral and integrating, we 
 have 
 
 -95 5 J* (TO) 
 
 THE ACCELERATION OF MOTION BY GIYEN 
 MOVING FOECES. 
 
 92. IF the forces applied to a moving body in the direc- 
 tion of its motion exceed those applied to it in the opposite 
 direction (both sets of forces being resolved in the direction 
 of a tangent to its path), the motion of the body will be ac- 
 celerated ; if they fall short of those applied in the opposite 
 direction, the motion will be retarded. In either case the 
 excess of the one set 'of forces above the other is called the 
 MOVING FORCE upon the body : it is measured by that single 
 pressure which being applied to the body in a direction op- 
 posite to the greater force, would just balance it ; or which, 
 had it been applied to the body (together with the other 
 forces impressed upon it) when in a state of rest, would have 
 maintained it in that state ; and which, therefore, if applied 
 when its motion had commenced, would have caused it to 
 pass from a state of variable to one of uniform motion. Thus 
 the moving force upon a body which descends freely by gra- 
 vity, is measured by its weight, that is, by the single force 
 which, being applied to the body before its motion had com- 
 menced in a direction opposite to gravity, would just have 
 supported it, and which being applied to it at any instant of 
 
 * Note (q) Ed. App. 
 
80 RELATIONS OF 
 
 its descent, would have caused its motion at that instant to 
 pass from a state of variable to a state of uniform motion. 
 If the resistance of the air upon its descent be taken into 
 account, then the moving force upon the body at any instant 
 is measured by that single pressure which, being applied up- 
 wards, would, together with the resistance of the air at that 
 instant, just balance the weight of the body. 
 
 A moving force being thus understood to be measured by 
 & pressure* being in fact the unbalanced pressure upon the 
 moving body, the following relations between the amount of 
 a moving force thus measured, and the degree of acceleration 
 produced by it will become intelligible. These are laws of 
 motion which have become known by experiment upon the 
 motions of the bodies immediately around us, and by obser- 
 vation upon those of the planets. 
 
 93. "When the moving force upon a body remains con- 
 stantly the same in amount (as measured by the equivalent 
 pressure) throughout the motion, or is a uniform moving 
 force, it communicates to it equal additions of velocity in 
 equal successive intervals of time. Thus the moving force 
 upon a body descending freely by gravity (measured by its 
 weight) being constantly the same in amount throughout its 
 descent (the resistance of the air being neglected), the body 
 receives from it equal additions of velocity in equal succes- 
 sive intervals of time, viz. 32 feet in each successive second 
 of time (Art. 44.). 
 
 94. The increments of velocity communicated to equal 
 todies by unequal moving forces (supposed uniform as above) 
 are to one another as the amounts of those moving forces 
 (measured by their equivalent pressures). 
 
 Thus let P and P 1 be any two unequal moving forces upon 
 two equal bodies, and let them act in the directions in which 
 the bodies respectively move ; let them be the only forces 
 tending to communicate motion to those bodies, and remain 
 constantly the same in amount throughout the motion. Also 
 let f and f t represent the additional velocities which these 
 two forces respectively communicate to those two equal 
 bodies in each successive second of time ; then it is a law of 
 the motion of bodies, determined by observation and experi- 
 ment, that P : P! ::/:/;. 
 
 * Pressure and moving force are indeed but different modes of the operation 
 of the same principle of force. 
 
PRESSURE AND MOTION. 81 
 
 if one of the moving forces, as for instance P n be the 
 weight "W of the body moved, then the value /j of the 
 increment of velocity per second corresponding to that 
 moving force is 32 l (Art. 44.) represented by <?, 
 
 =/ (71). 
 
 95. If the amount or magnitude of the moving, force does 
 not remain the same throughout the motion, or if it be at 
 variable moving force, then the increments, of velocity com- 
 municated by it in equal successive interval^ of time are not 
 equal; they increase continually if the moving force- 
 increases, and they diminish if it diminishes.. 
 
 If two unequal moving forces, one or both of them,, thus 
 variable in magnitude, become the moving- forces- of' two 
 equal bodies, the additional velocities which they would 
 communicate in the same interval of 'time to those bodies, 
 if at any period of the motion from variable they become 
 uniform, are to one another (Art. 94.) as the respective 
 moving forces at that period of the motion. 
 
 Thus let f and f t represent the -additional velocities which 
 would thus be communicated to two equal bodies in one 
 second of time, if at any instant the pressures P and P x , 
 which are at that instant the moving forces of those bodies, 
 were from variable to become constant pressures, then 
 (Art. 94.), 
 
 This being true of any two moving forces, is evidently true, 
 if one of them become a constant force. Let P, represent 
 the weight W of the body, then will f v be represented! 
 
 Let the moving force P be supposed to remain constant 
 during a number of seconds or parts of a second, repre- 
 sented by A, and let AY be the increment of velocity in 
 the time &t on this supposition. Now / represents the 
 increment of velocity in each second, and AY the increment 
 of velocity in A seconds : moreover the force P is supposed 
 constant (Tiring A, so that the motion is uniformly accele- 
 rated during that time (Art. 44.). 
 
 6 
 
82 RELATIONS OF 
 
 "Now this is true (if the supposition, that P remains constant 
 during the time A ? on which it is founded, be true), how- 
 ever small the time &t may be. But if this time be 
 infinitely small, the supposition on which it is founded is in 
 all cases true, for P may in all cases be considered to remain 
 the same during an infinitely small period of time, although 
 it does not remain the same during any time which is not 
 
 AY dV 
 
 infinitely small. Now when &t is infinitely small -=,- ; 
 
 : A dt 
 
 generally therefore f= -^ . 
 
 If Y increase as the time t increases, or if the motion be 
 
 dV 
 
 accelerated, then -^- is necessarily a positive quantity. If, 
 
 on the contrary, Y diminishes as the time increases, then 
 
 dV 
 
 -57 is negative ; so that, generally, 
 
 the sign being taken according as the motion is accele- 
 rated or retarded. Substituting mis value of f in the last 
 proportion we have in the case, in which P represents a 
 variable pressure, 
 
 The principles stated above constitute the fundamental rela- 
 tions of pressure and motion. 
 
 96. The velocity Y at any instant of a body moving with 
 a variable motion, being the space which it would describe 
 in a second of time, if at that instant its motion were to 
 become uniform, it follows, that if we represent by &t any 
 number of seconds or parts of a second, beginning from that 
 instant, and by AS, the space which the body would describe 
 
 * Note (r) Ed. App. 
 
PRESSURE AND MOTION. 
 
 in the time A, if its motion continued uniform from the com- 
 mencement of that time, then, 
 
 this is true if the motion remain uniform during the 
 time A, however small that time may be, and therefore if it 
 be infinitely small. But if the time &t be infinitely small, 
 the motion does remain uniform during that time, however 
 variable may be the moving force ; also when &t is infi- 
 
 nitely small, --3 = -JTT. Therefore, generally, 
 
 The equations (73) and (74) are the fundamental equations 
 of dynamics : they involve those dynamical results which 
 have been discussed on other principles in the preceding 
 parts of this work.* 
 
 THE DESCENT OF A BODY UPON A CURVE. 
 
 *97. If the moving force P upon a ~body varies (Rreetly as its 
 distance at any tome from a given point towards which it 
 falls, then the whole time of the body's falling to that 
 point will be the same, whatever may be the distance from 
 which it falls. 
 
 Let A be the point from which the body falls, and B a 
 point towards which it falls along the path 
 APB, which may be either curved or straight ; 
 also let the body be acted upon at each 
 point P of its path, by a force in the direc- 
 tion of its path at that point which varies as 
 
 * Thus if the latter equation be inverted, and multiplied by the former, we 
 obtain the equation 
 
 m t ya v a = 
 
 which is identical with equation (47). 
 
84 RELATIONS OF 
 
 its distance BP, measured along the path from B ; the time 
 of falling to B will be the same, whatever may be the dis- 
 tance of the point A from which the body falls. 
 
 For let BP S, and let the force impelling the body 
 towards B be represented by dS, where c is a constant quan- 
 tity ; suppose the body, instead of falling from A towards 
 B, to be projected with any velocity from B towards A, and 
 let v be the velocity acquired at ?, and Y that at A, and 
 let BA=S 15 then by equation (47), 
 
 Suppose now the velocity of projection from B to have 
 been such as would only just carry the body to A, so that 
 V=0, 
 
 Now by equation (74), 
 dtl 
 
 and if T represent the whole time in seconds occupied in 
 the ascent of the body from B to A, 
 
 2' 
 
 It is clear that the time required for the body's descent 
 from A to B is equal to that necessary for the ascent from 
 B to A, so that the whole time required to complete the 
 ascent and descent is equal to T, and is represented by the 
 formula 
 
 (T6). 
 
PKESSUKE AND MOTION. 85 
 
 Now this expression does not contain S 15 i. e. the distance 
 from which the body falls to B ; the time T is the same 
 therefore, whatever that distance may be. 
 
 THE SIMPLE PENDULUM. 
 
 98. If a heavy particle P ~be imagined to ~be suspended from a 
 point C ~by a thread without weight, and allowed to oscillate 
 freely, but so as to deviate out little on either side of the 
 vertical, then will its oscillations, so long as they are thus 
 small, be performed in the same time whatever their ampli- 
 tudes may be. 
 
 For let the inclination PCB of CP to the vertical be repre- 
 sented by 6, and let the weight w of the particle 
 P, which acts in the direction of the vertical VP, 
 be resolved into two others, one of which is in the 
 direction CP, and the other perpendicular to that 
 direction : the former will be wholly counteracted 
 by the tension of the thread CP, and the latter will 
 Jbe represented by w sin. VPC=w sin. & ; and, act- 
 ing in the direction in which the particle P moves, this will 
 be the whole impressed moving force upon it (Art. 92.) Now 
 so long as the arc $ is small, this arc does not differ sensibly 
 from its sine, so that for small oscillations the impressed mov- 
 
 ing force upon P is represented by wb, or by ^--, or by , 
 
 I L 
 
 if I represent the length CP of the suspending thread, and S 
 the length of the arc BP. Now in this expression w and I 
 are constant throughout the oscillation, the moving force va- 
 ries therefore as S. Hence by the last proposition, the small 
 oscillations on either side of CB are isochronous, since so long 
 as they are thus small, the impressed moving force in the 
 direction of the motion varies as the length of the path BP 
 from the lowest point B. Since in the last proposition the 
 moving force was assumed equal to cS, and that here it is 
 
 represented by yS, therefore in this case <?=y. Substitut- 
 ing this value in equation (76), 
 
 A single particle thus suspended by a thread without 
 
86 THE PARALLELOGRAM OF MOTION. 
 
 weight, is that which is meant by a SIMPLE PENDULUM. It is 
 evident that the time of oscillation increases with the length 
 I of the pendulum. 
 
 IMPULSIVE FORCE. 
 
 99. If any number of different moving forces be applied 
 to as many equal bodies, the velocities communicated to 
 them in the same exceedingly small interval of time, will be 
 to one another as the moving forces. For let P 1? P 2 , repre- 
 sent the moving forces, and f\, /" the additional velocities 
 they would communicate per second if each moving force 
 remained continually of the same magnitude (Art. 93.), then 
 would tf^ tfv be the whole velocities communicated on this 
 supposition in t seconds ; let these be represented by V,, Y 2 ; 
 therefore by Art. 94. 
 
 P, :?,::/ :/,::# : */,::, :V,. 
 
 The proposition is therefore true on the supposition that P t 
 and P 2 remain constant during the interval of time t ; but 
 if t be exceedingly small, then whatever the pressures P t 
 and P 2 may be, they may be considered to remain the same 
 during that time. Therefore the proposition is true generally, 
 when, as above, the moving forces are supposed to act on 
 equal bodies, or successively on the same body, through 
 equal exceedingly small intervals of time. 
 
 M oving forces thus acting through exceedingly small in- 
 tervals of time only, are called IMPULSIVE FORCES. 
 
 THE PARALLELOGRAM OF MOTION. 
 
 100. If two impulsive forces P 1? P 2 , whose directions are AB 
 and AC. he impressed at the same time upon 
 ^ a lody at A, which if made to act upon it 
 separately would cause it to move through 
 AB and AC in the same given time, then 
 will the hody he made, by the simultaneous action of these 
 impulsive forces, to describe in that time the diagonal AD 
 of the parallelogram, of which AB and AC are adjacent 
 sides. 
 
 For the moving forces Pj and P 3 acting separately upon 
 
INDEPENDENCE OF SIMULTANEOUS MOl'IONS. 87 
 
 the same body through equal infinitely small times, commu- 
 nicate to it velocities which are (Art. 99.) as those forces ; 
 therefore the spaces AB and AC described with these velo- 
 cities in any given time are also as those forces. Since then 
 AB and AC are to one another as the pressures "P l and P a , 
 therefore by the principle (Art. 2.) of the parallelogram of 
 pressures, the resultant K of P t and P 2 is in the direction of 
 the diagonal AD, and bears the same proportion to P x and 
 P 2 that AD does to AB and AC. 
 
 Therefore the velocity which the resultant K of P, and P 2 
 would communicate to the body in any exceedingly small 
 time is to the velocities which P, and P 2 would separately 
 communicate to it in the same time as AD to AB and AC 
 (Art. 99.), and therefore the spaces which the body would 
 describe uniformly with these three velocities in any equal 
 times are in the ratio of these three lines. But AB and AC 
 are the spaces actually described in the equal times by rea- 
 son of the impulses of Pj and P 2 . Therefore AD is the space 
 described in that time by reason of the impulse of R, that is, 
 by reason of the simultaneous impulses of P x and P a . 
 
 101. THE INDEPENDENCE OF SIMULTANEOUS MOTIONS. 
 
 It is evident that if the body starting from A had been 
 made to describe AB in a given time, and then 
 had been made in an equal time to describe 
 BD, it would have arrived precisely at the same 
 point D to which the simultaneous motions 
 AC and AB have brought it, so that the body is made to 
 move by these simultaneous motions precisely to the same 
 point to which it would have been brought by those motions, 
 communicated to it successively, but in half the time. The 
 following may be taken as an illustration of this principle of 
 the independence of simultaneous motions. Let a canal-boat 
 __ T* 3, be imagined to extend across the whole 
 
 = width of the canal, and let it be supposed 
 that a person standing on the one bank at 
 A is desirous to pass to a point D on the 
 
 opposite bank, and that for this purpose, as the boat passes 
 him, he steps into it, and walks across it in the direction 
 AB, arriving at the point B in the boat precisely at the in- 
 stant when the motion of the boat has carried it through 
 BD ; it is clear that he will be brought, by the joint effect 
 
88 THE POLYGON OF MOTION. 
 
 of his own motion across the boat and the 'booths motion 
 along the canal, to the point D (having in reality described 
 the diagonal AD), which point he would have reached in 
 double the time if he had walked across a bridge from A to 
 B in the same time that it took him to walk across the boat, 
 and had then in an equal time walked from B to D along 
 the opposite side. 
 
 THE POLYGON OF MOTION. 
 
 102. Let any number of impulses be communicated simul- 
 taneously to a body at O, one of which 
 would cause it to move from A to O in a 
 given time, another from B to O in the 
 same time, a third from C to O in that time, 
 and a fourth from D to O. Complete the 
 parallelogram of which AO and BO are ad- 
 jacent sides ; then the impulses AO and BO would simulta- 
 neously cause the body to move from E to O through the 
 diagonal EO in the time spoken of. Complete the parallelo- 
 gram EOCF, and draw its diagonal OF, then would the im- 
 pulses EO and CO, acting simultaneously, cause the body to 
 move through FO in the given time : but the impulse EO 
 produces the same eifect on the body as the impulses AO 
 and BO ; therefore the impulses AO, BO, and CO, will 
 together cause the body to move through FO in the given 
 time. In the same manner it may be shown that the im- 
 pulses AO, BO, CO, and DO, will together cause the body to 
 move through GO in a time equal to that occupied by the 
 body's motion through any one of these lines. 
 
 It will be observed that GD is the side which completes 
 the polygon OAEFG, whose other sides OA, AE, EF, FG, 
 are respectively equal and parallel to the directions 'O A, OB, 
 OC, and OD, of the simultaneous impulses. 
 
 Instead of the impulses AO, &c. taking place simultane- 
 ously., if they had been received successively, the body 
 moving first from O to A in a given time ; then through 
 AE, which is equal and parallel to OB, in an equal time ; 
 then through EF, which is equal and parallel to OC, in that 
 time ; and lastly through FG, which is equal and parallel to 
 OD, in that time, it would have arrived at the same point G. 
 to which these impulses have brought it simultaneously, but 
 after a period as many times greater as there are motions, so 
 
THE PRINCIPLE OF D 5 ALEMBERT. 89 
 
 that the principle of the independence of simultaneous 
 motions obtains, however great may be the number of such 
 motions. 
 
 THE PRINCIPLE OF D'ALEMBERT. 
 
 103. Let Wj, W 9 , W 8 , &c. represent the weights of any 
 number of bodies in motion, and P 15 P 2 , P 8 , &c. the moving 
 forces (Art. 92.) upon these bodies at any given instant of 
 the motion, i. e. the unbalanced pressures, or the pressures 
 which are wholly employed in producing their motion, and 
 pressures equal to which, applied in opposite directions, 
 would bring them to rest, or to a state of uniform motion. 
 
 WWW 
 
 Then (Art. 95.), P^- 1 /, P. = --/., P. = ^'Ac. 
 y y y 
 
 where f^f^ /,, &c. represent the additions of velocity which 
 the bodies would receive in each second of time, if the 
 moving force upon each were to become, at the instant at 
 which it is measured, an uniform moving force. Suppose 
 these bodies, whose weights are "W 15 W 2 , W 3 , &c. to form a 
 system of bodies united together by any conceivable mecha- 
 nical connection, on which system are impressed, in any 
 way, certain forces, whence result the unbalanced pressures 
 P P 2 , P 3 , &c. on the moving points of the system. Now 
 conceive that to these moving points of the system there are 
 applied pressures respectively equal to P x , P 2 , P 3 , &c. but 
 each in a direction opposite to that in which the motion of 
 the corresponding point is accelerated or retarded. Then 
 will the motion of each particular point evidently pass into 
 a state of uniform motion, or of rest (Art. 92.). The whole 
 system of bodies being thus then in a state of uniform 
 motion, or of rest, the forces applied to its different elements 
 must be forces in equilibrium. 
 
 "Whatever, therefore, were the forces originally impressed 
 upon the system, and causing its motion, they must, together 
 with the pressures P 15 P 2 , P s , &c. thus applied, produce a 
 state of equilibrium in the system ; so that these forces (ori- 
 ginally impressed upon the system, and known in Dynamics 
 as the IMPRESSED FORCES) have to the forces Pj, P 2 , P 3 , &c., 
 when applied in directions opposite to the motions of their 
 several points of application, the relation of forces in equili- 
 brium. The forces P n P 2 , P 3 , &c. are known in Dynamics 
 as the EFFECTIVE FORCES. Thus in any system of bodies 
 mechanically connected in any way, so that their motions 
 
90 
 
 may mutually influence one another, if forces equal to the 
 effective forces were applied in directions opposite to their 
 actual directions, these would be in equilibrium with the 
 impressed forces, which is the principle of D'Alembert. 
 
 104. The work accumulated in\ a moving ~body through any 
 space is equal to the worJc which must be done upon it, in 
 an opposite direction, to overcome the effective force upon 
 it through that space. 
 
 This is evident from Arts. 68. and 69., since the effective 
 force is the unbalanced pressure upon the body. 
 
 If the work of the effective force be said to be done upon 
 the body,* then the work of the effective force upon it is 
 equal to the work or power accumulated in it, and this work 
 of the effective force may be all said to be actually accu- 
 mulated in the body as in a reservoir. 
 
 MOTION OF TRANSLATION. 
 
 DEFINITION. When a body moves forward in space, with- 
 out at the same time revolving, so that all its. parts move 
 with the same velocity and in parallel directions, it is said to 
 move with a motion of translation only. 
 
 105. In order that a body may move with a motion of trans- 
 lation only, the resultant of the forces impressed upon it 
 must have its direction through the centre of gravity of 
 the body. 
 
 For let Wtf w^, w s , &c. represent the weights of the parts 
 or elements of the body, and let f represent the additional 
 velocity per second, which any element receives or would 
 receive if its motion were at any instant to become uniformly 
 accelerated. Since the motion is one of translation only, 
 the value of f is evidently the same in respect to every 
 other element. The effective forces Pj, P 2 , P 3 , &c. on the 
 different elements of the body are therefore represented by 
 
 ^/>7/>7/> &c - &c - 
 
 * This cannot perhaps be correctly said, since work supposes resistance. 
 
MOTION OF ROTATION. 91 
 
 Now the forces P a , P 9 , P 8 , &c. are evidently parallel pres- 
 sures. Let X be the distance of the centre (see Art. IT.) of 
 these parallel pressures from any given plane ; and let a?,, # 3 , 
 o? 8 , &c. be the perpendicular distances of the elements w w n 
 w z , &c. that is, of the points of application of P a , P a , P 8 , &c. 
 from the same plane. Therefore (by equation 18), 
 
 But this is the expression (Art. 19.) for the distance of the 
 centre of gravity from the given plane ; and this being true 
 of any plane, it follows that the centre of the parallel pres- 
 sures P 15 P a , P 3 , &c. which are the effective forces of the 
 system, coincides with the centre of gravity of the system, 
 and therefore that the resultant of the effective forces passes 
 through the centre of gravity. Now the resultant of the 
 effective pressures must coincide in direction with the result- 
 ant of the impressed pressures, since the effective pressures 
 when applied in an opposite direction are in equilibrium 
 with the impressed pressures (by D'Alembert's principle). 
 The resultant of the impressed pressures must therefore have 
 its direction through the centre of gravity. Therefore, &c. 
 
 MOTION OF ROTATION ABOUT A FIXED Axis. 
 
 106. Let a rigid body or system be capable of motion 
 about the axis A. Let m 1? m a , w 3 , &c. represent the volumes 
 of elements of this body, and ^ the weight of each unit 
 of volume. Also let /,/,/, &c. represent the increments 
 of velocity per second, communicated to these elements 
 respectively by the action of the forces impressed upon the 
 system. Let P,, P 2 , P 3 , &c. represent these impressed forces, 
 and Pup*, &c. the perpendicular distances from the axis at 
 which they are respectively applied. 
 
 Now since pm^ M-m 2 , v<m 3 , &c. are the weights of the ele- 
 ments, and/,/, &c. the increments of velocity they receive 
 
92 MOTION OF ROTATION. 
 
 per second, it follows that^/, f^/,, ^/,, &c. are 
 
 the effective forces upon them (Art. 103.). Let p,, p a , p 3 , &c. 
 represent the distances of these elements respectively from 
 the axis of revolution, then since their effective forces are 
 in directions perpendicular to these distances, the moments 
 
 of these effective forces about the axis are - *-/$ - - f^ 9 
 
 9 ' g 
 
 ! /aP & c - Also PI^, P^a, PS^S? &c. are the moments of 
 
 g 
 
 the impressed forces of the system about the axis. Now the 
 impressed forces P,, P 2 , P 3 , &c., together with the resistance 
 of the axis, which is indeed one of the impressed forces, are 
 in equilibrium with the effective forces by D'Alembert's 
 principle. Taking then the axis as the point from which the 
 moments are measured, the sum of the moments of P 15 P 2 , 
 &c. must equal the sum of the moments of the effective 
 forces, or 
 
 "Now let/" represent that value of f^f n &c. which corres- 
 ponds to a distance unity from the axis. Since the system 
 is rigid, and /*, f^ / &c. represent arcs described about 
 it in the same time at the different distances 1, p 1? p 2 , &c. it 
 follows that these arcs are as their distances, and therefore 
 that/^/p^^/pij/.^/p,, &c. Substituting these values 
 in tne preceding equation, we have 
 
 where I represents the moment of inertia of the mass about 
 its axis of revolution.* 
 
 * If a represent the angular velocity, or the velocity of an element at dis 
 tance unity, then by equation (72), /= -f ^, . . a -J = -f -y SPpa ; 
 
MOTION OF ROTATION. 
 
 93 
 
 AG by G, 
 
 If the impressed forces P be the weights of the parts 
 of the body and 6 be, in any position of 
 _ tf the body, the inclination to the vertical 
 Ay of the line AG, drawn from A to the 
 centre of gravity G, then since the sum of 
 the moments of the weights of the parts is 
 equal to the moment of the weight of the 
 whole mass collected in its centre of 
 gravity (Art. 17.), we have, representing 
 
 Mfx . G. sin. 
 
 MG 
 
 therefore ^equation I 78),f=g sin. 
 
 (79). 
 
 108. To find the resultant of the effective forces on a l>ody 
 which revolves about a fixed axis. 
 
 The resultant of the effective forces upon a body which 
 revolves about a fixed axis, is evidently equal to that single 
 force which would just be in equilibrium with these if there 
 were no resistance of the axis. Let R, be that single force, 
 then the moment of R about any point must equal the sum 
 of the moments of the effective forces about that point. 
 
 Take a point in the axis for the point 
 about which the moments are measured, 
 and let L be the perpendicular distance 
 from A of the resultant R. Now, as in 
 Art. 106. it appears that the sum of the 
 moments of the effective forces about A is 
 
 ii 
 represented by f- 
 
 o 
 Now pa is the velocity of a point at distance p, therefore "Ppa is the work 
 
 (Art. 50.) of the force P per second ; therefore / Ppadt is the work of P 
 
 
 (equation 40) in the time t, which is represented by U, therefore ai* a a a 
 
 =-f--^y- which corresponds with the result already obtained. See equation 
 (51)7 * 
 
94 MOTION OF KOTATION. 
 
 if* 
 
 (80). 
 
 To determine the value of E let it be observed that the 
 effective force -/m^ on any particle m a , acting in a direc- 
 
 tion n^m^ perpendicular to the distance Am, from the axis 
 A, may be resolved into two others, parallel to the two 
 rectangular axes Ay and Aa?, each of which is equal to the 
 product of this effective force, whose direction is n,m^ and 
 the cosine of the inclination of n,m, to the corresponding 
 axis. 'Now the inclination of mji, to Ax is the same as the 
 inclination of Am, to Ay, since these two last lines are per- 
 pendicular to the two former. The cosine of this inclination 
 
 equals therefore - i or ^i, if AN. y.. Similarly the cosine 
 Am, Pl 
 
 of the inclination of n^m, to Ay equals - ! or ^1 , if AM. l = x.. 
 
 Am, Pl 
 
 The resolved parts in the directions of Ax and Ay of the 
 
 effective force - fmj, are therefore - fm^, ^ and - fm^ 
 
 9 9 Pi 9 
 
 ?X or - fm.y, and - fm.x,. 
 
 p, f g 
 
 Similarly the resolved parts in the directions of Ax and 
 Ay of the effective force upon m a are -/^ 2 y s and - fm^ 
 
 ts t/ 
 
 and so of the rest. 
 
 The sums X and Y of the resolved forces in the directions 
 of Ax and Ay respectively (Art. 11.) are therefore 
 
 . . . =Y; 
 
 g ^ ff 
 
 and - / {m.x, + m^ + m 8 a? 3 -f ..... } = Y. 
 
 Now let G x and G, represent the distances G 2 G and G : G 
 of the centre of gravity of the body from Ay and Ax respec- 
 tively, and let the whole volume of the body be represented 
 byM, 
 
MOTION OF ROTATION. 
 
 (equation 18), MG 2 m^ 
 
 95 
 
 Now if G be the distance AG of the 
 centre of gravity from A, G= VGt* -f G a a , 
 
 /.K=-/MG (82). 
 
 y 
 
 Substituting in equation (82) the value off from equation 
 (78,) we have 
 
 And substituting in equation (80) for. R its value from 
 equation (82), 
 
 /-MGL=/-L 
 
 J g J g > 
 
 I 
 
 MG 
 
 where L is the distance of the point of application of the 
 resultant of the effective forces from the axis. 
 
 Now let A be the inclination of the resultant R to the 
 axis Aa?, 
 
 /. (Art. 11.), R cos. d=X, R sin. d=Y, 
 
 Y 
 
 /.tan. &=x'<> but by equations (81), 
 
 Y G 
 
 .-.tan. 4=tan. AGG /J= 
 
 The inclination of the resultant R to Ax is therefore 
 equal to the angle AGG X , but the perpendicular to AG is 
 evidently inclined to Ax at this same angle. Therefore the 
 direction of the resultant R is perpendicular to the line AG, 
 drawn from the axis to the centre of gravity. Moreover 
 
96 THE CENTRE OF OSCILLATION. 
 
 its magnitude and the distance of its point of application 
 from A have been before determined by equations (83) 
 and (84). 
 
 THE CENTRE OF PERCUSSION. 
 
 109. It is evident, that if at a point of the body through 
 which the resultant of the effective forces upon it passes, 
 there be opposed an obstacle to its motion, then there will 
 be produced upon that obstacle the same effect as though 
 the whole of the effective forces were collected in that 
 point, and made to act there upon the obstacle, so that the 
 whole of these forces will take effect upon the obstacle, and 
 there will be no effect of these forces produced else- 
 where, and therefore no repercussion upon the axis. 
 It is for this reason that the point O in the resultant, 
 where it cuts the line AG drawn from the axis to the 
 centre of gravity, is called the CENTRE OF PERCUSSION. 
 Its distance L from A is determined by the equation 
 
 I-=g- (85), 
 
 which is obtained from equation (84) by writing MK 2 for I 
 (Art. 80.), K being the radius of gyration. If at the centre 
 of percussion the body receive an impulse when at rest, 
 then since the resultant of the effective forces thereby pro- 
 duced will have its direction through the point where the 
 impulse is communicated, it follows that the whole impulse 
 will take effect in the production of those effective forces, 
 and no portion be expended on the axis. 
 
 THE CENTRE OF OSCILLATION. 
 
 110. It has been shown (Art. 98.) that in the simple pen- 
 dulum, supposed to be a single exceedingly small element 
 of matter suspended by a thread without weight, the time 
 of each oscillation is dependent upon the length of this 
 thread, or the distance of the suspended element from the 
 axis about which it oscillates. If therefore we imagine a 
 number of such elements to be thus suspended at different 
 distances from the same axis, and if we suppose them, after 
 having been at first united into a continuous body, placed 
 in an inclined position, all to be released at once from this 
 
THE CENTRE OF OSCILLATION. 97 
 
 union with one another, and allowed to oscillate freely, it is 
 manifest that their oscillations will all be performed in 
 different times. Now let all these elements again be con- 
 ceived united in one oscillating mass. All being then com- 
 pelled to perform these oscillations in the same time, whilst 
 all tend to perform them in different times, the motions of 
 some are manifestly retarded by their connexion with the 
 rest, and those of others accelerated, the former being those 
 which lie near to the axis, and the others those more remote ; 
 so that between the two there must be some point in the 
 body where the elements cease to be retarded and begin to 
 be accelerated, and where therefore they are neither accele-- 
 rated nor retarded by their connexion with the rest ;. an; elfe-- 
 ment there performing its oscillations precisely in> the same 
 time as it would do, if it were not connected, with the- rest,, 
 but suspended freely from the* axis by at thread without: 
 weight. This point in the body, at the distance of whieh 
 from the axis a single particle, suspended freely, would per- 
 form its oscillations precisely in the same time that the body- 
 does, is called the CENTRE OF OSCILLATION. 
 
 The centre of oscillation coincides with the centre of 
 percussion. 
 
 111. For (by equation 79) the increment of angular velo- 
 city per second f of a body revolving about an hori- 
 zontal axis, the forces impressed upon it being the 
 weights of its parts only, is represented by the for- 
 
 mula ^-y-sin. d, where d is the inclination to the ver- 
 tical of the line AG, drawn from the axis to its 
 centre of gravity. But (by equation 84), L=v, where L 
 
 is the distance AO of the centre of percussion from the 
 axis, 
 
 /. fL=g sin. 6 
 
 Now it has been shown (Art. 98.), that the impressed 
 moving force on a particle whose weight is w, suspended 
 from a thread without weight, inclined to the vertical at an 
 angle d, is represented by w sin. 6 ; moreover if/" represent 
 
 7 
 
98 THE CENTRE OF OSCILLATION. 
 
 the increment of velocity per second on this particle, then 
 f is the effective force upon it. Therefore by D'Alem- 
 bert's principle, 
 
 7/1 
 
 w Bin. =/", :.f=g sin. t, .; f=fl. 
 
 Now fL is the increment of velocity at the centre of 
 percussion, andy is that upon a single particle suspended 
 freely at any distance from the axis. If such a particle 
 were therefore suspended at a distance from the axis equal 
 to that of the centre of percussion, since it would receive, 
 at the same distance from the axis, the same increments of 
 velocity per second that the centre of percussion does, it 
 would manifestly move exactly as that point does, and per- 
 form its oscillations in the same time that the body does. 
 Therefore, &c. 
 
 112. The centres of suspension and oscillation are reci- 
 
 procal. 
 
 Let O represent the centre of oscillation of a body 
 when suspended from the axis A ; also let G be its 
 centre of gravity. Let AO=L, AG G, OG^G, ; 
 also let the radius of gyration about A be repre- 
 sented by K 2 , and that about G by & 2 . Therefore 
 (equation 59), K 2 =G 2 +& 2 ; 
 
 G a + 1 & 2 
 
 (equation 85), L= =G + ..... (8Y), 
 
 Now let the body be suspended from O instead of A ; 
 when thus suspended it will have, as before, a centre of 
 oscillation. Let the distance of this centre of oscillation 
 from O be L 1? 
 
 * 
 .*. by equation (8T), L^ 
 
PROJECTILES. 99 
 
 /. by equation (88), L^^r + G^L. 
 
 Since then the centre of oscillation in this second case is at 
 the distance L from O, it is in A'; what was before the 
 centre of suspension has now therefore become the centre 
 of oscillation. Thus when the centre of oscillation is con- 
 verted into the centre of suspension, the centre of suspen- 
 sion is thereby converted into the centre of oscillation. 
 This is what is meant, when it is said that the centres of 
 oscillation and suspension are reciprocal. 
 
 PROJECTILES. 
 
 113. To determine the path of a body projected obliquely 
 in vacuo. 
 
 Suppose the whole time, T seconds, of the flight of the 
 
 body to any given point P 
 ^--'''"| T of its path, to be divided 
 M.-"'''^,.---'?* into equal exceedingly small 
 
 *ir""^.,--\v'. i intervals, represented by 
 
 '"* "^ "'"' : -j AT, and conceive the whole 
 
 q \J B effect of gravity upon the 
 projectile during each one 
 of these intervals to be col- 
 lected into a single impulse at the termination of that inter- 
 val, so that there may be communicated to it at once, by 
 that single impulse, all the additional velocity which is in 
 reality communicated to it by gravity at the different periods 
 of the small time AT. 
 
 Let AB be the space which the projectile would describe, 
 with its velocity of projection alone, in the first interval of 
 time ; then will it be projected from B at the commence- 
 ment of the second interval of time in the direction ABT 
 with a velocity which would alone carry it through the dis- 
 tance BK= AB in that interval of time ; whilst at the same 
 time it receives from the impulse of gravity a velocity such 
 as would alone carry it vertically through a space in that in- 
 terval of time which may be represented by BF. By reason 
 of these two impulses communicated together, the body will 
 therefore describe in the second interval of time the diago- 
 nal BC of the parallelogram of which BK and BF are adja- 
 
100 PEOJECTILES. 
 
 cent sides. At the commencement of the third interval it 
 will therefore have arrived at C, and will be projected from 
 thence in the direction BOX, with a velocity which would 
 alone carry it through CX^BC in the third interval ; whilst 
 at the same time it receives an impulse from gravity com- 
 municating to it a velocity which would alone carry it 
 through a distance represented by CG=BF in that interval 
 of time. These two impulses together communicate there- 
 fore to it a velocity which carries it through CD in the third 
 interval, and thus it is made to describe all the sides of the 
 polygon ABCD ... P in succession. Draw the vertical PT, 
 and produce AB, BC, CD, &c. to meet it in T, N, O . . ., 
 and produce GC, HD, &c. to meet BT in K, L, &c. 
 
 Now, since BC is equal to CX, and CK is parallel to XL, 
 therefore KL is equal to BK or to AB. 
 
 Again, since CD is equal to DZ, and DL is parallel to ZM, 
 therefore LM is equal to KL or to AB ; and so of the rest. 
 
 If therefore there be n intervals of time equal to AT, so 
 that there are n sides AB, BC, CD, &c. of the polygon, and 
 n divisions AB, BK, &c. of the line AT, then AT,=7iAB and 
 
 Similarly CET=(w-2)CX, therefore ]Sr6=(^-2)DX= 
 (n 2)BF; and so of the remaining parts of TP. 
 
 these parts of TP are (nT) in number, therefore 
 -3)W+ ... \(nl) terms}; 
 
 Therefore, summing the series to (nV) terms. 
 TP={2(-l)-(-a)}=i . BF, 
 
 Now g represents the additional velocity which gravity 
 would communicate to the projectile in each second, if it 
 acted upon it alone. g&T is therefore the velocity which it 
 would communicate to it in each interval of AT seconds. 
 <?AT is therefore the velocity communicated to the body by 
 each of the impulses which it has been supposed to receive 
 from gravity. 
 
PROJECTILES. 101 
 
 Now BF is the space through which it would be carried 
 in the time AT by this velocity, 
 
 Also AT=-, 
 n 
 
 Now this is true, however small may be the intervals of 
 time AT, and therefore if they be infinitely small, that is, if 
 the impulses of gravity be supposed to follow one another at 
 infinitely small intervals, or if gravity be supposed to act, as 
 it really does, continuously. 
 
 But if the intervals of time AT be infinitely small, then 
 the number n of these intervals which make up the whole 
 finite time T, must be infinitely great. Also when n is infi- 
 
 nitely great, -=0. 
 
 In the actual case, therefore, of a projectile continually 
 deflected by gravity, the vertical distance TP between the 
 tangent to its path at the point of projection, and its position 
 P after the flight has continued T seconds, is represented by 
 the formula 
 
 TP=%gT ..... (89). 
 
 Moreover AT^TiAB, and AB is the space which the body 
 would describe uniformly with the velocity of projection in 
 the time AT, so that ?zAB is the space which it would de- 
 scribe in the time n . AT or T with that velocity. If there- 
 fore Y equal the velocity of projection, then 
 
 AT=Y . T . . . . (90) ; 
 
 so that the position of the body after the time T is the same 
 as though it had moved through that time with the velocity 
 of its projection alone, describing AT, and had then fallen 
 through the same time by the force of gravity alone, describ- 
 ing TP (see Art. 101.). 
 
 Let AM=x, MP=y, angle of 
 projection TAM^a, velocity of projec- 
 
 --*- * 
 
102 PROJECTILES. 
 
 x tan. a y=MT MF=P=i/r ..... (91). 
 Substituting the value of T from the preceding equation, 
 
 ., a? 2 sec. 2 a 
 tan. a = 
 
 <7 sec. a a 
 
 .-.y=0tan. a ~ 2y2 
 
 Let H be the height through which a body must fall freely 
 by gravity to acquire the velocity Y, or the height due to 
 
 that velocity ; then Y 2 2gTL (Art. 47.), therefore 4H= -- ; 
 therefore, by substitution, 
 
 (92). 
 
 115. To find the time of the fiight of a projectile. 
 
 It has been shown (equation 91), that if T represent the 
 time in seconds of the flight to a point whose co-ordinates 
 are x and y, then 
 
 2 
 x tan. a y, /. T= - \x tan. a y\ , 
 
 (93). 
 
 ..T=l^tan. a y nearly. 
 
 If the projectile descend again to the horizontal plane from 
 which it was projected, and T be the whole time of its flight; 
 and X its whole range upon the plane, then, since at the ex 
 piration of the time T, y=Q and a?=X, 
 
 -^X tan. a= Jy/Xtan. a nearly. 
 
PROJECTILES. 103 
 
 116. To find the greatest horizontal distance X, to which a 
 projectile ranges, having given the elevation a and the 
 velocity V of its projection. 
 
 When the projectile attains its greatest horizontal range, 
 its height y above the horizontal plane 
 becomes 0, whilst the abscissa x of the 
 
 / ! point P, which it has then reached in 
 
 its path, becomes X. Substituting 
 S^s^i?^ these values and X, for y and x in 
 
 ./ \ \ equation (92), we have 0=X tan. a 
 
 * JUi, ::> X 2 sec. 2 a 
 
 4H 
 
 .*. X=4H tan. a cos. 2 a=4H sin. a cos. a. 
 ..X=2Hsin. 2 (94). 
 
 If the body be projected at different angular elevations, 
 but with th same velocity, the horizontal range will be the 
 
 greatest when sin. 2a is the greatest, or when 2a=-, or a =j- 
 
 117. To find the greatest height which a projectile will 
 attain in its flight if projected with a given velocity , and 
 at a given inclination to the horizon. 
 
 Multiplying both sides of equation 
 (92) by 4H cos. 2 , we have 4H cos. 2 a 
 . y^R cos. 2 a tan. a . x x?=2T3. (2 
 cos. a sin. a) xx*=2TL sin. 2a . xx*. 
 Subtracting both sides of this equa- 
 2 sm - 2 2S we have 
 
 H 8 sin 2 2a 4H cos. 2 a . y=W sin. 2 2a 2H sin. 2a . 
 But sin. 2 2a=4: sin. 3 a cos. 2 a, 
 
 /.4H cos. 2 a{H sin. 2 a -y} --= JH sin. 2a-^ 8 . . . . (95). 
 
 E"ow the second member of this equation is always a 
 positive quantity, being a square. The first member is 
 therefore always positive ; therefore H sin. 2 a y is always 
 positive. "Whence it follows that y can never exceed H 
 sin. 2 a, so that it attains its greatest possible value when it 
 equals H sin. 2 a, a value which it manifestly attains when 
 
104 PROJECTILES. 
 
 the first member of the above equation vanishes, or when 
 a?z=H sin. 2, that is, when x becomes equal to half the 
 greatest horizontal range, as is apparent from the last pro- 
 position; so that the greatest height BD of the projectile 
 is represented by H sin. a a, a height which it attains when 
 AD equals half the horizontal range. 
 
 118. The path of a projectile in vacuo is a parabola. 
 
 Let B be the highest point in the 
 flight of the projectile, and BD its 
 greatest height. Draw PM t perpen- 
 dicular to BD. Let BM 1 =a? 1 , MJP 
 
 .-. aj^BD M 1 D=BD PM=H sin. 2 a y, 
 
 y 1 =DM=AM AD= - H sin. 2. 
 Substituting these values in equation (95), 
 
 y^-iH cos. 2 . x, ..... (96), 
 
 which is the equation to a porabola whose vertex is in 
 B, whose axis coincides with BD, and whose parameter is 
 4H cos. 2 . 
 
 The path of a projectile in vacuo is therefore a parabola, 
 whose vertex is at the highest point attained by the pro- 
 jectile, and whose axis is vertical. 
 
 119. To find the range of a projectile upon an inclined 
 
 plane. 
 
 Let ~R represent the range AP of a projectile upon an 
 inclined plane AB, whose inclination is 
 i. Then H and a being taken to repre- 
 , sent the same quantities as before, and 
 " c x, y being the co-ordinates of P to the 
 horizontal azis AC, we "have 
 
 x=AM=AP cos. PAM=K cos. i, 
 sin. PAM =R sin. . 
 
 ^ Substituting these values of x and y in the general equa- 
 tion (92) of the projectile we have 
 
PROJECTILES. 105 
 
 T, T-, R a cos. 3 i sec. 1 a 
 
 R sin. i=R cos. i tan. a -- -== - . 
 
 Dividing by R, multiplying by cos. , and transposing 
 
 R cos. 8 * sec. a 
 
 - JTJ -- =cos. i sin. a sm. cos. a=sm. (a i), 
 
 (97). 
 
 Now sin. (2a ) sin. =sin. {a + (a i)| sin. ja (a . 
 i)j =2 sin. (a ) cos. a. 
 
 Substituting this value of 2 sin. (a ) cos. a in the pre- 
 ceding equation, we have 
 
 ._. > 
 
 j 
 
 J^ow it is evident that if a be made to vary, < remaining 
 the same, R will attain its greatest value when sin. (2a ) 
 is greatest, that is when it equals unity, or when 2a 1= 
 
 o, or when a=- + -. This, then, is the angle of elevation 
 
 corresponding to the greatest range, with a given velocity 
 upon an inclined plane whose inclination is . 
 
 If in the preceding expression for the range we substitute 
 
 ( * I 
 
 ) o~( a ~') f f r a ? tne value of the expression will be found 
 
 to remain the same as it was before ; for sin. (2 i) will, by 
 this substitution, become sin. jtf 2(a ) } =sin. \t (2 
 1){ =sin. (2a i). The value of R remains therefore the 
 
 "Tf 
 
 same, whether the angle of elevation be a or s~( a ~')- 
 
 And the projectile will range the same distance on the 
 plane, whether it be projected at one of these angles of 
 elevation or the other. 
 
 Let BAG be the inclination of the plane on which the 
 projectile ranges, and AT the direc- 
 tion of projection. Take DAS equal 
 to BAT. Then BAT=TAC-BAC 
 =a-i. And SAC=:DAC-DAS= 
 
 - BAT=~( ) The ran g e AP 
 
 2 a 
 
 is therefore the same, whether TAG or SAO be the angle of 
 
106 CENTRIFUGAL FORCE. 
 
 elevation, and therefore whether AT or AS be the direction 
 of projection. 
 
 Draw AE bisecting the angle BAD, then the angle EAC 
 
 The angle EAC is therefore that corresponding to the 
 greatest range, and AE is the direction in which a body 
 should be projected to range the greatest distance on the 
 inclined plane AB. 
 
 It is evident that the directions of projection AS and AT, 
 which correspond to equal ranges, are equally inclined to 
 the direction AE corresponding to the greatest range. 
 
 120. The velocity of a projectile at different points of its 
 path. It has been shown (Art. 56.), that if a body move in 
 any curve acted upon by gravity, the work accumulated or 
 lost is the same as would be accumulated or lost, provided 
 the body, instead of moving in a curve, had moved in the 
 direction of gravity through a space equal to the vertical 
 projection of its curvilinear path. 
 
 thus a projectile moving from A to P will accumulate or 
 lose a quantity of work, which is equal to that which it would 
 accumulate or lose, had it moved vertically from M to P, or 
 from P to M, PM being the projection of its path on the 
 direction of gravity. 5" ow the work thus accumulated or 
 lost equals one half the difference between the vires vivce at 
 the commencement and termination of the motion. 
 
 Let Y equal the velocity at A, and v equal the velocity at 
 
 W "W 
 
 P, therefore the work -J V a -J- v*. Moreover, the work 
 
 9 9 
 
 done through PM=W . PM, therefore V ^-v*= 
 
 W . PM, therefore Y 2 v a =2^MP. Let PM=y, 
 
 /. v 3 Y a %gy (99), 
 
 which determines the velocity at any point of the curve. 
 
 CENTRIFUGAL FORCE. 
 121. Let a body of small dimensions move in any curvi- 
 
CENTRIFUGAL FOKCE. 1Q7 
 
 linear path AB, impelled continnally towards 
 a given point S (called a centre of force) by a 
 given force, whose amount, when the body 
 has reached the point P in its path, is repre- 
 sented by F.* Let PQ be an exceedingly 
 small portion of the path of the body, and 
 conceive the force F to remain constant and 
 parallel to itself, whilst this portion of its path is being de- 
 scribed. Then, if PR be a tangent at P, and QR be drawn 
 parallel to SP, PR is the space which the body would have 
 traversed in the time of describing PQ, if it had moved 
 with its velocity of projection from P alone, and had not been 
 attracted towards S, and RQ or PT (QT being drawn paral- 
 lel to RP) is the space through which it would have fallen 
 by its attraction towards S alone, or if it had not been pro- 
 jected at all from P.f Let v represent the velocity which 
 it would have acquired on this last supposition, when it 
 reached the point T. Therefore (Art. 66.), if w represent the 
 weight of the body, 
 
 Now the velocity v, which the body would have acquired in 
 falling through the distance PT by the action of the constant 
 1 orce F, is equal to double that which would cause it to de 
 scribe the same distance uniformly in the same time4 
 
 Representing therefore by Y the actual velocity of the 
 body in its path at P, we have 
 
 V~PE' 'PE' 
 
 9 
 
 Substituting this value of v in the preceding equation, 
 
 * The force here spoken of, and represented by F, is the moving force, or 
 pressure on the body (see Art. 92.), and is therefore equal to that pressure 
 which would just sustain its attraction towards S. 
 
 f See Art. 113. (equations 89 and 90) ; what is proved there of a body acted 
 upon by the force of gravity which is constant, and whose direction is con- 
 stantly parallel to itself, is evidently true of any other constant force similarly 
 retaining a direction parallel to itself. To apply the same demonstration to 
 any such case, we have only indeed to assume g to represent another number 
 than 32*. 
 
 1 If / represent the additional velocity per second which F would com- 
 municate to the body, and t the time of describing PT, then (Art. 44.) 
 
 =/*; but (Art. 46.) PT=^ 9 = *=|<; so that is the velocity with 
 which PT would be described uniformly in the time t. 
 
108 
 
 CENTRIFUGAL FORCE. 
 
 Now let a circle PQY be described having a common tan 
 gent with the curve AB in the point P, and passing through 
 the point Q. Produce PS to intersect the circumference of 
 this circle in Y, and join QY ; then are the triangles PQY 
 and QPR similar, for the angle RQP is equal to the angle 
 QPY (QE and YP being parallel), and the angle QPR is 
 equal to the angle Q YP in the alternate segment of the cir- 
 
 cle. Therefore = therefore QR=. Substi- 
 
 tuting this value of QR in the last equation, we have 
 
 QV 
 
 Now this is true, however much PQ may be diminished. 
 Let it be infinitely diminished, the supposed constant amount 
 and parallel direction of F will then coincide with the actual 
 case of a variable amount and inclination of that force, the 
 
 PQ 
 
 ratio ~~ will become a ratio of equality, and the circle 
 
 PQY will become the circle of curvature at P, and PY that 
 chord of the circle of curvature, which being drawn from P 
 passes through S. Let this chord of the circle of curvature 
 be represented by C, 
 
 The force or pressure F thus determined is manifestly 
 exactly equal to that force by which the body tends in its 
 motion continually to fly from the centre S, and may there- 
 fore be called its centrifugal force. This term is, however, 
 generally limited in its application to the case of a body re- 
 volving in a circle, and to the force with which it tends to 
 recede from the centre of that circle ; or if applied to the 
 case of motion in any other curve, then it means the force 
 with which the body tends to recede from the centre of the 
 circle of curvature to its path at the point through which it 
 is, at any time, moving. When the body revolves in a cir- 
 cular path, the circle of curvature to the path at any one 
 point evidently coincides with it throughout, and the chord 
 of curvature becomes one of its diameters. Let the radius 
 of the circle which the body thus describes be represented 
 by R, then C=2R ; 
 
CENTRIFUGAL FORCE. 109 
 
 > 
 
 Since in whatever curve a body is moving, it may be con- 
 ceived at any point of its path to be revolving in the circle 
 of curvature to the curve at that point, the force F, with 
 which it then tends to recede from the centre of the circle 
 of curvature is represented by the above formula, B. being 
 taken to represent the radius of curvature at the point of its 
 path through which it is moving. 
 
 If a be the angular velocity of the body's revolution about 
 the centre of its circle of curvature, then V=R 5 
 
 /.F=-a'R ..... (102). 
 9 
 
 122. From equation (100) we obtain 
 
 /y 
 
 Now (Art. 94.) ^ represents the additional velocity per 
 
 second f, which would be communicated to a body falling 
 towards 6, if the body fell freely and the force F remained 
 constant. Moreover, by Art. 47. it appears, that Y is the 
 whole velocity which the body would on this supposition 
 acquire, whilst it fell through a distance equal to JC, or to 
 one quarter of the chord of curvature. Thus, then, the velo- 
 city of a body revolving in any curve and attracted towards 
 a centre of force is, at any point of that curve, equal to that 
 which it would acquire in falling freely from that ppint to- 
 wards the centre of force through one quarter of that chord 
 of curvature which passes through the centre of force, if the 
 force which acted upon it at that point in the curve re- 
 mained constant during its descent. It is in this sense that 
 the velocity of a body moving in any curve about a centre 
 of force is said to be THAT DUE TO ONE .QUARTER THE CHORD 
 
 OF CURVATRE. 
 
 123. The centrifugal force of a mass of finite dimensions. 
 
 Let BC represent a thin lamina or slice 
 of such a mass contained between two planes 
 exceedingly near to one another, and both 
 perpendicular to a given axis A, about 
 which the mass is made to revolve. 
 
110 
 
 CENTRIFUGAL FORCE. 
 
 Through A draw any two rectangular axes Ax and Ay, 
 let m 1 be any element of the lamina whose weight is w^ and 
 let AM X and AN^ co-ordinates of m 1? be represented by x 1 
 and y y Then by equation (102), if a represent the angular 
 velocity of the revolution of the body, the centrifugal force 
 
 on the element m l is represented by w^Am^ Let now this 
 
 J/ 
 
 force, whose direction is Am 1 be resolved into two others, 
 whose directions are Ax and Ay. The former will be repre- 
 
 sentedby w^Am^ cos. xAm l7 or by wp^ and the latter 
 
 g 2 g 
 
 by w l Am l cos. yAm 1? or by w$ 1 ; and the centrifugal 
 
 9 <? 
 
 forces and all the other elements of the lamina being simi- 
 larly resolved, we shall have obtained two sets of forces, 
 those of the one set being parallel to A#, and represented 
 
 by w^ wjc^ WsPsi &c. and those of the other set 
 ' 
 
 parallel to Ay 1 represented by w^^ __w,,y 2 ,_i0 3 y 3 , &c. 
 
 ff 9 9 
 
 Now if X and Y represent the resolved parts parallel to 
 the directions of Ax and Ay, of the resultant of these two 
 sets of forces, then (Art. 11.) 
 
 if G- a and G 2 represent the co-ordinates AG! and AG 2 of the 
 centre of gravity G of the lamina, and W its weight 
 (Art. 18.). 
 
 Now the whole centrifugal force F on the lamina is the 
 resultant of these two sets of forces, and is represented by 
 
 * (Art. 11.), 
 
 
 /. F= T-W'G/ + W 2 G 2 3 = - W VG^ + eV, or 
 
 (103), 
 
 where G is taken to represent the distance AG of the centre 
 of gravity of the lamina from the axis of revolution. 
 
 Moreover, the direction of this resultant centrifugal force 
 
CENTRIFUGAL FORCE. 
 
 IB through A, since the direction of all its components are 
 through that point 
 
 124. From the above formula, it is apparent that if a body 
 revolving round a fixed axis be conceived to 
 be divided into laminae by planes perpendicu- 
 lar to the axis, then the centrifugal force of 
 each such laminae is the same as it would 
 have been if the whole of its weight had 
 been collected in its centre of gravity; so 
 o! that if the centres of gravity of all the laminae 
 
 be in the same plane passing through the 
 axis, then, since the centrifugal force on each lamina has its 
 direction from the axis through the centre of gravity of that 
 lamina, it follows that all the centrifugal forces of these 
 laminae are in the same plane, and that they are PARALLEL 
 forces, so that their resultant is equal to their sum, those 
 being taken with a negative sign which correspond to 
 laminae whose centres of gravity are on the opposite side of 
 the axis from the rest, and whose centrifugal forces are 
 therefore in the opposite directions to those of the rest. 
 Thus if F' represent the whole centrifugal force of such a 
 
 mass, then F'= 2WG. Now let W represent the weight 
 
 of the whole mass, and G' the distance of its centre of gra- 
 vity from the axis, therefore sWG=W'G'; 
 
 /. F'=-WG' (104). 
 
 Li the case, then, of a revolving body capable of being 
 divided into laminae perpendicular to the axis of revolution, 
 the centres of gravity of all of which laminae are in the 
 same plane passing through the axis, the centrifugal force is 
 the same as it would have been if the whole weight of the 
 body had been collected in its centre of gravity, the same 
 property obtaining in this case in respect to the whole body 
 as obtains in respect to each of its individual laminos. 
 Since, moreover, the centrifugal forces upon the laminae are 
 parallel forces when their centres of gravity are all in the 
 same plane passing through the axis of gravity, and since 
 their directions are all in that plane, it follows (Art. 16.), 
 that if we take any point O in the axis, and measure the 
 moments of these parallel forces from that point, arid call 
 a? the perpendicular distance OA of any lamina BC from 
 
112 THE PRINCIPLE OF VIRTUAL VELOCITIES. 
 
 that point, and H the distance of their resultant from that 
 point, then 
 
 The equations (104:) and (105) determine the amount and 
 the point of application of the resultant of the centrifugal 
 forces upon the mass, upon the supposition that it can be 
 divided into laminae perpendicular to the axis of revolution, 
 all of which have their centres of gravity in the same plane 
 passing through the axis. 
 
 It is evident that this condition is satisfied, if the body be 
 symmetrical as to a certain axis, and that axis be in the 
 same plane with the axis of revolution, and therefore if it 
 intersect or if it be parallel to the axis of revolution. 
 
 If, in the case we have supposed, 2WG O, that is, if the 
 centre of gravity be in the axis of revolution, then the cen- 
 trifugal force vanishes. This is evidently the case where a 
 body revolves round its axis of symmetry. 
 
 125. If the centres of gravity of the laminae into which 
 the body is divided by planes perpendicular to 
 the axis of revolution be not in the same plane 
 (as in the figure), then the centrifugal forces of 
 the different laminae will not lie in the same 
 plane, but diverge from the axis in different 
 directions round it. The amount and direction 
 of their resultant cannot in this case be deter- 
 mined by the equations which have been given 
 above. 
 
 THE PRINCIPLE OF VIRTUAL VELOCITIES. 
 
 126. If any pressure P, whose point of application A is 
 made to move through the straight line AB, ~be resolved 
 into three others X, Y, Z, in the directions of the three 
 rectangular axes, Ox, Oy, Qz / and if AC, AD, and AE, 
 be the projections of AB upon these axes, then the work of 
 P through AB is equal to the sum of the works of X, Y, Z, 
 through AC, AD, and AE respectively, or X . AC-f-Y . 
 
 E5+Z . AE=P . AM. 
 
THE PRINCIPLE OF VIKTtJAL VELOCITIES. 113 
 
 Let the inclinations of the direction 
 of P to the axes Ox, Oy, Oz respec- 
 tively, be represented by a, ft y, and 
 the inclniations of AB to the same 
 axes by a n ft, y n 
 
 /. (Art. 12.) X=P cos. a, Y=P cos. ft Z=P cos. y ; also AC 
 
 =AB cos. a,, AD=AB cos. ft, AE=rAB cos. y 15 
 /. X/AC=P. AB cos. a cos. 15 Y. AD^P. AB cos. /3 cos. ft, 
 
 Z . AE=P . AB cos. y cos. y,, 
 /.X . AC+Y . AD + Z . AE=P . AB {cos. a cos. a^cos. fr 
 
 cos. ft + cos. y cos. y^. 
 
 But by a well-known theorem of trigonometry, cos* eos. 
 Wj + cos. /3 cos. ft + cos. y cos. 7,= cos. PAB, 
 
 /.X . AC+Y . AD + Z . AE=P . AB cos. PAB; 
 
 but AB cos. PAB= AM ; 
 
 .-. X . AC+Y . AD + Z . AE=P . AM. 
 
 But (Art. 52.) the w^ork of P through AM is equal to its 
 work through AB. Therefore, &c.* 
 
 127. If any number of forces fie in equilibrium (being in 
 any way mechanically connected with one another), and if, 
 subject to that connection, their different points of appli- 
 cation "be made to move, each through any exceedingly small 
 distance, then the aggregate of the work of those forces, 
 whose points of application are made to move towards the 
 
 * This proposition may readily be deduced from Art. 53., for pressures equal 
 and opposite to X, Y, Z, would just be in equilibrium with P, and these tend- 
 ing to move the point A in one direction along the line AB, P tends to move 
 it in the opposite direction, therefore in the motion of the point A through AB, 
 the sum of the works of X, Y, Z, must equal the work of P. But the work of 
 X, as its point of application moves through AB, is equal (Art. 52.) to the 
 work of X through the projection of AB upon Aar, that is, through AC ; simi- - 
 larly the work of Y, as its point of application moves through AB, is equal to , 
 its work through the projection of AB upon Ay, or through AD; and so of Z. 
 The sum of the works of X, Y, and Z, as their point of application is made to 
 move through AB, is therefore equal to what would have been the sum of their - 
 works had their points of application been made to move separately through . 
 AC, AD, AE ; this last sum is therefore equal to the work of P through AB, 
 which is equal to the work of P through AM, AM being the projection of AB ; 
 upon the direction of P. 
 
 8 
 
114: PRINCIPLE OF VIRTUAL VELOCITIES. 
 
 directions in which the several forces applied to them act t 
 shall equal the aggregate of the work of those forces, the 
 motions of whose points of application are opposed to the 
 directions of the forces applied to them. 
 
 For let all the forces composing such a system be re- 
 solved into three sets of forces parallel to three rectangular 
 axes, and let these three sets of parallel forces be repre- 
 sented by A, B, and C respectively. Then must the result- 
 ant of the parallel forces of each set equal nothing. For if 
 any of these resultants had a finite value, then (by Art. 12.) 
 the whole three sets of forces would have a resultant, which 
 they cannot, since they are in equilibrium. 
 
 Now let the motion of the points of application of the 
 forces be conceived so small that the amounts and directions 
 -of tiie forces may be made to vary, during the motion, only 
 by aa exceedingly small quantity, and so that the resolved 
 forces upon any point of application may remain sensibly 
 unchanged. Also let u^ u^ u^ represent the works of these 
 resolved feces respectively on any point, and 2^ the sum 
 of all the works of the resolved forces of the set A, 2u^ the 
 ;sam of all the works of the forces of the set 13, and 2u s of the 
 ;set C. Now since the parallel forces of the set A have no 
 {resultant, therefore (Art. 59.) the sum of the works of those 
 forces of this set, whose points of application are moved 
 tom&rds the directions of their forces, is equal to the sum of 
 the works of those whose points of application are moved 
 jfrota> the directions of their forces, so that 2^=0, if the 
 values of u^ which compose this sum, be taken with the 
 positive or negative sign, according to the last mentioned 
 condition. 
 
 Similarly , 2w 2 = and 2^ 3 = 0, /. Zfa -\~u^ + ?/ 3 ) 0. 
 
 Now let U represent the actual work of that force P n the 
 works of whose components parallel to the three axes are 
 represented by u^ u^ u 3 ; then by the last proposition, 
 
 (106); 
 
 in which expression U is to be taken positively or negatively 
 according to the same condition as u^ u u 3 ; that is, accord- 
 ing as the work at each point is done in the direction of the 
 corresponding force, or in a direction opposite to it. Hence 
 therefore it follows, from the above equations, that the sum 
 
THE PRINCIPLE OF VIS YIVA. 115 
 
 of the works in one of these directions equals their sum in 
 the opposite direction. Therefore, &c. 
 
 The projection of the line described by the point of appli- 
 cation of any force npon the direction of that 'force is called 
 its VIRTUAL VELOCITY, so that the product of the force by its 
 virtual velocity is in fact the work of that force ; hence 
 therefore, representing any force of the system by P, and 
 its virtual velocity by p, we have Pp=tl, and therefore, 
 =iO, which is the principle of virtual velocities.* 
 
 128. If there be a system of forces such that their points of 
 application being moved through certain consecutive posi- 
 tions, those forces are in all such positions in equilibrium, 
 then in respect to any finite motion of the points of appli- 
 cation through that series of positions, the aggregate of the 
 work of those forces, which act in the directions in which 
 their several points of application are made to move, is equal 
 to the aggregate of the work in the opposite direction. 
 
 This principle has been proved in the preceding proposi- 
 tion, only when the motions communicated to the several 
 points of application are exceedingly small, so that the work 
 done by each force is done only through an exceedingly 
 small space. It extends, however, to the case in which each 
 point of application is made to move, and the work of each 
 force to be done, through any distance, however great, pro- 
 vided only that in all the different positions which the points 
 of application of the forces of the system are thus made to 
 take up, these forces be in equilibrium with one another ; for 
 it is evident that if there be a series of such positions 
 immediately adjacent to one another, then the principle 
 obtains in respect to each small motion from one of this set 
 of positions into the adjacent one, and therefore in respect 
 to the sum of all such small motions as may take place in the 
 system in its passage from any one position into any other, 
 that is, in respect to the whole motion of the system through 
 the intervening series of positions. Therefore, &c. 
 
 THE PRINCIPLE OF Yis YIVA. 
 
 129. If the forces of any system be not in equilibrium with 
 one another, then the difference between the aggregate work 
 
 * This proof of the principle of virtual velocities is given here for the first 
 time. 
 
116 THE PKINCTPLE OF VIS VIVA. 
 
 of those whose tendency is in the direction of the motions 
 of their several points of application, and those whose ten- 
 dency is in the opposite direction, is equal to one half the 
 aggregate vis viva of the system. 
 
 In each of the consecutive positions which the bodies com- 
 
 ring the system are made successively to take up, let there 
 applied to each body a force equal to the effective force 
 (Art. 103.) upon that body, but in an opposite direction; 
 every position will then become one of equilibrium. 
 
 Now, as the bodies which compose the system and the 
 various points of application of the impressed forces move 
 through any finite distances from one position into another, 
 let 2^ represent the aggregate work of those impressed 
 forces whose directions are towards the directions of the 
 motions of their several points of application, and let 2^ 2 
 represent the work of those impressed forces which act in the 
 opposite directions ; also let 2u 3 represent the aggregate 
 work of forces applied to the system equal and opposite to 
 the effective forces upon it ; the directions of these forces 
 opposite to the effective forces are manifestly opposite also 
 to the directions of the motions of their several points of 
 application, so that on the whole 2^ is the aggregate work 
 of those forces whose directions are towards the motions of 
 their several points of application, and 2w a + 5to 8 the aggre- 
 gate work opposed to them. Since therefore, by D'Alem- 
 berf s principle, an equilibrium obtains in every consecutive 
 position of the system, it follows by the last proposition, 
 that 
 
 (107). 
 
 Now u s is taken to represent the work of a force equal and 
 opposite to the effective force upon any body of the system ; 
 but the work of such a force through any space is equal to 
 the work which the effective force (being unopposed) accu- 
 mulates in the body through that space (Art. 69.), or it is 
 equal to one half the difference of the vires vivse of the body 
 at the commencement and termination of the time during 
 which that space is described (Art. 67.). Therefore 2^ 3 
 equals one half the aggregate difference of the vires vivce of 
 the system at the two periods ; 
 
 -O ..... (108). 
 
POSITION OF MAXIMUM OR MINIMUM VIS VIVA. 117" 
 
 Thus then it follows, that the difference between the aggre 
 gate work 2 u, of those forces, the tendency of each of which 
 is towards the direction of the motion of its point of applica- 
 tion, and that 2w 2 of those the direction of each of which is 
 opposed to the motion of its point of application (or, in other 
 words the difference between the aggregate work of the 
 accelerating forces of the system and that of the retarding 
 forces), is equal to one half the vis viva accumulated or lost 
 in the system whilst the work is being done, which is the 
 PRINCIPLE OF Yis VIVA. 
 
 130. One half the vis viva of the system measures its 
 accumulated work ; the principle of vis viva amounts, 
 therefore, to no more than this, that the entire difference 
 between the work done by those forces which tend to accele- 
 rate the motions of the parts of the system to which they 
 are applied, and those which tend to retard them, is accu- 
 mulated in the moving parts of the system, no work 
 whatever being lost, but all that accumulated which is done 
 upon it by the forces which* tend to accelerate its motion, 
 above that which is expended upon the retarding forces. 
 
 This principle has been proved generally of any mechani- 
 cal system ; it is therefore true of the most complicate 1 
 machin^. The entire amount of work done by the moving 
 power, whatever it may be, upon that machine, is yielde I 
 partly at its working points in overcoming the resistancos 
 opposed there to its motion (that is, in doing its useful 
 work), it is partly expended in overcoming the friction and 
 other prejudicial resistances opposed to the motion of the 
 machine between its moving and its working points, and all 
 the rest is accumulated in the moving parts of the machine, 
 ready to be yielded up under any deficiency of the moving 
 power, or to carry on the machine for a time, should the 
 operation of that power be withdrawn. 
 
 131. When the forces of any system (not in equilibrium in 
 every position which the parts of that system^ may be 
 made to take up] pass through a position of equilibrium, 
 the vis viva of the system 'becomes a maximum or a 
 minimum. 
 
 For, as in Art. 129., let the aggregate work done in the 
 directions of the motions of the several parts of the system 
 
US POSITION OF MAXIMUM OK MINIMUM VIS VIVA. 
 
 be represented by 2^ 15 and the aggregate work done in 
 directions opposed to the motions of the several parts by 
 2u^ then (Art. 129.), one half the acquired vis viva of 
 system =2^2^. E~ow as the system passes from any one 
 position to any other, each of the quantities 2^ and 2?^ a is 
 manifestly increased. If 2^ increases by a greater quan- 
 tity than 2^ 2 , then the ms viva is increased in this change 
 of position ; if, on the contrary, it is increased by a less 
 quantity than 2^ 2 , then the vis viva is diminished. Thus if 
 A2w l and A2?^ 2 represent the increments of 2^ and 2^ a in 
 this change of position, then (2^ 1 + A2^ 1 ) (2u 9 + A2u 9 ), or 
 (2^ x 2^ 2 ) -j- (A2u 1 A2t 2 ) 5 representing one half the vis 
 viva after this change of position, it is manifest that the vis 
 viva is increased or diminished by the change according a& 
 A2^ is greater or less than A2^ 2 ; and that if AI;^ be equa) 
 to A2^ 2 then no change takes place in the amount of the 
 vis viva of the system as it passes from the one position to 
 the other. 
 
 Now from the principle of virtual velocities (Art. 127.), 
 it appears, that precisely this case occurs as the system 
 passes through a position of equilibrium, the aggregate 
 work of those forces whose tendency is to accelerate the 
 motions of their points of application then precisely equal- 
 ling that of the forces whose tendency is opposed to these 
 motions. For an exceeding small change of position there- 
 fore, passing through a position of equilibrium, A2^=A2^ 3J 
 an equality which does not, on the other hand, obtain, 
 unless the body do thus pass through a position of equili- 
 brium. 
 
 Since then the sum 2^ 2^ 25 and therefore the aggregate 
 vis viva of the system, continually increases or diminishes 
 up to a position of equilibrium, and then ceases (for a cer- 
 tain finite space at least) to increase or diminish, it follows, 
 that it is in that position a maximum or a minimum. 
 Therefore, &c. 
 
 132. When the forces of any system pass through a position 
 of equilibrium, the vis viva "becomes a maximum or a 
 minimum, according as that position is one of stable or 
 unstable equilibrium. 
 
 For it is clear that if the vis viva be a maximum in any 
 position of the equilibrium of the system, so that after it 
 Has passed out of that position into another at some finite 
 
STABLE AND UNSTABLE EQUILIBRIUM. 
 
 distance from it, the acquired vis viva may have become 
 less than it was before, then the aggregate work of the 
 forces which tend to accelerate the ^notion between these 
 two positions must have been less than that of the forces 
 which tend to retard the motion (Art. 131.). Now suppose 
 the body to have been placed at rest in this position of 
 equilibrium, and a small impulse to have been communi- 
 cated to it, whence has resulted an aggregate amount of 
 vis viva represented by 2mY 2 . In the transition from the 
 first to the second position, let this vis viva have become 
 2mv* ; also let the aggregate work of the forces which have 
 tended to accelerate the motion, between the two positions, 
 be represented by 2U,, and that of the forces which have 
 tended to retard the motion by 2U 2 ; then, for the reasons 
 assigned above, it appears that 2U ? is greater than 21^. 
 Moreover, by the principle of vis viva, 
 
 in which equation the quantity 2(2 TJ 2 2U,) is essentially 
 positive, in respect to the particular range of positions 
 through which the disturbance is supposed to take place.* 
 
 For every one of these positions there must then be a 
 certain. value of 2mV 2 , that is, a certain original impulse 
 and disturbance of the system from its position of equili- 
 brium, which will cause the second member of the above 
 equation, and therefore its first member 2mv*, to vanish. 
 Now every term of the sum ^mv z is essentially positive ; 
 this sum cannot therefore vanish unless each term of it 
 vanish, that is, unless the velocity of each body of the 
 system vanishes, or the whole be brought to- rest. This 
 repose of the system can, however, only be instantaneous ; 
 for, by supposition, the position into which it has been dis- 
 placed is not one of equilibrium. Moreover, the displace- 
 ment of the system cannot be continued in the direction in 
 which it has hitherto taken place, for the negative term in 
 the second member of the above equation would yet fkrther 
 be increased so as to exceed the positive term, and the first 
 
 * The disturbance is of course to be limited to that particular range of 
 positions to which the supposed position of equilibrium stands in the relation 
 of a position of maximum vis viva. If there be other positions of equili- 
 brium of the system, there will be other ranges of adjacent positions, in 
 respect to each of which there obtains a similar relation of maximum or mini- 
 mum vis viva. 
 
i20 STABLE AND UNSTABLE EQUILIBRIUM. 
 
 member 2mv* would thus become negative, which ia 
 impossible. 
 
 The motion of the system can then only be continued by 
 the directions of the motions of certain of the elements 
 which compose it being changed, so that the corresponding 
 quantities by which SU, and 2U 2 are respectively increased 
 may change their signs, and the whole quantity ^Uj 2U 2 
 which before increased continually may now continually 
 diminish. This being the case, 2mv* will increase again 
 until, when st^ 2U 2 =0, it becomes again equal to %m V s ; 
 that is, until the system acquires again the vis viva with 
 which its disturbance commenced. 
 
 Thus, then, it has been shown, that in respect to every 
 one of the supposed positions of the system* there is a cer- 
 tain impulse or amount of vis viva, which being communi- 
 cated to the system when in equilibrium, will just cause it 
 to oscillate as far as that position, remain for an instant at 
 rest in it, then return again towards its position of equili- 
 brium, and re-acquire the vis viva with which its displace- 
 ment commenced. Now this being true of every position 
 of the supposed range of positions, it follows that it is true 
 of every disturbance or impulse which will not carry the 
 system beyond this supposed range of positions ; so that, 
 having been displaced by any such disturbance or impulse, 
 the system will constantly return again towards the position 
 of equilibrium from which it set out, and is STABLE in 
 respect to that position. 
 
 On the other hand, if the supposed position of equili- 
 brium be one in which the vis viva is a minimum, then the 
 aggregate work of the forces which tend to accelerate the 
 motion must, after the system has passed through that posi- 
 tion, exceed that of the forces which tend to retard the 
 motion ; so that, adopting the same notation as before, 2U, 
 must be greater than 2U 25 and the second member of the 
 equation essentially positive. Whatever may have been the 
 original impulse, and the communicated vis viva 2mY 2 , 
 Sm^ 2 must therefore continually increase ; so that the whole 
 system can never come to a position of instantaneous repose ;f 
 but on the contrary, the motions of its parts must continu- 
 ously increase, and it must deviate continually farther from 
 its position of equilibrium, in which position it can never 
 
 * That is, of that range of positions over which the supposed position of 
 equilibrium holds the relation of a position of maximum vis viva. 
 
 f Within that range of positions over which the supposed position of 
 equilibrium holds the relation of minimum vis viva. 
 
DYNAMICAL 'STABILITY. 121 
 
 rest. The position is thus one of unstable equilibrium 
 Therefore, &c. 
 
 DYNAMICAL STABILITY.* 
 
 If a body be made, by the action of certain disturbing 
 forces, to pass from one position of equilibrium into another, 
 and if in each of the intermediate positions these forces are 
 in excess of the forces opposed to its motion, it is obvious 
 that, by reason of this excess, the motion will be continually 
 accelerated, and that the body will reach its second position 
 with a certain finite velocity, whose eifect (measured under 
 the form of vis viva) will be to carry it beyond that position. 
 This however passed, the case will be reversed, the resist- 
 ances will be in excess of the moving forces, and the body's 
 velocity being continually diminished and eventually de- 
 stroyed, it will, after resting for an instant, again return 
 towards the position of equilibrium through which it had 
 passed. It will not however finally rest in this position until 
 it has completed other oscillations about it. Now the am- 
 plitude of the first oscillation of the body beyond the posi- 
 tion in which it is finally to rest, being its greatest ampli- 
 tude of oscillation, involves practically an important condi- 
 tion of its stability ; for it may be an amplitude sufficient to 
 carry the body into its next adjacent position of equilibrium, 
 which being, of necessity, a position of unstable equilibrium, 
 the motion will be yet further continued and the body 
 overturned. Different bodies requiring moreover different 
 amounts of work to be done upon them to produce in all the 
 same amplitude of oscillation, that is (relatively to that am- 
 plitude) the most stable which requires the greatest amount 
 of work to be so done upon it. It is this condition of stabi- 
 lity, dependent upon dynamical considerations, to which, in 
 the following paper, the name of dynamical, stability is 
 given. * 
 
 I cannot find that the question has before been considered 
 in this point of view, but only in that which determines 
 whether any given position be one of stable, unstable, or 
 mixed equilibrium ; or which determines what pressure is 
 necessary to retain the body at any given inclination from 
 such a position. 
 
 * Extracted from a paper " On Dynamical Stability, and on the Oscillations 
 of Floating Bodies," by the author of this work, published in the Transactions 
 of the Royal Society, Part. II. for 1850. The remainder of the paper will be 
 found in the Appendix. 
 
122 DYNAMICAL STABILITY. 
 
 1. To the discussion of the conditions of the dynamicaj 
 stability of a body the principle of vis viva readily lends 
 itself. That principle,* when translated into a language 
 which the labours of M. PONCELET have made familiar to 
 the uses of practical science, may be stated as follows : 
 
 " When, being acted upon by given forces, a body or sys- 
 tem of bodies has been moved from a state of rest, the differ- 
 ence between the aggregate work of those forces whose 
 tendencies are in the directions in which their points of 
 application have been moved, and that of the forces whose 
 tendencies are in the opposite direction, is equal to one-half 
 the vis viva of the system." 
 
 Thus, if 2^ be taken to represent the aggregate work of 
 the forces by which a body has been displaced from a posi- 
 tion in which it was at rest, and 2^ Q the aggregate work 
 (during this displacement) of the other forces applied to it ; 
 and if the terms which compose 2^ and 2^ 2 be understood 
 to be taken positively or negatively, according as the ten- 
 dencies of the corresponding forces are in the directions in 
 which their points of application have been made to move 
 or in the opposite directions ; then representing the aggre- 
 gate vis viva of the body by - 2wv*. 
 
 t/ 
 
 2^ + 2^ =^2^', (!'). 
 
 Now 2i 2 representing the aggregate work of those forces 
 which acted upon the body in the position from which it has 
 been moved, may be supposed to the known ; 2^ may there- 
 fore be determined in terms of the vis viva, or conversely. 
 
 2. In the extreme position into which the body is made to 
 oscillate and from which it begins to return, it, for an instant, 
 rests. In this position, therefore, its vis viva disappears, and 
 we have 
 
 2^+2^=0 (2'). 
 
 This equation, in which 2-^ and 2^ 2 are functions of the 
 impressed forces and of the inclination, determines the ex- 
 treme position into which the body is made to roll by the 
 action of given disturbing forces ; or, conversely, it deter- 
 mines the forces by which it may be made to roll into a 
 given extreme position. 
 
 * See Art. 129. 
 
DYNAMICAL STABILITY. 123 
 
 3. The position in. which it will finally rest is determined 
 by the maximum value of 2^ + 2^ in equation (I/) ; for, by 
 a well-known property, the vis viva of a system* attains a 
 maximum value when it passes through a position of stable, 
 and a minimum, when it passes through a position of unstable 
 equilibrium. The extreme position into which the body 
 oscillates is therefore essentially different from that in which 
 it will finally rest. 
 
 4. Different bodies, requiring different amounts of work to 
 be done upon them to bring them to the same given inclina- 
 tion, that is (relatively to that inclination) the most stable 
 which requires the greatest amount of work to be so done 
 upon it, or in respect to which ^u { is the greatest. If, in- 
 stead of all being, brought to the same given inclination, each 
 is brought into a position of unstable equilibrium, the corre- 
 sponding value of 2^ represents the amount of work which 
 must be done upon it to overthrow it, and may be considered 
 to measure its absolute, as the former value measures its 
 relative dynamical stability, f The absolute dynamical sta- 
 bility of a body thus measured I propose to represent by the 
 symbol U, and its relative dynamical stability, as to the 
 inclination 0, by U(0). 
 
 The measure of the absolute dynamical stability of a body 
 is the maximum value of its relative stability, or U the max- 
 imum of U(d) ; for whilst the body is made to incline from 
 its position of stable equilibrium, it continually tends to 
 return to it until it passes through a position of unstable 
 equilibrium, when it tends to recede from it ; the aggregate 
 amount of work necessary to produce this inclination must 
 therefore continually increase until it passes through that 
 position and afterwards diminish. 
 
 5. The work opposed by the weight of a body to any 
 change in its position is measured by the product of the 
 vertical elevation of its centre of gravity by its weight.;): 
 Kepresenting therefore by W the weight of the body, and 
 by AH the vertical displacement of its centre of gravity 
 when it is made to incline through an angle 0, and observ- 
 ing that the displacement of this point is in a direction oppo- 
 site to that in which the force applied to it acts, we have 
 
 , and by equation (2'), 
 
 * Art. 132. 
 
 f It is obvious that the absolute dynamical stability of a body may be 
 greater than that of another, whilst its stability, relatively to a given inclina- 
 tion, is less ; less work being required to incline it than the other at that 
 angle, but more, entirely to overthrow it. 
 
 f Art. 60. 
 
FKICTION. 
 
 (3). 
 
 If therefore no other force than its weight be opposed to a 
 body's being overthrown, its absolute dynamical stability, 
 when resting on a rigid surface, is measured by the product 
 of its weight hy the height through which its centre of gravity 
 must he raised to bring it from a stable into an unstable 
 position of equilibrium. 
 
 6. The Dynamical Stability of Floating Bodies. The 
 action of gusts of wind upon a ship, or of blows of the sea, 
 being measured in their 'eifects upon it by their work, that 
 vessel is the most stable under the influence of these, or will 
 roll and pitch the least' (other things being the same), which 
 requires the greatest amount of work to be done upon it to 
 bring it to a given inclination ; or, in respect to which the 
 relative dynamical stability U (4) is the greatest for a given 
 value of 0. In another sense, that ship may be said to be the 
 most stable which would require the greatest amount of work 
 to be done upon it to bring it into a position from which it 
 would not again right itself, or whose absolute dynamical 
 stability U is the greatest. Subject to the one condition, 
 the ship will roll the least, and subject to the other, it will 
 be the least likely to roll over. 
 
 Thus the theory of dynamical stability involves a question 
 of naval construction. It will be found discussed in its ap- 
 plication to this question in the Appendix. 
 
 FBICTIOK 
 
 133. It is a matter of constant experience, that a certain 
 resistance is opposed to the motion of one body on the sur- 
 face of another under any pressure, however smooth may be 
 the surfaces of contact, not only at the first commencement, 
 but at every subsequent period of the motion ; so that, not 
 only is the exertion of a certain force necessary to cause the 
 one body to pass at first from a state of rest to a state of mo- 
 tion upon the surface of the other, but that a certain force is 
 further requisite to keep up this state of motion. The resist- 
 ance thus opposed to the motion of one body on the surface 
 of another when the two are pressed together, is called fric- 
 
FRICTION. 125 
 
 tion ; that which opposes itself to the transition from a state 
 of continued rest to a state of motion is called the friction 
 of quiescence that which continually accompanies the state 
 of motion is called the friction of motion. 
 
 The principal experiments on friction have been made by 
 Coulomb*, Vince, G. Kennief, K "Wood;):, and recently 
 (at the expense of the French Government) by Morin. 
 They have reference, first, to the relation of the friction 
 of quiescence to the friction of motion ; secondly, to the 
 variation of the friction of the same surfaces of contact under 
 different pressures / thirdly, to the relation of the friction to 
 the extent of the surface of contact ; fourthly, to the relation 
 of the amount of the friction of motion to the velocity of the 
 motion ; fifthly, to the influence of unguents on the laws of 
 friction, and on its amount under the same circumstances of 
 pressure and contact. The following are the principal facts 
 which have resulted from these experiments ; they consti- 
 tute the laws of friction. 
 
 1st. That the friction of motion is subject to the same 
 laws with the friction of quiescence (about to be stated), but 
 agrees with them more accurately. That, under the same 
 circumstances of pressure and contact, it is nevertheless dif- 
 ferent in amount. 
 
 2ndly. That, when no unguent is interposed, the friction 
 of any two surfaces (whether of quiescence or of motion) is 
 directly proportional to the force with which they are pressed 
 perpendicularly together (up to a certain limit of that pres- 
 sure per square inch), so that, for any two given surfaces 
 of contact, there is a constant ratio of the friction to the per- 
 pendicular pressure of the one surface upon the other, 
 Whilst this ratio is thus the same for the same surfaces of 
 contact, it is different for different surfaces of contact. The 
 particular value of it in respect to any two given surfaces 
 of contact is called the CO-EFFICIENT of friction in re- 
 spect to those surfaces. The co-efficients of friction in respect 
 to those surfaces of contact, which for the most part form the 
 moving surfaces in machinery, are collected in a table, which 
 will be found at the termination of Art. 140. 
 
 3rdly. That, when no unguent is interposed, the amount 
 of the friction is, in every case, wholly independent of the 
 extent of the surfaces of contact, so that the force with which 
 two surfaces are pressed together being the same, and 
 
 Mem. des Sav. Etrang. 1781. t Phil - Trans - 1829. 
 
 A Practical Treatise on Rail-roads, 3d ed. chap. 76. 
 Mem. de 1'Institut. 1833, 1834, 1838. 
 
126 FRICTION. 
 
 not exceeding a certain limit (per square inch), their friction 
 is the same whatever may be the extent of their surfaces of 
 contact. 
 
 4thly. That the friction of motion is wholly independent 
 of the velocity of the motion.* 
 
 Stilly. That where unguents are interposed, the co-efficient 
 of friction depends upon the nature of the unguent, and upon 
 the greater or less abundance of the supply. In respect to 
 the supply of the ungent, there are two extreme cases, that 
 in which the surfaces of contact are but slightly rubbed with 
 the unctuous matterf , and that in which, by reason of the 
 abudant supply of the unguent, its viscous consistency, and 
 the extent of the surfaces of contact in relation to the insist- 
 ent pressure, a continuous stratum of unguent remains con- 
 tinually interposed between the moving surfaces, and the 
 friction is thereby diminished, as far as it is capable of being 
 diminished, by the interposition of the particular unguent 
 used. In this state the amount of friction is found (as might 
 be expected) to be dependent rather upon the nature of the 
 unguent than upon that of the surfaces of contact ; accord- 
 ingly M. Morin, from the comparison of a great number of 
 results, has arrived at the following remarkable conclusion, 
 easily fixing itself in the memory, and of great practical 
 value : " that with unguents, hog's lard and olive oil, inter- 
 posed in a continuous stratum between them, surfaces of wood 
 on metal, wood on wood, metal on wood, and metal on metal 
 (when in motion), have all of them very nearly the same co- 
 efficient of friction, the value of that co-efficient being in all 
 cases included between '07 and '08. 
 
 " For the unguent tallow, the co-efficient is the same as for 
 the other unguents in every case, except in that of metals upon 
 metals. This unguent appears, from the experiments of Mo- 
 rin, to be less suited to metallic substances than the others, 
 and gives for the mean value of its co-efficient under the same 
 circumstances -10." 
 
 134. Whilst there is a remarkable uniformity in the results 
 thus obtained in respect to the friction of surfaces, between 
 which a perfect separation is effected throughout their whole 
 extent by the interposition of a continuous stratum of the 
 
 * This result, of so much importance in the theory of machines, is fully esta- 
 blished by the experiments of Morin. 
 
 \ As, for instance, with an oiled or greasy cloth. 
 
FRICTION. 127 
 
 unguent, there is an infinite variety in respect to those states 
 of unctuosity which occur between the extremes, of which 
 we have spoken, of surfaces merely unctuous* and the most 
 perfect state of lubrication attainable by the interposition 
 of a given unguent. It is from this variety of states of the 
 unctuosity of rubbing surfaces, that so great a discrepancy 
 has been found in the experiments upon friction with ungu- 
 ents, a discrepancy which has not probably resulted so much 
 from a difference in the quantity of the unguent supplied to 
 the rubbing surfaces in different experiments, as in a diffe- 
 rence of the relation of the insistent pressures to the extent 
 of rubbing surface. It is evident, that for every description 
 of unguent there must correspond a certain pressure per 
 square inch, under which pressure a perfect separation of 
 two surfaces is made by the interposition of a continuous 
 stratum of that unguent between them, and which pressure 
 per square inch being exceeded, that perfect separation can- 
 not be attained, however abundant may be the supply of the 
 unguent. 
 
 The ingenious experiments of Mr. Nicholas Woodf, con- 
 firmed by those of Mr. G. Rennie^,. have fully established 
 these important conditions of the friction of unctuous surfaces. 
 It is much to be regretted that we are in possession of no 
 experiments directed specially to the determination of that 
 particular pressure per square inch, which corresponds in 
 respect to each unguent to the state of perfect separation, 
 and to the determination of the co-efficients of frictions in 
 those different states of separation which correspond to pres- 
 sures higher than this. 
 
 It is evident, that where the extent of the surface sustain- 
 ing a given pressure is so great as to make the pressure per 
 square inch upon that surface less than that which corres- 
 ponds to the state of perfect separation, this greater extent of 
 surface tends to increase the friction by reason of that adhe- 
 siveness of the unguent, dependent upon its greater or less 
 viscosity, whose effect is proportional to the extent of the 
 surfaces between w T hich it is interposed. The experiments 
 of Mr. Wood exhibit the effects of this adhesiveness in a 
 remarkable point of view. 
 
 * Or slightly rubbed with the unguent. 
 
 t Treatise on Rail-roads, 3rd ed. p. 399. 
 
 i Trans. Royal Soc. 1829. 
 
 5 It is evident that, whilst by extending the unctuous surface which sustains 
 any given pressure, we diminish the co-efficient of friction up to a certain 
 limit, we at the same time increase that adhesion of the surfaces which results 
 
128 FRICTION. 
 
 It is perhaps deserving of enquiry, whether in respect to 
 those considerable pressures under which the parts of the 
 larger machines are accustomed to move upon one another, 
 the adhesion of the unguent to the surfaces of contact, and 
 the opposition presented to their motion by its viscidity, are 
 causes whose influence may be altogether neglected as com- 
 pared with the ordinary friction. In the case of lighter 
 machinery, as for instance that of clocks and watches, these 
 considerations evidently rise into importance. 
 
 135. The experiments of M. Morin show the friction of 
 two surfaces which have been for a considerable time in con- 
 tact, to be not only different in its amount from the friction 
 of surfaces in continuous motion, but also, especially in this, 
 that the laws of friction (as stated above) are, in respect to 
 the friction of quiescence, subject to causes of variation and 
 uncertainty from which the friction of motion is exempt. 
 This variation does not appear to depend upon the extent of 
 the surfaces of contact, in which case it might be referred to 
 adhesion ; for with different pressures the co-efficient of the 
 friction of quiescence was found, in certain cases, to vary 
 exceedingly, although the surfaces of contact remained the 
 same.* The uncertainty which would have been introduced 
 into every question of construction by this consideration, is 
 removed by a second very important fact developed in the 
 course of the same experiments. It is this, that by the 
 slightest jar or shock of two bodies in contact, their friction 
 is made to pass from that state which accompanies quiescence 
 
 from the viscosity of the unguent, so lhat there may be a point where the gain 
 on the one hand begins to be exceeded by the loss on the other, and where 
 the surface of minimum resistance under the given pressure is therefore 
 attained. 
 
 Mr. Wood considers the pressure per square inch, which corresponds to the 
 minimum resistance, to be 90lbs. in the case of axles of wrought iron turning 
 upon cast iron, with fine neat's foot oil. The experiments of Mr. Wood, whilst 
 they place the general results stated above in full evidence, can scarcely how- 
 ever be considered satisfactory as to the particular numerical values of the con- 
 stants sought in this inquiry. In those experiments, and in others of the same 
 class, the amount of friction is determined from the observed space or time 
 through which a body projected with a given velocity moves before all its 
 velocity is destroyed, that is, before its accumulated work is exhausted. This 
 is an easy method of experiment, but liable to many inaccuracies. It is much 
 to be regretted that the experiments of Morin were not extended to the fric- 
 tion of unctuous surfaces, reference being had to the pressure per square 
 inch. 
 
 * Thus in the case of oak upon oak with parallel fibres, the co-efficient of 
 the friction of quiescence varied, under different pressures upon the same sur- 
 face, from -55 to '76. 
 
FRICTION. 129 
 
 to that which accompanies motion ; and as every machine or 
 structure, of whatever kind, may be considered to be subject 
 to such shocks or imperceptible motions of its surfaces of 
 contact, it is evident that the state of friction to be made 
 the basis on which all questions of statics are to be deter- 
 mined, should be that which accompanies continuous motion. 
 The laws stated above have been shown, by the experiments 
 of Morin, to obtain, in respect to that friction which accom- 
 panies motion, with a precision and uniformity never before 
 assigned to them ; they have given to all our calculations in 
 respect to the theory of machines (whose moving surfaces 
 have attained their proper bearings and ; been worn to their 
 natural polish) a new and unlooked-for certainty, and; may 
 probably be ranked amongst the most accurate and:valuable- 
 of the constants of practical science. 
 
 It is, however, to be observed, that all these experiments; 
 were made under comparatively small insistent pressures a& 
 compared with the extent of the surface pressed (pressures, 
 not exceeding from one to two kilogrammes per square- cen>- 
 timeter, or from about 14*3 to 28*6 Ibs. per square 'inch:.} In 
 adopting the results of M. Morin, it is of importance to bear 
 this fact in mind, because the experiments of Coulomb, and 
 particularly the excellent experiments of Mr. G\ Rennie, car- 
 ried far beyond these limits of insistent pressure*, have fully 
 shown the co-efficient of the friction of quiescence to increase 
 rapidly, from some limit attained long before the surfaces 
 abrade. In respect to some surfaces, as, for instance, wrought 
 iron upon wrought iron, the co-efficient nearly tripled itself 
 as the pressure advanced to the limits of abrasion. It is 
 greatly to be regretted that no experiments have yet been 
 directed to a determination of the precise limit about which 
 this change in the value of the co-efficient begins to take 
 place. It appears, indeed, in the experiments of Mr. Ren- 
 nie in respect to some of the soft metals, as, for instance, tin 
 upon tin, and tin upon cast iron ; but in respect to the harder 
 metals, his experiments passing at once from a pressure of 
 32 Ibs. per square inch to a pressure of 1*66 cwt. per square 
 inch, and the co-efficient (in the case of wrought iron for in- 
 stance) from about -148 to '25, the limit which we seek is 
 lost in the intervening chasm. The experiments of Mr. Ren- 
 nie have reference, nowever, only to the friction of qui- 
 escence. It seems probable that the co-efficient of the fric- 
 
 * Mr. Rennie's experiments were carried, in some cases, to from 5 cwt. to 
 7 cwt. per square inch. 
 
 9 
 
130 FRICTION. 
 
 tion of motion remains constant tinder a wider range of pres- 
 sure than that of quiescence. It is moreover certain, that 
 the limits of pressure beyond which the surfaces of contact 
 begin to destroy one another or to abrade, are sooner reached 
 when one of them is in motion upon the other, than when 
 they are at rest: it is also certain that these limits are not in- 
 dependent of the velocity of the moving surface. The dis- 
 cussion of this subject, as it connects itself especially with 
 the friction of motion, is of great importance ; and it is to be 
 regretted, that, with the means so munificently placed at his 
 disposal by the French Government, M. Morin did not ex- 
 tend his experiments to higher pressures, and direct them 
 more particularly to the circumstances of pressure and velo- 
 city under which a destruction of the rubbing surfaces first 
 begins to show itself, and to the amount of the destruction 
 of surface or wear of the material which corresponds to the 
 same space traversed under different pressures and different 
 velocities. Any accurate observer who should direct his 
 attention to these subjects would greatly promote the inter- 
 ests of practical science. 
 
 SUMMARY OF THE LAWS OF FRICTION. 
 
 136. From what has here been stated it results, that if P 
 represent the perpendicular or normal force by which one 
 body is pressed upon the surface of another, F the friction of 
 the two surfaces, or the force, which being applied parallel 
 to their common surface of contact, would cause one of them 
 to slip upon the surface of the other, and/* the co-efficient of 
 friction, then, in the case in which no unguent is interposed, 
 f represents a constant quantity, and (Art. 133.) 
 
 F=/P (109); 
 
 a relation which obtains accurately in respect to the friction 
 of motion, and approximately in respect to the friction of 
 quiescence. 
 
 137. The same relation obtains, moreover, in respect to 
 unctuous surfaces when merely rubbed with the unguent, or 
 where the presence of the unguent has no other influence 
 than to increase the smoothness of the surfaces of contact 
 without at all separating them from one another. 
 
 In unctuous surf aces partially lubricated, or between which 
 
THE LIMITING ANGLE OF RESISTANCE. 131 
 
 a stratum of unguent is partially interposed, the co-efficient 
 of friction/ 1 is dependent for its amount upon the relation of 
 the insistent pressure to the extent of the surface pressed, 
 or upon the pressure per square inch of surface. This 
 amount, corresponding to each pressure per square inch in 
 respect to the different unguents used in machines, has not 
 yet been made the subject of satisfactory experiments. 
 
 The amount of the resistance F opposed to the sliding of 
 the surfaces upon one another is, moreover, as well in this 
 case as in that of surfaces perfectly lubricated, influenced by 
 the adhesiveness of the unguent, and is therefore dependent 
 upon the extent of the adhering surface ; so that, if S repre- 
 sent the number of square units in this surface, and a the 
 adherence of each square unit, then aS represents the whole 
 adherence opposed to the sliding of the surfaces, and 
 
 (110); 
 
 where f is a function of the pressure per square unit ^-, and 
 
 a is an exceedingly small factor dependent on the viscosity 
 of the unguent. 
 
 THE LIMITING ANGLE OF RESISTANCE. 
 
 We shall, for the present, suppose the parts of a solid body 
 to cohere so firmly, as to be incapable of separation by the 
 action of any force which may be impressed upon them. 
 The limits within which this suposition is true will be dis- 
 cusse,d hereafter. 
 
 It is not to this resistance that our present inquiry has 
 reference, but to that which results from the friction of the 
 surface of bodies on one another, and especially to the direc- 
 tion of that resistance. 
 
 138. Any pressure applied to the surface of an immoveable 
 solid body by the intervention of another body moveable 
 upon it, will be sustained by^ the resistance of t/ie surfaces 
 of contact, whatever be its direction, provided only the an- 
 gle which that direction makes with the perpendicular to 
 the surfaces of contact do not exceed a certain angle called 
 
 the LIMITING ANGLE OF RESISTANCE of those SURFACES. 
 
132 THE LIMITING ANGLE OF RESISTANCE. 
 
 This is true, however great the pressure may he. Also, if 
 the inclination of the pressure to the perpendicular exceed 
 the limiting angle of resistance, then this pressure will not 
 he sustained by the resistance of the surfaces of contact y 
 and this is true, however small the pressure may ~be. 
 
 Let PQ represent the direction in which the surfaces of 
 two bodies are pressed together at Q, and let 
 QA be a perpendicular or normal to the sur- 
 faces of contact at that point, then will the pres- 
 sure PQ be sustained by the resistance of the 
 surfaces, however great it may be, provided its 
 direction lie within a certain given angle AQB, 
 called the limiting angle of resistance ; and it will not be sus- 
 tained, however small it may be, provided its direction lie 
 without that angle. For let this pressure be represented by 
 PQ, and let it be resolved into two others AQ and RQ, of 
 which AQ is that by which it presses the surfaces together 
 perpendicularly, and RQ that by which it tends to cause 
 them to slide upon one another, if therefore the friction F 
 produced by the first of these pressures exceed the second 
 pressure RQ, then the one body will not be made to slip 
 upon the other by this pressure PQ, however great it may 
 be ; but if the friction F, produced by the perpendicular 
 pressure AQ, be less than the pressure RQ, then the one 
 body will be made to slip upon the other, however small PQ 
 may be. Let the pressure in the direction PQ be repre- 
 sented by P, and the angle AQP by 6, the perpendicular 
 pressure in AQ is then represented by P cos. d, and therefore 
 the friction of the surfaces of contact by/T cos. 0, f repre- 
 senting the co-efficient of friction (Art. 136.). Moreover, the 
 resolved pressure in the direction RQ is represented by P 
 sin. &. The pressure P will therefore be sustained by the 
 friction of the surfaces of contact or not, according as 
 
 P sin. & is less or greater than fP cos. 6 ; 
 
 or, dividing both sides of this inequality by P cos. d, ac 
 cording as 
 
 tan. 6 is less or greater than f. 
 
 Let, now, the angle AQB equal that angle whose tangent is 
 f, and let it be represented by 0, so that tan. 0=/". Substi- 
 tuting this value of f in the last inequality, it appears that 
 the pressure P will be sustained by the friction of the s^ 
 faces of contact or not, according as 
 
THE TWO STATES BORDERING UPON MOTION. 133 
 
 tan. & is greater or less than tan. 0, 
 that is, according as 
 
 6 is less or greater than 0, 
 or according as 
 
 AQP is less or greater than AQB. 
 Therefore, &c. [Q. E. D.] 
 
 THE CONE OF RESISTANCE. 
 
 139. If the angle AQB be conceived to revolve about the 
 axis AQ, so that BQ may generate the surface of 
 a cone BQC, then this cone is called the CONE OP 
 RESISTANCE i it is evident, that any pressure, how- 
 ever great, applied to the surfaces of contact at 
 Q will be sustained by the resistance of the sur- 
 faces of contact, provided its direction be any 
 where within the surface of this cone ; and that it will not 
 be sustained, however small it may be, if its direction lie any 
 where without it. 
 
 THE Two STATES BORDERING UPON MOTION. 
 
 140. If the direction of the pressure coincide with the sur- 
 face of the cone, it will be sustained by the friction of the 
 surfaces of contact, but the body to which it is applied will 
 be upon the point of slipping upon the other. The state of 
 the equilibrium of this body is then said to be that BORDER- 
 ING UPON MOTION. If the pressure P admit of being applied 
 in any direction about the point Q, there are evidently an 
 infinity of such states of the equilibrium bordering upon mo- 
 tion, corresponding to all the possible positions of P on the 
 surface of the cone. 
 
 If the pressure P admit of being applied only in the same 
 plane, there are but two such states, corresponding to those 
 directions of P, which coincide with the two intersections of 
 this plane with the surface of the cone ; these are called the 
 superior and inferior states bordering upon motion. In the 
 case in which the direction of P is limited to the plane AQB, 
 BQ and CQ represent its directions corresponding to the 
 
134 THE TWO STATES BORDERING UPON MOTION. 
 
 two states bordering on motion. Any direction of P within 
 the angle BQC corresponds to a state of equilibrium ; any 
 direction, without this angle, to a state of motion. 
 
 141. Since, when the direction of the pressure P coincides 
 with the surface of the cone of resistance, the equilibrium is 
 in the state bordering upon motion ; it follows, conversely, 
 and for the same reasons, that this is the direction of the 
 pressure sustained by the surfaces of contact of two bodies 
 whenever the state of their equilibrium is that bordering upon 
 motion. This being, moreover, the direction of the pressure 
 of the one body upon the other is manifestly the direction of 
 the resistance opposed by the second body to the pressure of 
 the first at their surface of contact, for this single pressure 
 and this single resistance are forces in equilibrium, and there- 
 fore equal and opposite. All that has been said above of the 
 single pressure and the single resistance sustained by two 
 surfaces of contact, is manifestly true of the resultant of any 
 number of such pressures, and of the resultant of any num- 
 ber of such resistances. Thus then it follows, that when any 
 number of pressures applied to a body movedble upon another 
 which is fixed, are sustained by the resistance of the surfaces 
 of contact of the two bodies, and are in the state of equilibrium 
 bordering upon motion, then the direction of the resultant of 
 these pressures coincides with the surface of the cone of resist- 
 ance, as does that also of the resultant of the resistances of the 
 different points of the surfaces of contact*, that is, they are 
 both inclined to the perpendicular to the surfaces of contact 
 (at the point where they intersect it), at an angle equal to the 
 limiting angle of resistance. 
 
 * The properties of the limiting angle of resistance and the 
 ance, were first given by the author of this work in a paper published m the 
 Cambridge Philosophical Transactions, vol. v. 
 
FRICTION. 
 
 135 
 
 TABLE I. 
 Friction of Plane Surfaces, when they have 
 
 some time in Contact. 
 
 Surfaces in Contact. 
 
 Disposition of 
 the Fibres. 
 
 State of the 
 Surfaces. 
 
 Co-efficient 
 of Friction. 
 
 Limiting 
 Angle of 
 Resist- 
 ance. 
 
 EXPERIMENTS OF M. MORIN. 
 
 
 
 
 
 
 parallel 
 
 without 
 unguent 
 
 0-62 
 
 31 48' 
 
 
 ditto 
 
 rubbed with 
 dry soap 
 
 0-44 
 
 23 45 
 
 Oak noon oak 
 
 perpendicu- 
 lar 
 ditto 
 
 without ) 
 unguent f 
 with water 
 
 0-54 ' 
 0-71 
 
 28 22 
 35 23 
 
 
 endways of "] 
 
 
 
 
 
 one upon | 
 the flat V 
 
 without ) 
 
 0'43 
 
 23 16 
 
 
 surface of 
 
 unguent ) 
 
 
 
 
 the other J 
 
 
 
 
 Oak upon elm 
 
 parallel 
 
 ditto 
 
 0-38 
 
 20 49 
 
 C 
 
 ditto 
 
 ditto 
 
 0-69 
 
 34 37 
 
 Elm upon oak - < 
 
 ditto 
 
 rub tfed with 
 dry soap 
 
 0'41 
 
 22 18 
 
 V 
 
 perpendicu- 
 
 without 
 
 0'57 
 
 29 41 
 
 \ 
 
 lar 
 
 unguent 
 
 
 
 Ash, fir, beech, service- ) 
 tree, upon oak \ 
 
 parallel 
 
 ditto 
 
 0-53 
 
 27 56 
 
 \ 
 
 the leather 
 flat 
 
 ditto 
 
 0-61 
 
 31 23 
 
 Tanned leather upon oak^ 
 
 the leather 1 
 length- 1 
 
 ditto 
 
 0-43 
 
 23 16 
 
 ' 1 
 
 ways, but f 
 sideways J 
 
 steeped in ) 
 water j 
 
 0-79 
 
 38 19 
 
 , , fupon a plane 
 
 B * ack . surface of - 
 dressed Qak 
 
 parallel j 
 
 without ) 
 unguent ) 
 
 0-74 
 
 36 30 
 
 leathe M upon a round- 
 er strap ^ d gurface 
 
 leather f , 
 
 perpendicu- ) 
 lar J 
 
 ditto 
 
 0-47 
 
 25 11 
 
 1 oi oak 
 
 
 
 
 
 Hemp matting upon oak -I 
 
 parallel 
 ditto 
 
 ditto 
 steeped in ) 
 water J 
 
 0-50 
 
 0-87 
 
 26 34 
 41 2 
 
 Hemp cords upon oak - 
 
 ditto 
 
 without ) 
 unguent ) 
 
 0-80 
 
 38 40 
 
 
 
 ditto 
 
 0-62 
 
 31 48 
 
 Iron upon oak 
 
 ditto 
 
 steeped in 
 water 
 
 0-65 
 
 33 2 
 
 Cast-iron upon oak 
 
 ditto 
 
 ditto 
 
 0-65 
 
 33 2 
 
 Copper upon oak - 
 
 ditto | 
 
 without 
 unguent 
 
 0-62 
 
 31 48 
 
 
 | 
 
 steeped in 
 
 0-62 
 
 31 48 
 
 Ox-hide as a piston sheath ) 
 
 flat or side-] 
 
 water 
 with oil, 
 
 
 
 upon cast-iron ) 
 
 ways 
 
 tallow, or 
 
 0'12 
 
 6 51 
 
 
 hog's lard 
 
 
 
136 
 
 FRICTION. 
 
 
 
 
 
 Limiting 
 
 Surfaces in Contact. 
 
 Disposition of 
 the Fibres. 
 
 State of the 
 Surfaces. 
 
 Co-efficien 
 of Friction 
 
 Angle of 
 Resist- 
 
 
 
 
 
 ance. 
 
 ['EXPERIMENTS OF M. MORIN. 
 
 
 
 
 
 continued. 
 
 
 
 
 
 Black dressed leather, or \ 
 strap leather, upon a > 
 cast-iron pulley ) 
 
 flat ) 
 
 without ) 
 unguent ) 
 steeped 
 
 0-28 
 0-38 
 
 15 39' 
 20 49 
 
 Cast-iron upon cast-iron - 
 
 ditto -j 
 
 without ) 
 unguent j" 
 
 0-16 
 
 9 6 
 
 Iron upon cast-iron 
 
 ditto 
 
 ditto 
 
 0-19 
 
 10 46 
 
 Oak, elm, yoke elm, iron, "| 
 cast-iron, and brass 1 
 sliding two and two, [ 
 one upon another J 
 
 ditto 
 
 with tallow 
 with oil, or ) 
 hog's lard j" 
 
 o-iof 
 
 0-15^: 
 
 5 43 
 8 32 
 
 Calcareous oolite stone ) 
 upon calcareous oolite ) 
 
 ditto 
 
 without ) 
 unguent j 
 
 0-74 
 
 36 30 
 
 Hard calcareous stone, ) 
 
 
 
 
 
 called muschelkalk, > 
 
 ditto 
 
 ditto 
 
 0-75 
 
 36 52 
 
 upon calcareous oolite ) 
 
 
 
 
 
 Brick upon calcareous ) 
 oolite J 
 
 ditto 
 
 ditto 
 
 0-67 
 
 33 50 
 
 Oak upon calcareous j 
 oolite ( 
 
 wood end- ) 
 ways f 
 
 ditto 
 
 0-63 
 
 32 13 
 
 Iron upon calcareous oolite 
 
 flat 
 
 ditto 
 
 0-49 
 
 26 7 
 
 i Hard calcareous stone, or 
 
 
 
 
 
 muschelkalk, upon 
 
 ditto 
 
 ditto 
 
 0'70 
 
 35 
 
 muschelkalk 
 
 
 
 
 
 'Calcareous oolite stone 
 upon muschelkalk 
 
 ditto 
 
 ditto 
 
 0-75 
 
 36 52 
 
 Brick upon muschelkalk - 
 
 ditto 
 
 ditto 
 
 0-67 
 
 33 50 
 
 Iron upon muschelkalk 
 
 ditto 
 
 ditto 
 
 0-42 
 
 22 47 
 
 Oak upon muschelkalk - 
 
 ditto 
 
 ditto 
 
 0-64 
 
 32 38 
 
 
 
 with a coat- "1 
 
 
 
 
 
 ing of mor- 
 
 
 
 Calcareous oolite stone ) 
 upon calcareous oolite J" 
 
 ditto 
 
 tar,ofthree 
 parts of fine V 
 sand and 
 
 0'74 
 
 36 80 
 
 
 
 one part of 
 
 
 
 
 
 slack lime J 
 
 
 
 * The surfaces retaining some unctuousness. 
 
 f When the contact has not lasted long enough to express the grease. 
 \ When the contact has lasted long enough to express the grease and bring 
 back the surfaces to an unctuous state. 
 
 After a contact of from ten to fifteen minutes. 
 
FRICTION. 
 
 137 
 
 Nature of Bodies and Unguents. 
 
 Co-efficient 
 of Friction. 
 
 Limiting 
 Angle. 
 
 Soft calcareous stone, well dressed, upon the same 
 
 0-74 
 
 36 30' 
 
 Hard calcareous stone, ditto .... 
 
 0-75 
 
 36 52 
 
 Common brick, ditto - 
 
 0-67 
 
 33 50 
 
 Oak, endways, ditto ..... 
 
 0-63 
 
 32 13 
 
 Wrought iron, ditto ..... 
 
 0-49 
 
 26 7 
 
 Hard calcareous stone, well dressed, upon hard calcare- 
 
 
 
 ous stone ...... 
 
 0-70 
 
 35 
 
 Soft, ditto ...... 
 
 0-75 
 
 36 52 
 
 Common brick, ditto ..... 
 
 0-67 
 
 83 50 
 
 Oak, endways, ditto 
 
 0-64 
 
 32 37 
 
 Wrought iron, ditto ..... 
 
 0-42 
 
 22 47 
 
 Soft calcareous stone upon soft calcareous stone, with 
 
 
 
 fresh mortar of fine sand .... 
 
 0-74 
 
 36 30 
 
 EXPERIMENTS BY DIFFERENT OBSERVERS. 
 
 
 
 Smooth free-stone upon smooth free-stone, dry (Rennie) 
 
 0-71 
 
 35 23 
 
 Ditto, with fresh mortar (Rennie) 
 
 0-66 
 
 33 26 
 
 Hard polished calcareous stone upon hard polished cal- 
 
 
 
 careous stone - .... 
 
 0-58 
 
 30 7 
 
 Calcareous stone upon ditto, both surfaces being made 
 
 
 
 rough with a chisel (Bonchardi) ... 
 
 0-78 
 
 37 58 
 
 Well dressed granite upon rough granite (Rennie) 
 
 0-66 
 
 33 26 
 
 Ditto, with fresh mortar, ditto (Rennie) - 
 
 0-49 
 
 26 7 
 
 Box of wood upon pavement (Regnier) - 
 
 0-58 
 
 30 7 
 
 Ditto upon beaten earth (Herbert) - 
 
 0-33 
 
 18 16 
 
 Libage stone upon a bed of dry clay - 
 
 0-51 
 
 27 2 
 
 Ditto, the clay being damp and soft ... 
 
 0-34 
 
 18 47 
 
 Ditto, the clay being equally damp, but covered with 
 
 
 
 thick sand (Greve) - 
 
 0-40 
 
 21 48 
 
138 
 
 FfilOTIOX. 
 
 TABLE II. 
 friction of Plane Surfaces, in Motion one upon the other. 
 
 Surfaces in Contact. 
 
 Disposition of 
 the Fibres. 
 
 State of the 
 Surfaces. 
 
 Co-efficient 
 of Friction. 
 
 Limiting 
 Angle of 
 Resist- 
 ance. 
 
 EXPERIMENTS OP M. MORIN. 
 
 
 
 
 
 
 
 parallel 
 
 without 
 unguent 
 
 0-48 
 
 25 C 39' 
 
 > 
 
 ditto 
 
 rubbed with 
 
 j 
 
 0-16 
 
 9 6 
 
 
 
 dry soap 
 
 
 
 
 perpendicu- 
 lar 
 
 without 
 unguent 
 
 0-34 
 
 18 47 
 
 Oak upon oak 
 
 ditto 
 
 steeped in 
 
 0-25 
 
 14 3 
 
 
 
 water 
 
 
 
 
 wood endO 
 
 
 
 
 
 ways on 
 wood j- 
 length- 
 
 without ) 
 unguent J 
 
 0-19 
 
 10 46 
 
 
 ways J 
 
 
 
 
 r 
 
 parallel 
 
 ditto 
 
 0-43 
 
 23 17 
 
 Ehn upon oak - < 
 
 perpendicu- ) 
 lar J 
 
 ditto 
 
 0-45 
 
 24 14 
 
 [ 
 
 parallel 
 
 ditto 
 
 0-25 
 
 14 3 
 
 Ash, fir, beech, wild pear- } 
 tree, and service-tree, V 
 upon oak ) 
 
 ditto 
 
 ditto 
 
 0-36 to 
 0-40 
 
 ) 19 48 
 }21 49 
 
 
 
 ditto 
 
 0-62 
 
 31 48 
 
 
 
 with water 
 
 0-26 
 
 14 35 
 
 Iron upon oak 
 
 ditto 
 
 rubbed with 
 dry soap 
 
 0-21 
 
 11 52 
 
 
 
 without 
 
 
 
 
 
 unguent 
 
 0'49 
 
 26 7 
 
 
 
 with water 
 
 0-22 
 
 12 25 
 
 Cast-iron upon oak 
 
 ditto 
 
 rubbed with 
 dry soap 
 
 0-19 
 
 10 46 
 
 Copper upon oak - 
 
 ditto j 
 
 without 
 ' unguent 
 
 0-62 
 
 31 48 
 
 Iron upon elm 
 
 ditto 
 
 ditto 
 
 0-25 
 
 14 3 
 
 Cast-iron upon elm - 
 
 ditto 
 
 ditto 
 
 0-20 
 
 11 19 
 
 Black dressed leather ) 
 upon oak 
 
 ditto 
 
 ditto 
 
 0-27 
 
 15 7 
 
 r 
 
 flat, or 1 
 
 
 
 
 Tanned leather upon oak j 
 
 ** u > v * 
 
 length- 1 
 ways, and ( 
 edgeways J 
 
 ditto -j 
 with water 
 
 3-30 to 
 0-35 
 0-29 
 
 16 42 
 19 18 
 16 11 
 
 
 ' 
 
 without 
 
 
 
 
 
 unguent 
 
 0-56 
 
 29 15 
 
 Tanned leather upon ) 
 cast-iron and brass J 
 
 ditto 
 
 steeped in 
 water 
 greased and 
 
 0-36 
 
 19 48 
 
 
 
 steeped in 
 
 0-23 
 
 12 58 
 
 
 
 water 
 
 
 
 . 
 
 . 
 
 with oil 
 
 0-15 
 
 8 32 
 
FRICTION. 
 
 139 
 
 Surfaces in Contact. 
 
 Disposition of 
 the Fibres. 
 
 State of the 
 Surfaces. 
 
 Coefficient 
 of Friction. 
 
 Limiting i 
 Angle of i 
 Resist- 
 ance. 
 
 EXPERIMENTS OF M. MORIN. 
 
 
 
 
 
 
 
 continued. 
 
 
 
 
 
 
 
 Hemp, in threads or in] 
 
 parallel 
 
 
 without 
 unguent 
 
 
 0-52 
 
 2729' 
 
 cord, upon oak 
 
 perpendicu- 
 lar 
 
 i 
 
 with water 
 
 
 0-33 
 
 18 16 
 
 Oak and elm upon cast- \ 
 iron ) 
 
 parallel 
 
 
 without 
 unguent 
 
 
 0-38 
 
 20 49 
 
 Wild pear-tree, ditto 
 
 ditto 
 
 
 ditto 
 
 
 0-44 
 
 23 45 
 
 Iron upon iron 
 
 ditto 
 
 
 ditto 
 
 
 0-44 
 
 23 45* 
 
 Iron upon cast-iron and ) 
 brass [ 
 
 ditto 
 
 
 ditto 
 
 
 o-isf 
 
 10 13 
 
 Cast-iron, ditto 
 
 ditto 
 
 
 ditto 
 
 
 0-15 
 
 8 32 
 
 {upon brass - 
 
 ditto 
 
 
 ditto 
 
 
 0-20 
 
 11 19 
 
 upon cast-iron 
 
 ditto 
 
 
 ditto 
 
 
 0-22 
 
 12 25 
 
 upon iron - 
 
 ditto 
 
 
 ditto 
 
 
 0'16| 
 
 9 6 
 
 
 
 
 greased in ~ 
 
 
 
 
 
 
 
 the usual 
 
 
 
 
 Oak, elm, yoke elm, wild "1 
 pear, cast-iron, wrought 
 iron, steel, and moving V 
 
 ditto 
 
 
 way with 
 tallow, 
 hog's lard, 
 oil soft 
 
 
 0-07 to 
 0-08 
 
 U 1 
 14 35 
 
 one upon another, or on 
 themselves 
 
 
 
 gom 
 slightly ; 
 
 
 
 
 
 
 
 greasy to 
 
 w 
 
 0-15 
 
 8 32 
 
 
 
 
 the touch 
 
 
 
 
 Calcareous oolite stone 
 upon calcareous oolite 
 
 
 ^ 
 
 without 
 unguent 
 
 : 
 
 0-64 
 
 82 37 
 
 Calcareous stone, called 
 
 
 
 
 
 
 
 muschelkalk, upon cal- 
 
 ditto 
 
 
 ditto 
 
 
 0-67 
 
 33 50 
 
 careous oolite 
 
 
 
 
 
 
 
 Common brick upon cal- 
 
 ditto 
 
 
 ditto 
 
 
 0-65 
 
 33 2 
 
 careous oolite 
 
 
 
 
 
 
 
 Oak upon calcareous j 
 oolite \ 
 
 wood end- 
 ways 
 
 
 ditto 
 
 
 0-38 
 
 20 49 
 
 Wrought iron, ditto 
 
 parallel 
 
 
 ditto 
 
 
 0-69 
 
 34 37 
 
 Calcareous stone, called 
 
 
 
 
 
 
 
 muschelkalk,upon mus- 
 
 ditto 
 
 
 ditto 
 
 
 0-38 
 
 20 49 
 
 chelkalk 
 
 
 
 
 
 
 
 Calcareous oolite stone 
 upon muschelkalk 
 
 ditto 
 
 
 ditto 
 
 
 0-65 
 
 33 2 
 
 Common brick, ditto 
 
 ditto 
 
 
 ditto 
 
 
 0'60 
 
 30 58 
 
 Oak upon muschelkalk \ 
 
 wood end- 
 ways 
 
 \ 
 
 ditto 
 
 
 0-38 
 
 20 49 
 
 
 
 i ditto 
 
 
 0-24 
 
 
 Iron upon muschelkalk - 
 
 
 saturated 
 with water 
 
 
 0-30 
 
 16 42 
 
 * The surfaces wear when there is no grease. 
 
 jr The surfaces still retaining a little unctuousness. \ Ibid. 
 
 When the grease is constantly renewed and uniformly distributed, thia 
 proportion can be reduced to 0'05. 
 
140 
 
 FEICTIOK. 
 
 TABLE III. 
 
 friction of Gudgeons or Axle-ends, in Motion, upon their Bearings. 
 (From the experiments of Morin.) 
 
 Surfaces in Contact. 
 
 State of the Surfaces. 
 
 Co-efficient of Friction when 
 the Grease is renewed. 
 
 Limiting 
 Angle of 
 Resistance. 
 
 In the usual 
 Way. 
 
 Continuously. 
 
 
 coated with oil of 
 
 
 
 ( 4 0' 
 
 
 olives, with hog's 
 lard, tallow, anc 
 
 0-07 to 0-08 
 
 0-054 
 
 \ 4 35 
 ( 3 6 
 
 Cast-iron axles 
 in cast-iron - 
 bearings 
 
 soft goin 
 with the same, 
 and water 
 coated with as- 
 
 0-08 
 
 0-28 
 
 4 35 
 
 
 phaltum 
 
 0-054 
 
 0-19 
 
 3 6 
 
 
 greasy 
 
 0-14 
 
 . 
 
 7 58 
 
 
 greasy and wetted 
 
 0-14 
 
 
 
 7 58 
 
 
 coated with oil of 
 
 
 
 ( 4 
 
 Cast-iron axles, 
 ditto 
 
 olives, with hog's 
 lard, tallow, and 
 soft gom 
 
 0-07 to 0-08 
 
 0-054 
 
 \ 4 35 
 ( 3 6 
 
 
 greasy 
 
 0-16 
 
 - 
 
 9 6 
 
 
 greasy and damped 
 
 0-16 
 
 * 
 
 9 6 
 
 
 scarcely greasy 
 
 0-19 
 
 
 10 46 
 
 
 without unguent 
 
 0-18 
 
 . 
 
 10 12 
 
 Cast-iron axles 
 in lignum vit,ae< 
 bearings 
 
 with oil or hog's 
 lard 
 greasy with ditto 
 greasy, with a 
 mixture of hog's 
 lard and molyb- 
 
 o-io 
 
 I 0-14 
 
 0-090 
 
 5 9 
 5 43 
 
 7 58 
 
 
 daena 
 
 J 
 
 
 
 Wrought-iron 
 axles in cast-K 
 iron bearings 
 
 coated with oil 
 of olives, tallow, 
 hog's lard, or 
 soft gom 
 
 L 0-07 to 0-08 
 
 0-054 
 
 ( 4 
 \ 4 35 
 ( 3 6 
 
 ' 
 
 coated with oil of 
 
 ) 
 
 
 i4 
 * v 
 
 
 olives, hog's lard, 
 
 [ 0-07 to 0-08 
 
 0-054 
 
 4 35 
 
 [ron axles in 
 brass bearings " 
 
 or tallow 
 coated with hard 
 gom 
 
 ) 
 
 [ 0-09 
 
 . 
 
 3 6 
 5 9 
 
 
 greasy and wetted 
 
 0-19 
 
 . 
 
 10 46 
 
 
 scarcely greasy 
 
 0-25 
 
 * 
 
 14 2 
 
 [ron axles in 
 
 coated with oil, 
 
 ) nil 
 
 
 
 lignum vitae 
 
 or hog's lard 
 
 h O'll 
 \ 
 
 * - 
 
 6 17 
 
 bearings 
 
 greasy 
 
 0-19 
 
 . 
 
 10 46 
 
 Brass axles in 
 
 coated with oil 
 
 o-io 
 
 
 
 5 43 
 
 brass bearings 
 
 with hog's lard 
 
 0-09 
 
 m 
 
 5 9 
 
 Brass axles in 
 cast-iron bear- 
 ings 
 
 coated with oil 
 or tallow 
 
 
 0-045 to 0-052 
 
 j 2 35 
 { 2 59 
 
 * The surfaces beginning to wear. 
 
FRICTION. 
 
 
 
 Co-efficient of Friction when 
 
 
 Surfaces in Contact. 
 
 State of the Surfaces. 
 
 the Grease is renewed. 
 
 Limiting 
 Angle of 
 
 
 
 In the usual 
 Way. 
 
 Continuously. 
 
 Resistance. 
 
 Lignum vitse j 
 axles, ditto 1 
 
 coated with hog's 
 lard 
 greasy 
 
 I 0-12 
 0-15 
 
 
 
 6051' 
 8 82 
 
 Lignum vitas "I 
 
 
 
 
 
 axles in lig- I 
 num vitas j 
 
 coated with hog's 
 lard 
 
 1- - 
 
 0-07 
 
 4 
 
 bearings J 
 
 
 
 
 
 TABLE IV. 
 
 Co-efficients of friction under Pressures increased continually up to the 
 
 Limits of Abrasion. 
 (From the experiments of Mr. G. Rennie.*) 
 
 
 Co-efficients of Friction. 
 
 Pressure per 
 Square Inch. 
 
 Wrought-iron 
 upon 
 Wrought-iron. 
 
 Wrought-iron 
 upon 
 Cast-iron. 
 
 Steel upon 
 Cast-iron. 
 
 Brass upon 
 Cast-iron. 
 
 32- 51bs. 
 
 140 
 
 174 
 
 166 
 
 157 
 
 1-66 cwt. 
 
 250 
 
 275 
 
 300 
 
 225 
 
 2'00 
 
 271 
 
 292 
 
 333 
 
 219 
 
 2-33 
 
 285 
 
 321 
 
 340 
 
 214 
 
 2-66 
 
 297 
 
 329 
 
 344 
 
 211 
 
 3-00 
 
 312 
 
 333 
 
 347 
 
 215 
 
 3-33 
 
 350 
 
 351 
 
 351 
 
 206 
 
 3-66 
 
 376 
 
 353 
 
 353 
 
 205 
 
 4-00 
 
 376 
 
 365 
 
 354 
 
 208 
 
 4-33 
 
 395 
 
 366 
 
 356 
 
 221 
 
 4-66 
 
 403 
 
 366 
 
 357 
 
 223 
 
 5-00 
 
 409 
 
 367 
 
 358 
 
 233 
 
 5-33 
 
 
 367 
 
 359 
 
 234 
 
 5-66 
 
 
 367 
 
 367 
 
 235 
 
 6-00 
 
 
 376 
 
 408 
 
 283 
 
 6-33 
 
 
 484 
 
 
 234 
 
 6 -60 
 
 
 
 
 235 
 
 7-00 
 
 
 
 
 282 
 
 7-88 
 
 
 
 
 273 
 
 Phil. Trans. 1829, table 8. p. 159. 
 
142 THE RIGIDITY OF COEDS. 
 
 THE RIGIDITY OF COEDS. 
 
 
 
 142. It is evident that, by reason of that resistance to 
 r~ , deflexion which constitutes the ri- 
 gidity of a cord, a certain force or 
 pressure must be called into action 
 whenever it is made to change its 
 rectilineal direction, so as to adapt 
 itself to the form of any curved sur- 
 face over which it is made to pass ; 
 as, for instance, over the circumfe- 
 rence of a pulley or wheel. Sup- 
 pose such a cord to sustain tensions represented by P a and 
 P 2 , of which P x is on the point of preponderating, and let 
 the friction of the axis of the pulley be, for the present, 
 neglected. It is manifest that, in order to supply the force 
 necessary to overcome the rigidity of the cord and to pro- 
 duce its deflection at B, the tension P 1 must exceed P 2 ; 
 whereas, if there were no rigidity, P, would equal P 2 ; so 
 that the effect of the rigidity in increasing the tension P, is 
 the same as though it had, by a certain quantity, increased 
 the tension P 2 . Now, from a very numerous series of 
 experiments made by Coulomb upon this subject, it appears 
 that the quantity by which the tension P 2 may thus be con- 
 sidered to be increased by the rigidity, is partly constant 
 and partly dependent on' the amount of P 2 ; so as to be 
 represented by an algebraical formula of two terms, one 
 of which is -i constant quantity, and the other the product 
 of a constant quantity by P 2 . Thus if D represent the 
 constant part of this formula, and E the constant factor 
 of P 2 , then is the effect of the rigidity of the cord the same 
 as though the tension P 2 were increased by the quantity 
 D+E.P,. 
 
 When the cord, instead of being bent, under different 
 pressures, upon circular arcs of equal radii, was bent upon 
 circular arcs of different radii, then this quantity D + E . P 2 ; 
 by which the tension P 2 may be considered to be increased 
 by the rigidity, was found to vary inversely as the radii 
 of the arcs ; so that, on the whole, it may be represented 
 by the formula 
 
THE EIGIDITY OF COKDS. 143 
 
 D+E . P, 
 
 R 
 
 (HI), 
 
 where E represents the radius of the circular arc over which 
 the rope is bent. Tims it appears that the yielding tension 
 P 2 may be considered to have been increased by the rigidity 
 of the rope, when in the state bordering upon motion, so as 
 to become 
 
 This formula applies only to the bending of the same cord 
 under different tensions upon different circular arcs : for dif- 
 ferent cords, the constants D and E vary (within certain 
 limits to be specified) as the squares of the diameters or of the 
 circumferences of the cords, in respect to new cords, wet or 
 dry / 111 respect to old cords they vary nearly as the power f 
 of the diameters or circumferences. 
 
 Tables have been furnished by Coulomb of the values of 
 the constants I) and E. These tables, reduced to English 
 measures, are given on the next page.* 
 
 * The rigidity of the cord exerts its influence to increase resistance^ only at 
 that point where the cord winds upon the pulley ; at the point where it leaves 
 the pulley its elasticity favours rather, and does not perceptibly affect, the 
 conditions of the equilibrium. 
 
 In all calculations of machines, in which the moving power is applied by the 
 intervention of a rope passing over a pulley, one-half the diameter of rope is 
 to be added to the radius of the pulley, or to the perpendicular on the direction 
 of the rope from the point whence the moments are measured, the pressure 
 applied to the rope producing the same effect as though it were all exerted 
 along the axis of the rope. > 
 
144 
 
 THE KIGIDITY OF CORDS. 
 
 TABLE V. RIGIDITY OF ROPES. 
 
 Table of the values of the constants D and E, according to the experiments of 
 Coulomb (reduced to English measures}. The radius R of the pulley is to be 
 taken in feet. 
 
 No. 1. New dry cords. Rigidity proportional to the square of the 
 circumference. 
 
 Circumference of 
 the Rope in Inches. 
 
 Value of D in Ibs. 
 
 Value of E in Ibs. 
 
 1 
 2 
 4 
 8 
 
 131528 
 526108 
 2-104451 
 8-413702 
 
 033533 
 
 023030 
 073175 
 368494 
 
 Squares of propor- 
 tions of the in- 
 termediate cir- 
 cumferences to 
 those of the 
 table. 
 
 No. 2. New ropes dipped in water. Rigidity proportional 
 to the square of the circumference. 
 
 Circumference of 
 the Rope in Inches. 
 
 Value of D in Ibs. 
 
 Value of E in Ibs. 
 
 1 
 2 
 4 
 
 8 
 
 263053 
 1-052217 
 
 4-208902 
 16-835606 
 
 0057576 
 0230303 
 0731755 
 3684860 
 
 No. 3. 
 
 Dry half-worn ropes. Rigidity proportional to the square root 
 of the cube of the circumference. 
 
 Circumference of 
 the Rope in Inches. 
 
 Value of D in Ibs. 
 
 Value of E in Ibs. 
 
 1 
 2 
 4 
 8 
 
 146272 
 413656 
 1-169641 
 
 3-308787 
 
 0064033 
 0180827 
 0512115 
 1448238 
 
 . 
 
 Square roots of the 
 cubes of propor- 
 tions of the in- 
 termediate cir- 
 cumferences to 
 those of the 
 table. 
 
 No. 4. Wetted half-worn cords. Rigidity proportional 
 to the square root of the cube of the circumference. 
 
 Circumference of 
 the Rope in Inches. 
 
 Value of D in Ibs. 
 
 Value of E in Ibs. 
 
 1 
 2 
 4 
 8 
 
 292541 
 827328 
 2-339675 
 6-616589 
 
 006401 
 018107 
 051212 
 144822 
 
THE RIGIDITY OF COEDS. 1 
 
 No. 5. Tarred rope. Rigidity proportional to the number of strands. 
 
 Number of Strands. 
 
 Value of D in Ibs. 
 
 Value of E in Ibs. 
 
 6 
 15 
 30 
 
 0-33390 
 0-17212 
 1-25294 
 
 0-009305 
 0-021713 
 0-044983 
 
 To determine the constants D and E for ropes whose circumferences are 
 intermediate to those of the tables, find the ratio of the given circumference 
 to that nearest to it in the tables, and seek this ratio or proportion in the first 
 column of the auxiliary table to the right of the page. The corresponding 
 number in the second column of this auxiliary table is a factor by which the 
 values of D and E for the nearest circumference in the principal tables being 
 multiplied, their values for the given circumference will be determined.* 
 
 * Note (*) Ed. App. 
 
 10 
 
146 THE THEOEY OF MACHINES. 
 
 T III. 
 
 THE THEOEY OF MACHINES. 
 
 143. THE parts of a machine are divisible into those which 
 receive the operation of the moving power immediately, those 
 which operate immediately upon the work to be performed, 
 and those which communicate between the two, or which 
 conduct the power or work from the moving to the working 
 points of the machine. The first class may be called RECEIV- 
 ERS, the second OPERATORS, and the third COMMUNICATORS of 
 work. 
 
 THE TRANSMISSION OF WORK BY MACHINES. 
 
 144. The moving power divides itself whilst it operates in 
 a machine, first, Into that which overcomes the prejudicial 
 resistances of the machine, or those which are opposed by 
 friction and other causes uselessly absorbing the work in its 
 transmission. Secondly, Into that which accelerates the 
 motion of the various moving parts of the machine ; so long 
 as the work done by the moving power upon it exceeds that 
 expended upon the various resistances opposed to the motion 
 of the machine (Art. 129.). Thirdly, Into that which over- 
 comes the useful resistances, or those which are opposed to 
 the motion of the machine at the working point or points 
 by the useful work which is to be done by it. Thus, then, 
 the work done by the moving power upon the moving points 
 of the machine (as distinguished from the working points) 
 divides itself in the act of transmission, first, Into the work 
 expended uselessly upon the friction and other prejudicial 
 resistances opposed to its transmission. Secondly, Into that 
 accumulated in the various moving elements of the machine, 
 and reproducible. Thirdly, Into the useful work, or that 
 done by the operators, whence results immediately the useful 
 products of the machine. 
 
THE THEORY OF MACHINES. 
 
 145. The aggregate number of units of useful works yielded 
 ly any machine at its working points is less than the num- 
 ber received upon the machine directly from the moving 
 power, ly the number of units expended ujpon the prejudi- 
 cial resistances and ~by the number of units accumulated 
 in the moving parts of the machine whilst the work is being 
 done.* 
 
 For, by the principle of vis viva (Art. 129.), if 2U 1 repre- 
 sent the number of units of work received upon the machine 
 immediately from the operation of the moving power, %u 
 the whole number of such units absorbed in overcoming the 
 prejudicial resistances opposed to the working of the ma- 
 chine, 2U 2 the whole useful work of the machine (or that 
 done by its operators in producing the useful effect), and 
 
 ^-Zw^v*) one half the aggregate difference of the vires 
 
 vivae of the various moving parts of the machine at the 
 commencement and termination of the period during which 
 the work is estimated, then, by the principle of vis VIVA 
 (equation 108), 
 
 in which v l and v 9 represent the velocities at the commence- 
 ment and termination of the period, during which the work 
 is estimated, of that moving element of the machine whose 
 weight is w. Now one-half the aggregate difference of the 
 vires vivse of the moving elements represents the work accu- 
 mulated in them during the period in repect to which the 
 work is estimated (Art. 130.). Therefore, &c. 
 
 146. If the same velocity of every part of the machine re- 
 turn after any period of time, or if the motion ~be periodical, 
 then is the whole work received upon it from, the moving power 
 during that time exactly equal to the sum of the useful work 
 done, and the work expended upon the prejudicial resistances. 
 For the velocity being in this case the same at the com- 
 mencement and expiration of the period during which the 
 work is estimated, 2w(v* -y 2 fl )=0, so that 
 
 * Note (0 Ed. App. 
 
148 THE MODULUS OF A MACHINE. 
 
 (113). 
 
 Therefore, &c. 
 
 The converse of this proposition is evidently true. 
 
 147. If the prime mover in a machine be throughout the 
 motion ^n equilibrium with the useful and the prejudicial 
 resistances, then the motion of the machine is uniform. 
 For in this case, by the principle of virtual velocities 
 (Art. 127.), 2U, 2U 3 +2^; therefore (equation 112) 
 2w(v* v*)= 0; whence it follows that (in the case sup- 
 posed) the velocities v l and v 2 of any moving element of the 
 machine are the same at the commencement and termi- 
 nation of any period of the motion however small, or that 
 the motion of every such element is a uniform motion. 
 Therefore, &c. 
 
 The converse of this proposition is evidently true. 
 
 THE MODULUS OF A MACHINE MOVING WITH A UNIFORM OR 
 PERIODICAL MOTION. 
 
 148. The modulus of a machine, in the sense in which the 
 term is used in this work, is the relation between the work 
 constantly done upon it by the moving power ', and that con- 
 stantly yielded at the working points, when it has attained 
 a state of uniform motion if it admit of such a state of 
 motion ; or if the nature of its motion be periodical, then 
 is its modulus the relation between the work done at its 
 moving and at its working points in the interval of time 
 which it occupies in passing from any given velocity to the 
 same velocity again. 
 
 The modulus is thus, in respect to any machine, the parti- 
 cular form applicable to that machine of equation (113), and 
 being dependent for its amount upon the amount of work ^u 
 expended upon the friction and other prejudicial resistances 
 opposed to the motion of the various elements of the ma- 
 chine, it measures in respect to each such machine the loss 
 of work due to these causes, and therefore constitutes a true 
 standard for comparing the expenditure of moving power ne- 
 cessary to the production of the same effects by different ma- 
 
THE MODULUS OF A MACHINE. 14:9 
 
 chines: it is thus a measure of the working qualities of 
 machines.* 
 
 Whilst the particular modulus of every^ differently con- 
 structed machine is thus different, there is nevertheless a 
 general algebraical type or formula to which the moduli of 
 machines are (for the most part and with certain modifica- 
 tions) referable. That form is the following, 
 
 U^A . U 2 +B . S (114), 
 
 where U 1 is the work done at the moving point of the ma- 
 chine through the space S, TJ 2 the work yielded at the work- 
 ing points, and A and B constants dependent for their value 
 upon the construction of the machine : that is to say, upon 
 the dimensions and the combinations of its parts, their 
 weights, and the co-efficients of friction at their various rub- 
 bing surfaces. 
 
 It would not be difficult to establish generally this form of 
 the modulus under certain assumed conditions. As the mo- 
 dulus of each particular machine must however, in this work, 
 be discussed and determined independently, it will be better 
 to refer the reader to the particular moduli investigated in 
 the following pages. He will observe that they are for the 
 most part comprised under the form above assumed; sub- 
 ject to certain modifications which arise out of the discus- 
 sion of each individual case, and which are treated at length. 
 
 149. There is, however, one important exception to this 
 general form of the modulus : it occurs in the case of ma- 
 chines, some of whose parts move immersed in fluids. It is 
 only when the resistances opposed to the motion of the parts 
 of the machine upon one another are, like those of friction, 
 proportional to the pressures, or when they are constant re- 
 sistances, that this form of the modulus obtains. If there^be 
 resistances which, like those of fluids in which the moving 
 parts are immersed (the air, for instance), vary with the velo- 
 city of the motion, and these resistances be considerable, 
 then must other terms be added to the modulus. This sub- 
 ject will be further discussed when the resistances of fluids 
 are treated of. It may here, however, be observed, that^if 
 the machine move uniformly subject to the resistance of^a 
 fluid during a given time T, and the resistance of the fluid 
 
 * The properties of the modulus of a machine are here, for the first time, 
 discussed. 
 
150 THE MODULUS OF A MACHINE. 
 
 be supposed to vary as the square of the velocity Y, then 
 will the work expended on this resistance vary as Y 2 . S, or 
 as Y 3 . T, since S= Y . T. If then II, and IT, represent the 
 work done at the moving and working points during the 
 time T, then does the modulus (equation 114) assume, in this 
 case, the form 
 
 TJ^A . U a +B . Y . T-f C . Y 3 /T (115). 
 
 THE MODULUS OF A MACHINE MOVING WITH AN ACCELERATED 
 OK RETARDED MOTION. 
 
 150. In the two last articles the work IT,, done upon the 
 moving point or points of the machine, has been supposed to 
 be just that necessary to overcome the useful and prejudi- 
 cial resistances opposed to the motion of the machine, either 
 continually or periodically ; so that all the work may be ex- 
 pended upon these resistances, and none accumulated in the 
 moving parts of the machine as the work proceeds, or else 
 that the accumulated work may return to the same amount 
 from period to period. Let us now suppose this equality to 
 cease, and the work U 1 done by the moving power to exceed 
 that necessary to overcome the useful and prejudicial resist- 
 ances ; and to distinguish the work represented by U 1 in the 
 one case from that in the other, let us suppose the former, 
 (that which is in excess of the resistances) to be represented 
 by U 1 ; also let U 2 be the useful work of the machine, done 
 through a given space S a , and which is supposed the same 
 whatever may be the velocity of the motion of the machine 
 whilst that space is being described ; moreover, let Sj be the 
 space described by the moving point, whilst the space S a is 
 being described by the working point. 
 
 Now since Uj is the work which must be done at the 
 moving point just to overcome the resistances opposed to 
 the motion of that point, and U 1 is the work actually done 
 upon that point by the power, therefore U 1 U^ is the excess 
 of the work done by the power over that expended on the 
 resistances, and is therefore equal to the work accumulated 
 in the machine (Art. 130.) ; that is, to one half of the 
 increase of the vis viva through the space S t (Art. 129.) ; so 
 that, if ^ represent the velocity of any element of the 
 machine (whose weight is w) when the work U 1 began to be 
 done, and v 9 its velocity when that work has been com- 
 pleted, then (Art. 129.), 
 
THE VELOCITY OF A MACHINE. 
 
 Now by equation (114) U^ 
 .-. T?=A . 
 
 If instead of the work I? done by the power exceeding that 
 Uj expended on the resistances it had been less than it, then, 
 instead of work being accumulated continually through the 
 space S 1? it would continually have been lost, and we should 
 have had the relation (Art. 129.), 
 
 so that in this case, also, 
 
 The equation (116) applies therefore to the case of a 
 retarded motion of the machine as well as to that of an 
 accelerated motion, and is the general expression for the 
 modulus of a machine moving with a variable motion. 
 Whilst the co-efficients A and B of the modulus are depen- 
 dent wholly upon the friction and other direct resistances to 
 the motion of the machine, the last term of it is wholly 
 independent of all these resistances, its amount being deter- 
 mined solely by the velocities of the various moving ele- 
 ments of the machines and their respective weights. 
 
 THE TELOCITY OF A MACHINE MOVING WITH A VARIABLE 
 
 MOTION. 
 
 151. The velocities of the different parts or elements of 
 every machine are evidently connected with one another by 
 certain invariable relations, capable of being expressed by 
 algebraical formulae, so that, although these relations are 
 different for different machines, they are the same for ail 
 circumstances of the motion of the same machine. In a 
 great number of machines this relation is expressed by a 
 constant ratio. Let the constant ratio of the velocity v, of 
 any element to that V 1 of the moving point in such a 
 
152 THE VELOCITY OF A MACHINE. 
 
 machine be represented by X, so that v l =\'V l , and let v 9 and 
 V, be any other values of v l and Yj ; then -y =XY a . Sub- 
 stituting these values of v 1 and v a in equation (116), we 
 have 
 
 TJ'=A . TT,+B . S I+ (V a '- 
 
 in which expression 2i0X a represents the sum of the weights 
 of all the moving elements of the machine, each being mul- 
 tiplied by the square of the ratio X of its velocity to that of 
 the point where the machine receives the operation of its 
 moving power. For the same machine this co-efficient 2i0X a 
 is therefore a constant quantity. For different machines it 
 is different. It is wholly independent of the useful or pre- 
 judicial resistances opposed to the motion of the machine, 
 and has its value determined solely by the weights and 
 dimensions of the moving masses, and the manner in which 
 they are connected with one another in the machine. 
 transforming this equation and reducing, we have 
 
 by which equation the velocity Y 2 of the moving point of 
 the machine is determined, after a given amount of work 
 IP has been done upon it by the moving power, and a given 
 amount U 2 expended on the useful resistances ; the velocity 
 of the moving point, when this work began to be done 
 being given and represented by Y r 
 
 It is evident that the motion of the machine is more 
 equable as the quantity represented by 2wX 2 is greater. 
 This quantity, which is the same for the same machine and 
 different for different machines, and which distinguishes 
 machines from one another in respect to the steadiness of 
 their motion, independently of all considerations arising out 
 of the nature of the resistances useful or prejudicial opposed 
 to it, may with propriety be called the CO-EFFICIENT OF 
 EQUABLE MOTION.* The actual motion of the machine is 
 more equable as this co-efficient and as the co-efficients A 
 and B (supposed positive) are greater. 
 
 * The co-efficient of equable motion is here, for the first time, introduced 
 into the consideration of the theory of machines. 
 
CO-EFFICIENTS OF THE MODULUS. 153 
 
 To DETERMINE THE Co-EFFICIENTS OF THE MODULUS OF A 
 
 MACHINE. 
 
 152. Let that relation first be determined between the 
 moving pressure P, upon the machine and its working pres- 
 sure P 2 , which obtains in the state bordering upon motion by 
 the preponderance of P,. This relation will, in all cases 
 where the constant resistances to the motion of the machine 
 independently of P 2 are small as compared with P 2 , be 
 found to be represented by formulae of which the following 
 is the general type or form : 
 
 P,=P 3 .*,+*, ..... (119); 
 
 where *, and $ 2 represent certain functions of the friction 
 and other prejudicial resistances in the machine, of which 
 the latter disappears when the resistances vanish and the 
 former does not; so that if *j(> and * a () represent the 
 values of these functions when the prejudicial resistances. 
 vanish, then $>W=Q and * 1 ()= a given finite quantity - 
 dependent for its amount on the composition of the machine... 
 Let P/> represent that value of the pressure Pj which wxmlftj 
 be in equilibrium with the given pressure P 2 , if there- ware 
 no prejudicial resistances opposed to the motion off tbie 
 machine. Then, by the last equation, P/)=P 2 . Qffi. 
 
 But by the principle of virtual velocities (Art.,12T:),\if 
 we suppose the motion of the machine to be unifdrm-j . so 
 that Pj and P 2 are constantly in equilibrium upon it, and if 
 we represent by S, any space described by the point of 
 application of P,, or the projection of that space on . the 
 direction of P, (Art. 52.), and by S 2 the corresponding 
 space or projection of the space described by P 2 , then 
 P/ ) . S 1 ^P 2 . S 2 . Therefore, dividing this, equation by 
 the last, we have 
 
 Multiplying this equation by equation (119), , 
 
154: AXES. 
 
 which is the modulus of the machine, so that the constant 
 
 <f> 
 A in equation (114) is represented by j^, and the constant 
 
 B by * r 
 
 The above equation has been proved for any value of S 15 
 provided the values of P, and P 2 be constant, and the 
 motion of the machine uniform ; it evidently obtains, there- 
 fore, for an exceedingly small value of S 15 when the motion 
 of the machine is variable. 
 
 GENERAL CONDITION OF THE STATE BORDERING- UPON MOTION 
 
 IN A BODY ACTED UPON BY PRESSURES IN THE SAME PLANE, 
 AND MOVEABLE ABOUT A CYLINDRICAL AxiS. 
 
 153. If any number of pressures P,, P 2 , P 3 , <&c. applied in 
 the same plane to a body moveable about a cylindrical 
 axis, be in the state bordering upon motion, then is the 
 direction of the resistance of the axis inclined to its radius, 
 at the point where it intersects the circumference, at an 
 angle equal to the limiting angle of resistance. 
 
 For let R represent the resultant of P, P 2 , &c. Then, 
 since these forces are supposed to be upon the 
 11 point of causing the axis of the body to turn 
 upon its bearings, their resultant would, if made 
 to replace them, be also on the point of causing 
 the axis to turn on its bearings. Hence it fol- 
 lows that the direction of this resultant R cannot 
 be through the centre C of the axis ; for if it 
 f were, then the axis would be pressed by it in the 
 direction of a radius, that is, perpendicularly 
 upon its bearings, and could not be made to turn upon them 
 by that pressure, or to be upon the point of turning upon 
 them. The direction of 11 must then be on one side of C, 
 so as to press the axis upon its bearings in a direction RL, 
 inclined to the normal CL (at the point L, where it inter- 
 sects the circumference of the axis) at a certain angle RLC. 
 Moreover, it is evident (Art. 141.), that since this force R 
 pressing the axis upon its bearings at L is upon the point of 
 causing it to -slip upon them, this inclination RLC of R to 
 the perpendicular CL is equal to the limiting angle of 
 
THE WHEEL AND AXLE. 155 
 
 resistance of the axis and its bearings.* Now the resistance 
 of the axis is evidently equal and opposite to the resultant 
 R of all the forces P 1? P 2 , &c. impressed upon the body. 
 This resistance acts, therefore, in the direction LR, and is 
 inclined to CL at an angle equal to the limiting angle of 
 resistance. Therefore, &c. 
 
 THE WHEEL AND AXLE. 
 
 154. The pressures P, and P 2 applied ver- 
 tically by means of parallel cords to a 
 wheel and axle are in the state bordering 
 upon motion by the preponderance of P 15 
 it is required to determine a relation 
 between P, and P 2 . 
 
 The direction LR of the resistance of the axis is on that 
 side of the centre which is towards P 15 and is inclined to the 
 perpendicular CL at the point L, where it intersects the 
 axis at an angle CLR equal to the limiting angle of resist- 
 ance. Let this angle be represented by <p, and the radius 
 CL of the axis by p ; also the radius CA of the wheel by a,, 
 and that CB of 1jie axle by & 2 ; and let "W be the weight of 
 the wheel and axle, whose centre of gravity is supposed to 
 be C. Now, the pressures P n P 2 , the weight W of the 
 wheel and axle, and the resistance R of the axis, are pres- 
 sures in equilibrium. Therefore, by the principle of the 
 equality of moments (Art. 7.), neglecting the rigidity of the 
 cord, and observing that the weight W may be supposed to 
 act through C, we have, 
 
 P, . CA=P 2 . CB + R . Cm. 
 
 If, instead of P, preponderating, it had been on the point 
 of yielding, or P 2 had been in the act of preponderating, 
 then R would have fallen on the other side of C, and we 
 should have obtained the relation P, . CA=P 2 . CB 
 R . Cm; so that, generally, P, . CA=P 2 . CBR .^Cm; 
 the sign being taken according as P, is in the superior or 
 inferior state bordering upon motion. 
 
 Now CA=a t , CB=# 2 , Cm= CL sin. CLR=p sin. <p, and 
 
 * The nide of C on which RL falls is manifestly determined by the direction 
 towards which the motion is about to take place. In this case it is supposed 
 about to take place to the right of C. If it had been to the left, the direc- 
 tion of R would have been on the opposite side of C. 
 
156 THE WHEEL AND AXLE. 
 
 "W; the sign being taken according as the 
 weight W of the wheel and axle acts in the same direction 
 with the pressures Y 1 and P 2 , or in the opposite direction ; 
 that is, according as the pressures P x and P a act vertically 
 downwards (as shown in the figure) or upwards / 
 
 P sn. 9, 
 /.PX^ p sin. 9)=P a (# a + p sin. <p)Wp sin. 9. 
 
 Now the effect (Art. 142.) of the rigidity of the cord BP a 
 is the same as though it increased the tension upon that cord 
 
 (~r\ i "U* ~p \ 
 P,+ - a ): allowing, therefore, for the 
 # a ' 
 
 rigidity of the cord, we have finally 
 
 (T)-I-"R P \ 
 P 3 + - -^~ -J ( 2 + p sin. <p)"W p sin. 9, 
 
 or reducing, 
 
 aa^ p sm. 9 a l p sin. 9 
 
 which is the required relation between P x and P a in the 
 state bordering upon motion. 
 
 p p 
 
 sin. 9 and sin. 9 are in all cases exceedingly small ; 
 
 a, a, 8 J 
 
 we may therefore omit, without materially affecting the 
 result, all terms involving powers of these quantities above 
 the first, we shall thus obtain by reduction 
 
 155. The modulus of uniform motion m the wheel and axle. 
 
 It is evident from equation (122), that, in the case of the 
 wheel and axle, the relation assumed in equation 
 
 / E \& a 4- p sin. 9 
 
 obtains, if we take ^=1 1 -f I- -. I ; 
 1 V aja,o sm. e ' 
 
THE WHEEL AND AXLE. 
 
 157 
 
 and 
 
 a l - p sin. 9 
 
 ^observing that * 1 () represents the value of $, when 
 the prejudicial resistances vanish (or when 9=0 andE=0) 
 
 T_ * /m ^ 
 
 we have &W=. 
 
 \ p sin. 
 Therefore by equation (121), 
 
 
 _/P\ sin. 9 ' 
 \aj 
 
 (Si sin - 
 
 1 sin. 9 
 
 P sin. 9 
 
 .(124), 
 
 which is the modulus of the wheel and axle. 
 
 Omitting terms involving dimensions of sin. 9, and 
 
 Cd l 
 
 sin. 9, and above the first, we have 
 
 ' 
 
 156. The modulus of variable motion in the wheel and axle. 
 
 If the relation of P, and P 2 be not that of either state 
 bordering upon motion, then the motion will be continually 
 accelerated or continually retarded, and work will continu- 
 ally accumulate in the moving parts of the machine, or the 
 work already accumulated there will continually expend 
 
158 
 
 THE WHEEL AND AXLE. 
 
 itself until tlie whole is exhausted, and the machine is 
 brought to rest. The general expression for the modulus in 
 this state of variable motion is (equation 116) 
 
 ^ 
 
 ^Now in this case of the wheel and axle, if 'V l and Y 2 re- 
 present the velocities of P l at the 
 commencement and completion 
 of the space S,, and the angular 
 velocity of the revolution of the 
 wheel and axle ; if, moreover, the 
 pressures P t and P 2 be supposed 
 to be supplied by weights sus- 
 
 \ / pended from the cords ; then, 
 
 since the velocity of P 2 is repre- 
 sented by -? -, we have 
 a, 
 
 \ represent the moment of inertia of the revolving wheel, 
 and I a that of the revolving axle, (Art. 75.), and if f\ repre- 
 sent the weight of a unit of the wheel and f* 2 of the axle ; 
 since 2wv* represents the sum of the weights of all the mov- 
 ing elements of the machine, each being multiplied by the 
 square of its velocity, and that (by Art. 75.) a> 1 I 1 represents 
 this sum in respect to the wheel, and -V 2 I 2 in respect to the 
 axle. Now, Y 1 =a^ 1 , 
 
 a. 
 
 Y 3 ! lCt/l + r * a * "T 
 Similarly 2^v 2 2 =Y a a 
 
 Substituting in the general expression (equation 116), we 
 have 
 
THE WHEEL AND AXLE. 159 
 
 TJ'=AU,+BS 1 +1(V,"-V,') 
 
 which is the modulus of the machine in the state of variable 
 motion, the co-efficients A and B being those already deter- 
 mined (equation 124), whilst the co-efficient 
 
 ig the co . efficient SwX , ( tion 
 
 i 
 
 117) of equalle motion. If the wheel and axle be each of them 
 a solid cylinder, and the thickness of the wheel be & and the 
 length of the axle 5 2 , then (Art. 85.) I l =faeb l a\, I 2 =i^ 2 & 2 4 . 
 Now if W l and W 2 represent the weights of the wheel and 
 axle respectively, then W l =ifa l 9 b l ^ W 2 =tf# 2 2 & 2 M< 2 ; therefore 
 M-Ji 2-W^x 2 , M- a I a =i'W' a a a *. Therefore the co-efficient of 
 equable motion is represented by the equation 
 
 or 
 
 (137), 
 
 157. To determine the velocity acquired through a given 
 space when the r elation o the weights P, and P 2 , suspended 
 from a wheel and axle, is not that of the state bordering 
 upon motion* 
 
 Let Sj be the space through which the weight P a moves 
 whilst its velocity passes from Y x to V 2 : observing that 
 
 !?=?&, and that U a =P 2 S a =P 9 ^- 2 , substituting in equa- 
 
 a l 
 
 tion (126), and solving that question in respect to Y a , we 
 have 
 
 .0*); 
 
 * Note (w) Ed. App. 
 
160 
 
 THE PULLEY. 
 
 making the same suppositions as in formula 127, and repre- 
 senting the ratio by ra, we have 
 
 THE PULLEY. 
 
 158. If the radius of the axle be taken equal to that of the 
 wheel, the wheel and axle becomes a pul- 
 ley. Assuming then in equation 122, 
 
 of the 
 state 
 when 
 the strings are parallel, 
 
 ill ^* Assuming then in equation 
 
 rfpk M Ir I 1 = 3 =,we obtain for the relation < 
 
 I H Wk I moving pressures Y l and P 2 , in the 
 
 rH C5 \ t) r( i erm g upon motion in the pulley, 
 
 ..(129); 
 
 and by equation 124 for the value of the modulus, 
 
 p 
 
 l + -sm. 9 
 a 
 
 w) P sin. 
 
 Ob I 
 
 ? sn. 9 
 
 . . . (130); 
 
 in which the sign is to be taken according as the pressures 
 P t and P 2 act downwards, as in the first pulley of the pre- 
 ceding figure ; or upwards, as in the second. Omitting 
 
 p 
 
 dimension of - sin 9, - sin. 9, and above the first, we have 
 a a a 
 
 by equations (123, 125) 
 
SYSTEM OF ONE FIXED ONE MOVEABLE PULLET. 
 
 161 
 
 E 2 P sin.< 
 - + 
 
 TT TT . , , 
 
 U S =TT. 1+- + - +~ 
 
 Also observing that ,=, and I,=0, the modulus of varia- 
 ble motion (equation 126) becomes 
 
 (133), 
 
 ~ij 
 
 and the velocity of variable motion (equations 118, 128) is 
 determined by the equation 
 
 in which two last equations the values of A and B are those 
 of the modulus of equable motion (equation 125).. 
 
 SYSTEM OF ONE FIXED AND ONE MQIOJABUE 
 
 159. In the last article (equation 131) it was 
 shown that the relation between the tensions 
 P! and P 2 upon the two parts of a string pass- 
 ing over a pulley and parallel to one another, 
 was, in the state bordering upon motion by the 
 preponderance of P,, represented by an expres- 
 sion of the form P l =aP 9 + l>, where a and b are 
 constants dependent upon the dimensions of the 
 pulley and its axis, its weight, and the rigidity 
 of the cord, and determined in terms of these 
 elements by equation 131 ; and in which ex- 
 pression 5 has a different value according as the 
 tension upon the cord passing over any pulley 
 acts in the same direction with the weight of that pulley (as-. 
 in the first pulley of the system shown in the figure), or in. 
 the opposite direction (as in the second pulley) : let these 
 different values of 5 be represented by I and J,. Now it is, 
 evident that before the weight P 2 can be raised by means of 
 a system such as that shown in the figure, composed of one 
 fixed and one moveable pulley, the state of the equilibrium 
 of both pulleys must be that bordering upon motion, which 
 is described in the preceding article ; since both must be 
 upon the point of turning upon their axes before the weight 
 P a can begin to be raised. If then T and t represent the 
 tensions upon the two parts of the string which pass round, 
 the moveable pulley, we have 
 
162 SYSTEM OF ONE FIXED AND ANY 
 
 and T= 
 
 Now the tensions T and t together support the weight P,, 
 and also the weight of the moveable pulley, 
 
 Adding aT to both sides of the second of the above equa- 
 tions, and multiplying both sides by a, we have 
 
 Also multiplying the first equation by (1 + a\ 
 
 Now if there were no friction or rigidity, a would evi- 
 dently become 1 (see equation 121), and <t>= * would 
 
 become-; the co-efficients of the modulus (Art. 148.) are 
 
 / a? \ 
 =2( 1 ), 
 
 \l + ar 
 
 . -^ 
 
 therefore A=2( and B 
 
 which is the modulus of uniform motion to the single move- 
 able pulley.* 
 
 If this system of two pulleys had been 
 arranged thus, with a different string passing 
 over each, instead of with a single string, as 
 shown in the preceding figure, then, represent- 
 ing by t the tension upon the second part of 
 the string to which P 1 is attached, and by T 
 that upon the first part of the string to which 
 P 2 is attached, we have 
 
 * The modulus may be determined directly from equation (135); for it is 
 evident that if Si and S 2 represent the spaces described in the same time by 
 PI and P 2 , then S 1 = 2S 2 . Multiplying both sides of equation (135) by this 
 equation, we have, 
 
 now P 1 S 1 ==U 1 , and P 2 S 2 = U2, therefore &c. 
 
NUMBER OF MOVEABLE PULLEYS. 
 
 163 
 
 Multiplying the last of these equations by #, and adding it 
 to the first, we have Y l (I + a)+wa=Ta+b==a*'Pt + (l + ajb' 9 
 
 and for the modulus (equation 121), 
 
 It is evident that, since the co-efficient of the second term 
 of the modulus of this systen is less than that of the first 
 system (equation 136) (the quantities a and b being essen- 
 tially positive), a given amount of work U 2 may be done by 
 a less expense of power TJ,, or a gived weight P 2 may be 
 raised to a given height with less work, by means of this 
 system than the other ; an advantage which is not due 
 entirely to the circumstance that the weight of the move- 
 able pulley in this case acts m favour of the power, whereas 
 in the other it acts against it ; and which advantage would 
 exist, in a less degree, were the pulleys without weight. 
 
 A SYSTEM OF ONE FIXED AND ANY J^UMBEK OF MOVEABLE 
 
 PULLEYS. 
 
 160. Let there be a system of n moveable 
 pulleys and one fixed pulley combined as 
 shown in the figure, a separate string passing 
 over each moveable pulley ; and let the ten- 
 sions on the two parts of the string which 
 passes over the first moveable pulley be re- 
 presented by Tj and those upon the two 
 parts of the string which passes over the 
 second by T 2 and ,, &c. Also, to simplify 
 the calculation, let all the pulleys be sup- 
 posed of equal dimensions and weights, and 
 the cords of equal rigidity ; 
 
 /.T^^ + fct, and T i +W=T 1 + * 1 ; 
 
 /.eliminating, T = 
 
 (139). 
 
 Let the co-efficients of this equation be represented by 
 and /3 
 
SYSTEM OF ONE FIXED AND ANY 
 
 164 
 
 Similarly, T 3 =aT 8 +/3, T s =aT 4 +/3, T 4 =aT 6 +/3, & c .=&c.. 
 
 Multiplying these equations successively, beginning from 
 the second, by , a a , a 3 , &c., a**- 1 , adding them together, and 
 striking out terms common to both sides of the resulting 
 equation, we have 
 
 .... -f a-l/3; 
 
 or summing the geometrical progression in the second 
 member, 
 
 (140); 
 
 Substituting for a and (3 their values from equation (139), 
 and reducing 
 
 \ j 
 
 Whence observing, that, were there no friction, a would 
 
 become unity, and(--^-j = (-) . We have (equation 121) 
 for the modulus of this system, 
 
 161. If each cord, instead of having one of 
 its extremities attached to a fixed obstacle, had 
 been connected by one extremity to a move- 
 able bar carrying the weight P 2 to be raised 
 (an arrangement which is shown in the second 
 figure), then, adopting the same notation as 
 before, we have 
 
 Adding these equations together, striking out 
 terms common to both sides, and solving in 
 respect to T 1? we have 
 
NUMBER OF MOVEABLE PULLEYS. 165 
 
 in which equation it is to be observed, that the symbol b 
 does not appear ; that element of the resistance (which is 
 constant), affecting the tensions t l and 2 equally, and there 
 
 fore eliminating with T, and T 2 . Let ^ be represented 
 by , then 
 
 a^ -W. Similarly, ,=*, -W, 
 
 
 (143). 
 
 Eliminating between these equations precisely as between 
 the similar equations in the preceding case (equation 140), 
 observing only that here (3 is represented by oW, and that 
 the equations (143) are n 1 in number intead of n, we have 
 
 Also adding the preceding equations (143) together, we have 
 
 aiW 
 
 Now the pressure P, is sustained by the tensions &,, &c. 
 of the different strings attached to the bar which carries it. 
 Including P 2 , therefore, the weight of the bar, we have 
 
 Substituting this value of t n in equation (144), 
 
 n n ^W 
 
 , i=(1 _ aK _ lps+a%+(w _ lf _w 
 
 Transposing and reducing, 
 
 2 
 
 W( 
 
 2 
 
166 TACKLE OF ANY NUMBER OF SHEAVES. 
 
 = ,!+-; 
 
 r-1 
 
 a n 
 
 f\ - 
 
 8 ~ 
 
 g-ip. Wj n j 
 
 '""(l + a- 1 )*-! a \(l + a- l ) n -l ) * 
 
 - -- al+b. . (145). 
 
 a--i 
 
 Whence observing that when a=l, ^l + a" 1 ) 7 * 1} = 2 W 1 ; 
 we obtain for the modulus of uniform motion (equation 
 121), ' 
 
 A TACKLE OF ANY XUMBER OF SHEAVES. 
 
 162. If an number of pulleys (called in this case sheaves) 
 be made to turn on as many different centres in the same 
 block A, and if in another block B there be simi- 
 larly placed as many others, the diameter of each 
 of the last being one half that of a correspond- 
 ing pulley or sheave in the first ; and if the same 
 cord attached to the first block be made to pass 
 in succession over all the sheaves in the two 
 blocks, as shown in the figure, it is evident that 
 the parts of this cord 1, 2, 3, &c. passing between 
 the two blocks, and as many in number as there 
 are sheaves, will be parallel to each other, and 
 will divide between them the pressure of a weight 
 P 2 suspended from the lower block : moreover, 
 that they would divide this pressure between 
 them equally were it not for the friction of the 
 ** sheaves upon their bearings and the rigidity of 
 
 the rope ; so that in this case, if there were n sheaves, the 
 
 tension upon each would be -P a ; and a pressure ^ l of that 
 
 n 
 
TACKLE OF ANY NUMBER OF SHEAVES. 
 
 167 
 
 amount applied to the extremity of the cord would be suffi- 
 cient to maintain the equilibrium of the state bordering upon 
 motion. Let T 15 T 2 , T 3 , &c. represent the actual tensions 
 upon the strings in the state bordering on motion by the pre- 
 ponderance of Tj, beginning from that which passes from P t 
 over the largest sheaf; then 
 
 P^o.T.+S,, T,=a,T I+ J s , T,=a il T,+5, 
 &c.=&c.,T lr _ 1 =a.T. + &.; 
 
 where a^ # 2 , &c., 5,, & 2 , &c. represent certain constant co- 
 efficients, dependent upon the dimensions of the sheaves and 
 the rigidity of the rope, and determined by equation (131). 
 Moreover, since the weight P 2 is supported by the parallel 
 tensions of the different strings, we have 
 
 P T 4-T 
 
 -L 2 - -Lj-f- J. a 
 
 4-T 
 
 -f i 
 
 It will be observed that the above equations are one more 
 in number than the quantities T,, T 2 , T 3 , &c. ; the latter may 
 therefore be eliminated among them, and we shall thus ob- 
 tain a relation between the weight P 2 to be raised and that 
 Pj necessary to raise it, and from thence the 
 modulus of the system. 
 
 To simplify the calculation, and to adapt 
 it to that form of the tackle which is com- 
 monly in use, let us suppose another ar- 
 rangement of the sheaves. Instead of their 
 being of different diameters and placed all 
 in the same plane, as shown in the last 
 figure, let them be of equal diameter and 
 placed side by side, as in the accompanying 
 figure, which represents the common tackle. 
 The inconvenience of this last mode of ar- 
 rangement is, that the cord has to pass from 
 the plane of a sheaf in one block to the plane 
 of the corresponding sheaf in the other ob- 
 liquely, so that the parts of the cords be- 
 tween the blocks are not truly parallel to 
 one another, and the sum of their tensions is not truly equal 
 to the weight P 2 to be raised, but somewhat greater than it. 
 So long, however, as the blocks are not very near to one an- 
 other, this deflection of the cord is inconsiderable, and the 
 error resulting from it in the calculation may be neglected. 
 Supposing the different parts of the cord between the blocks 
 then to be parallel, and the diameters of all the sheaves and 
 
168 TACKLE OF ANY NUMBER OF SHEAVES. 
 
 their axes to be equal, also neglecting the influence of the 
 weight of each sheaf in increasing the friction of its axis, 
 since these weights are in this case comparatively small, the 
 co-efficients 15 a 9 , a z will manifestly all be equal ; as also 
 
 .., . , 23 , . 
 
 &c.=&c., T._ 1 =aT. + & f ' 
 
 also P 2 =T 1 +T 2 +T 3 + ..... +T n . 
 
 Multiplying equations (147) successively (beginning from the 
 second) by 0, # 2 , 3 , and a n ~ l ; then adding them together, 
 striking out the terms common to both sides, and summing 
 the geometric series in the second member (as in equation 
 140), we have 
 
 Cb - 1 
 
 Adding equations (147), and observing that 
 . . . . +T n =P 3 , and that P 1 +T 1 +T a + .... +T ir _ 1 = 
 2 T w , we have 
 
 Eliminating T n between this equation and the last, 
 
 To determine the modulus let it be observed, that, neglect- 
 ing friction and rigidity, a becomes unity ; and that for this 
 
 ^(/r _ ~\\ 
 value of 0, - - becomes a vanishing fraction, whose 
 
 d 1 
 
 value is determined by a well known method to be -*. 
 Hence (Art. 152.), 
 
 * Dividing numerator and denominator of the fraction by (a 1) it becomes 
 a -i + a v i -- :TI which evidently equals - when a=l. The modulus 
 
 may readily be determined from equation (148). Let Si and S 2 represent the 
 spaces described by P x and P 2 in any the same time ; then, since when the 
 blocks are made to approach one another by the distance S a , each of the n por- 
 tions of the cord intercepted between the two blocks is shortened by this dis- 
 
THE MODULUS OF A COMPOUND MACHINE. 169 
 
 nba n I 
 
 Hitherto no account lias been taken of the work expended 
 in raising the rope which ascends with the ascending weight. 
 The correction is, however, readily made. By Art. 60. it 
 appears that the work expended in raising this rope (diffe- 
 rent parts of which are raised different heights) is precisely 
 the same as though the whole quantity thus raised had been 
 raised at one lift through a height equal to that through 
 which its centre of gravity is actually raised. Now the cord 
 raised is that which may be conceived to lie between two 
 positions of P 2 distant from one another by the space S a , so 
 that its whole length is represented by nS^ ; and if j* repre- 
 sent the weight of each foot of it, its whole weight is repre- 
 sented by fwS, : also its centre of gravity is evidently raised 
 between the first and second positions of P 2 by the distance 
 S 2 ; so that the whole work expended in raising it is repre- 
 
 sented by JfwS a * or by i^-, since S 1 =7iS a . Adding this 
 
 work expended in raising the rope to that which would be 
 necessary to raise the weight P 2 , if the rope were without 
 weight, we obtain* 
 
 TT a (a 1) TT ( nba n 
 
 u-^irr 11 ^ i ^i- 
 
 which is the MODULUS of the tackle. 
 
 THE MODULUS OF A COMPOUND MACHINE. 
 
 163. Let the work of a machine be transmitted from one 
 to another of a series of moving elements forming a com- 
 pound machine, until from the moving it reaches the working 
 point of that machine. Let P be the pressure under which 
 the work is done upon the moving point, or upon the first 
 moving element of the machine ; Pj that under which it is 
 
 tance S 2 , it is evident that the whole length of cord intercepted between the 
 two blocks is shortened by wS 2 ; but the whole of this cord must have passed 
 over the first sheaf, therefore Si=wS 2 . Multiplying equation (148) by this 
 equation, and observing that TJ^PiS 1 and U 2 =P 2 S 2 , we obtain the modulus 
 as given above. 
 
 * A correction for the weight of the rope may be similarly applied to the 
 modulus of each of the other systems of pulleys. The effect of the weight of 
 the rope in increasing the expenditure of work on the friction of the pulleys if 
 neglected as unimportant to the result. 
 
170 MODULUS OF A COMPOUND MACHINE. 
 
 F* elded from the first to the second element of the machine ; 
 t from the second to the third element, &c. ; and P w the 
 pressure under which it is yielded by the last element upon 
 the useful product, or at the working point of the machine. 
 Then, since each element of the compound machine is a sim- 
 ple machine, the relation between the pressures applied to 
 that element when in the state bordering on motion will be 
 found to present itself under the form of equation (119) 
 (Art. 152), in all cases where the pressure under which the 
 work upon each element is done is great as compared with 
 the weight of that element (see Art. 166.). 
 
 Kepresenting, therefore, by 1? a^ a 3 . . . 5 15 5 2 , J 3 . . ., cer- 
 tain constants, which are given in terms of the forms and 
 dimensions of the several elements and the prejudicial resist- 
 ances, we have 
 
 &c.=&c., ?_!=?.+&. 
 
 Eliminating the n 1 quantities P 1? P Q , P s . . ., P B _ 1 , between 
 these n equations, we obtain an equation, of the form, 
 
 P=P.+&' ..... (151); 
 where a=^a^a z . . . a n , and 
 
 If the only prejudicial resistance to which each element is 
 subjected be conceived to be friction, and the limiting angle 
 of resistance in respect to each be represented by <p ; then 
 considering each of the quantities a^ J 1? a a , 5 2 , as a function 
 of 9, expanding each by Maclaurin's theorem into a series 
 ascending by powers of that variable, and neglecting terms 
 which involve powers of it above the first, we nave 
 
 \ C#P / V WP / 
 
 where, a(\ 5/ ), 2 (), 5 2 (), represent the values of a b a 
 
 6 2 , &c., when 9=0 and ( ^-| , ( -+- } , &c. represent 
 \ d$ / \ d$ I 
 
 the similar values of their first differential co-efficients. 
 
AXES. 
 
 171 
 
 Let 
 
 Therefore fl^tf^ (! + *,) &i=&i (0) (1 +&)> 9 =a,<> (1+aN 
 ,=^>(l-fj$ t ), &c:=&c.; where a J5 ft, f , /3 f , & c ., each 
 involving the factor <p, are exceedingly small. Substituting 
 the values of a 19 a 25 &c. in the expression for <z, and neglect- 
 ing terms which involve dimensions of a 1? a a) &c. above the 
 first, we have 
 
 a=a<P> aj . . . a^ Jl + a l+ a a + a 3 + ---- + a j ---- (153). 
 
 Now the co-eficient of the first term of the modulus is 
 represented (equation 121) by -, a representing the co- 
 
 efficient of the first term of equation (119), also substituting 
 the value of a from equation (153), and observing that 
 
 a,(o) .... ^(o) ? we have -.= 
 
 ..U={l-f.a 1 + a 4 + a 1 4- .... + ajU n + 5.S .... (154), 
 which is the modulus of a compound machine of n elements, 
 U representing the work done at the moving point, U w 
 that at the working point, S the space described by the 
 moving point, and 5 a constant determined by equation 
 (152). 
 
 164. THE CONDITIONS OF THE EQUILIBRIUM OF ANY TWO PRES- 
 SURES Pj AND P 2 APPLIED IN THE SAME PLANE TO A BODY 
 MOVEABLE ABOUT A FIXED AXIS OF GIVEN DIMENSIONS. 
 
 In. fig. 1. the pressure P 1 and P a are shown acting on oppo- 
 site sides of the axis 
 whose centre is C, and 
 in fig. 2. upon the same 
 side. Let the direc- 
 tion of the resultant 
 of P! and P a be repre- 
 sented, in the first 
 case, by IR, and in 
 the second by El. It 
 
172 
 
 AXES. 
 
 is in the directions of these lines that the axis is, in the two 
 cases, pressed upon its bearings. Suppose the relation 
 between P 1 and P 2 to be such that the body is, in both 
 cases, upon the point of turning in the direction in which 
 Pj acts. This relation obtaining between P, and P 2 , it is 
 evident that, if these pressures were replaced by their re- 
 sultant, that resultant would also be upon the point of caus- 
 ing the body to turn in the direction of P r The direction 
 TR of the resultant, thus acting alone upon the body, lies, 
 therefore, in the first case, upon the same side of the centre 
 C of the axis as P l does, and in the second case it lies upon 
 the opposite side ;* and in both cases, it is inclined to the 
 radius CK at the point K, where it intersects the axis at an 
 angle CKK, equal to the limiting angle of resistance (see 
 Art. 153.). Now, the resistance of the axis acts evidently in 
 both cases in a direction opposite to the resultant of P, and 
 P 8 , and is equal to it ; let it be represented by R. Upon 
 the directions of P 15 P 2 , and R, let fall the perpendiculars 
 CA 15 CA 2 , and CL, and let them be represented by a l9 a 
 and \ Then, by the principle of the equality of moments, 
 since P 1? P 2 , and R are pressures in equilibrium, 
 
 If Pj had been upon the point of yielding, or P 2 on the 
 point of preponderating, then R would have had its direction 
 (in both cases) on the other side of C ; so that the last equa- 
 tion would have become 
 
 According, therefore, as Pj is in the superior or inferior 
 state bordering upon motion, 
 
 And if we assume X to be taken with the sign -f or , ac- 
 cording as P! is about to preponderate or to yield, then 
 generally 
 
 Now, since the resistance of the axis is equal to the resultant 
 of P x and P 2 , if we represent the angle PJP 3 by if, we have 
 (Art. 13.) 
 
 * The arrows in the figure represent, not the directions of the resultants 
 but of the resistances of the axis, which are opposite to the resultants. 
 
 f Care must be taken to measure this angle, so that PI and P 2 may have 
 
AXES. 173 
 
 Substituting this value of R in the preceding equation, and 
 squaring both sides, 
 
 (P A -PA)r=x'(P 1 < +2P 1 P 1 cos. 
 transposing and dividing by P 2 2 , 
 
 W (a '~ x ' } ~ 2 (^) (aA+x ' cos-i)= ~ (a '~ xs) ; 
 
 solving this quadratic in respect to ( -^ ) , 
 
 \-t 2/ 
 
 P 1 _fa^ a +^ a cos, i) 4/pA+ x2 cos. *)* fa 8 - 1 * 8 ) X a ) 
 P." a/-* ; 
 
 cos, i) X 4/(<^ 1 2 + 2^ 1 ^ 2 cos. + <% 2 a ) X a sin. a <. 
 
 ISTow let the radius CK of the axis be represented by p, 
 and the limiting angle of resistance CKR by 9 ; therefore 
 X=CL=CK sin. CKR:=p sin. 9. Also draw a straight line 
 from Aj to A 2 in both figures, and let it be represented by L ; 
 :.a^^a^ cos. A l CA li -\-a^=lu. Now, since the angles at 
 A x and A 2 are right angles, therefore the angles AJA, and 
 AiCAj are together qqual to two right angles, or A^CAj + i 
 =*; therefore AjCA a * , and cos. AjCA 2 = cos. ; 
 therefore L 2 = 1 3 + 2<x 1 2 cos. + ^ 2 2 : substituting these values 
 of L 2 and X in the preceding equation, 
 
 p _(X<2 2 + p 3 cos. i sin. V) + p sin. 9 (L 2 p a sin. 2 sin. *<p)* 
 
 ( 1 a - P 2 sin. a 9) 
 . P 2 . . (156). 
 
 The two roots of the above equation are given by positive 
 and negative values of X, they correspond therefore (equa- 
 tion 155) to the two states bordering upon motion. These 
 two values of X are, moreover, given by positive and nega- 
 tive values of 9 ; assuming therefore 9 to be taken positively 
 or negatively, according as P, preponderates or yields, we 
 may replace the ambiguous by the positive sign. The 
 
 their directions both towards or both from the angular point I (as shown in the 
 figure), and not one of them towards that point and the other from it. Thus, 
 in the second figure, the inclination L of the pressures P t and P 2 is not the 
 angle AjIPj, but the angle PJPi. It is of importance to observe this distinc- 
 tion (see note p. 194.). 
 
AXES. 
 
 relation above determined between P 4 and P 2 evidently 
 satisfies the conditions of equation (119). We obtain there- 
 fore for the modulus (equation 121) 
 
 COS> 
 
 - P sn. 
 U 2 . . . (157). 
 
 If terms involving powers of | 1 sin. <p above the first be 
 
 neglected, that quantity being in all cases exceedingly 
 small, we have 
 
 165. Jb determine the resultant R o/" any number of pres- 
 sures P 15 P 2 , P 3 . . . ., in terms of those pressures, and the 
 cosines of their inclinations to one another. 
 
 Let 15 a 25 a g5 &c. represent the inclinations 
 I AC, IBC, &c. of the several pressures P,, 
 P 2 , &c. to any given axis CA in the same 
 plane; arid let 12 , i 13 , i aa , &c. represent the 
 inclinations of these pressures severally to one 
 another. 
 Now / AIB^ ZIBC- ZIAC (Euc. I. 32.) ; 
 
 /. 12 2 a i5 /.cos. i 12 =cos. a x cos. a 3 -fsin. a x sin. a a . 
 Similarly, cos. 13 =cos. x cos. 8 + sin. a x sin. a; ? 
 cos. i 23 =cos. 2 cos. a 8 -fsin. a 2 sin. 3 . 
 
 Now E 2 =(P 1 cos. ,+?, cos. a 2 +P 3 cos. 3 + . . . ) 2 + (P 1 
 
 sin. ^4-P, sin. a 2 + P 3 sin. 3 + . . . ) 2 , (equation 9, Art. 11.). 
 
 Squaring the two terms in the second member, adding the 
 results, and observing that cos. X-j-sin. X=l, 
 
 E a =P a 2 +P 2 2 + P 3 2 . . . +2P.P, (cos. , cos. a a + s in. a sin. a a ) 
 +2P,P 8 (cos. x cos. a s -fsin. a x sin. 3 )+ . . . . ; 
 
AXES. 
 
 175 
 
 + 2P a P 8 cos. I M + &c 
 
 (160). 
 
 166. THE CONDITIONS OF THE EQUILIBRIUM OF THREE PRES 
 SURES, Pj, P 2 , P 3 , IN THE SAME PLANE APPLIED TO A BOD? 
 
 MOVEABLE ABOUT A FIXED AXIS, THE DIRECTION OF ONE OF 
 THEM, P 3 , PASSING THROUGH THE CENTRE OF THE AXIS, AND 
 THE SYSTEM BEING IN THE STATE BORDERING UPON MOTION 
 BY THE PREPONDERANCE OF P. 
 
 represent the inclinations of the directions of 
 the pressures P,, P 2 , P 3 to one 
 another, a^ and $ 2 the perpen- 
 diculars let fall from the cen- 
 tre of the axis upon P 1 and P 2 , 
 and X the perpendicular let 
 fall from the same point upon 
 the resultant R of P 1? P 2 , P 3 . 
 Then, since R is equal and 
 opposite to the resistance of 
 the axis (Art. 153.), we have, 
 by the principle of the equality of moments, P,^ P 2 & 2 
 XR, for P 3 passes through the centre of the axis, and its 
 moment about that point therefore vanishes. 
 
 Substituting the value of Rfrom equation (160), 
 
 .. cos. i lt + 
 
 2P 1 P 3 cos.i 13 +2P 2 P 3 cos. 23 .p 
 
 Squaring both sides of this equation, and transposing, 
 P 1 -(a 1 -X-)-2P l JP^A-X'CP, cos.. 12 +P 3 cos. .)} = 
 
 -P> 2 2 + ^ 2 JP 2 2 + P 3 2 + 2P 2 P 3 cos. i,,} . 
 If this quadratic equation be solved in respect to P and 
 
 * In which expression it is to be understood that the inclination j 12 of the 
 directions of any two forces is taken on the supposition that both the forces 
 act from or both act towards the point in which they intersect, 
 and not one towards and the other from that point; so that in 
 the case represented in the accompanying figure, the inclina- 
 tion t l2 of the two forces PI and P 2 represented by the arrows, 
 is not the angle PiIP 2 , but the angle QlPi, since IQ and IPi are 
 directions of these two forces, both tending from their point 
 of intersection ; whilst the directions of P 2 I and I?! are one 
 of them towards that point, and the other from it. 
 
176 
 
 terms which involve powers of X above the first be omitted, 
 we shall obtain the equation 
 
 2 cos.i w + O+P 3 V+2P 2 P A(> s cos.1,,4-^ cos. 23 ) ; 
 
 or representing (as in Art. 164.) the line which joins the 
 feet of the perpendiculars, a^ and & 3 by L, and the function 
 #, ( 2 cos. *! + #! cos. i 33 ) by M, and substituting for X its 
 value p sin. 9, 
 
 p= 
 
 ^Representing (as in Art. 152.) the value of P t when the 
 prejudicial resistances vanish, or when 9=0, by P^ 0) , we 
 
 haveP^ ^ I IP 2 . Also by the principle of virtual velo- 
 \ d> l l 
 
 cities P/> . 8,=?, . S,. Eliminating P/ ) between these 
 equations, we have S t = I I S 3 . Multiplying equation (161) by 
 
 Substituting U l for P^, U 2 for P 2 S 2 , and observing that 
 
 }*.... (162.) 
 
 which is the MODULUS of the system. 
 
 If P 3 be so small as compared with P 2 that in the expan- 
 sion of the binomial radical (equation 161), terms involving 
 
 p 
 
 powers of -p^ above the first may be neglected ; then, 
 " 
 
 * It will be shown in the appendix, that this equation is but a particular 
 case of a more general relation, embracing the conditions of the equilibrium 
 of any number of pressures applied to a body moveable about a cylindrical 
 axis of given dimensions. 
 
AXES. 
 
 which equation may be placed under the form 
 
 "Whence observing that the direction of P 3 being always 
 through the centre of the axis, the point of application of 
 that force does not move, so that the force P 3 does not work 
 as the body is made to revolve by the preponderance of P, ;. 
 observing, moreover, that in this case the conditions of 
 equation (119) (Art. 152.) are satisfied, we obtain for the? 
 modulus 
 
 167. The conditions of the equilibrium of two pressures ~P l 
 and P 2 applied to a body moveable about a cylindrical axis, 
 taking into account the weight of the body and supposing it 
 to be symmetrical about its axis. 
 
 The body being symmetrical about its axis, its centre of 
 gravity is in the centre of its axis, and its weight produces 
 the same effect as though it acted continually through the 
 centre of its axis. In equation (161.) let then P 2 be taken to 
 represent the weight W of the body, and i ja , i aa the inclina- 
 tions of the pressures P t and P 2 to the vertical. Then 
 
 P,= (%,+ (P:) | P,'L'+2P,WM+WV \ *. - (165.) 
 \<V \ a 1 I ( 
 
 Also by the equation (162) we find for the modulus 
 U,=U.+ 
 
 And in the case in which P 2 is considerable as compared 
 with W, by equations (163, 164). 
 
 12 
 
178 THE DIRECTION OF THE 
 
 /.\ J -. , L P ' 
 
 , = 1 I 1 1 H -am. 
 
 \aj ( a& 
 
 U i= 1 1+^-sin. 9 | U 2 + (J^)^nS, sin. ... (168.) 
 
 168. A MACHINE TO- WHICH ABE APPLIED ANY TWO PRESSURES 
 P t AND P 2 , AND WHICH IS MOVEABLE ABOUT A CYLINDRICAL 
 AXIS, IS WORKED WITH THE GREATEST ECONOMY OF POWER 
 WHEN THE DIRECTIONS OF THE PRESSURES ARE PARALLEL, 
 AND WHEN THEY ARE APPLIED ON THE SAME SIDE OF THE 
 AXIS, IF THE WEIGHT OF THE MACHINE ITSELF BE SO SMALL 
 THAT ITS INFLUENCE IN INCREASING THE FRICTION MAY BE 
 NEGLECTED. 
 
 For, representing the weight of such a machine by "W, and 
 neglecting terms involving W sin. 9, it appears by equation 
 (168) that the modulus is 
 
 whence it follows that the work U,, which must be done at 
 
 the moving point to yield a given amount TJ 2 at the working 
 
 point, is less as L is less. 
 
 Now L represents 
 the distance A^ be- 
 tween the feet of the 
 perpendiculars CA, and 
 CA 2 , which distance is 
 evidently least when P, 
 and P 2 act on the same 
 side of the axis, as in 
 fig. 2, and when CA t 
 
 and CA 2 are in the same straight line ; that is, when P t and 
 
 P 2 are parallel. 
 
 169. A MACHINE TO WHICH ARE APPLIED TWO GIVEN PRES- 
 SURES P! AND P 2 AND WHICH IS MOVEABLE ABOUT A CYLIN- 
 DRICAL AXIS, IS WORKED WITH THE GREATEST ECONOMY OF 
 POWER, THE INFLUENCE OF THE WEIGHT OF THE MACHINE 
 BEING TAKEN INTO THE ACCOUNT, WEEN THE TWO PRESSURES 
 
GREATEST ECONOMY OF POWER. 179 
 
 ARE APPLIED ON THE SAME SIDE OF THE AXIS, AND WHEN 
 THE DIRECTION OF THE MOVING PRESSURE P x IS INCLINED TO 
 THE VERTICAL AT A CERTAIN ANGLE WHICH MAT BE DETER- 
 MINED. 
 
 Let P 3 be taken to represent the weight of the machine, 
 and let its centre of gravity coincide with the centre of its 
 axis, then is its modulus represented (equation 166.) by 
 
 ^) +P 3 'S 1 V 1 *; 
 
 (Zj / 
 
 in which expression the work Uj, which must be done at the 
 moving point to yield a given amount U 2 of work at the 
 working point, is shown to be greater than that which must 
 have been done upon the machine to yield the same amount 
 of work if there had been no friction by the quantity 
 
 r^L j u.-L'+aujp.s.Mh +P.-S.V 1 4 
 
 a^ \q,i 
 
 The machine is worked then with the greatest economy of 
 power to yield a given amount of work, U 2 , when this func- 
 tion is a minimum. Substituting for L 2 its value 
 
 0/4- 2^0, cos. 12 + # 2 2 , and for M its value a, {# 2 cos. i 18 -f 
 a^ cos. ag } (see Art. 166.), also for Sj ) its value S 9 . it be- 
 
 \Cb l f 
 
 comes 
 
 (169.) 
 
 let us suppose that the perpendicular distance # 2 from 
 the centre of the axis at which the work is done, and the in- 
 clination i a8 of its direction to the vertical, are both given, as 
 also the space S 2 through which it is done, so that the work 
 is given in every respect ; let also the perpendicular distance 
 j at which the power is applied, and, therefore, the space S x 
 though which it is done, be given ; and let it be required to 
 determine that inclination i ia of the power to the work which 
 will under these circumstances give to the above function its 
 minimum value, and which is therefore consistent with the 
 most economical working of the machine. 
 
 Collecting all the terms in the function (169.) which con- 
 
180 THE DIRECTION OF THE 
 
 tain (on the above suppositions) only constant quantities, and 
 representing their sum by C, it becomes 
 
 -55^ { 2a A U 2 (U 2 cos. , 12 +P 3 S, cos. + } k 
 
 "Now C being essentially positive, this quantity is a mini- 
 mum when 20 1 a 3 TJ s (U a cos. ' 19 +P,S ? cos. i 13 is a minimum ; or, 
 observing that U 2 P 3 S 3 and dividing by the constant factor 
 when 
 
 P a cos. i 12 +P 3 cos. 13 is a minimum. 
 
 From the centre of the axis C let lines Cp 1 
 Cps be drawn parallel to the directions of the 
 pressures P^ respectively ; and whilst C/? 2 
 and Cj? 3 retain their positions, let the angle 
 ^jCPg or 18 be conceived to increase until P x 
 attains a position in which the condition 
 P 3 cos. i ia + P 3 cos. 13 =a minimum is satisfied. 
 Now ^ ^jOP,=p 1 Cp 1 #,CP 3 , or ^ I M =I H ; 23 ; 
 Substituting which value of 23 this condition 
 becomes 
 
 P 2 cos. 12 +P 8 cos. () a minimum, 
 
 or P 2 cos. ' 12 + P 3 cos. J2 cos. 23 +P 3 sin. < 13 sin. 23 a minimum, 
 
 or (P a +P 3 cos. i 23 ) cos. ia +P 8 sin. i 28 sin. ia a minimum. 
 
 P 3 sin. i 29 
 Let now^^ ^ - = = tan. 7, 
 
 /. (P 2 +P 3 cos. 23 ) cos. i M + (P t +P, cos. I M ) tan. 7 sin. i 12 is a mi- 
 nimum, or dividing by the constant quantity (P 2 +P 3 cos. i 23 ) 
 and multiplying by cos. 7, 
 
 cos. i 12 cos. 7 + sin. 12 sin. 7=cos. ( 12 7) is a minimum. 
 
 i 
 
 To satisfy the condition of a minimum, the angle p$p* 
 must therefore be increased until it exceeds 180 by that 
 
 angle 7, whose tangent is represented by p 3 p " . To 
 
 Jr 2 -j- Jr 3 cos. 23 
 
 determine the actual direction of Pj produce then pfj to <?, 
 make the angle qCr equal to 7 ; and draw Cm perpendicular 
 
GREATEST ECONOMY OF POWER. 181 
 
 to O, and equal to the given perpendicular distance a x of 
 the direction of P, from the centre of the axis. If raP* be 
 then drawn through the point m parallel to O, it will be in 
 the required direction of P x ; so that being applied in this 
 direction, the moving pressure P x will work the machine with 
 a greater economy of power than when applied in any other 
 direction round the axis. 
 
 It is evident that since the value of the angle 1U or pf/p l9 
 which signifies the condition of the greatest economy of 
 power, or of the least resistance, is essentially greater than 
 two right angles, Pj and P 2 must, TO SATISFY THAT CONDITION, 
 
 BOTH BE APPLIED ON THE SAME SIDE OF THE AXIS. It is then 
 
 a condition necessary to the most economical working of any 
 machine (whatever may be its weight) which is moveable about 
 a cylindrical axis under two given pressures, that THE MOV- 
 ING PRESSURE SHOULD BE APPLIED ON THAT SIDE OF THE AXIS 
 OF THE MACHINE ON WHICH THE RESISTANCE IS OVERCOME, OR 
 
 THE WORK DONE. It is a further condition of the greatest 
 economy of power in such a machine, that the direction in 
 which the moving pressure is applied should he inclined to 
 the vertical at an angle 12 , whose tangent is determined hy 
 equation (170.). 
 
 When i a3 =0, or when the work is done in a vertical 
 direction, tan. y 0; therefore ,=*, whence it follows that 
 the moving power also must in this case be applied in a ver- 
 tical direction and on the same side of the axis as the work. 
 
 "When ' 23 =o or when the work is done horizontally, tan. 
 P 
 
 The moving power must, therefore, in this case, be applied 
 on the same side of the axis as the work, and at an incli- 
 nation to the horizon whose tangent equals the fraction 
 obtained by dividing the weight of the machine by the 
 working pressure. 
 
 3* 
 Since the angle < 12 is greater than * and less than -^ 
 
 cos. i, a is negative ; and, for a like reason, cos. 13 is also in 
 certain cases negative. Whence it is apparent that the 
 function (169.) admits of a minimum value under certain 
 
182 
 
 THE PULLET. 
 
 conditions, not only in respect to the inclination of the 
 moving pressure, but in respect to the distance a l of its 
 direction from the centre of the axis. If we suppose the 
 space Sj through which the power acts whilst the given 
 amount of work U 2 is done to be given, and substitute in 
 that function for the product S 2 ^ its value S^, and then 
 assume the differential of the function in respect to a t to 
 vanish, we shall obtain by reduction 
 
 U 2 a 
 
 , cos. . 13 +P 3 2 S 1 2 
 
 U 2 2 cos. ^ 
 
 (in.) 
 
 If we proceed in like manner assuming the space S 2 instead 
 of Sj to be constant and substituting in the function (169.) 
 for S # a its value S a a,, we shall obtain by reduction 
 
 = _ _ _ 
 
 P 2 COS. I 12 + P 3 COS. 13 . 
 
 It is easily seen that if, when the values of i ia and 23 deter- 
 mined by equation (170.) are substituted in these equations, 
 the resulting values of a : are positive, they correspond in 
 the two cases to minimum values of the function (169.), and 
 determine completely the conditions of the greatest economy 
 of power in the machine, in respect to the direction of the 
 moving pressure applied to it. 
 
 170. THE PULLEY, WHEN THE TENSIONS UPON THE TWO 
 EXTREMITIES OF THE COED HAVE NOT VERTICAL DIRECTIONS. 
 
 L 
 
 In the case in which the two parts of the 
 string which pass over a pulley are not 
 parallel to one another, the relations estab- 
 lished in Article 158. no longer obtain ; 
 and we must have recourse to equation 
 (167.) to establish a relation between the 
 tensions upon them in the state bordering 
 upon motion. Calling "W the weight of 
 the pulley, a its radius, and observing 
 that the effect of the rigidity of the cord, 
 in increasing the tension P 1? is the same 
 as though it caused the tension P 2 to be- 
 
 "F i T) 
 
 come P 2 (l + I + (Art. 142.), we have 
 \ a j a 
 
THE PULLET. 183 
 
 D DL MW 
 
 -+-rpsiri.9 + -w .p sin. 9 ; 
 
 Gu Ob w it 
 
 or. 
 
 where L represents the chord AB of the arc embraced by 
 the string, and M=& 2 (cos. i ls + cos. i 23 ), 13 and 23 represent- 
 ing the inclinations of P 1 and P 2 to the vertical: which 
 inclinations are measured by the angles P^Pj and P 2 FP 3 , 
 or their supplements, according as the corresponding pres- 
 sures Pj and P 2 act downwards, as shown in the figure, or 
 upwards (see note to Article 165.) ; so that if both these 
 pressures act upwards: then the cosines of both the angles 
 become negative, and the value of M becomes negative ; 
 whilst if one only acts upwards, then one term only of the 
 value of M becomes negative. 
 
 Substituting this value for M, observing that L=20 cos. , 
 where 2 represents the inclination of the two parts of the 
 cord to one another (so that 2i= 13 -H 23 ), and omitting terms 
 which involve products of two of the exceedingly small 
 
 D E o . 
 
 quantities , , and -sin. 9 we have 
 a a' a 
 
 E 2 P D 
 
 cos.* 
 
 Wp(cos. i ia 4-cos. < 23 ) sin 
 20 cos. i 
 
184 
 
 THE PULLET. 
 
 which last equation is the modulus to the pulley, when the 
 two parts of the string are inclined to the vertical and to 
 one another. 
 
 171. If both the strings be inclined at equal angles to the 
 vertical, on opposite sides of it ; or if i 13 =u as =, so that cos. 
 i ]8 + cos. ' 23 =2 cos. , then equations (172.) and (173.) become 
 
 P,= 
 
 sn. 
 
 +i,,. 9 . . . (174), 
 
 U = |l+-+- p cos. sin. <e I U,+ |-+ P sin. 9 [s, . . . (175.) 
 
 / Ui Cti ' ) { (M Cb J 
 
 172. If both parts of the cord passing over a pulley be in 
 the same horizontal straight line, so that the 
 pulley sustains no pressure resulting from the 
 tension upon the cord, but only bears its 
 
 weight) then < ^, and the term involving 
 
 cos. * in each of the above equations vanishes. It is, how- 
 ever, to be observed that the weight bearing upon the axis 
 of the pulley is in this case the weight of the pulley 
 increased by the weight of cord which it is made to support. 
 So that if the length of cord supported by the pulley be 
 represented by s, and the weight of each foot of cord by ^, 
 then is the weight sustained by the axis of the pulley repre- 
 sented by "W+fxs. Substituting this value for "W in equa- 
 tion (175.), and assuming cos. <=0, we have 
 
 P sin. 
 
 (176.) 
 
 173. Let us now suppose that there are n equal pulleys 
 
 sustaining each the same length 
 s of cord, and let U w represent 
 the work yielded by the rope 
 (through the space S,) after it 
 has passed over the n**, or last 
 pulley of the system, U, being 
 that done upon it before it 
 passes over the first pulley ; 
 then by Art. 163., equations 
 
THE PULLEY. 185 
 
 152. 154. and 176., neglecting terms involving powers of 
 
 "IP TV * 
 
 . , - sin. <p above the first, and observing that #.=#,= 
 a a a 
 
 TT "R* ~\ ( 
 
 &C.=1+-, a i: =a a :=&c.= , ^=^=&c. = -\ 
 
 Ctr Of Gi \ 
 
 p sin. <p > , we have 
 
 Representing the whole weight of the cord sustained by the 
 pulleys by w, and observing that pns^=w, we have 
 
 ) ? sin. 9 | S, . . . (ITT.) 
 
 In the above equations it has been supposed, that although 
 the direction of the rope on either side of each pulley is so 
 nearly horizontal that cos. < may be considered = 0, yet that 
 it does so far lend itself over each pulley as to cause the 
 surface of the rope to adapt itself to the circumference of 
 the pulley, and thereby to produce the whole of that resist- 
 ance which is due to the rigidity of the cord. If the tension 
 were so great as to cause the cord to rest upon the pulley 
 only as a rigid rod or bar would, then must we assume E=0 
 and D in the preceding equations. 
 
 174. If one part of the cord passing over a pulley have a 
 horizontal, and the other a vertical direction, as, for instance, 
 when it passes into the shaft of a mine over the sheaf or 
 wheel which overhangs its mouth ; then one of the 
 
 angles 13 or 23 (equation 173.) becomes -, and the 
 
 other or or, according as the tension on the ver- 
 tical cord is downwards or upwards, so that cos. 
 13 + cos. a8 =l, the sign being taken according 
 as the tension upon the vertical cord is downwards 
 or upwards. Moreover, in this case (Art. 169.) 
 
 =- and cos. - ; therefore (equation 173.) 
 4 4/2 
 
 (W8), 
 
186 
 
 THE PULLET. 
 
 174. The modulus of a system of any number of pulleys, over 
 one of which the rope passes vertically, and over the rest 
 horizontally. 
 
 Let Uj repre- 
 sent the work 
 done upon the 
 rope through 
 the space S x be- 
 fore it passes 
 horizontally 
 over the first 
 pulley of the 
 system, and let 
 it pass horizon- 
 tally over n such pulleys; and then, after having passed 
 over another pulley of different dimensions, let it take a 
 vertical direction, descending, for instance, into a shaft. Let 
 U 2 be the work yielded by it through the space Sj immedi- 
 ately that it has assumed this vertical direction : also let u^ 
 represent the work done upon it in the horizontal direction 
 immediately before it passed over this last pulley of the 
 system. Then, by equation (179.), 
 
 - + 
 
 sm. 9 
 
 Also, by equation (177.) representing the radius of each of 
 the pulleys which carry the rope horizontally by #, the radius 
 of its axis by p,, and its weight by W 1? and obse 
 
 ! is here the power and u : the work, we have 
 
 observing that 
 
 
 sn. 
 
 Eliminating the value of u^ between these equations, and 
 
 neglecting powers above the first in , &c., we have 
 
 a 
 
THE PIVOT. 187 
 
 . . (180.) 
 
 . 
 
 ) 
 
 175. If the strings be parallel, and their common 
 inclination to the vertical be represented by , so 
 that i ia = i M = i; then, since in this case L=2#, we 
 have (equation 172.), neglecting terms of more than 
 
 one dimension in and^., 
 a a 
 
 in which equation * is to be taken greater or less than -, and 
 
 2i 
 
 therefore the sign of cos. is to be taken (as before explained) 
 positively or negatively, according as the tensions on the 
 cords act downwards or upwards. If the tensions are verti- 
 cal, =0 or *, according as they act upwards or downwards, 
 so that cos. i 1. The above equations agree in this case, 
 as they ought with equations (131.) and (132.). If the par- 
 
 allel tensions are horizontal, then i=-, and the terms inyolr- 
 ing cos. in the above equations vanish. 
 
 176. THE FKICTION OF A PIVOT. 
 
 When an axis rests upon its bearings, 
 not by its convex circumference, but by 
 its extremity, as shown in the accompany- 
 ing figure, it is called a pivot. Let W 
 represent the pressure borne by such a 
 pivot supposed to act in a direction per- 
 pendicular to its surface, and to press 
 
188 THE PIVOT. 
 
 equally upon every part of it ; also let p x represent the 
 radius of the pivot ; then will *?? represent the area of the 
 
 W 
 
 pivot, and - the pressure sustained by each unit of that 
 
 "?! 
 area. And if f represent the co-efficient of friction (Art. 
 
 133.), Z will represent the force which must be applied 
 
 parallel to the surface of the pivot to overcome 
 the friction of each such unit. JSTow let the dot- 
 ted lines in the accompanying figure represent 
 an exceedingly narrow ring of the area of the pivot, and let 
 p and p+Ap represent the extreme radii of this ring; then 
 will its area be represented by *(p -f- Ap) 2 tfp a , or by it j2p(Ap) -f 
 (Ap) a j , or, since Ap is exceedingly small as compared with p, 
 by 2tfpAp. Now the friction upon each unit of this area is 
 
 W/ 
 
 represented by ^ ; therefore the whole friction upon the 
 
 *fc 
 
 ring is represented by ^- . 2tfpAp ) or by ^-p^p, and the 
 
 ^Pi 3 Pi 2 
 
 moment of that friction about the centre of the pivot by 
 
 at . p a A, and the sums of the moments of the frictions of 
 
 Pi 2 
 all such rings composing the whole area of the pivot by 
 
 V 2W , 2W , 2W X , 
 
 2 f- . P 2 A P , or by ^-2p a A P , or by f- I P 2 4, or by 
 
 Pi Pi Pi % 
 
 i 3 , or by |W/ Pl ....... (183.); 
 
 whence it appears that the friction of the pivot produces the 
 same effect to oppose the revolution of the mass which rests 
 upon it, as though the whole pressure which it sustains were 
 collected over a point distant ~by two-thirds of its radius from 
 its centre. 
 
 If 6 represent the angle through which the pivot is made 
 to revolve, then $-p/ will represent the space described by 
 the point last spoken of ; so that the work expended upon 
 the resistance Wf acting there, would be represented by 
 "fWpi/^ which therefore represents the work expended upon 
 the friction of the pivot, whilst it revolves through the angle 
 
189 
 
 6 ; so that the work expended on each complete revolution 
 of the pivot is represented by 
 
 ITT. If the pivot be hollow, or its surface be an annular 
 instead of a continuous circular area, then 
 representing its internal radius by p a , and 
 observing that its area is represented by 
 <7r (pi a P 3 a )> an( i therefore the pressure upon 
 
 each unit of it by . a _ a , and the fric- 
 
 tion of each such unit by . a a , we obtain, as before, 
 
 *\Pi ~Pa ) 
 
 for the friction of each elementary annulus the expression 
 
 r z-. pAp t and for the sum of the moments of the frictions 
 
 Pi P 2 
 
 of all the elements of the pivot s- / ^ 
 
 or 
 
 Let r represent the mean radius of the pivot, i. e. let 
 7 l =i(p ] + p a ) ; and let I represent one half the breadth of the 
 ring, i. e. let l=^ l 2 ); therefore p l =r+l and vj=.rl. 
 These values of p t and p 2 being substituted in the above for- 
 mula, it becomes 
 
 or 
 
 
 
 (185.); 
 
 whence it follows that the friction of an annular pivot pro- 
 duces the same effect as though the whole pressure were col- 
 
 lected over a point in it distant ly T* | l+i(-j | from it* 
 
 centre, where r represent its mean radius and I one half its 
 fyreadth. From this it may be shown, as before, that the 
 
190 AXES. 
 
 whole work expended upon each complete revolution of the 
 annular pivot is represented by the formula, 
 
 1Y8. To DETERMINE THE MODULUS OF A SYSTEM OF TWO PRES- 
 SURES APPLIED TO A BODY MOVEABLE ABOUT A FIXED AXIS, 
 WHEN THE POINT OF APPLICATION OF ONE OF THESE PRES- 
 SURES IS MADE TO REVOLVE WITH THE BODY, THE PERPEN- 
 DICULAR DISTANCE OF ITS DIRECTION FROM THE CENTRE RE- 
 MAINING CONSTANTLY THE SAME. 
 
 Let the pressures Pj and P 2 , instead of retaining constantly 
 T, (as we have hitherto supposed them to do) 
 the same relative positions, be now conceived 
 ^ continually to alter their relative positions by 
 " the revolution of the point of application of 
 P! with the body, that pressure nevertheless 
 retaining constantly the same perpendicular 
 distance a from the centre of the axis, whilst 
 the direction of P 2 and its amount remain 
 constantly the same. 
 
 It is evident that as the point A 1 thus continually alters its 
 position, the distance AjAjj or L will continually change, so 
 that the value of P, (equation 158.) will continually change. 
 ]N"ow the work done under this variable pressure during one 
 revolution of P a is represented (Art. 51.) by the formula 
 
 ^, if 6 represent the angle A X CA described at 
 
 o 
 
 any time about C, by the perpendicular QA,, and therefore 
 aj, the space S described in the same time by the point of 
 application A 1 of P l (see Art. 62.). 
 
 Substituting, therefore, for ~P 1 its value from equation 
 (158.), we have 
 
 27T 
 
AXES. 
 
 Let now P s be assumed a constant quantity ; 
 
 27T 27T 
 
 Now L=A 1 A^= {a'+Sa^ cos. 
 
 2;r 2;r 
 
 o 
 
 27T 
 
 neglecting powers of (-_|_^) ' above the first, since in all 
 ' 
 
 cases its value is less than unity. Integrating this quantity 
 between the limits and 2* the second term disappears, so 
 that 
 
 1 f\, 1 1 IV* 
 J Ldd = - 3 +- J 2* nearly; 
 
 .-.PA . 
 
 
 
 27T 
 
 o 
 
 since 2-au, is the space through which the point of applica- 
 tion of the constant pressure r a is made to move in each re- 
 
AXES. 
 
 volution. Therefore by equation (187), in the case in which 
 P a is constant, 
 
 U 1 =U 9 j 1+ (i+^ a )*P Bin. 9 } ..... (188). 
 
 179. If the pressure P, be supplied by the tension of a 
 rope winding upon a drum whose radius is # (as in the cap- 
 Btan), then is the effect of the rigidity of the rope (Art. 142.) 
 the same as though P a were increased by it so as to become 
 
 Now, assuming P a to be constant, and observing that 
 U a =2tfP a a a , we have, by equation (1ST), 
 
 Substituting in this equation the above value for P a , 
 
 ( \ aj 2 ) 1 <V^ *^ 
 
 Performing the actual multiplication of these factors, ob- 
 serving that is exceedingly small, and omitting the term 
 
 involving the product of this quantity and , we have 
 
 o 
 Whence performing the integration as before, we obtain 
 
 ^ + lVp sin. 9 \ 
 a l a l / 
 
 If this equation be multiplied by n, and if instead of U, and 
 U a representing the work done during one complete revolu- 
 tion, they be taken to represent the work done through n 
 such revolutions, then 
 
AXES. 193 
 
 ) (189), 
 
 which is the MODULUS. 
 
 180. If the quantity (~+ 1 be not so small that terms 
 
 \ C* a U/ l I 
 
 of the binomial expansion involving powers of that quan- 
 tity above the first may be neglected, the value of the 
 
 ^27T 
 
 definite integral /Ld& may be determined as follows .:-* 
 o 
 
 /27T /27T 
 
 J (a* + Sauces J + a?y<ttJ {(a, + ay i %a*0 
 
 
 
 27T 27T 
 
 E l (*), where 
 
 represents the complete elliptic function of the second order, 
 whose modulus is &* The value of this function is given 
 for all values of ^ in a table which will be found at the end 
 of this work. 
 
 Substituting in equation (187), 
 
 . E,(t)t . P,=TT.+ 
 
 * See Encyc. Met. art. DBF. INT. theorem 2. 
 
 f An approximate value of EI(&) is given when & is small by the formula 
 
 =(] +K- 1 ), where K=T. ( See Encyc. Met. art. DEP. INT. equation 
 
THE CAPSTAN. 
 
 . . . . (190). 
 
 THE CAPSTAN. 
 
 181. The capstan, as used on shipboard, is represented in 
 
 the accompanying figure. 
 It consists of a solid timber 
 CO, pierced through the 
 greater part of its length by 
 an aperture AD, which 
 receives the upper portion 
 of a solid shaft AB of great 
 strength, whose lower ex- 
 tremity is prolonged, and 
 strongly fixed into the tim- 
 ber framing of the ship. The piece (DC, into the upper por- 
 tion of which are fitted the moveable 
 arms of the capstan, turns upon the shaft 
 AB, resting its weight upon the crown of 
 the shaft, coiling the cable round its cen- 
 tral portion CC, and sustaining the ten- 
 sion of the cable by the lateral resistance 
 of the shaft. Thus the capstan combines 
 the resistances of the pivot and the axis, 
 so that the whole resistance to its motion 
 is equal to the sum of the resistances due separately to the 
 axis and the pivot, and the whole work expended in turning 
 it equal to the whole work which would be expended in 
 turning it upon its pivot were there no tension of the cable 
 upon it, added to the whole work necessary to turn it upon 
 its axis under the tension of the cable were there no friction 
 of the pivot. Now, if U 1 represent the work to be done 
 upon the cable in n complete revolutions, the work which 
 must be clone upon the capstan to yield this work upon (he 
 cable is represented (equation 189.) by 
 
THE CAPSTAN. 195 
 
 where a i represents the length of the arm, and a 2 the radius 
 of that portion of the capstan on which the cable is winding. 
 Moreover (Art. 175.), the work due to the friction of the 
 
 4 
 pivot in n complete revolutions is represented by o^Pi/'W. 
 
 On the whole, therefore, it appears that the work Uj 
 expended upon n complete revolutions of the capstan is 
 represented by the formula 
 
 which is the MODULUS of the capstan. 
 
 A single pressure P x applied to a single arm has been 
 supposed to give motion to the capstan ; in reality, a num- 
 ber of such pressures are applied to its different arms when 
 it is used to raise the anchor of a ship. These pressures, 
 however, have in all cases, except in one particular case 
 about to be described, a single resultant. It is that single 
 resultant which is to be considered as represented by I*, 
 and the distance of its point of application from the axis 
 by flj, when more than one pressure is applied to move the 
 capstan. 
 
 The particular case spoken of above, in which the pres- 
 sures applied to move the capstan have no resultant, or can- 
 not be replaced by any single pressure, is that in which 
 they may be divided into two sets of pressure, each set hav- 
 ing a resultant, and in which these two resultants are equal, 
 act in opposite directions, on opposite sides of the centre, 
 perpendicular to the same straight line passing through the 
 centre, and at equal distances from it.* 
 
 Suppose that they may be thus compounded into the 
 equal pressures B^ and K 2 , and let them be replaced by 
 these. The capstan will then be acted upon by four pres- 
 sures, the tension P 2 of the cable, the resistance K of the 
 shaft or axis, and the pressures K x and K a . Now these pres- 
 sures are in equilibrium. If moved, therefore, parallel to 
 their present directions, so as to be applied to a single point, 
 
 * Two equal pressures thus placed constitute a STATICAL COUPLE. The pro- 
 perties of such couples have been fully discussed by M. Poinsot, and by Mr. 
 Pritchard in his Treatise on Statical Couples ; some account of them will be 
 found in the Appendix to this work. 
 
196 THE CAPSTAN. 
 
 they would be in equilibrium about that point (Art. 8.), 
 But when so removed, Rj and R a will act in the same 
 straight line and in opposite directions. Moreover, they 
 are equal to one another ; R x and R 2 will therefore sepa- 
 rately be in equilibrium with one another when applied to that 
 point ; and therefore P 2 and R will separately be in equili- 
 brium ; whence it follows, that R is equal to P 2 or the whole 
 pressure upon the axis, equal in this case to the whole tension 
 P 2 upon the cable. So that the friction of the axis is repre- 
 sented in every position of the capstan by P 2 tan. 9 (tan. 9 
 being equal to the co-efficient of friction (Art. 138.)), and 
 the work expended on the friction of the axis, whilst the 
 capstan revolves through the angle d by P 2 pd tan. 9, or by 
 
 I J tan. 9, or by U 2 | j tan. 9 ; so that, on the whole, 
 
 introducing the correction for rigidity and for the friction of 
 the pivot, the modulus (equation 191) becomes in this case 
 
 j D+|p,/W } . . . . (192). 
 
 This is manifestly the least possible value of the modulus, 
 being very nearly that given (equation 191) by the value 
 infinity of a r * 
 
 Tims, then, it appears generally from equation (191), that 
 the loss by friction is less as a l is greater, or as P x is applied 
 at a greater distance from the axis ; but that it is least of all 
 when the pressures are so distributed round the capstan as 
 to be reducible to a COUPLE, that case corresponding to the 
 value infinity of a t . This case, in which the moving pres- 
 sures upon the capstan are reducible to a couple, manifestly 
 occurs when they are arranged round it in any number of 
 pairs, the two pressures of each pair being equal to one an- 
 other, acting on opposite sides of the centre, and perpendi- 
 cular to the same line passing through it. This symmetrical 
 distribution of the pressures about the axis of the capstan is 
 therefore the most favourable to the working of it, as well 
 as to the stability of the shaft which sustains the pressure 
 upon it. 
 
 * being exceedingly small, tan. # is very nearly equal to sin. <f>> 
 
AXES. 197 
 
 182. THE MODULUS OF A SYSTEM OF THREE PRESSURES APPLIED 
 TO A BODY MOVEABLE ABOUT A CYLINDRICAL AXIS, TWO OF 
 
 THESE PRESSURES BEING GIVEN IN DIRECTION AND PARAL- 
 LEL TO ONE ANOTHER, AND THE DIRECTION OF THE THIRD 
 CONTINUALLY REVOLVING ABOUT THE AXIS AT THE SAME 
 PERPENDICULAR DISTANCE FROM IT. 
 
 Let P 3 and P 3 represent the parallel pressures of the sys- 
 V^ tern, and 'P l the revolving pressure. 
 
 /f'\ From the centre of the axis C, let fall 
 /'/ v the perpendiculars CA^ CA 2 , CA 3 upon 
 H/d the directions of the pressures, and let 
 
 L-U. & represent the inclination of CA X to 
 
 C A 3 -at any period of the revolution of 
 t*. ai H ' PI- Let P x be the preponderating 
 
 pressure, and let P 2 act to turn the 
 
 system in the same direction as P 1? and P 3 in the opposite 
 direction ; also let R represent the resultant of P 2 and P 3 , 
 and r the perpendicular distance CA of its direction from C. 
 Suppose the pressures P 2 and P 8 to be replaced by R ; the 
 conditions of the equilibrium of P t throughout its revolu- 
 tion, and therefore the work of P x will remain unaltered by 
 this change, and the system will now be a system of two 
 pressures P, and R instead of three ; of which pressures R 
 is given in direction. The modulus of this system is there- 
 fore represented (equation 187) by the formula 
 
 (193); 
 
 where U r represents the work of R, and L represents the dis- 
 tance AA, between the feet of the perpendiculars r and a l9 
 so that Va^Zay cosJ-^-r t =(ar cos. d) 2 +7* a sin. 2 d ; 
 
 /. R 2 L a =(R^-R^ cos. 
 Now, R= 
 
 [Now if 'the relations of a, to # 3 are such that 
 | (P 3 + P a X_(P 3 a 3 -P 2 <) cos. 6 | 2 >(P 3 3 -P 2 <)sin. 9 4 
 then the value of R 2 L 2 will be represented by the sum of the 
 
198 AXES. 
 
 squares of two quantities the first of which is greater than 
 the second. ED.] Therefore, extracting the square root by 
 Poncelet's theorem, (see Appendix B.) 
 
 RL=a}(P I + P 1 )a l _(P 1 l _p s 9 ) cos. &} +/3(P 3 a 9 -P 2 A 2 ) sin. 6 
 very nearly ; or, 
 
 -/3smJ). . . .(194). 
 
 \ - 
 
 cos. - sn. 
 o 
 e 6 
 
 
 
 (a cos. d /3 sin. fy#. . . . (195). 
 
 If P 2 and P 3 be constant, the integral in the second member 
 of this equation becomes (~P a a 3 P 2 & 2 ) ( sin. 6 + /3 cos. 0) ; 
 
 , . . -D P.a.d P,a,d TJ 3 U 3 
 
 whence observing that P s a 3 P 2 <^ 2 = 8 3 ^ 2-^-= L.- ?; 
 
 also, that IJ r ==dEr==^P t fl^-r-*P 1 a t =IJ, TJ,, and substituting 
 in equation (193), we have 
 
 U.^TJ.-U^+p sin. 9 I (-+-') - 
 ( \a 9 a z I 
 
 (52\( s in. 4+/S cos. 6) \ . . . . (196) ; 
 
 \ df I 
 
 for complete revolution making 0=2*, we have 
 
 n-u.-u. + ,-. 
 
 reducing, 
 
 which is the modulus of the system where a and j3 are to be 
 de'ermined, as in Note B, (Appendix.) 
 
THE CHINESE CAPSTAN. 199 
 
 183. If the pressure P 3 be supplied by the tension of a cord 
 which winds upon a cylinder or drum at the point A 3 , then 
 allowance must be made for the rigidity of the cord, and a 
 correction introduced into the preceding equation for that 
 purpose. To make this correction let it be observed 
 (Art. 142.) that the effect of the rigidity ^ of the cord at A 8 is 
 the same as though it increased the tension there from 
 
 D 
 
 or (multiplying both sides of this inequality by a,, and inte- 
 grating in respect to d,) as though it increased 
 
 27T 27T 27T 
 
 dA to l-f- 
 
 or,U 3 to(l+-)U 
 
 \ & z i 
 
 Thus the effect of the rigidity of the rope to which P 3 is ap- 
 plied upon the work U a of that force is to increase it to 
 
 (l + ) U 3 + 2*-D. Substituting this value for U 3 in equa- 
 
 tion (197), and neglecting terms which involve products of 
 
 ,, ,. .. ,, .... E P sin. <p p sin. 9 __j pv 
 
 the exceedingly small quantities,^ - ,- -- ,andlJ, 
 
 we have 
 
 To determine the modulus for n revolutions we must sub- 
 stitute in this expression w* for if. 
 
 THE CHINESE CAPSTAN. 
 184. This capstan is represented in the accompanying 
 
200 THE CHINESE CAPSTAN. 
 
 figure under an exceed- 
 ingly portable and con- 
 venientform.* The axle 
 or drum of the capstan 
 is composed of two parts 
 of different diameters. 
 One extremity of the 
 cord is coiled upon one 
 of these, and the other, in an oposite direction, upon the 
 other ; so that when the axle is turned, and the cord is 
 wound upon one of these two parts of the drum, it is, at the 
 same time, wound off the other, and the intervening cord is 
 shortened or lengthened, at each revolution, by as much as 
 the circumference of the one cylinder exceeds that of the 
 other. In thus passing from one part of the drum to the 
 other, the cord is made to pass round a moveable pulley 
 which sustains the pressure to be overcome. 
 
 To determine the modulus of this machine, let u^ and u 3 
 represent the work done upon the two parts of the cord 
 respectively, whilst the work U^ is done at the moving point 
 of the machine, and U 2 yielded at its working point. 
 
 Then, since in this case we have a body moveable about a 
 cylindrical axis, and acted upon by three pressures, two of 
 which are parallel and constant, viz. the tensions of the two 
 parts of the cord ; and the point of application of the third 
 is made to revolve about the axis, remaining always at the 
 same perpendicular distance from it ; it follows (by equation 
 198), that, for n revolutions of the axis, 
 
 (199); 
 where 
 
 in. 9 (---) Land 
 \a a %najt] \ 
 
 # 2 and a s representing the radii of the two parts of the drum, 
 #! the constant distance at which the power is applied, and p 
 the radius of the axis. 
 
 * A figure of the capstan with a double axle was seen by Dr. 0. Gregory 
 among some Chinese drawings more than a century old. It appears to have 
 been invented under the particular form shown in the above figure by Mr. G. 
 Eckhardt and by Mr. M'Lean of Philadelphia. (See Professor Robinson's Mech. 
 Phil. vol. ii. p. 255.) 
 
THE CHINESE CAPSTAN. 
 
 201 
 
 Also, since the two parts of the cord pass over a pulley, 
 
 and the pulley is made 
 to revolve under the ten- 
 sions of the two parts of 
 the cord, ? 3 being the 
 work of that tension 
 which preponderates, we 
 have (by equation 181), 
 if S represents the length 
 of cord which passes 
 over the pulley, 
 
 where 
 
 E 
 
 a 
 
 and 
 
 7 
 
 D 
 
 . ) 
 m '* ' 
 
 
 
 a representing the radius of the pulley, Pl the 
 axis, W its weight, and the inclination of the direction of 
 the tensions of the two parts of the cord to the vertical, the 
 axis of the pulley being supposed horizontal, and the two 
 
 parts of the cord parallel. Now t 3 = U * , L =-^ . Sub- 
 
 2n*0. 2wr0 a 
 stituting these values, and multiplying by 2n^a^ we have 
 
 '' , 
 
 te-v- . - 
 
 (200). 
 
 Since the tensions 4 and t s of the two parts of the cord, 
 and the pressure P 2 overcome by the machine, are pressures 
 applied to the pulley and in equilibrium, and that the points 
 of application of ? and P 2 are made to move in directions 
 opposite to those in which those pressures act, whilst the 
 point of application of t 5 is made to move in the same direc- 
 tion ; therefore (Art. 59.), 
 
 Eliminating u z and 
 (200), we have 
 
 between this equation and equation 
 
202 THE HORSE CAPSTAN. 
 
 Substituting these values in equation (199), and reducing, 
 
 a* \ 
 Substituting their values for A, A,, B, B 1? neglecting terms 
 
 "T^ 
 
 involving more than one dimension of - , , &c. and 
 reducing, we obtain for the MODULUS of the machine, 
 
 E , . (2a / 0v ft ) 
 
 hpsm. <p \ (1 - V 
 
 ! <h ( a 3 \ aj %rw.f ) 
 
 ., a E 2p. . 
 
 1 -| (--^ sin. 9 
 
 -D 
 
 2p 1 sin. 
 
 .. (201). 
 
 From which expression it is apparent that when the radii # a 
 and $ 3 of the double axle are nearly equal, a great sacrifice 
 of power is made, in the use of this machine, by reason of 
 the rigidity of the cord. 
 
 THE HOUSE CAPSTAN, OR THE WHIM GIN. 
 
 185. The whim is a form of the capstan, used in the first 
 operations of mining, for raising materials from the shaft and 
 levels by the power of horses, before the quantity excavated 
 is so great as to require the application of steam power, or 
 before the valuable produce of the mine is sufficient to give 
 a return upon the expenditure of capital necessary to the 
 erection of a steam engine. The construction of this machine 
 will be sufficiently understood from the accompanying figure. 
 It will bo observed that there are two ropes wound upon the 
 drum in opposite directions, and which traverse the space 
 
THE HORSE CAPSTAN. 203 
 
 between the capstan 
 and the mouth of the 
 shaft. One of these 
 carries at its extrem- 
 ity the descending 
 (empty) bucket, and is 
 continually in the act 
 of winding off the drum of the capstan as it revolves ; whilst 
 the other, from whose extremity is suspended the ascending 
 (loaded) bucket, continually winds on the drum. The pres- 
 sure exerted by the horses is that necessary to overcome the 
 friction of the different bearings, and the other prejudicial 
 resistances, and to balance the difference between the weight 
 of the ascending load, bucket, and rope, and that of the 
 descending bucket and rope. The rope, in passing from the 
 capstan to the shaft, traverses (sometimes for a considerable 
 distance) a series of sheaves or pulleys, such as those shown 
 in the accompanying figure. 
 
 Let now # 2 represent the radius of the drum on which the 
 rope is made to wind, and n the number of resolutions 
 which it must make to wind up the whole cord ; also let f* 
 represent the weight of each foot of cord, and d the angle 
 which the capstan has described between the time when the 
 ascending bucket has attained any given position in the 
 shaft and that when it left the bottom ; then does a repre- 
 sent the length of the ascending rope wound on the drum, 
 and therefore of the descending rope wound off it. Also, 
 let "W represent the whole weight of the rope ; then does 
 W pajb represent the weight of the ascending rope, and 
 f*0 a d that of the descending rope, both of which hang sus- 
 pended in the shaft. Let P 2 represent the load raised at 
 each lift in the bucket, and w the weight of the bucket ; 
 then is the tension upon the ascending rope at the mouth of 
 the shaft represented by W M-a^+x^+w, and that upon 
 the descending rope by pa+w. 
 
 Let, moreover, p 3 and j!? 2 represent the tensions upon these 
 ropes after they have passed from the mouth of the shaft, 
 over the intervening pulleys, to the circumference of the 
 capstan. 
 
 Now, since the tension upon the ascending rope, which is 
 "V\T (x& 2 d-fP a -{-w at the mouth of the shaft, is increased to 
 p s at the capstan, and that the tension upon the descending 
 rope, which is p^ at the capstan, is increased to pa+w at 
 the mouth of the shaft, if we represent by (1 -f- A) and B the 
 constants which enter into equation 180 (Art. 174.), we have. 
 
204 THE HORSE CAPSTAN. 
 
 by that equation (observing that U 1 =P 1 S 1 and U a =P 2 Sj so 
 that S, disappears from both sides of it), 
 
 ^ 3 =(l+A)(W+P 2 +w-^) + B, . . . .(202), 
 and f*a 1 +w=(l+A)j> 1 +B .... (203). 
 
 Multiplying the former of the above equations by 
 adding them, transposing, dividing by (1 -h A), and neglect- 
 ing terms of more than one dimension in A and B, 
 
 Now U r in equation (193) represents the work of J;he 
 resultant of p z and p z during n revolutions of the capstan, it 
 therefore equals the difference between the work of p 3 and 
 that of p z (see p. 198). 
 
 27T 2tt7T 
 
 9 p,) d6 
 
 2717T 
 
 o 
 { (1 + A) (W + P 2 ) 4- 2 Aw + 2B } (2 wa.) f 
 
 observing that 2nrra 2 =:S 2 , and that P 2 S 2 =U 2 . 
 
 Now, let it be observed that the pressures applied to the 
 capstan are three in number ; two of them, p s and p^ being 
 parallel and acting at equal distances a. 2 from its axis ; and 
 the third, P 1? being made to revolve at the constant distance 
 a t from the axis (P, representing the pressure of the horses, 
 or the resultant of the pressures of the horses, if there be 
 more than one, and a^ the distance at which it is applied) ; 
 BO that equation 193 (Art. 182.) obtains in respect to the 
 pressures P., p^ p z ; a 3 being assumed equal to a v 
 
 Substituting^ and^? 8 for P 2 and P 8 in equation (194), 
 
 a i (^ 8 -j-^ a )_ a 9 (p 3 p^) (a cos. p sin. 0) ; 
 
 
 
 (a cos. 6 (3 sin. 6) dO. 
 
THE HORSE CAPSTAN. 
 
 205 
 
 Now, the terms of equation (180), represented in the above 
 equations by A and B, are all of one dimension in the exceed- 
 ingly small quantities D, E, sin. <p. If, therefore, the values 
 of p^ and p z given by these equations be substituted in the 
 
 27T 
 
 value of LSL$/EL<# (equation 193), then all the terms 
 a v t/ 
 
 o 
 
 of that expression which involve the quantities A and B will 
 be at least of two dimensions in D, E, sin. 9, and may be ne- 
 glected. Neglecting, therefore, the values of A and B in 
 equations (202, 203), we obtain 
 
 Similarly, 
 
 f 2 / (Pi jpyaeos. 
 / 
 
 
 
 c$ P, 
 
 representing by S 2 the 
 space described by the 
 load, and by U 3 the 
 useful work done upon 
 it, during n revolu- 
 tions of the capstan. 
 
 2n7T 
 
 - )8 sin. 0)dO = a 
 
 (a cos. 0P sin. 6)de= 
 
 2nir 
 
 a cos. 6 (3 sin. 
 
 
 a cos. ^^ sin. 
 
206 THE HORSE CAPSTAN. 
 
 Now Aa cos. 0(3 sin. 6)dO=(3, and Aa cos. (3 sin. 
 
 ,jp, />,)(* cos. sin. fyZ0==00, 
 
 =/ 3 a 2 ^+^ 2 (W-2fxS 2 ) ; observing that P 2 
 
 2717T 
 
 Substituting this value, and also that of U r (equation 204) 
 in equation (193), and assuming 
 
 and C a = 
 we have 
 
 00 
 
 / /* /* 
 
 * For I 0cos. 0f?0=0sin. / sin. 0d0*=0sin.0 vers. 0; also I sin. 0d0 
 
 00 
 
 
 
 =0 cos. + /cos. 6dO*=6 cos. 0-f-sin. 0. Now, substituting 2w7r for 0, 
 
 
 these integrals become respectively and 2mr. 
 
 * Church's Diff. and Int. Cal. Art. 140. 
 
THE FRICTION OF CORDS. 207 
 
 + i- g _ 
 a, I ' , 
 
 which is the MODULUS of the machine, all the various ele- 
 ments, whence a sacrifice of power may arise in the working 
 of it, being taken into account. 
 
 THE FRICTION OF CORDS. 
 
 186. Let the polygonal line ABC . . . YZ, of an infinite 
 number of sides, be taken to represent 
 , the curved portion of a cord embracing 
 1 any arc of a cylindrical surface (whe- 
 ther circular or not), in a plane per- 
 pendicular to the axis of the cylinder ; 
 also let Aa, B5, Cc,- &c., be normals 
 or perpendiculars to the curve, inclined 
 to one another at equal angles, each 
 represented by Ad. Imagine the surface of the cylinder to 
 be removed between each two of the points A, B, &c., in 
 succession, so that the cord may be supported by a small 
 portion only of the surface remaining at each of those 
 points, whilst in the intermediate space it assumes the direc- 
 tion of a straight line joining them, and does not touch the 
 surface of the^ cylinder. Let P, represent the tension upon 
 the cord before "it has passed over the point A ; T 1 the ten- 
 sion upon it after it has passed over that point, or before it 
 passes over the point B ; T 2 the tension upon it after it has 
 passed over the point B, or before it passes over C ; T 8 that 
 after it has passed over C ; and let P 2 represent the tension 
 upon the cord after it has passed over the nth or last 
 point Z. 
 
 Now, any point B of the cord is held at rest by the ten- 
 sions T t and T 2 upon it at that point, in the directions BC 
 and BA, and by the resistance K of the surface of the Cylin- 
 der there ; and, if we conceive the cord to be there in the 
 state bordering upon motion, then (Art, 138.) the direction 
 of this resistance K is inclined to the perpendicular 5B to 
 the surface of the cylinder at an angle RB& equal to the 
 limiting angle of resistance <p. 
 
208 THE FKICTION OF CORDS. 
 
 Now T,, T 2 , and E are pressures in equilibrium ; there- 
 fore (Art. 14.) 
 
 T,_jaiiL TJBE 
 T a ~sin. T,BE ' 
 
 but T 1 BE=AB5-EBJ=i(*-A^B)-EB5)= ~-~ -9, 
 
 2 A 
 if Ad 
 
 sm.^- 
 
 "T 
 
 COS. \-7T 
 
 . Ad . 
 
 2 sin. sin. 9 
 
 Ad . Ad . 
 
 cos. cos. 9 sin. sin. 9 
 
 or dividing numerator and denominator of the fraction in the 
 
 Ad i 
 
 second member by cos. -~- cos. 9, 
 
 T.-T, 
 
 Ad 
 
 2 tan. tan. 9 
 
 " a 1 tan.- tan. 9 
 
 Suppose now the angles A#5, BZ>C, &c., each of which 
 equals Ad, to be exceedingly small, and therefore the points 
 A, B, C, &c., to be exceedingly near to one another, and 
 exceedingly numerous. By this supposition we shall mani- 
 festly approach exceedingly near to the actual case of an in- 
 finite number of such points and a continuous surface ; and 
 
THE FRICTION OF COEDS. 209 
 
 if we suppose Ad infinitely small, our supposition will coincide 
 with that case. Now, on the supposition that Ad is exceed- 
 
 Ad 
 
 ingly small, tan. -^ . tan. 9 is exceedingly small, and may 
 
 be neglected as compared with unity ; it may therefore be 
 neglected in the denominator of the above fraction. More- 
 
 over Ad being exceedingly small, tan. -^- = 
 
 ,- S 
 
 * Ta = tan. 9 . Ad* ; .-. T 1= T 3 (1 + tan. 9 . Ad). 
 
 Now the number of the points A, B, C, &c. being repre- 
 sented by 7i, and the whole angle AdZ between the extreme* 
 normals at A and Z by d, it follows (Euclid, i. 32.) that 
 
 6n. Ad; therefore Ad =-; 
 7i 
 
 n 
 
 Similarly, P^T, (1 +-tan. 9) 
 
 n 
 
 T,=T, (l+tan. ? ), 
 
 ^P, (1+tan. 9). 
 
 Multiplying these equations together, and striking out fac- 
 tors common to both sides of their product, we haye 
 
 * If we consider the tension T as a function of 6, of which any consecutive 
 values are represented by Tj and T 2 , and their difference or the increment of 
 
 L * rn - AT* 
 
 T by AT, then ^ = tan. 0. A0 ; therefore - . ^ = tan. ; therefore, 
 
 passing to the limit - = tan. 0, and integrating between the limits 
 and 6, observing that at the latter Limit T=P 2 , and that at the former it equals 
 PI, we have log. ( 1 = - 6 tan. 0; therefore P 1 =P^ 6 tan " *. 
 
 \TI/ 
 
 H 
 
210 THE FKICTION OF CORDS. 
 
 ( 
 
 or P 1 =P a \ 1 
 
 \ 
 
 n In 26' \ 
 
 - a -- r -tan.V + &c. f; 
 
 ( !-; 
 
 orP 1 =P a |l+4 tan. 9 + ^ a tan. a <p + 
 
 Now this relation of P x and P 3 obtains however small Ad 
 be taken, or however great n be taken. Let n be taken 
 infinitely great, so that the points A, B, C, &c. may be 
 infinitely numerous and infinitely near to each other. The 
 supposed case thus passes into the actual case of a con- 
 
 i 2 Q 
 tinuous surface, the fractions -, -, -, &c. vanish, and the 
 
 above equation becomes 
 
 6 tan. 9 d a tan. "9 &* tan. '<p ) 
 
 But the quantity within the brackets is the well known ex- 
 pansion (by the exponential theorem) of the function efltaa-d, 
 
 ** (205). 
 
 Since the length of cord S,, which passes over the point 
 A, is the same with that S 2 -which passes over the point Z, 
 it follows that the modulus (Art.* 152.) of such a cylindrical 
 surface considered as a machine, and supposed to \>t fixed 
 and to have a rope pulled and made to slip over it, is 
 
 U^-Q^tan.0 ____ (206). 
 
 It is remarkable that these expressions are wholly inde- 
 pendent of the form and dimensions of the surface sustain- 
 ing the tension of the rope, and that they depend exclu- 
 sively upon the inclination 6 or AeZ of the normals to the 
 points A and Z, where the cord leaves the surface, and upon 
 the co-efficient of friction (tan. 9), of the material of which 
 the rope is composed and the material of which the surface 
 is composed. It matters not, for instance, so far as ihefric- 
 
THE FKICTION OF CORDS. 211 
 
 t-ion of the rope or band is concerned, whether it passes 
 over a large pulley or drum, or a small one, provided the 
 angle subtended by the arc which it embraces is the same, 
 and the materials of the pulley and rope the same. 
 
 In the case in which a cord is made to pass m times round 
 such a surface, G=9m7r ; 
 
 T> p c2m TT tan. & 
 **) x <r r 
 
 And this is true whatever be the form of the surface, so 
 that the pressure necessary to cause a cord to slip when 
 wound completely round such a cylindrical surface a given 
 number of times is the same (and is always represented by 
 this quantity), whatever may be the form or dimension of 
 the surface, provided that its material be the same. It 
 matters not whether it be square, or circular, or elliptical. 
 
 1ST. If P/, P/', P/", &c. represent the pressures which 
 must be applied to one extremity of a rope to cause it to 
 slip when wound once, twice, three times, &c. round any 
 such surface, the same tension P 2 being in each case sup- 
 posed to be applied to the other extremity of it, we have 
 
 So that the pressures P/, P/', P/", &c. are in a geome- 
 trical progression, whose common ratio is e^tan.^ which 
 ratio is always greater than unity. Thus it appears by the 
 experiments of M. Morin (p. 135.), that the co-efficient of 
 friction between hempen rope and oak free from unguent is 
 33, when the rope is wetted. In this case tan. 9= -33 and 
 27rtan. <p=2 x3-U159x -33=2-07345. The common ratio 
 of the progression is therefore in this case e 2 ' 07345 , or it is the 
 number whose hyperbolic logarithm is 2-07345. This num- 
 ber is 7*95 ; so that each additional coil increases the fric- 
 tion nearly eight times. Had the rope been dry, this 
 proportion would have been much greater. If an additional 
 half coil had been supposed continually to be put upon the 
 rope instead of a whole coil, the friction would have been 
 found in the same way to increase in geometrical progres- 
 sion, but the common ratio would in this case have 
 been e 71 "** 11 -? instead of e 2 7rtan.^ j n ^ ne above example the 
 value of this ratio would for each half coil have been 
 2-82. 
 
 The enormous increase of friction which results from 
 
212 THE FRICTION OF COEDS. 
 
 each additional turn of the cord upon a capstan or drum 
 may from these results be understood. 
 
 188. We may, from what has been stated above, readily 
 explain the reason why a knot connecting the two extremi- 
 ties of a cord effectually resists the action of any force 
 tending to separate them. If a wetted cord be wound round 
 Fig. i. mg. 2. Fig. s. a cylinder of oak as 
 
 in fig. 1., and its ex- 
 tremities be acted 
 upon by two forces P 
 and R, it has been 
 shown that P will not 
 overcome R, unless it 
 be equal to some- 
 where about eight times that force. Now if the string to 
 which R is attached be brought underneath the other string 
 so as to be pressed by it against the surface of the cylinder, 
 as at m,fig. 2.; then, provided the friction produced by 
 this pressure be not less than one eighth of P, the string will 
 not move even although the force R cease to act. And if 
 both extremities of the string be thus made to pass between 
 the coil and the cylinder, as in fig. 3., a still less pressure 
 upon each will be requisite. Now, by diminishing the 
 radius of the cylinder, this pressure can be increased to any 
 extent, since, by a known property of funicular curves, it 
 varies inversely as the radius.* We may, therefore, so far 
 diminish the radius of a cylinder, as that no force, however 
 great, shall be able to pull away a rope coiled upon it, as 
 represented in fig. 3., even although one extremity were 
 loose, and acted upon by no force. 
 
 Fig - 4t Let us suppose the rope to be 
 
 doubled as in fig. 4., and coiled 
 as before. Then it is apparent, 
 from what has been said, that 
 the cylinder may be made so 
 small, that no forces P and P' 
 applied to the extremities of 
 either of the double cords will 
 be sufficient to pull them from 
 it, in whatever directions these are applied. 
 
 * This property will be proved in that portion of the work which treats of 
 the THEORY OF CONSTRUCTION. 
 
THE FRICTION BREAK. 213 
 
 Now let the cylinder be removed. The cord then being 
 drawn tight, instead of being coiled round the cylinder, wifl 
 be coiled round portions of itself, at the points m and n ; 
 and instead of being pressed at those points upon the cylin- 
 der, by a force acting on one portion of its circumference, it 
 will be pressed by a greater force acting all round its cir- 
 cumference. All that has been proved before, with regard 
 to the impossibility of pulling either of the cords away from 
 the coil when the cylinder is inserted, will therefore now 
 obtain in a greater degree ; whence it follows that no forces 
 P and P' acting to pull the extremities of the cords asunder, 
 may be sufficient to separate the knot. 
 
 THE FRICTION BREAK. 
 
 189. There are certain machines whose motion tends, at 
 certain stages, to a destructive acceleration ; as, for instance, 
 a crane, which, having raised a heavy weight in one position 
 of its beam, allows it to descend by the action of gravity in 
 another ; or a railway train, which, on a certain portion of 
 its line of transit, descends a gradient, having an inclination 
 greater than the limiting angle of resistance. In each of 
 these cases, the work done by gravity on the descending 
 weight exceeds the work expended on the ordinary resist- 
 ance due to the friction of the machine ; and if some other 
 resistance were not, under these circumstances, opposed to 
 its motion, this excess (of the work done by gravity upon it 
 over that expended upon the friction of its rubbing surfaces) 
 would be accumulated in it (Art. 130.) under the form of 
 vis viva, and be accompanied by a rapid acceleration and a 
 destructive velocity of its moving parts. The extraordinary 
 resistance required to take up its excess of work, and to 
 prevent this accumulation, is sometimes supplied in the 
 crane by the work of the laborer, who, to let the weight 
 down gradually, exerts upon the revolving crank a pressure 
 in a direction opposite to that which he used in raising it. 
 It is more commonly supplied in the crane, and always in 
 the railway train, without any work at all of the laborer, by 
 a simple pressure of his hand or foot on the lever of the fric- 
 tion break, which useful instrument is represented in the 
 figure under the form in which it is com 
 
-l THE FRICTION BEEAK. 
 
 monly applied to the crane, a form of it which may serve 
 to illustrate the principle of its application nnder every 
 
 other. BC represents a wheel 
 fixed commonly upon that 
 axis of the machine to which 
 the crank is attached, and 
 which axis is carried round 
 by it with greater velocity 
 than any other. The peri- 
 phery of this wheel, which is 
 usually of cast iron, is em- 
 braced by a strong band* ABCE of wrought iron, fixed 
 firmly by its extremity A to the frame of the machine, and 
 by its extremity E to the short arm AE of a bent lever PAE, 
 which turns upon a fixed axis or fulcrum, at A, and whose 
 arm PA, being prolonged, carries a counterpoise D just 
 sufficient to overbalance the weight of the arm AP, and to 
 relieve the point E of all tension, and loosen the strap from 
 the periphery of the wheel, when no force P is applied to the 
 extremity of the arm AP, or when the break is out of 
 action. 
 
 It is evident that a pressure P applied to the extremity of 
 the lever will produce a pressure upon the point E, and a 
 tension upon the band in the direction ABCE, and that 
 being fixed at its extremity A, the band will thus be tight- 
 ened upon the wheel, producing by its friction a certain 
 resistance upon the circumference of the wheel. 
 
 Moreover, it is evident that this resistance of friction upon 
 the circumference of the wheel is precisely equal to the 
 tension upon the extremity A of the band, being, indeed, 
 wholly borne by that tension ; and that it is the same 
 whether the wheel move, as in this case it does, under the 
 band at rest, or whether the band move (under the same 
 tensions upon its extremities, but in the opposite direction) 
 over the wheel at rest. Let R and Q represent the tensions 
 upon the extremities A and E of the band ; then if we sup- 
 pose the wheel to be at rest, and the band to be drawn over 
 it in the direction ECB by the tension R, and & to represent 
 the angle subtended at the centre of the wheel by that part 
 of its circumference which the band embraces, we have 
 (equation 205) 
 
 * Blocks of wood arc interposed between the band, the periphery of the 
 break wheel. This case will be discussed in the Appendix. 
 
THE BAND. 215 
 
 Let &J represent the length of the arm AP, and # a the 
 length of the perpendicular let fall from A upon the direc- 
 tion of a tangent to that point in the circumference of the 
 wheel where the end EC of the band leaves it. 
 
 Then, neglecting the friction of the axis A, we have 
 (Art. 5.) 
 
 P . a = . 
 
 (207). 
 
 If Pj represent any pressure applied to the circumference^ of 
 the break wheel, and P 2 a pressure applied to the working 
 point of the machine, whatever it may be, to which the 
 break is applied, and if Y l =aP^-\-b (Art. 152.) represent the 
 relation between P t and P 2 in the inferior state bordering 
 upon motion by the preponderance of P 2 ; then, when P 2 is 
 taken in this expression to represent the pressure W, whose 
 action upon the working point of the machine the break is 
 intended to control, 'P l will represent that value K of the 
 friction upon the break which must be produced by the 
 intervention of the lever to control the action of the pressure 
 W upon the machine; so that taking E to represent the 
 same quantity as in equation (207), we have 
 
 Eliminating E- between this equation and equation (207), 
 and solving in respect to P, 
 
 .... (208). 
 a. % 
 
 THE BAND. 
 
 1 90. When the circular motion of any shaft in a machine, 
 and the pressure which accompanies that motion, consti- 
 tuting together with it the work of the shaft, are to be com- 
 municated to any other distant shaft, this communication is 
 
216 THE BAND. 
 
 usually established by means of a band of 
 leather, which passes round drums fixed upon 
 the two shafts, and has its extremities drawn 
 together with a certain pressure and united, 
 so as to produce a tension, which should be 
 just that necessary to prevent the band from 
 slipping upon the drums, subject to the pres- 
 sure under which the work is transferred. 
 The facility with which this communication 
 of rotatory motion may be established or 
 broken at any distance and under almost 
 every variety of circumstance, has brought 
 the band so extensively into use in machinery, 
 that it may be considered as a principal chan- 
 nel through which work is made to flow in its distribution 
 to the successive stages of every process of mechanism, 
 carried on in the same workshop or manufactory. 
 
 191. The sum of the tensions upon the two parts of a hand 
 is the same, whatever he the pressure under which the hand 
 is d/riven, or the resistance overcome, the tension of the 
 driving part of the hand being always increased hy just so 
 much as that of the driven part is diminished. 
 
 This principle was first given by M. Poncelet ; it has since 
 been amply confirmed by the experiments of M. Morin.* It 
 may be proved as followsf : In the very commencement of 
 the motion of that drum to which the driving pressure is 
 applied, no motion is communicated by it to the other drum. 
 Before any such motion can be communicated to the latter, 
 a difference must be produced between the tensions of the 
 two parts of the band sufficient to overcome the resistance, 
 whatever it may be, which is opposed to the revolution of 
 the driven drum. Now, an increase of the tension on the 
 driving side of the band must be followed by an elongation 
 of that side of the band (since the band is elastic), and by 
 the revolution of the circumference of the driving drum 
 through a space precisely equal to this elongation. Sup- 
 posing, then, the other, or driven side of the band, to 
 remain extended, as before, in a straight line between its two 
 points of contact with the drums, this portion of the band 
 
 * Nouvelles Experiences sur le Froltement, &c. Metz. 
 
 f No demonstration appears to have been given of it by M. Poncelet, 
 
THE BAND. 
 
 217 
 
 must evidently have contracted by precisely the length 
 through which the circumference of the driving drum has 
 revolved, or the driving side of the band elongated. Thus, 
 the elongation of the driving side of the band is precisely 
 equal to the contraction of the driven side. Now, the band 
 being supposed perfectly elastic, the increase or dimi- 
 nution of its tension is exactly proportional to the increase 
 or diminution of its length. The increase of tension on the 
 one side, produced by a given elongation, is therefore pre- 
 cisely equal to the diminution of tension produced by a con- 
 traction equal to that elongation on the other side. Thus, 
 if T represent the tension upon each side of the band before 
 the driving pressure, whatever it may be, was applied, 
 and if T, and T 2 represent the tensions upon the driving 
 and the driven sides of the band after that pressure is 
 applied; then, since T x T represents the increase of tension 
 on the one side, and T T 2 the diminution of tension on the 
 other, ^-T^T-T,; 
 
 (209). 
 
 It is a great principle of the economy of power in the use 
 of the band to adjust this initial tension T, so that it may 
 just be sufficient to prevent the band from slipping upon 
 the drum under any pressure which it is required to transmit. 
 The means of making this adjustment will be explained 
 hereafter. 
 
 THE MODULUS OF THE BAND. 
 
 192. For simplifying the consideration of this important 
 element in machinery, we shall first consider a particular 
 case of its application. Let the two drums^ whose axis are 
 G! and C 2 , be supposed equal to one another, so that the two 
 parts of the band which pass round them may be parallel. 
 
 Let, moreover, the centres of the 
 two drums be in the same verti- 
 cal straight line, so that the two 
 parts of the band may be verti- 
 cal. 
 
 Let Pj and P 2 be pressures ap- 
 plied, in vertical directions, to 
 turn the drums, and at perpen- 
 dicular distances from their cen- 
 tres, represented by G X P> and 
 C 2 P 2 ; of which pressures P 2 is 
 the working or driven pressure. 
 
 Fig 2 
 
218 THE BA^D. 
 
 or that which is upon the point of yielding by the prepon- 
 derance of the other P,. In fig. 1. P 2 is seen applied on 
 the same side of the centre of the drums as P 15 and in jig. 
 2. on the opposite side. Let T x and T 2 represent the tensions 
 upon the two parts of the band, T x being that on the driving, 
 and T 2 that on the driven side. 
 
 ^=0^, 2 =C 2 P 2 , 
 
 r= radius of each drum, 
 
 W= weight of each drum, 
 
 p= radius of axis of each drum, 
 
 R! and R 2 = resistances of axes of drums, 
 
 <p limiting angle of resistance. 
 
 Now, the parallel pressures P,, W, T 15 T 2 , R n applied to the 
 lower drum, are in equilibrium ; therefore (Art. 16.), 
 
 or substituting for T^+T, its value 2T (equation 209), 
 E 1 =(2T P 1 W) ..... (210). 
 
 The sign being taken according as 2T is greater or less 
 than Pj+Wj, or according as the axis of the lower drum 
 presses upon the upper surface of its bearings, as shown in 
 fig. 1., or upon the lower surface, as shown v&fig. 2. In like 
 manner, the pressures P 2 , W, T 1? T 2 R 2 , applied to the upper 
 drum, being in equilibrium, 
 
 or (equation 209) R 2 =2TTP a +W .... (211), 
 
 where the sign ^p is to be taken according as P 2 is applied 
 on the same side of the axis as P,, or on the opposite side. 
 
 Since, moreover, Rj and R 2 act, in the state bordering 
 upon motion, at perpendicular distances from the centre of 
 the axis, which are each represented by p sin. <p (Art. 153.), 
 we have, by the principle of the equality of moments, 
 
 P A +T,r=T l r+:R l psm.9 
 
 observing that the resultant of all the pressures applied to 
 each drum (excepting only the resistance of its axis) must be 
 such as would alone communicate motion to it in the direc- 
 tion in which it actually moves, and therefore that the re 
 sistance of the axis, which is opposite to this resultant, must 
 tend to communicate motion to the drum in a direction oppo- 
 site to that in which it actually moves. 
 
THE BAND. .219 
 
 Subtracting the above equations, and transposing, 
 P 1 ,-P A =(R I +R 2 ) p sin. 9. 
 
 Substituting the values of R x and R a from equations (210) 
 and (211), we obtain, in the case in which the negative sign 
 of Rj is to be taken, or in which 2T is less than P a + W, the 
 axis C x resting upon the lower surface of its collar as shown 
 in fig. 2., 
 
 sin. 9 ; 
 
 and in the case in which the positive sign of 3^ is to be 
 taken, 2T being greater than P X +W, and the axis C t press- 
 ing against the upper surface of its collar, as shown iRJtg. 1., 
 
 sn. 9. 
 
 Transposing and reducing, we obtain for the relation be- 
 tween the driving and driven pressures in these two cases 
 respectively, 
 
 P P /sSL2i\ 2W P Bin.,> . . (213), 
 
 \a 1 p sin. 9; a t psm. 9 
 
 p p . (214) 
 
 \a 1 + psm. 97 
 
 and therefore (by equation 121), for the moduli in the two 
 cases a 
 
 ! p sn. 9 
 
 sin. 
 
 In all which equations the sign =F is to be taken according 
 as P 2 is applied on the same side of the line OjC,, joining the 
 axis as P 15 or on the opposite side. 
 
 193. To determine the initial tension*^ upon the land, so that 
 it may not slip upon the surface of the drum when sub- 
 jected to the given resistance opposed to its motion by the 
 work. 
 
220 
 
 THE BAND. 
 
 Suppose the maximum resistance which may, during the 
 action of the machine, be opposed to the motion 
 of the drum to be represented by a pressure P 
 applied at a given distance a from its centre C 2 . 
 Suppose, moreover, that the band has received 
 such an initial tension T as shall just cause it to 
 be on the point of slipping when the motion of 
 the drum is subjected to this maximum resist- 
 ance ; and let ^ and t z be the tensions upon 
 the two parts of the band when it is thus 
 Just in the act of slipping and of overcoming the resistance 
 r. Now, the two parts of the band being parallel, it em 
 braces one half of the circumference of each drum ; the rela- 
 tion between t l and 2 is therefore expressed (equation 205) 
 by the equation 
 
 TT tan. ^ 
 
 t = t a e*-tan. 'whence we obtain - 1 J-=-- ~ . But . + ,= 
 
 1 * ' -r I / , - to^ A. * * 
 
 2T (equation 209), 
 
 (TT tan. 
 e ^ 
 it tan. ^ 
 e + I 
 
 Also, the relation between the resistance P, opposed to the 
 motion of the upper drum, and the tensions ^ and 2 upon 
 the two parts of the band, when this resistance is on the 
 point of being overcome, is expressed (equation 212) by the 
 equation 
 
 or substituting the value of R a (equation 211), and transpos- 
 ing 
 
 P+(2T=f P-f W)p sin. 9=)^ t^r ; 
 
 whence, substituting the value of ^4, determined above, 
 and transposing, we have 
 
 {/e"-'\\ } 
 
 P(aqFpsin. 9)H-"Wpsin. 9=2T^ | -^^-^ \r psin.9 j. ; 
 
 e + I 
 
THE BAND. 
 
 T _ - J P(a=Fp sin. 0) + Wpsin.0 ) 
 
 * i / TT tan. \ 
 I 8 1\ 
 
 I -tS^ l^-psm. </> 
 
 221 
 
 (217). 
 
 194. The modulus of the band under its most general form, 
 
 The accompanying figure represents an. elastic band pass- 
 ing round drums of unequal radii, the 
 line joining whose centres 0, and C 2 
 is inclined at any angle to the vertical, 
 and which are acted upon by any 
 given pressures P, and P 2 , P 1 being 
 supposed to be upon the point of giv- 
 ing motion to the system. 
 
 Let T x and T a represent the tensions 
 upon the two parts of the band, T x be- 
 ing that on the driving side. 
 # 15 # 2 perpendiculars upon the directions of P a and P a re- 
 spectively. 
 
 1? 2 the inclinations of the directions of P, and P 2 to the 
 line 0,0,. 
 
 TV r 2 the radii of the drums. 
 W j, W 2 the weights of the drums. 
 
 the inclination of the line Cfl 9 to the vertical, and2 aj the 
 inclination of the two parts of the band to one another, 
 p! p 2 the radii of the axes of the drums. 
 the limiting angle of resistance between the axis of the 
 drum and its collar. 
 
 K 15 R 2 the resistances of the collars in which the axes of 
 the drums turn in the state bordering upon motion, or the 
 resultants of the pressures upon these axes. The perpendi- 
 cular distances at which these resistances act from the cen- 
 tres of the axes are (Art. 153.) p : sin. and p 2 sin. 0. Since 
 the pressures acting upon the lower drum are T 15 T 2 , P 1? W iy 
 and Kj, and that these pressures are in equilibrium, W l act- 
 ing through the centre of the axis, and T, and E t acting to 
 turn the drum in one direction about the axis, and P x and T, 
 to turn it in the opposite direction ; we have, by the princi- 
 ple of the equality of moments (Art. 153.), 
 
 PA+T^^T^ + K^sin. 9. 
 And since T 1? T 2 , P 2 , W 2 , E a are similarly in equilibrium 
 
222 THE BAND. 
 
 on the upper drum, W 2 acting through the centre, and P 2 , 
 R 2 , T 2 acting to turn it in one direction, whilst T l acts to 
 turn it in the opposite direction, 
 
 sin. 9= 
 
 /.PA-^-T^K^sin.? ) 
 
 p A _( Tl -T 2 K=-K 2 p 2 sin. 9 T 
 Let Tj-T^B*, and T.+T^^T, 
 
 ^^R^ sin. 9 ) ,01 c^ 
 
 -2r f =- R 2 p, sin. 9 (' 
 
 To determine the values of 1^ and R 2 let the pressures 
 applied to each drum be resolved (Art. 11.) in directions 
 parallel and perpendicular to the line CjC, ; tnose applied to 
 the lower drum which, being thus resolved, are parallel to 
 00 are 
 
 ! COS. 15 +T a COS. a i5 P! COS. ^, W l COS. I, 
 
 those pressures being taken positively which tend to move 
 the axis of the drum from G l towards 2 , and those nega- 
 tively whose tendency is in the opposite direction. 
 
 In like manner the pressures resolved perpendicular to 
 0,0, are 
 
 T x sin. a l? -j-T 2 sin. a l? -f P 4 sin. ^, "W 1 sin. , 
 
 those pressures being taken negatively whose tendency 
 when thus resolved perpendicular to C^ is to bring that 
 line nearer to a vertical direction, and those positively whose 
 tendency is in the opposite direction. 
 
 Observing that K 1 is the resultant of all these pressures, 
 we have (Art. 11.) 
 
 os - a i- p i cos - *i-Wi cos - '}* + 
 
 |P X sin. ^-(T.-T^sin. a.-W, sin. i} 2 . 
 
 Proceeding similarly in respect to the pressures applied to 
 the upper drum, we shall obtain 
 
 E a 3 = {(T.+T,) cos. , P, cos. ^ a +W 2 cos. i} f + 
 
 |P 3 sin. ^+(T 1 T 3 ) sin. a, W 2 sin. J 3 : 
 or substituting 2T for T^T,, and 2# for T, T 2 
 
 K^ |2T cos. , P, cos. *! W, cos. } 2 
 {P, sin. ^-2^ sin. ^-W, sin. { 3 
 
 R a 9 = J2T cos. ^-P, cos. ^ a + W 2 cos. 1} '+ ' 
 }P a sin. d 2 + 2 sin. a, W 2 sin. 1} 2 
 
THE BAND. 
 
 By eliminating R 15 R 2 , and t between the four equations 
 (218) and (219), a relation is determined between the three 
 quantities P 15 P 2 , T. To simplify this elimination let us sup- 
 pose that the preceding hypothesis in respect to the direc- 
 tions in which the pressures are to be taken positively and 
 negatively is so made, that the expressions enclosed within 
 the brackets in the above equations (219) and squared may, 
 each of them, represent a positive quantity. Let us, more- 
 over, suppose t\iQ first of the two quantities squared in each 
 equation to be considerably greater than the second, or the 
 pressure upon the axis of each drum in the direction of the 
 line C a C 2 joining their centres, greatly to exceed the pres- 
 sure upon it in a direction perpendicular to that line ; an 
 hypothesis which will in every practical case be realised. 
 These suppositions being made, we obtain, with a sufficient 
 degree of approximation, by Poncelet's Theorem*, 
 
 R i: =aj2T cos. a x P t cos. 0,W, cos. i{ + 
 /3 ? sin. 62t sin. a W sin. 
 
 R 2 =a{2T cos. a l P 2 cos. 3 +W 3 cos. 1} 
 /3 JP 2 sin. 2 +22 sin. a x W 2 sin. 1} . 
 
 Substituting these values of Rj and R a in equation (218), 
 and reducing, we have 
 
 PA 2(r\ /3p l sin. a t sin. 9)= 
 Pl |2T cos. ai -PA-W lTl l sin. 9 
 
 P 2 a 2 -2^ 2 -/3p 2 sin. ttl sin. 9)= 
 -p 9 {2a T cos. a.-PA+'W^I sin. 9 
 
 where ft=(a cos. X ]8 sin. X ), 
 ft=(. co, ^_sin *,), 
 y 1= (a cos. i + P Sin. <), 
 y 3 =(a cos. i sin. i). 
 
 Eliminating # between these equations, and neglecting 
 terms above the first dimension in p 1 sin. 9 and p 2 sin. 9, 
 
 ( H-P^Xn p a sin. a x sin. 9) | _ 
 ( P^r, p a sin. a, sin. 9) j " 
 
 + Pl r,(2aT cos. ^-PA-W^) ) . 
 
 T,) [ Sm ' 
 
 + P2 rX2aT cos. a.- 
 , being for the most part exceedingly small, the terms 
 
 * See Appendix. 
 
24 
 
 THE BAND. 
 
 /3pj sin. ctj sin. 9 and /3p a sin. a z sin. 9 may be neglected ; we 
 shall then obtain by transposition and reduction 
 
 +2*T(p 1 r i H-p t r 1 )cos. a t 
 
 -sin. 9 ... (222). 
 
 If this equation be compared with equation (214), it will 
 be found to agree with it, mutatis mutandis, except that 
 the co-efficient 2a is in that equation 2. This difference 
 manifestly results from the approximate character of the 
 theorem of Poncelet. 
 
 Substituting the latter co-efficient for the former, multiply- 
 
 p (3 
 
 ing both sides of the equation by (1 f sin. 9), neglecting 
 
 AI 
 
 terms of more than two dimensions in f , , and sin. 9, and 
 reducing, 
 
 which is the relation between the moving and working 
 pressures in the state bordering upon motion. From this 
 relation we obtain for the MODULUS of the band (equation 
 121) 
 
 ' 
 
 If the angle a be conceived to increase until it exceed 
 x, P, will pass to the opposite side of C,0 a , and (3 t will 
 become negative; whence it is apparent, that equation 
 
THE BAND. 225 
 
 (224) agrees with equation (214) in other respects, and in 
 the condition of the ambiguous sign. It is moreover appa- 
 rent, from the form assumed by the modulus in this case 
 and in that of the preceding article, that the greatest 
 economy of power is obtained by applying the moving and 
 the working pressures on the same side of the line C^ join- 
 ing the axes of the drums. This is in fact but a particular 
 case of the general principle established in Art. 168. 
 
 195. The initial tension T of the band may be deter- 
 mined precisely as in the former case (equation 217). 
 Representing by the angle sub- 
 tended by the circumference which 
 the band embraces on the second, 
 or driven drum, by P the maxi-- 
 mum resistance opposed to its mo** 
 tion at the distance a, by $ t\& 
 limiting angle of resistance between 
 the band and the surface of the 
 drum, and by ^ and t t the tensions 
 upon the two parts of the band, 
 when its maximum resistance being opposed, it is upon the 
 point of slipping; observing, moreover, that in this case 
 
 e tan. $ 
 
 2ft ,) or 2 is represented (Art. 193.) by 2T ^~ ^; then 
 
 e +" 1 
 
 substituting in the second of equations (220) this value for 
 2, and P and a for P a and # and neglecting the exceed- 
 ingly small term which involves the product sin. a, sin. 9, 
 we have 
 
 (0tan. $ ^ 
 e -6^1T V a =-p 9 {2aTco8. ai -P/3 a +W 2 r a } sin 9. 
 e + I/ 
 
 Also, since a a represents the inclination of the two parts of 
 the band to one another ; since, moreover, these touch the 
 surfaces of the drums, and that 6 represents the inclination 
 of the radii drawn from the centre of the lesser drum to the 
 touching points, therefore O=TC a r Substituting this value 
 of in the above equation, and solving it in respect to T, we 
 have 
 
 15 
 
226 raE BAND. 
 
 P(<3 p 2 /3 2 sin. </>)-f W 2 p 2 7 2 sin. 
 
 ((TT-OI) tan. 4>\ 7* a p a a COS, a sin. 
 e - l ] 
 (Tr-aO tan. <|> I 
 
 (225), 
 
 196. The modulus of the hand when the two parts of it, 
 which intervene between the drums, are made to cross one 
 another. 
 
 If the directions of the two parts of the band be made 
 to cross, as shown in the accompanying 
 figure, the moving pressure Tj upon the 
 second drum is applied to it on the side 
 opposite to that on which it is applied 
 when the bands do not cross ; so that in 
 this case, in order that the greatest eco- 
 nomy of power may be attained (Art. 
 168.), the working pressure or resistance 
 P 2 should be applied to it on the side 
 opposite to that in which it was applied 
 in the other case, and therefore on the side of the line CjCg, 
 opposite to that on which the moving pressure P t upon the 
 first drum is applied. This disposition of the moving and 
 working pressures being supposed, and this case being inves- 
 tigated by the same steps as the preceding, we shall arrive 
 at precisely the same expressions (equations 223 and 224) 
 for the relation of the moving and the working pressures, 
 and for the modulus. 
 
 In estimating the value of the initial tension T (equation 
 225) it will, however, be found, that the angle d, subtended 
 at the centre C 2 of the second drum by the arc KML, which 
 is embraced by the band, is no longer in this case repre- 
 sented by if a, but by * + ,. This will be evident if we 
 consider that the four angles of the quadrilateral figure 
 C 2 KIL being equal to four right angles, and its angles at K 
 and L being right angles, the remaining angles KIL and 
 KC 2 L are equal to tw r o right angles, so that KC 2 L=tf a t ; 
 but the angle subtended by KML equals 2^ KC 2 L; it 
 equals therefore * + ,. If this value be substituted for ta t 
 in equation (225) it becomes 
 
THE TEETH OF WHEELS. 227 
 
 If P(a-pA sin. <p)+W 2 p 2 sin. <p 
 
 2 
 
 A"' *1\ 
 
 I ~, \ tT n P O ot cos. & sin 
 
 I CTT + CI; tan. (p > i^ 
 
 V + i/ 
 
 (226.) 
 
 Now the fraction in the denominator of this expression 
 being essentially greater in value than that in the denomi- 
 nator of the preceding (equation 225), it follows that the 
 initial tension T, which must be given to the band in order 
 that it may transmit the work from the one drum to the 
 other under a given resistance P, is less when the two parts 
 of the band cross than when they do not, and, therefore, that 
 the modulus (equation 224) is less; so that the band is 
 worked with the greatest economy of power (other things 
 'being the same) when the two parts of it which intervene 
 between the drums are made to cross one another. Indeed it 
 is evident, that since in this case the arc embraced by the 
 band on each drum subtends a greater angle than in the 
 other case, a less tension of the band in this case than in the 
 other is required (Art. 185.) to prevent it from slipping 
 under a given resistance, so that the friction upon the axis 
 of the drums which results from the tension of the band is 
 less in this case than the other, and therefore the work 
 expended on that friction less in the same proportion. 
 
 THE TEETH OF WHEELS. 
 
 197. Let A, B represent two circles in contact at D, and 
 moveable about fixed centres at C, and C 2 . It 
 is evident that if by reason of the friction of 
 these two circles upon one another at D any 
 motion of rotation given to A be communicated 
 to B, the angles PC^D and QC 2 D described in 
 the same time by these two circles, will be such 
 as will make the arcs PD and QD which they 
 subtend at the circumferences of the circles equal to one 
 another. Let the angle PCjD* be represented by d,, and the 
 angle QC 2 D by d 2 ; also let the radii C^D and C 2 D of the cir- 
 cles be represented by ?\ and r 9 . Now, arc PD=r^^ arc 
 r&; and since PD=QD, therefore ^A =/1 A j 
 
 * Or rather the arc which this angle subtends to radius unity. 
 
228 THE TEETH OF WHEELS. 
 
 (227). 
 
 The angles described, in the same time, by two circles 
 which revolve in contact are therefore inversely proportional 
 to the radii of the circles, so that their angular velocities 
 (Art. 74.) bear a constant proportion to one another ; and if 
 one revolves uniformly, then the other revolves uniformly ; 
 if the angular revolution of the one varies in any proportion, 
 then that of the other varies in like proportion. 
 
 When the resistance opposed to the rotation of the driven 
 circle or wheel B is considerable, it is no longer possible to 
 give motion to that circle by the friction on its circum- 
 ference of the driving circle. It becomes therefore neces- 
 sary in the great majority of cases to cause the rotation of 
 the driven wheel by some other means than the friction of 
 the circumference of the driving wheel. 
 
 One expedient is the band already described, by means of 
 which the weels may be made to drive one another at any 
 distances of their centres, and under a far greater resistance 
 than they could by their mutual contact. When, however, 
 the pressure is considerable, and the wheels may be brought 
 into actual contact, the common and the more certain 
 method is to transfer the motion 
 from one to the other by means of 
 projections on the one wneel called 
 TEETH, which engage in similar pro- 
 jections on the other. 
 
 In the construction of these teeth 
 the problem to be solved is, to give 
 such shapes to their surfaces of mu- 
 tual contact, as that the wheels shall 
 be made to turn by the intervention 
 of their teeth precisely as they would by the friction of 
 their circumferences. 
 
 198. That it is possible to construct teeth which shall 
 answer this condition may thus be shown. 
 Let mn and m'n' be two curves, the one 
 described on the plane of the circle A, and 
 the other on. the plane of the circle B ; and 
 let them be such that as the circle A re- 
 volves, carrying round with it the circle B, 
 by their mutual contact at D, these two 
 curves mn and m'n' may continually touch 
 
THE TEETH OF WHEELS. 229 
 
 one another, altering of course, as they will do continually, 
 their relative positions and their point of contact T. 
 
 It is evident that the two circles would be made to 
 revolve by the contact of teeth whose edges were of the 
 forms of these two curves mn and m'n r precisely as they 
 would by their friction upon the circumferences of one 
 another at the point D; for in the former case a certain 
 series of points of contact of the circles (infinitely near to 
 one another) at D, brings about another given series of points 
 of contact (infinitely near to one another) of the curves mn 
 and m'n' at T ; and in the latter case the same series of 
 points in the curves mn and m'n' brought into contact neces- 
 sarily produces the contact of the same series of points in 
 the two circumferences of the two circles at D. 
 
 To construct teeth whose surfaces of contact shall possess 
 the properties here assigned to the curves mn and m'n' is 
 the problem to be solved. Of the solution of this problem 
 the following is the fundamental principle : 
 
 199. In order that two circles A and B may "be made to 
 revolve by the contact of the surfaces mn and m'n' of their 
 teeth, precisely as they would ly the friction of their cir- 
 cumferences, it is necessary, and it is suf- 
 ficient, that a line drawn from the point 
 of contact T of the teeth to the point of 
 contact D of the circumferences should, in 
 every position of the point T, he perpendi- 
 cular to the surfaces in contact there, i. e., 
 a normal to loth the curves mn and m'n'. 
 
 To prove this principle, we must first establish the follow- 
 ing LEMMA : If two circles M and N be made to revolve 
 about the fixed centres E and F by their mu- 
 tual contact at L, and if the planes of these 
 circles be conceived to be carried round with 
 them in this revolution, and a point P on the 
 plane of M to trace out a curve PQ on the 
 plane of N" whilst thus revolving, then is this 
 curved line PQ precisely the same as would 
 have been described on the plane of N by the same point P, 
 if the latter plane, instead of revolving, had remained at 
 rest, and the centre E of the circle M having been released 
 
230 THE TEETH OF WHEELS. 
 
 from its axis, that circle had been made to. roll (carrying its 
 plane with it) on the circumference of N. For conceive O 
 to represent a third plane on which the centres of E and F 
 are fixed. It is evident that if, whilst the circles M and K 
 are revolving by their mutual contact, the plane O, to which 
 their centres are both fixed, be in any way moved, no change 
 will thereby be produced in form of the curve PQ, which the 
 point P in the plane of M is describing upon the plane of !N", 
 such a motion being common to both the planes M and E".* 
 Now let the direction in which the circle N is revolving be 
 that shown by the arrow, and its angular velocity uniform ; 
 and conceive the plane O to be made to revolve about F with 
 an angular velocity (Art. 74) which is equal to that of N, 
 but in an opposite direction, communicating 
 this angular velocity to M and !N". these re- 
 volving meantime in respect to one another, 
 and by their mutual contact, precisely as they 
 did before.f 
 
 It is clear that the circle E" being carried 
 round by its own proper motion in one direc- 
 tion, and by the motion common to it and the plane O with 
 the same angular velocity in the opposite direction, will, in 
 reality rest in space ; whilst the centre E of the circle M, 
 having no motion proper to itself, will revolve with the 
 angular velocity of the plane O, and the various other points 
 in that circle with angular velocities, compounded of their 
 proper velocities, and those which they receive in common 
 with the plane O, these velocities neutralising one another 
 at the point L of the circle, by which point it is in contact 
 with the circle N". So that whilst .M revolves round 1ST, the 
 point L, by which the former circle at any time touches the 
 other, is at rest ; this quiescent point of the circle M never- 
 theless continually varying its position on the circumferences 
 of both circles, and the circle M being in fact made to roll 
 on the circle N at rest. 
 
 Thus, then, it appears, that by communicating a certain 
 common angular velocity to both the circles M and !N" about 
 
 * Thus for instance, if the circles M and N continue to revolve, we may 
 evidently place the whole machine in a ship under sail, in a moving carriage, 
 or upon a revolving wheel, without in the least altering the form of the curve, 
 which the point P, revolving with the plane of the circle M, is made to trace 
 on the plane of N, because the motion we have communicated is common to 
 both these circles. 
 
 f M and N may be imagined to be placed upon a horizontal wheel 0, first at 
 rest, and then made to revolve backwards in respect to the motion of N. 
 
THE TEETH OF WHEELS. 231 
 
 the centre F, the former circle is made to Toll upon the other 
 at rest ; and, moreover, that this common angular velocity 
 does not alter the form of the curve PQ, which a point P in 
 the plane of the one circle is made to trace upon the plane 
 of the other, or, in other words, that the curve traced under 
 these circumstances is the same, whether the circles revolve 
 round fixed centres by their mutual contact, or whether the 
 centre of one circle be released, and it be made to roll upon 
 the circumference of the other at rest. 
 
 This lemma being established, the truth of the proposition 
 stated at the head of this article becomes evident ; for if M 
 roll on the circumference of N, it is evident that P will, at 
 any instant, be describing a circle about their point of con- 
 tact L.* 
 
 Since then P is describing, at every instant, a circle about 
 L when M rolls upon N, !N" being fixed, and since the curve 
 described by P upon this supposition is precisely the same 
 as would have been traced by it if the centres of both cir- 
 cles had been fixed, and they had turned by their mutual 
 contact, it follows that in this last case (when the circles 
 revolve about fixed centres by their mutual contact) the 
 point P is at any instant of the revolution describing, during 
 that instant, an exceedingly small circular arc about the 
 point L ; whence it follows that PL is always a perpendicu- 
 lar to the cur,ve PQ at the point P, or a normal to it. 
 
 ISTow let p be a point exceedingly near to T in the curve 
 raW, which curve is fixed upon the plane 
 of the circle A. It is evident that, as the 
 point p passes through its contact with the 
 curve mn at T (see Art. 195.), it will be 
 made to describe, on the plane of the circle 
 B, an exceedingly small portion of that 
 curve TTiTi. But the curve which it is 
 (under these circumstances) at any instant 
 describing upon the plane of B has been shown to be 
 always perpendicular to the line DT ; the curve mn is there- 
 fore at the point T perpendicular to the line DT ; whence it 
 follows that the curve m'n' is also perpendicular to that line, 
 and that DT is a normal to ~both those curves at T. This is 
 the characteristic property of the curves mn and raW, so that 
 they may satisfy the condition of a continual contact with 
 
 * For either circle may be imagined to be a polygon of an infinite number 
 of sides, on one of the angles of which the rolling circle will, at any instant, 
 be in the act of turning. 
 
232 THE TEETH OF WHEELS. 
 
 one another, whilst the circles revolve by the contact of 
 their circumferences at D, and therefore conversely, so that 
 these curves may, by their mutual contact, give to the cir- 
 cles the same motion as they would receive from the contact 
 of their circumferences. 
 
 200. To describe, ~by means of circular arcs, the form of a 
 tooth on one wheel which shall work truly with a tooth of 
 any given form on another wheel. 
 
 Let the wheels be required to revolve by the action of 
 their teeth, as they would by the 
 contact of the circles ABE and 
 ADF, called their primitive ov pitch 
 circles. Let AB represent an arc 
 of the pitch circle ABE, included 
 between any two similar points A. 
 and B of consecutive teeth, and let 
 AD represent an arc of the pitch 
 circle ADF equal to the arc AB, so 
 that the points D and B may come 
 simultaneously to A, when the cir- 
 cles are made to revolve by their 
 mutual contact. AB and AD are 
 called the pitches of the teeth of the two wheels. Divide 
 each of these pitches into the same number of equal parts 
 in the points a, b, &c., a', b', &c. ; the points a and a', b and 
 b', &c., will then be brought simultaneously to the point A. 
 Let mn represent the form of the face of a tooth on the 
 wheel, whose centre is C,, with which tooth a corresponding 
 tooth on the other wheel is to work truly ; that is to say, 
 the tooth on the other wheel, whose centre is C 2 , is to be cut, 
 so that, driving the surface mn* or being driven by it, the 
 wheels shall revolve precisely as they would by the con- 
 tact of their pitch circles ABE and ADF at A. From A 
 measure the least distance A to the curve mn, and with 
 radius A and centre A describe a circular arc /3 on the 
 plane of the circle whose centre is C a . From a measure, in 
 like manner, the least distance a*', to the curve mn, and 
 with this distance a*' and the centre a, describe a circular 
 arc 7, intersecting the arc a/3 in ft. From the point b 
 measure similarly the shortest distance W to mn, and with 
 
THE TEETH OF WHEELS. 233 
 
 the centre V and this distance W l describe a circular arc 
 7(?, intersecting 7 in 7, and so with the other points of 
 division. A curve touching these circular arcs a/3, ^7, 7^ 
 &c., will give the true surface or boundary of the tooth.* 
 
 In order to prove this let it be observed, that the shortest 
 distance afl! from a given point a to a given curve mn is a 
 normal to the curve at the point a' in which it meets it ; and 
 therefore, that if a circle be struck from this point a with this 
 least distance as a radius, then this circle must touch the 
 curve in the point ', and the curve and circle have a com- 
 mon normal in that point. 
 
 Now the points a and a will be brought by the revolution 
 of the pitch circles simultaneously to the point of contact A, 
 and the least distance of the curve mn from the point A will 
 then be #a', so that the arc ^7 will then be an arc struck 
 from the centre A, with this last distance for its radius. This 
 circular arc ^7 will therefore touch the curve mn in the point 
 a' and the line aa', which will then be a line drawn from 
 the point of contact A of the two pitch circles to the point 
 of contact ' of the two curves mn and m'n ', will also be a 
 normal to both curves at that point. The circles will there- 
 fore at that instant drive one another (Art. 196.) by the con- 
 tact of the surfaces mn and m'n' , precisely as they would by 
 the contact of their circumferences. And as every circular 
 arc of the curve m'n similar to ^7 becomes in its turn the 
 acting surface of the tooth, it will, in like manner, at one 
 point work truly with a corresponding point of mn, so that 
 the circles will thus drive one another truly at as many 
 points of the surfaces of their teeth, as there have been taken 
 points of division a, 5, &c. and arcs a/3, 7. &c.f 
 
 * This method of describing, geometrically, the forms of teeth is given, without 
 demonstration, by M. Poncelet in his Mecanique Industrielle, 3 me partie, Art. 60. 
 
 f The greater the number of these points of division, the more accurate the 
 form of the tooth. It appears, however, to be sufficient 
 in most cases, to take three points of division, or even 
 two, where no great accuracy is required. M. Poncelet 
 (Mee. Indust. 3 me partie, Art. 60.) has given the following, 
 yet easier, method by which the true form of the tooth 
 may be approximated to with sufficient accuracy in most 
 cases. Suppose the given tooth N upon the one wheel to 
 be placed in the position in which it is first to engage or 
 disengage from the required tooth on the other wheel, 
 and let Aa and A6 be equal arcs of the pitch circles of 
 the two wheels whose point of contact is A. Draw Aa 
 the shortest distance between A and the face of the tooth 
 N ; join aa; bisect that line in m, and draw mn perpendi- 
 cular to aa intersecting the circumference Aa in n. If 
 from the centre, n a circular arc be described passing 
 through the points a and a, it will give the required form 
 of the tooth nearly. 
 
234: 
 
 INVOLUTE TEETH. 
 
 INVOLUTE TEETH. 
 
 201. The teeth of two wheels will work truly together if they 
 he hounded hy curves of the form traced out hy the extremity 
 of a flexible line, unwinding from the circumference of a 
 circle, and called the involute of a circle, provided that the 
 circles of which these are the involutes he concentric with 
 the pitch circles of the wheels, and have their radii in the 
 same proportion with the radii of the pitch circles. 
 
 Let OE and OF represent the pitch circles of two wheels, 
 AG and BH two circles concentric with 
 them and having their radii C t A and C.,B 
 in the same proportion with the radii C X O 
 and C 2 O of the pitch circles. Also let mn 
 and m'n' represent the edges of teeth on the 
 two wheels struck by the extremities of flexi- 
 ble lines unwinding from the circumferences 
 of the circles AG and BH respectively. Let 
 these teeth be in contact, in any position 
 of the wheels, in the point T, and from the 
 point T draw TA and TB tangents to the 
 generating circles GA and BH in the points 
 A and B. Then does AT evidently represent the position of 
 the flexible line when its extremity was in the act of gene- 
 rating the point T in the curve mn / w r hence it follows, that 
 AT is a normal to the curve mn at the point T* ; and in 
 like manner that BT is a normal to the curve m'n' at the 
 same point T. Now the two curves have a 
 common tangent at T; therefore their nor- 
 mals TA and TB at that point are in the same 
 straight line, being both perpendicular to their 
 tangent there. Since then ATB is a straight 
 line, and that the vertical angles at the point 
 o where AB and CjC, intersect are equal, as 
 also the right angles at A and B, it follows 
 that the triangles AoC^and B<?0 2 are similar, 
 and that 0,o : C t o :: C,A : C 2 B. But C,A : 
 C Q B :: 0,0 : C 2 O ; .-. C t o : 0,0 :: 0,0: 
 C 2 O ; therefore the points O and o coincide, 
 and the straight line AB, which passes through the point of 
 
 * For if the circle be conceived a polygon of an infinite number of sides, it 
 is evident that the line, when in the act of unwinding from it at A, is turning 
 upon one of the angles of that polygon, and therefore that its extremity is, 
 through au infinitely small angle, describing a circular arc about that point. 
 
INVOLUTE TEETH. 235 
 
 contact T of the two teeth, and is perpendicular to the sur- 
 faces of both at that point, passes also through the point of 
 contact O of the pitch circles of the wheels. Now this is 
 true, whatever be the positions of the wheels, and whatever, 
 therefore, be the points of contact of the teeth. Tims then 
 the condition established in Art. 199. as that necessary and 
 sufficient to the true action of the teeth of wheels, viz. " that 
 a line drawn from the point of contact to the pitch circles tc 
 the point of contact of the teeth should be a normal to their 
 surfaces at that point, in all the different positions of the 
 teeth," obtains in regard to involute teeth.* 
 
 The point of contact T of the teeth moves along the straight- 
 line AB, which is drawn touching the generating circles BEL 
 and AG of the involutes ; this line is what is called the locus 
 of the different points of contact. Moreover, this property 
 obtains, whatever may be the number of teeth in contact at 
 once, so that all the points of contact of the teeth, if there 
 be more than one tooth in contact at once, lie always in this 
 
 line ; which is a characteristic, and a most important pro- 
 perty of teeth of the involute form. Thus in the above 
 
 * The author proposes the following illustration of the action of involute 
 teeth, which he believes to be new. Conceive AB to represent a band passing 
 round the circles AG and BH, the wheels would evidently be driven by this 
 band precisely as they would by the contact of their pitch circles, since the 
 radii of AG and BH are to one another as the radii of the pitch circles. Con- 
 ceive, moreover, that the circles BH and AG carry round with them their 
 planes as they revolve, and that a tracer is fixed at any point T of the band, 
 tracing, at the same time, lines mn and m'n', upon both planes, as they revolve 
 beneath it. It is evident that these curves, being traced by the same point, 
 must be in contact in all positions of the circles when driven by the band, and 
 therefore when driven by their mutual contact. The wheels would therefore 
 be driven by the contact of teeth of the forms mn and m'n' thus traced by the 
 point T of the band precisely as they would by the contact of their pitch cir- 
 cles. NoV it is easily seen, that the curves mn and m'n', thus described by the 
 point T of the band, are involutes of the circles AG and BH. 
 
236 EPICYCLOIDAL AND HYPCCYCLOIDAL TEETH. 
 
 figure, which represents part of two wheels with involute 
 teeth, it will be seen that the points r s of contact of the 
 teeth are in the same straight line touching the base* of one 
 of the involutes, and passing through the point of contact A 
 of the pitch circles, as also the points A and J in that touch- 
 ing the base of the other. 
 
 EPICYCLOIDAL AND HYPOOYCLOIDAL TEETH. 
 
 202. If one circle be made to roll externally on the cir- 
 cumference of another, and if, whilst this mo- 
 tion is taking place, a point in the circumfe- 
 rence of the rolling circle be made to trace 
 out a curve upon the plane of the fixed circle, 
 the curve so generated is called an EPICYCLOID, 
 the rolling circle being called the generating 
 circle of the epicycloid, and the circle upon 
 which it rolls its oase. 
 
 If the generating circle, instead of rolling 
 on the outside or convex circumference of its 
 base, roll on its inside or concave circumfe- 
 rence, the curve generated is called the HYPOCYCLOID. 
 
 Let PQ and Pit be respectively an epicycloid and a hypo- 
 cycloid, having the same generating circle APH, and 
 having for their bases the pitch circles AF and AE of two 
 wheels. If teeth be cut upon these wheels, whose edges 
 coincide with the curves PQ and PR, they will work truly 
 with one another ; for let them be in contact at P, and let 
 their common generating circle APH be placed so as to 
 touch the pitch circles of both wheels at A , then will its cir- 
 cumference evidently pass through the point of contact P 
 of the teeth : for if it be made to roll through an exceed- 
 ingly small angle upon the point A, rolling there upon the 
 circumference of loth circles, its generating point will 
 traverse exceedingly small portions of both curves; since 
 then a given point in the circumference of the circle APII 
 is thus shown to be at one and the same time in th perime- 
 ters of both the curves PQ and PR, that point must of 
 necessity be the point of contact P of the curves ; since, 
 
 * The circles from which the involutes are described are called their 
 This cut and that at page 237. are copied from Mr. Hawkins' edition of Camus 
 on the Teeth of Wheels. 
 
EPICYCLOIDAL AND HYPOCYCLOIDAL TEETH. 
 
 237 
 
 moreover, when the circle APH rolls upon the point A, its 
 generating point traverses a small portion of the perimeter 
 of each of the curves PQ and PR at P, it follows that the 
 line AP is a normal to both curves at that point ; for whilst 
 the circle APH is rolling through an exceedingly small 
 angle upon A, the point P in it, is describing a circle about 
 that point whose radius is AP.* Teeth, therefore, whose 
 edges are of the forms PQ and PR satisfy the condition 
 that the line AP drawn from the point of contact of the 
 pitch circles to any point of contact of the teeth is a normal 
 to the surfaces of both at that point, which condition has been 
 shown (Art. 199.) to be that necessary and sufficient to the 
 correct working of the teeth.f 
 Thus then it appears, that if an epicycloid be described 
 
 * The circle AFH may be considered a polygon of an infinite number of 
 sides, on one of the angles of which polygon it may at any instant be con- 
 ceived to be turning. 
 
 f The entire demonstration by which it has been here shown that the 
 curves generated by a point in the circumference of a given generating circle 
 APH rolling upon the convex circumference of one of the pitch circles, and 
 upon the concave circumference of the other are proper to form the edges of 
 contact of the teeth, is evidently applicable if any other generating curve be 
 substituted for APH. It may be shown precisely in the same manner, that 
 the curves PQ and PR generated by the rolling of any such curve (not being 
 a circle) upon the pitch circles, possess this property, that the line PA drawn 
 from any point of their contact to the point of contact of their pitch circles 
 is a normal to both, which property is necessary and sufficient to their correct 
 action as teeth. This was first demonstrated as a general principle of the con- 
 struction of the teeth of wheels by Mr. Airy, in the Cambridge Phil. Trans, 
 vol. ii. He has farther shown, that a tooth of any form whatever being cut 
 upon a wheel, it is possible to find a curve which, rolling upon the pitch circle 
 of that wheel, shall by a certain generating point traverse the edge of the 
 given tooth. The curve thus found being made to roll on the circumference 
 of the pitch circle of a second wheel, will therefore trace out the form of a 
 tooth which will work truly with the first. This beautiful property involves 
 
238 EPICYCLOID AL AND HYPOCYCLOIDAL TEETH. 
 
 on the plane of one of the wheels with any generating 
 circle, and with the pitch circle of that wheel for its base ; 
 and if a hypocycloid be described on the plane of the other 
 wheel with the pitch circle of that wheel for its base ; and 
 if the faces or acting surfaces of the teeth on the two 
 weeels be cut so as to coincide with this epicycloid and this 
 hypocycloid respectively, then will the wheels be driven 
 correctly by the intervention of these teeth. Parts of two 
 wheels having epicycloidal teeth are represented in the pre- 
 ceding figure. 
 
 203. LEMMA. If the diameter of the generating circle of a 
 hypocycloid equal the radius of its ~base, the hypocycloid 
 becomes a straight line having the direction of a radius of 
 its base. 
 
 Let D and d represent two positions of the centre of such 
 a generating circle, and suppose the 
 generating point to have been at A in 
 the first position, and join AC ; then 
 will the generating point be at P in the 
 second position, i. e. at the point where 
 CA intersects the- circle in its second 
 position; for join Ca and P<#, then 
 / Yda= Z PCW+ / CP^=2ACa. Also 
 
 ACa ; .'. arc A# =arc P#. Since then the arc aP equals 
 the arc A, the point P is that which in the first position 
 coincided with A, i. e. P is the generating point ; and this 
 is true for all positions of the generating circle ; the gene- 
 rating point is therefore always in the straight line AC. 
 The edge, therefore, of a hypocycloidal tooth, the diameter 
 of whose generating circle equals half the diameter of the 
 pitch circle of its wheel, is a straight line whose direction 
 is towards the centre of the wheel.* 
 
 the theoretical solution of the problem which Poncelet has solved by the 
 geometrical construction given to Article 200. If the rolling curve be a 
 logarithmic spiral, the involute form of tooth will be generated. 
 
 * The following very ingenious application has been made of this property 
 of the hypocycloid to convert a circular into an alternate rectilinear motion. 
 AB represents a ring of metal, fixed in position, and having teeth cut upon ita 
 
TO SET OUT THE TEETH OF WHEELS. 
 
 To SET OUT THE TEETH OF WHEELS. 
 
 204. All the teeth of the same wheel are constructed of 
 the same form and of equal dimensions : it would, indeed, 
 evidently be impossible to construct two wheels with dif- 
 ferent numbers of teeth, which should work truly with one 
 another, if all the teeth on each wheel were not thus alike. 
 
 All the teeth of a wheel are therefore set out by the work- 
 man from the same pattern or model, and it is in determining 
 the form and dimensions of this single pattern or model of 
 one or more teeth in reference to the mechanical effects 
 which the wheel is to produce, when all its teeth are cut out 
 upon it and it receives its proper place in the mechanical 
 combination of which it is to form a part, that consists the 
 art of the description of the teeth of wheels. 
 
 The mechanical function usually assigned to toothed wheels 
 is the transmission of work under an increased or diminished 
 velocity. If CD, DE, &c., represent arcs of the pitch circle 
 
 concave circumference. C is the centre of 
 a wheel, having teeth cut in its circum- 
 ference to work with those upon the circum- 
 ference of the ring, and having the diame- 
 ter of its pitch circle equal to half that of 
 the pitch circle of the teeth of the ring. 
 This being the case, it is evident, that if the 
 pitch circle of the wheel C were made to 
 roll upon that of the ring, any point in its 
 circumference would describe a straight line 
 passing through the centre D of the ring; 
 but the circle C would roll upon the ring by 
 the mutual action of their teeth as it wo ild 
 by the contact of their pitch circles ; if the 
 circle C then be made to roll upon the ring 
 by the intervention of teeth cut upon both, any point in the circumference of 
 C will describe a straight line passing through D. Now, conceive C to be thus 
 made to roll round the ring by means of a double or forked link CD, between 
 the two branches of which the wheel is received, being perforated at tlu-ii- 
 extremities by circular apertures, which serve as bearings to the solid axis of 
 the wheel. At its other extremity D, this forked link is rigidly connected 
 with an axis passing through the centre of the ring, to which axis is commu- 
 nicated the circular motion to be converted by the instrument into an alter- 
 nating rectilinear motion. This circular motion will thus be made to carry 
 the centre C of the wheel round the point D, and at the same time, cause it to 
 roll upon the circumference of the ring. Xow, conceive the axis C of the 
 wheel, which forms part of the wheel itself, to be prolonged beyond the collar 
 in which it turns, and to have rigidly fixed upon its extremity a bar CP. It is 
 evident that a point P in this bar, whose distance from the axis C of the wheel 
 equals the radius of its pitch circle, will move precisely as a point in the pitch circle 
 of the wheel moves, and therefore that it will describe continually a straight 
 line passing through the centre D of the ring. This point P receives, there 
 fore, the alternating rectilinear motion which it was required to communicate. 
 
240 
 
 TO SET OUT THE TEETH OF WHEELS. 
 
 of a wheel intercepted between similar points of consecutive 
 teeth (the chords of which arcs are called the pitches of the 
 teeth), it is evident that all these arcs must be equal, since 
 the teeth are all equal and similarly placed ; so that each 
 tooth of either wheel, as it passes through its contact with a 
 corresponding tooth of the other, carries its pitch line through 
 the same space CD, over the point of contact C of the pitch 
 lines. Since, therefore, the pitch line of the one wheel is 
 carried over a space equal to CD, and that of the other over 
 a space equal to cd by the contact of any two of their teeth, 
 and since the wheels revolve by the contact of their teeth 
 as they would by the contact of their pitch circles at C, it 
 follows that the arcs CD and cd are equal. Now let T\ and 
 r 2 represent the radii of the pitch circles of the two wheels, 
 then will 2^ and 2irr 9 represent the circumferences of their 
 pitch circles ; and if n^ and n 9 represent the numbers of 
 
 teeth cut on them respectively, then CD= l and cd = - 2 , 
 
 ,1 . . 
 
 therefore, --- > 
 
 n a 
 
 (227); 
 
 Therefore the radii of the pitch circles of the two wheels 
 must be to one another as the numbers of teeth to be cut 
 upon them respectively. 
 
 Again, let m, represent the number of revolutions made 
 by the first wheel, whilst m 2 revolutions are made by the 
 second ; then will ^irr l m l represent the space described by 
 
A TKAIN OF WHEELS. 24:1 
 
 the circumference of the pitch circle of the first wheel while 
 these revolutions are made, and 27rr 2 m 2 that described by the 
 circumference of the pitch circle of the second; but the 
 wheels revolve as though their pitch circles were in contact, 
 therefore the circumferences of these circles revolve through 
 equal spaces, therefore 27ir 1 m 1 =2'nT a m 2 ; 
 
 (228). 
 
 The radii of the pitch circles of the wheels are therefore 
 inversely as the numbers of revolutions made in the same 
 time by them. 
 
 Equating the second members of equations (227) and; (223)) 
 
 (229).. 
 
 The numbers of revolutions made by the wheel's in the same 
 time are therefore to one another inversely as the numbers 
 of teeth. 
 
 205. In a train of wheels, to determine how many revolutions 
 the last wheel makes whilst the first is making any given 
 number of revolutions. 
 
 When a wheel, driven by another, carries its axis round 
 
 with it, on which axis a third 
 wheel is fixed, engaging with and 
 giving motion to ^fourth, which, 
 in like manner, is fixed upon its 
 axis, and carries round with it a 
 fifth wheel fixed upon the same 
 axis, which fifth wheel engages 
 with a sixth upon another axis, 
 and so on as shown in the above figure, the combination 
 forms a train of wheels. Let n l9 n n . . . n^ represent the 
 numbers of teeth in the successive wheels forming such a 
 train of p pairs of wheels ; and whilst the first wheel is 
 making m revolutions, let the second and third (which revolve 
 together, being fixed on the same axis) make m^ revolutions ; 
 the fourth and fifth (which, in like manner, revolve together) 
 w 2 revolutions, the sixth and seventh m^ and so on ; and let 
 the last or 2p th wheel thus be made to revolve m p times whilst 
 
 16 
 
242 A TKAIN OF WHEELS. 
 
 the first revolves m times. Then, since the first wheel which 
 has n t teeth gives motion to the second which has n t teeth, 
 and that whilst the former makes m revolutions the latter 
 
 makes m, revolutions, therefore (equation 229), i = ; 
 
 and since, while thfe third wheel (which revolves with the 
 second, makes m a revolutions, the fourth makes ra 2 revolu- 
 tions ; therefore, 2 = . Similarly, since while the fifth 
 i m l n 4 
 
 wheel, which has n 6 teeth, makes m a revolutions (revolving 
 with the fourth), the sixth, which has n 6 teeth, makes m, revo- 
 lutions ; therefore - 3 = . In like manner 4 = , &c. &c. 
 m a n 6 m z nj 
 
 . Multiplying these equations together, and 
 
 striking out factors common to the numerator and denomi- 
 nator of the first member of the equation which results from 
 their multiplication, we obtain 
 
 m p n t . n z . n 6 . . . . n Zp -i f 
 
 = .... (230). 
 
 m n^ . n A . n 6 . . . . n^ p 
 
 The factors in the numerator of this fraction represent the 
 numbers of teeth in all the driving wheels of tfyis train, 
 and those in the denominator the numbers of teeth in the 
 driven wheels, or followers as they are more commonly 
 called. 
 
 If the numbers of teeth in the former be all equal and 
 represented by n^ and the numbers of teeth in the latter 
 also equal and represented by 7i a , then 
 
 >- = . (231). 
 
 m \n z / 
 
 Having determined what should be the number of teeth 
 in each of the wheels which enter into any mechanical 
 combination, with a reference to that particular modification 
 of the velocity of the revolving parts of the machine, which 
 is to be produced by that wheel,* it remains next to consider, 
 what must be the dimensions of each tooth of the wheel, so 
 
 * The reader is referred for a more complete discussion of this subject (which 
 belongs more particularly to descriptive mechanics) to Professor Willis's Prin- 
 ciples of Mechanism, chap, vii., or to Camus on the Teeth of Wheels, by Haw- 
 kins, p. 90. 
 
THE STRENGTH OF TEETH. 243 
 
 that it may be of sufficient strength to transmit the work 
 which is destined to pass through it, under that velocity, or 
 to bear the pressure which accompanies the transmission of 
 that work at that particular velocity ; and it remains further 
 to determine, what must be the dimensions of the wheel 
 itself consequent upon these dimensions of each tooth, and 
 this given number of its teeth. 
 
 206. To -determine the pitch of the teeth of a wheel, knowing 
 the work to be transmitted by the wheel. 
 
 Let U represent the number of units of work to be trans- 
 mitted by the wheel per minute, m the number of revolutions 
 to be made by it per minute, n the number of the teeth to 
 be cut iri it, T the pitch of each tooth in feet, P the pressure 
 upon each tooth in pounds. 
 
 Therefore nT represents the circumference of the pitch 
 circle of the wheel, and mnT represents the space in feet 
 described by it per minute. Now U represents the work 
 
 transmitted by it through this space per minute, therefore 1~ 
 
 represents the mean pressure under which this work is trans- 
 mitted (Art. 50.) ; 
 
 The pitch T of the teeth would evidently equal twice the 
 breadth of each tooth, if the spaces between the teeth were 
 equal in width to the teeth. In order that the teeth of 
 wheels which act together may engage with one another and 
 extricate themselves, with facility, it is however necessary 
 that the pitch should exceed twice the breadth of the tooth 
 by a quantity which varies according to the accuracy of the 
 construction of the wheel from T Vth to T V th of the breadth.* 
 
 Since the pitch T of the tooth is dependant upon its 
 breadth, and that the breadth of the tooth is dependant, by 
 the theory of the strength of materials, upon the pressure P 
 which it sustains, it is evident that the quantity P in the 
 above equation is a function of T. This functionf may be 
 assumed of the form 
 
 * For a full discussion of this subject see Professor Willis's Principles of 
 Mechanism, Arts. 107-112. 
 
 f See Appendix, on the dimensions of wheels. 
 
244 THE STRENGTH OF TEETH. 
 
 T=c 4/P (233) ; 
 
 where c is a constant dependant for its amount upon the 
 nature of the material out of which the tooth is formed. 
 Eliminating P between this equation and the last, and solving 
 in respect to T, 
 
 The number of units of work transmitted by any machine 
 per minute is usually represented in horses' power, one 
 horse's power being estimated at 33,000 units, so that the 
 number of horses' power transmitted by the machine means 
 the number of times 33,000 units of work are transmitted by 
 it every minute, or the number of times 33,000 must be 
 taken to equal the number of units of work transmitted by 
 it every minute. If therefore H represent the number of 
 horses' power transmitted by the wheel, then U=33,OOOH. 
 Substituting this value in the preceding equation, and repre- 
 senting the constant 33,000c 2 by C 8 , we have 
 
 '. . (234). 
 
 mn 
 
 The values of the constant C for teeth of different mate- 
 rials are given in the Appendix. 
 
 207. To determine the radius of the pitch circle of a wheel 
 which shall contain n teeth of a given pitch. 
 
 Let AB represent the pitch T of a tooth, 
 and let it be supposed to coincide with its 
 chord AMB. Let E represent the radius AC 
 of the pitch circle, and n the number of teeth 
 to be cut upon the wheel. 
 
 Now there are as many pitches in the cir- 
 cumference as teeth, therefore the angle ACB 
 
 subtended by each pitch is represented by . 
 
 Also T=2AM=2AC sin. ACB=2Esin. - ; 
 
 n 
 
 "" (235). 
 
TO DESCRIBE EPICYCLOIDAL TEETH. 245 
 
 208. To make the pattern of an epicycloidal tooth. 
 
 Having determined, as above, 
 the pitch of the teeth, and the 
 radius of the pitch circle, strike 
 an arc of the pitch circle on a 
 thin piece of oak board or me- 
 tal plate, and, with a fine saw, 
 cut the board through along 
 the circumference of this cir- 
 cle, so as to divide it into two 
 parts, one having a convex and 
 the other a corresponding con- 
 cave circular edge. Let EF 
 represent one of these portions 
 of the board, and GH another. 
 
 Describe an arc of the pitch circle upon a second board or 
 plate from which the pattern is to be cut. Let MN repre- 
 sent this arc. Fix the piece GH upon this board, so that its 
 circular edge may accurately coincide with the circumference 
 of the arc SOT. Take, then, a circular plate D of wood or 
 metal, of the dimensions which it is proposed to give to the 
 generating circle of the epicycloid ; and let a small point of 
 steel P be fixed in it, so that this point may project slightly 
 from its inferior surface, and accurately coincide with its cir- 
 cumference. Having set off the width AB of the toothj so 
 that twice this width increased by from T Vth to T Vth of that 
 width (according to the accuracy of workmanship to be 
 attained) may equal the pitch, cause the circle D to roll upon 
 the convex edge GK of the board GH, pressing it, at the 
 same time, slightly upon the surface of the board on which 
 the arc IVCST is described, and from which the pattern is to he 
 cut, having caused the steel point in its circumference first 
 of all to coincide with the point A ; an epicycloidal arc AP 
 will thus be described by the point P upon the surface MN". 
 Describe, similarly, an epicycloidal arc BE through the point 
 B, and let them meet in E. 
 
 Let the board GHnow be removed, and let EF be applied 
 and fixed, so that its concave edge may accurately coincide 
 with the circular arc MK With the same circular plate D 
 pressed upon the concave edge of EF, and made to roll upon 
 it, cause the point in its circumference to describe in like 
 manner, upon the surface of the board from which the pat- 
 tern is to be cut, a hypococloidal arc BH passing through the 
 
246 
 
 TO DESCKIBE EPICYCLOIDAL TEETH. 
 
 point B, and another AI passing through the point A. HEI 
 will then represent the form of a tooth, which will work cor- 
 rectly (Art. 202.) with the teeth similarly cut upon any other 
 wheel ; provided that the pitch of the teeth so cut upon the 
 other wheel be equal to the pitch of the teeth upon this, and 
 provided that the same generating circle D l)e used to strike 
 the curves upon the two wheels. 
 
 209. To determine the proper lengths of epicycloidal teeth. 
 
 The general forms of the teeth of wheels being determined 
 by tlie method explained in the preceding article, it remains 
 to cut them off of such lengths as may cause them succes- 
 sively to take up the work from one another, and transmit it 
 under the circumstances most favourable to the economy of 
 its transmission, and to the durability of the teeth. 
 
 In respect to the economy of the power in its transmission, 
 it is customary, for reasons to be assigned hereafter, to pro- 
 vide that no tooth of the one wheel should come into action 
 with a tooth of the other until both are in the act of passing 
 through the line of centres. This condition may be satisfied 
 in all cases where the numbers of teeth on neither of the 
 wheels is exceedingly small, by properly adjusting the 
 lengths of the teeth. Let two of the teeth of the wheels be 
 in contact at the point A in the line CD, joining the centres 
 of the two wheels ; and let the wheel whose centre is C be 
 the driving wheel. Let AH be a portion of the circumfe- 
 rence of the generating circle of the teeth, then will the 
 points A and !>, where this circle intersects the edges of the 
 
 
 /\ 
 
 
 X 
 
 f 1* \ 
 
 ^==: Sit cf^ 
 
 
 
 &". -:_ 
 
 PI* 
 
 H* 
 
 v \ 
 
 teeth O and K of the driving wheel, be points of contact 
 
TO DESCRIBE EPICYCLOIDAL TEETH. 24:? 
 
 with the edges of the teeth M and L of the driven wheel 
 (Art. 202.). JSTow, since each tooth is to come into action 
 only when it comes into the line of centres, it is clear that 
 the tooth L must have been driven by K from the time when 
 their contact was in the line of centres, until they have come 
 into the position shown in the figure, when the point of con- 
 tact of the anterior face of the next tooth O of the driving 
 wheel with the flank* of the next tooth M of the driven 
 wheel has just passed into the line of centres ; and since the 
 tooth O is now to take up the task of impelling the driven 
 wheel, and the tooth K to yield it, all that portion of the 
 last-mentioned tooth which lies beyond the point B may evi- 
 dently be removed ; and if it ~be thus removed, then the tooth 
 K, passing out of contact, will manifestly, at that period of 
 the motion, yield all the driving strain to the tooth O, as it 
 k required to do. In order to cut the pattern tooth of the 
 
 proper length, so as to satisfy 
 the proposed condition, we have 
 only then to take A.a (see the 
 accompanying figure) equal to 
 the pitch of the tooth, and to 
 bring the convex circumference 
 of the generating circle, so as 
 to touch the convex circumfe- 
 rence of the arc MK in that 
 point a ; the point of intersec- 
 tion e of this circle with the 
 N 8 face AE of the tooth will be 
 the last acting point of the tooth ; and if a circle be struck 
 from the centre of the pitch circle passing through that 
 point, all that portion of the tooth which lies beyond this cir- 
 cle may be cut off.f 
 
 The length of the tooth on the wheel intended to act with 
 this, may be determined in like manner. 
 
 210. In the preceding article we have supposed the same 
 generating circle to be used in striking the entire surfaces 
 of the teeth on both wheels. It is not however necessary to 
 
 * That portion of the edge of the tooth which is without the pitch circle is 
 called its face, that within it its flank. 
 
 f The point e thus determined will, in some cases, fall beyond the extremity 
 E of the tooth. In such cases it is therefore impossible to cut the tooth of 
 such a length as to satisfy the required conditions, viz. that it shall drive only 
 after it has passed the line of centres. A full discussion of these impossible 
 cases will be found in Professor Willis's work (Arts. 102-104.). 
 
24:8 
 
 TO DESCRIBE EPICYCLOIDAJL TEETH. 
 
 the correct working of the teeth, that the same circle should 
 thus be used in striking the entire surfaces of ttwo teeth 
 which act together, but only that the generating circle of 
 every two portions of the two teeth which come into actual 
 contact should be the same. Thus the flank of the driving 
 tooth and the face of the driven tooth being in contact at 
 
 P in the accompanying figure,* this face of the one tooth 
 and flank of the other must be respectively an epicycloid 
 and a hypocycloid struck with the same generating circle. 
 Again, the face of a driving tooth and the flank of a driven 
 tooth being in contact at Q, these, too, must be struck by 
 the same generating circle. But it is evidently unnecessary 
 that the generating circle used in the second case should be 
 the same as that used in the first. Any generating circle 
 will satisfy the conditions in either case (Art. 202.), provided 
 it be the same for the epicycloid as for the hypocycloid 
 which is to act with it. 
 
 According to a general (almost a universal) custom among 
 mechanics, two different generating circles are thus used for 
 striking the teeth on two wheels which are to act together, 
 the diameter of the generating circle for striking the faces 
 of the teeth on the one wheel being equal to the radius of 
 the pitch circle of the other wheel. Thus if we call the 
 wheels A and B, then the epicycloidal faces of the teeth on 
 A, and the corresponding hypocycloidal flanks on B, are 
 generated by a circle whose diameter is equal to the radius 
 of the pitch circle of B. The hypocycloidal flanks of the 
 teeth on B thus become straight lines (Art. 203.), whose 
 directions are those of radii of that wheel. In like manner, 
 
 * The upper wheel is here supposed to drive the lower. 
 
TO DESCRIBE EPICYCLOIDAL TEETH. e 249 
 
 the epicycloidal faces of the teeth on B, and the correspond- 
 ing hypocycloidal flanks of the teeth on A, are struck by a 
 circle whose diameter is equal to the radius of the pitch cir- 
 cle of A ; so that the hypocycloidal flanks of the teeth of A 
 become in like manner straight lines, whose directions are 
 those of radii of the wheel A. By this expedient of using 
 two different generating circles, the flanks of the teeth on 
 both wheels become straight lines, and the faces only are 
 curved. The teeth shown in the above figure are of this 
 form. The motive for giving this particular value to the 
 generating circle appears to be no other than that saving of 
 trouble which is offered by the substitution of a straight for 
 a curved flank of the tooth. A more careful consideration 
 of the subject, however, shows that there is no real economy 
 of labour in this. In the first place, it renders necessary 
 the use of two different generating circles or templets for 
 striking the teeth of any given wheel or pinion, the curved 
 portions of the teeth of the wheel being struck with a circle 
 whose diameter equals half the diameter of the pinion, and 
 the curved portions of the teeth of the pinion with a circle 
 whose diameter equals half that of the wheel. Now, one 
 generating circle would have done for both, had the work- 
 man been contented to make the flanks of his teeth of the 
 hypocycloidal forms corresponding to it. But there is yet a 
 greater practical inconvenience in this. method. A wheel 
 and pinion thus constructed will only work with one another / 
 neither will work truly any third wheel or pinion of a differ- 
 ent number of teeth, although it have the same pitch. Thus 
 the wheels A and B having each a given number of teeth, 
 and being made to work with one another, will neither of 
 them work truly with C of a different number of teeth of 
 the same pitch. For that A may work truly with C, the 
 face of its teeth must be struck with a generating circle, 
 whose diameter is half that of C : but they are struck with 
 a circle whose diameter is half that of B ; the condition of 
 uniform action is not therefore satisfied. Now let us sup- 
 pose that the epicycloidal faces, and the hypocycloidal flanks 
 of all the teeth A, B, and C had been struck with the same 
 generating circle, and that all three had been of the same 
 pitch, it is clear that any one of them would then have 
 worked truly with any other, and that this would have been 
 equally true of any number of teeth of the same pitch. 
 Thus, then, the machinist may, by the use of the same gen- 
 erating circle, for all his pattern wheels of the same pitch, so 
 construct them, as that any one wheel of that pitch shall 
 
250 TO DESCRIBE EPICYCLOIDAL TEETH. 
 
 work with any other. This offers, under many circumstances 
 great advantages, especially in the very great reduction of 
 the number of patterns which he will be required to keep. 
 There are, moreover, many cases in which some arrange- 
 ment similar to this is indispensable to the true working of 
 the wheels, as when one wheel is required (which is often 
 the case) to work with two or three others, of different num- 
 bers of teeth, A for instance to turn B and C ; by the ordi- 
 nary method of construction this combination would be 
 impracticable, so that the wheels should work truly. Any 
 generating circle common to a whole set of the same pitch, 
 satisfying the above condition, it may be asked whether 
 there is any other consideration determining the best dimen- 
 sions of this circle. There is such a consideration arising 
 out of a limitation of the dimensions of the generating circle 
 of the hypocycloidal portion of the tooth to a diameter not 
 greater than half that of its base. As long as it remains 
 within these limits, the hypocycloidal generated by it is of 
 that concave form by which the flank of the tooth is made 
 to spread itself, and the base of the tooth to widen ; when 
 it exceeds these limits, the flank of the tooth takes the con- 
 vex form, the base of the tooth is thus contracted, and its 
 strength diminished. Since then, the generating circle 
 should not have a diameter greater than half that of any of 
 the wheels of the set for which it is used, it will manifestly 
 be the greatest which will satisfy this condition when its 
 diameter is equal to half that of the least wheel of the set. 
 Now no pinion should have less than twelve or fourteen 
 teeth. Half the diameter of a wheel of the proposed pitch, 
 which has twelve or fourteen teeth, is then the true diame- 
 ter or the generating circle of the set. The above sugges- 
 tions are due to Professor Willis.* 
 
 * Professor Willis has suggested a new and very ingenious method of 
 striking the teeth of wheels by means of circular arcs. A detailed description 
 of this method has been given by him in the Transactions of the Institution 
 of Civil Engineers, vol. ii., accompanied by tables, &c., which render its prac 
 tical application exceedingly simple and easy. 
 
TO DESCEIBE INVOLUTE TEETH. 
 
 251 
 
 211. To DESCRIBE INVOLUTE TEETH. 
 
 Let AD and AG represent the pitch circles of 
 two wheels intended to work together. Draw a 
 straight line FE through the point of contact A 
 of the pitch circles and inclined to the line of 
 centres CAB of these wheels at a certain angle 
 FAG, the influence of the dimensions of which 
 on the action of the teeth will hereafter be ex- 
 plained, but which appears usually to be taken 
 not less than 80.* Describe two circles eEK 
 and /*FL from the centres B and C, each touching the 
 straight line EF. These circles are to be taken as the bases 
 from which the involute faces of the teeth are to be struck. 
 It is evident (by the similar triangles ACF and AEB) that 
 their radii CF and BE will be to one another as the radii 
 CA and BA of the pitch circles, so that the condition neces- 
 sary (Art. 201.) to the correct action of the teeth of the 
 wheels will be satisfied, provided their faces be involutes to 
 
 these two circles. Let AG and AH in the above figure 
 represent arcs of the pitch circles of the wheels on an 
 enlarged scale, and 0E, /X, corresponding portions of the 
 circles eEK and yFL of the preceding figure. Also let A.a 
 represent the pitch of one of the teeth of either wheel. 
 Through the points A and a describe involutes ef and mn.\ 
 
 * See Camus on the Teeth of Wheels, by Hawkins, p. 168. 
 
 f Mr. Hawkins recommends the following as a convenient method of striking 
 involute teeth, in his edition of " Camus on the Teeth of Wheels," p. 166. Take 
 a thin board, or a plate of metal, and reduce its edge MX so as accurately to 
 
252 TO DESCEIBE INVOLUTE TEETH. 
 
 Let 5 be the point where the line EF intersects the involute 
 mn ; then if the teeth on the two wheels are to be nearly of 
 the same thickness at their bases, bisect the line AJb in c ; or 
 if they are to be of different thicknesses, divide the line Ab 
 in c in the same proportion*, and strike through the point c 
 an involute curve hg, similar to ef, but inclined in the oppo- 
 site direction. If the extremity^ of the tooth be then cut 
 off so that it may just clear the circumference of the circle 
 y*L, the true form of the pattern involute tooth will be 
 obtained/ 
 
 There are two remarkable properties of involute teeth, by 
 the combination of which they are distinguished from teeth 
 of all other forms, and cceteris paribus rendered greatly pre- 
 ferable to all others. The lirst of these is, that any two 
 wheels having teeth of the involute form, and of the same 
 pitch,t will work correctly together, since the forms of the 
 teeth on any one sucli wheel are entirely independent of 
 those on the wheel which is destined to work with it (Art. 
 201.) Any two wheels with involute teeth so made to work 
 together will revolve precisely as they would by the actual 
 contact of two circles, whose radii may be found by divid- 
 ing the line joining their centres in the proportion of the 
 radii of the generating circles of the involutes. This pro- 
 perty involute teeth possess, however, in common with the 
 epicycloidal teeth of different wheels, all of which are struck 
 with the same generating circle (Art. 210.) The second no 
 less important property of involute teeth a property which 
 distinguishes them from teeth of all other forms is this, 
 that they work equally well, however far the centres of the 
 
 coincide with the circular arc 
 - JB eE, and let a piece of thin 
 watch-spring OR, having two 
 projecting points upon it as 
 shown at P, and which is of a 
 width equal to the thickness of the plate, be fixed upon its edge by means of 
 a screw 0. Let the edge of the plate be then made to coincide with the arc 
 eE in such a position that, when the spring is stretched, the point P in it may 
 coincide with the point from which the tooth is to be struck ; and the spring 
 being kept continually stretched, and wound or unwound from the circle, the 
 involute arc is thus to be described by the point P upon the face of the board 
 from which the pattern is to be cut. 
 
 * This rule is given by Mr. Hawkins (p. 170.); it can only be an approxima- 
 tion, but may be sufficiently near to the truth for practical purposes. It is to 
 be observed that the teeth may have their bases in any other circles than 
 those, /L and eE, from which the involutes are struck. 
 
 f The teeth being also of equal thicknesses at their bases, the method of 
 ensuring which condition has been explained above. 
 
THE TEETH OF A BACK AND PINION. 253 
 
 wheels are removed asunder from one another / so that the 
 action of the teeth of two wheels is not impaired when 
 their axes are displaced by that wearing of their brasses or 
 collars, which soon results from a con- 
 tinued and a considerable strain. The 
 existence of this property will readily be 
 admitted, if we conceive AG and BH to 
 represent the generating circles o/bases 
 of the teeth, and these to be placed with 
 their centres Cj and C 2 any distance 
 asunder, a band AB (p. 235., note) passing 
 round both, and a point T in this band 
 generating a curve mn, m' n' on the plane 
 of each of the circles as they are made to 
 revolve under it. It has been shown that 
 these curves mn and m r n' will represent the faces of two 
 teeth which will work truly with one another ; moreover, 
 that these curves are respectively involutes of the two 
 circles AG and BH, and are therefore wholly independent 
 in respect to their forms of the distances of the centres of 
 the circles from one another, depending only on the dimen- 
 sions of the circles. Since then the circles would drive at 
 any distance correctly by means of the band ; since, more- 
 over, at every such distance they would be driven by the 
 curves mn and m'n' precisely as by the band ; and since 
 these curves would in every such position be the same 
 curves, viz. involutes of the two circles, it follows that the 
 same involute curves inn and m'n' would drive the circles 
 correctly at whatever distances their centres were placed ; 
 and, therefore, that involute teeth would drive these wheels 
 correctly at whatever distances the axes of those wheels 
 were placed. 
 
 THE TEETH or A KAOK AND PINION. 
 
 212. To determine the pitch circle of the pinion. Let H 
 represent the distance through which the rack is to be 
 moved by each tooth of the pinion, and let these teeth be 
 N in number ; then will the rack be moved through the 
 space N . H during one complete revolution of the wheel. 
 !N"ow the rack and pinion are to be driven by the action of 
 their teeth, as they would by the contact of the circum- 
 
254 
 
 THE TEETH OF A HACK AND PTNIOX. 
 
 ference of the pitch circle of the 
 pinion with the plane face of the 
 rack, so that the space moved through 
 by the rack during one complete 
 revolution of the pinion must pre- 
 cisely equal the circumference of the 
 pitch circle of the pinion. If, there- 
 fore we call R the radius of the 
 pitch circle of the pinion, then 
 
 213. To describe the teeth of the 
 pinion, those of the rack being 
 straight. The properties which have 
 been shown to belong to involute 
 teeth (Art. 201.) manifestly obtain, 
 however great may be the dimensions of the pitch circle 
 of their wheels, or whatever disproportion 
 may exist between them. Of two wheels 
 OF and OE with involute teeth which 
 work together, let then the radius of the 
 pitch circle of one OF become infinite, its 
 circumference will then become a straight 
 line represented by the face of a rack. 
 Whilst the radius C 2 O of the pitch circle 
 OF thus becomes infinite, that C 2 B of the 
 circle from which its involute teeth are 
 struck (bearing a constant ratio to the first) 
 will also become infinite, so that the invo- 
 lute m'n' will become a straight line* perpendicular to the 
 line AB given in position. The involute teeth on the 
 wheel OF will thus become straight teeth (see fig. 1.), hav- 
 ing their faces perpendicular to the line AB determined by 
 drawing through the point O a tangent to the circle AC, 
 from which the involute teeth of the pinion are struck. If 
 the circle AC from which the involute teeth of the pinion 
 are struck coincide with its pitch circle, the line AB becomes 
 
 * For it is evident that the extremity of a line of infinite length unwinding 
 itself from the circumference of a circle of infinite diameter will describe, 
 through a finite space, a straight line perpendicular to the circumference of 
 the circle. The idea of giving an oblique position to the straight faces of the 
 teeth of a rack appears first to have occurred to Professor Willis. 
 
THE TEETH OF A RACK AND PINION. 
 
 255 
 
 parallel to the face of the rack, and the edges of the teeth 
 of the rack perpendicular to its face (fig. 2.). 
 
 Now, -the involute teeth of the one wheel have remained 
 unaltered, and the truth of their action with teeth of the 
 other wheel .has not been influenced by that change in the 
 dimensions of the pitch circle of the last, which has con- 
 verted it into a rack, and its curved into straight teeth. 
 Thus, then, it follows, that straight teeth upon a rack, work 
 truly with involute teeth upon a pinion. Indeed it is evi- 
 
 ct-) 
 
 (2.) 
 
 dent, that if from the point of contact P (fig. 2.) of such an 
 involute tooth of the pinion with the straight tooth of a 
 rack we draw a straight line *PQ parallel to the face ab of 
 the rack, that straight line will be perpendicular to the 
 surfaces of both the teeth at their point of contact P, and 
 that being perpendicular to the face of the involute tooth, 
 it "will also touch the circle of which this tooth is the invo- 
 lute in the point A, at which the face ab of the rack would 
 touch that circle if they revolved by mutual contact. Thus, 
 then, the condition shown in Art. 199. to be necessary and 
 sufficient to the correct action of the teeth, namely, that a 
 line drawn from their point of contact, at any time, to the 
 point of contact of their pitch circles, is satisfied in respect 
 to these teeth. Divide, then, the circumference of the 
 pitch circle, determined as above (Art. 212.), into N equal 
 
256 
 
 THE TEETH OF A BACK AND PINION. 
 
 parts, and describe (Art. 211.) a pattern involute tooth from 
 the circumference of the pitch circle, limiting the length of 
 the face of the tooth to a little more than the length 5P of 
 the involute curve generated by unwinding a length AP of 
 the flexible line equal to the distance H through which the 
 rack is to be moved by each tooth of the pinion. The 
 straight teeth of the rack are to be cut of the same length, 
 and the circumference of the pitch circle and the face ao of 
 the rack placed apart from one another by a little more 
 than this length. 
 
 It is an objection to this last application of the involute 
 form of tooth for a pinion working with a rack, that the 
 point P of the straight tooth of the rack upon which it acts 
 is always the same, being determined by its intersection with 
 a line AP touching the pitch circle, and parallel to the face 
 of the rack. The objection does not apply to the preceding, 
 the case (fig. 1.) in which the straight faces of each tooth of 
 the rack are inclined to one another. By the continual 
 action upon a single point of the tooth of the rack, it is 
 liable to an excessive wearing away of its surface. 
 
 214. To describe the teeth of the pinion, the teeth of the rack 
 leing curved. 
 
 This may be done by giving to the face of the tooth of 
 
 the rack a cycloidal form, and making the face of the tooth 
 of the pinion an epicycloid, as will be apparent if we con- 
 K. ceive the diameter of the circle whose 
 
 centre is C (see fig. p. 236.) to become 
 infinite, the other two circles remain- 
 ing unaltered. Any finite portion of 
 the circumference of this infinite circle 
 will then become a straight line. Let 
 AE in the accompanying figure repre- 
 
THE TEETH OF A WHEEL WITH A LANTERN. 257 
 
 sent such a portion, and let PQ and PR represent, as 
 before, curves generated by a point P in the circle whose 
 centre is D, when all three circles revolve by their mutual 
 contact at A. Then are PR and PQ the, true forms of the 
 teeth which would drive the circles as they are driven by 
 their mutual contact at A (Art. 202). Moreover, the curve 
 PQ is the same (Art. 199.) as would be generated by the 
 point P in the circumference of APH ; if that circle rolled 
 upon the circumference AQF, it is therefore an epicycloid / 
 and the curve PR is the same as would be generated by the 
 point P, if the circle APH rolled upon the circumference 
 or straight line AE, it is therefore a cycloid. Thus then it 
 appears, that after the teeth have passed the line of centres, 
 when the face of the tooth of the pinion is driving the flank 
 of the tooth of the rack, the former must have an epicy- 
 cloidal, and the latter a cycloidal form. In like manner, by 
 transferring the circle APH to the opposite side of AE, it 
 may be shown, that before the teeth have passed the line of 
 centres when the flank of the tooth of the pinion is driving 
 the face of the tooth of the wheel, the former must have a 
 hypocycloidal, and the latter a cycloidal form, the cycloid 
 having its curvature in opposite directions on the flank and 
 the face of the tooth. The generating circle will be of the 
 most convenient dimensions for the description of the teeth 
 when its diameter equals the radius of the pitch circle of 
 the pinion. The hypocycloidal flank of the tooth of the 
 pinion will then pass into a straight flank. The radius of 
 the pitch circle of the pinion is determined as in Art. 212., 
 and the method of describing its teeth is explained in 
 Art. 208. 
 
 15. THE TEETH OF A WHEEL WORKING WITH A LANTERN OR 
 
 TRUNDLE. 
 
 In some descriptions of mill work the ordinary form of 
 the toothed wheel is replaced by a contrivance called a lan- 
 tern or trundle, formed by two circular discs, which are con- 
 nected with one another by cylindrical columns called 
 staves, engaging, like the teeth of a pinion, with the teeth 
 of a wheel which the lantern is intended to drive. This 
 combination is shown in the following figure. 
 
 It is evident that the teeth on the wheel which works with 
 the lantern have their shape determined by the cylindrical 
 
258 
 
 THE TEETH OF A WHEEL WITH A LANTEBN. 
 
 shape of the staves. Their forms may readily be found by 
 the method explained in Art. 200. 
 
 Having determined npon the dimensions of the staves in 
 reference to the strain they are to be subjected to, and upon 
 the diameters of the pitch circles of the lantern and wheel, 
 and also upon the pitch of the teeth ; strike arcs AB and 
 AC of these circles, and set off upon them 
 the pitches A.a and A5 from the point of 
 contact A of the pitch circles (if the teeth 
 are first to come into contact in the line 
 of centres, if not, set them off from the 
 points behind the line of centres where 
 the teeth are first to come into contact). 
 Describe a circle #<?, having its centre in 
 AB, passing through #, and having its 
 diameter equal to that of the stave, and divide each of the 
 pitches Aa and A5 into the same number of equal parts 
 (say three). From the points of division A, a, /3 in the 
 pitch A#, measure the shortest distances to the circle #0, and 
 with these shortest distances, respectively, describe from the 
 points of division 7, <$ of the pitch A5, circular arcs inter- 
 secting one another ; a curve ah touching all these circular 
 arcs will give the true face of the tooth (Art. 200.). The 
 opposite face of the tooth must be struck from similar cen- 
 tres, and the base of the tooth must be cut so far within the 
 pitch circle as to admit one half of the stave ae when that 
 stave passf*s the line of centres. 
 
PRESSURES UPON WHEELS. 
 
 259 
 
 216. THE RELATION BETWEEN TWO PRESSURES P t AND P a 
 
 APPLIED TO TWO TOOTHED WHEELS IN THE STATE BORDER- 
 ING- UPON MOTION BY THE PREPONDERANCE OF P a . 
 
 Let the influence of the weights of the wheels be in the 
 first place neglected. Let B and C represent the centres of 
 the pitch circles of the wheels, A their point of contact, P 
 the point of contact of the driving and driven teeth at any 
 period of the motion, KP the direction of the whole 
 resultant pressure upon the teeth at their point of contact, 
 which resultant pressure is equal and opposite to the resist- 
 ance B- of the follower to the driver, BM and CN perpen- 
 diculars from the centres of the axes of the wheels upon IIP ; 
 and BD and CE upon the directions of P l and P a . 
 
 p i5 p 2 =radii of axes of wheels. 
 
 <p l5 <p 2 =r limiting angles of resistance between the axes of 
 the wheels and their bearings. 
 
 Then, since P a and B, applied to the wheel whose centre is 
 
260 RELATION OF THE DRIVING AND WORKING 
 
 B are in the state bordering upon motion by the preponder- 
 ance of Pj, and since a t and m^ are the perpendiculars on 
 the directions of these pressures respectively, we have (equa- 
 tion 158) 
 
 where L x represents the length of the line DM joining the 
 feet of the perpendiculars BM and BD. 
 
 Again, since B. and P 2 , applied to the wheel whose centre 
 is C, are in the state bordering upon motion by the yielding 
 
 of P 3 (Art. 164.), 
 
 where L 2 represents the distance NE between the feet of the 
 perpendiculars CE and GN. Eliminating B. between these 
 equations, we have 
 
 F .= (^H ' .? ' 'k (238). 
 
 Now let it be observed, that the line AP, drawn from the 
 point of contact A of the pitch circles to the point of contact 
 P of the teeth is perpendicular to their surfaces at that point 
 P, whatever may be the forms of the teeth, provided that 
 they act truly with one another (Art. 199.) ; moreover, that 
 when the point of contact P has passed the line of centres, 
 as shown in the figure, that point is in the act of moving on 
 the driven surface Ppfrom the centre 0, or from P towards 
 p, so that the friction of that surface is exerted in the opposite 
 direction, or from p towards P ; whence it follows that the 
 resultant of this friction, and the perpendicular resistance aP 
 of the driven tooth upon the driver, lias its direction rP 
 within the angle aPp and that it is inclined (Art. 141.) to the 
 perpendicular aP at an angle aPr equal to the limiting angle 
 of resistance. Now this resistance is evidently equal and 
 opposite to the resultant pressure upon the surfaces of the 
 teeth in the state bordering upon motion ; whence it follows 
 that the angle EPA is equal to the limiting angle of resist- 
 ance between the surfaces of contact of the teeth. Let this 
 angle be represented by 9, and let AP=X. Also let the 
 
PRESSURES UPON WHEELS. 
 
 261 
 
 inclination PAC of AP to the line of centres BC be repre- 
 sented by 0. Through A draw An perpendicular to KP, and 
 sAt parallel to it. Then, 
 
 m 1 =EM=Kt + m==Et+An=BA sin. BArf+ AP sin. APE- 
 Also BAz5=BOE=:PAC+ APE=d+9 ; 
 
 /. m v r^ sin. (d + 9)-f X sin. 9 (239); 
 
 m,=CN=Cs-s^=Cs-An=CA sin. CAs-AP sin. APE. 
 But As is parallel to PE, therefore CAs=BOE=0+9; 
 
 /. m a =7* a sin. (^4-9) X sin. 9 (240.). 
 
 Substituting these values of ra x and ra a in the preceding 
 equation, 
 
 -te\H 
 
 l ~\aj 
 
 . 9+ 
 
 a, 
 
 Xsin.9 _?sin.9 3 
 
 217. In the preceding investigation the point of contact P 
 
262 RELATION OF THE DRIVING AND WORKING 
 
 of the teeth of the driving and driven wheels is supposed to 
 have passed the line of centres, or to be behind that line ; 
 let us now suppose it not to have passed the line of centres, 
 or to be before that line. 
 
 It is evident that in this case the point of contact P is ir 
 the act of moving upon the surface pPq of the driven tooth 
 towards the centre C, or from P towards , as in the other 
 case it is from the centre, or from P towards^. In this case, 
 therefore, the friction of the driven surface is exerted in the 
 direction qP ; whence it follows, that in this state bordering 
 upon motion the direction of the resistance R of the driven 
 upon the driving tooth must lie on the other side of the 
 normal APQ, being inclined to it at an angle APN equal to 
 the limiting angle of resistance. Thus the inclination of B, 
 to the normal APQ is in both cases the same, but its position 
 in respect to that line is in the one case the reverse of its 
 position in the other case.* 
 
 The same construction being made as before, 
 
 m 1 =BM=B+M;=:B + A^=BA. sin. BA+ AP. sin. APO. 
 Also BA=BOK:=BAP APO=4 9 ;f 
 /. m 1 =r 1 sin. (6 <?)+* sin. <p, 
 
 m,=C'N=Cs-sN=Cs A.n=CA. sin. CAs AP. sin. APO. 
 But As is parallel to PN, 
 
 /. m a =r 2 sin. (6 9) X sin. 9. 
 Substituting these values of m 1 and ra 2 in equation (238), 
 
 ^sin. (d 9)+Xsin. 9 + I ijsin. <p 4 
 p _ /a,\ I * . ; 
 
 x ~ fe/ 
 
 r a sin. (d 9) Xsin. 9 i^-^)sin. 9 2 
 
 \ ^2 ' 
 
 This expression differs from the preceding (equation 241) 
 ily in the substitution of (d 9) foi 
 of the numerator and denominator. 
 
 only in the substitution of (d 9) for (d-f?) in the first terms 
 th 
 
 * Hence it follows, that when the point of contact is in the act of crossing 
 the line of centres, the direction of the resultant pressure R is passing from 
 one side to the other of the perpendicular APQ ; and therefore that when the 
 point of contact is in the line of centres, the resultant pressure is perpendicu- 
 lar to that line, and the angle BOR a right angle ; a condition which cannot 
 however be assumed to obtain approximately in respect to positions of any 
 point of contact exceedingly near to the line of centres. 
 
 f The angle 6 being here taken as before to represent the inclination BAP 
 of the line AP, joining the point of contact of the pitch circles with the point 
 of contact of the teeth, to the line of centres. 
 
PRESSURES UPON WHEELS. 
 
 263 
 
 Dividing numerator and denominator of the fraction in 
 the second member of that equation by sin. (0 + <p), and 
 throwing out the factors r l and r a , we have 
 
 X sin. 9 + J - 
 
 Xsin. 9 +(^)sm. ? 2 
 
 1 * a * ' 
 
 r 2 sin. (0 + <p) 
 
 sin. 
 
 Now it is evident, that if in this fractional expression & 9 
 be substituted for $+<p the numerator will be increased and 
 the denominator diminished, so that the value of Pj corre- 
 sponding to any given value of P 2 will be increased. Whence 
 it follows, that the resistance to the motion of the wheels by 
 the friction of the common surfaces of contact of their teeth 
 and of the bearings of their axes is greater when the contact 
 of their teeth takes place before than when it takes place, 
 at an equal angular distance, behind the line of centres a 
 principle confirmed by the experience of all practical me- 
 chanists. 
 
 218. To DETERMINE THE RELATION OF THE STATE BORDERING 
 UPON MOTION BETWEEN THE PRESSURE P t APPLIED TO THE 
 DRIVING WHEEL AND THE RESISTANCE P 2 OPPOSED TO THE 
 
 MOTION OF THE DRIVEN WHEEL, THE WEIGHTS 
 WHEELS BEING TAKEN INTO THE ACCOUNT. 
 
 OF THE 
 
 Now let the influence of the weights W l and W 2 of the 
 two wheels be taken into the account. The pressures applied 
 to each wheel being now three in number instead of two, the 
 relations between P x and R, and P 3 and K are determined 
 by equation (163) instead of equation (158). Substituting 
 "W, and W 2 for P 3 in the two cases, we obtain, instead of 
 equations (236) and (237), the following, 
 
 -...(243); 
 
 in which equations M, and M, represent certain functions 
 
264: RELATION OF THE DRIVING AND WORKING 
 
 \ 
 
 determined (Art. 166.) by the inclinations of the pressures 
 P x and P 2 to the vertical. 
 
 Eliminating B, between the above equations, neglecting 
 terms above the first dimensions in sin. 9, and sin. 9 2 , and 
 multiplying by a^ 
 
 ,-^sin. 9. } -PA I m t + ~ i sin. ?, J = 
 
 #2 I d>! 
 
 M 3 W 3 
 
 "Xpi sm. 9J + V m J* sm - ^ 
 
 Substituting the values of m l and w a from equations (239) 
 and (240), and neglecting the products of sin. 9, sin. 9 1 and 
 sin. 9 9 , we obtain 
 
 x sin. 9 -- sin. 9 3 \ 
 n. 9+- sin. 9, = 
 
 A /-..sn. 
 
 sn. 
 
PRESSURES UPON WHEELS. 265 
 
 (245.) 
 
 M 
 
 Now (Art. 166.) -=m l cos. 'js + flj cos. , where 13 repre- 
 
 i 
 
 eents the inclination W,FP 1 of "P l to the vertical, and < a3 the 
 inclination ErF of E to the vertical.* 
 
 Let the inclination W,BD of the perpendicular upon P x to 
 the vertical be represented by a 1? that angle being so mea- 
 sured that the pressure P x may tend to increase it ; let a, re- 
 present, in like manner, the inclination EGG of CE to the 
 vertical; and let (3 represent the inclination ABr of the 
 line of centred to the vertical, 
 
 .-. . Ii =W l FP l =W 1 BD-BDF= 1 - J 
 
 i 23 =E7-F=BOE OBr=d+9 ft ; 
 
 M 
 
 V l =m l sin. aj+flj cos. (0+9 /3). 
 
 a \ 
 Similarly ?=ra a cos. P a GH+ 2 cos. E^.f Now 
 
 ; and E^W^rf ErF, and 
 
 ErF was before shown to be equal to (04-9 (3 
 
 M 
 
 V -= w a sin. a a a cos. (4 + 9 p) 
 
 Substituting the values of m : and w a , from equations (239) 
 and (240), 
 
 M 
 
 =r l sin. (0+9) sin. a t +X sm. a! sin. 9 + 
 a, 
 
 M 
 
 /*., sin. (0 + 9) sin. a a + A. sin. a a sin. 9 
 
 a, cos. (4+9 
 
 * See note, p. 172. 
 
 j- It is to be observed that the direction of the arrow in the figure repre- 
 sents that of the resistance opposed by the driven wheel to the motion of the 
 driving wheel, so that the direction of the pressure of the driving upon the 
 driven wheel is opposite to that of the arrow. 
 
266 RELATION OF THE DRIVING AND WORKING 
 
 Let it be supposed that the distances DM and EN", repre- 
 sented by L, and L 2 , are of finite dimensions, the directions 
 of neither of the pressures P 1 and P 2 approaching to coinci- 
 dence with the direction of R, a supposition which has been 
 virtually made in deducing equation (163) from equation 
 (161), on the former of which equations, equations (243) de- 
 pend. And let it be observed that the terms involving sin. 9 
 in the above expressions (equations 246) will be of two di- 
 mensions in <p 1? 9 2 and 9, when substituted in equation (245), 
 and may therefore be neglected. Moreover, that in all cases 
 the direction of HP is BO nearly perpendicular to the line 
 of centres BC, that in those terms of equation (245), which 
 are multiplied by sin. <p x and sin. 9,, the angle$ + 9, or BOK, 
 
 may be asssumed= - ; any error which that supposition in- 
 
 2i 
 
 volves, exceedingly small in itself, being rendered exceed- 
 ingly less by that multiplication. Equations (246> will then 
 become 
 
 M, M, 
 
 *=r. sin. c^-htfj sm. p, -= r 3 sm. a a a z sm. p. 
 
 a, a, 
 
 Substituting these values in the first factor of the second 
 member of- equation (245), and representing that factor by 
 Nr we have 
 
 .rjs'j^r^ (/\ sin. a.-i-a, sin. (3) sin. 9, 
 -L-i 
 
 w 
 
 -T-* ^iPafo sin. a a -f- a, sin. /3) sin. 9, ; 
 
 ^2 
 
 and dividing by r^ 
 
 lsr*=- p -i(sin. ,+- sin. /3)sin. 9,- 
 
 i(sin.a,^^ sin. /3) sin. 9, . . . . ( 247). 
 
 * If the direction of PI be that of a tangent at the point of contact A of 
 the wheels, a case of frequent occurrence, the value of In ^vanishing, that of N 
 would appear to become infinite in this expression. The difficulty will however 
 be removed, if we consider that when aj becomes, as in this case, equal to r if 
 and the point M is supposed to coincide with A, Lj becomes a chord of the pitch 
 
PRESSURES UPON WHEELS. 
 
 267 
 
 Substituting NT/, for the factor, which it represents in 
 equation (245), we have 
 
 PAto sin. (d+9) x sin. 9 L p a sin. 9,} PA K sin. 
 
 X sin. 9 + sin. 9,} ='Nr 1 r. t sin. (6 +9) (248). 
 
 Ob 
 
 Solving this equation in respect to P 15 
 
 X sin. 9 H sin. 9 t 
 
 1 + 
 
 X sin. 9 + sin. 9. 
 a, 
 
 1 
 
 TV sin. (6 + 9) 
 
 a, 
 
 X sin. 9 H sin. 9 a 
 
 Cb n 
 
 * 2 sin. 
 
 "Whence, performing actual division by the denominators of 
 the fractions in the second member of the equation, and 
 omitting terms of two dimensions in sin. 9 15 sin. <p a , sin. 9 
 (observing that !N" is already of one dimension in those vari- 
 ables), we have 
 
 r 
 
 circle, and is therefore represented by 2ri sin. $DBA, or 2r a sin. \ (ai-f-/?) s 
 sin. a\-\ - sin. 3 
 
 that 
 
 in. 2 sin. 
 
 cos. !(<Zi-{-/?)_ 
 
 2r 1 sin. 
 
 2fi sin. 
 
 cos. 
 
 If, therefore, we take the angle ai = (3, so as to give to PI the direction of 
 
 a tangent at A, this expression will assume the value, cos. 0, or- ; so that 
 
 TI r 
 
 in this case 
 
 sin. 0i 
 
 
 ein. /? sin. 
 
268 THE MODULUS OF A SYSTEM 
 
 cosec. (4+<puP a + (249). 
 
 la this expression it is assumed that the contact of the teeth 
 is behind the line of centres. 
 
 219. THE MODULUS OF A SYSTEM OF TWO TOOTHED WHEELS. 
 
 Let n^ and n^ represent the numbers of teeth in the 
 driving and driven wheels respectively, and let it be ob- 
 served that these number are one to another as the radii of 
 the pitch circles of the wheels ; then, multiplying both sides 
 
 v 
 of equation (249) by a t , we shall obtain 
 
 cosec. 
 
 Now let A4> represent an exceedingly small increment of 
 the angle 4>, through which the driven wheel is supposed to 
 have revolved, after the point of contact P has passed the 
 line of centres ; and let it be observed that the first member 
 
 M /\-j. flf* 
 
 of the above equation is equal to PA -f , and that A-^ 
 
 r l A4 TI 
 
 represents the angle described by the driving wheel (Art. 
 204.), whilst the driven wheel describes the angle A^; 
 
 whence it follows (Art. 50.) that P^J A^l represents the 
 
 W , / 
 
 work AlJ l done by the driving pressure P 1? whilst this angle 
 A^ is described by the driven wheel, 
 
 cosec. 
 
 Let now A-s^ be conceived infinitely small, so that the first 
 member of the above equation may become the differential 
 co-efficient of U,, in respect to 4^. Let the equation, then, 
 be integrated between the limits and 4/ ; P a , L,, and L 2 . 
 and therefore ]$" (equation 247) being conceived to remain 
 
OF TWO TOOTHED WHEELS. 269 
 
 constant, whilst the angle 4> is described; we shall then 
 obtain the equation 
 
 cosec. (6 +<p) ^+^T . S ____ (250), 
 
 where S is taken to represent the arc r^ described by the 
 pitch circle of the driven wheel, and therefore by that of the 
 driving wheel also, whilst the former revolves through the 
 angle 4/. 
 
 220. THE MODULUS OF A SYSTEM OF TWO TOOTHED WHEELS, 
 
 THE NUMBER OF TEETH ON THE DRIVEN WHEEL BEING CON- 
 SIDERABLE, AND THE WEIGHTS OF THE WHEELS BEING TAKEN 
 INTO ACCOUNT. 
 
 It is evident that the space traversed by the point of con- 
 tact of two teeth on the face of either of them is, in this case, 
 small as compared with the radius of its pitch circle, and 
 that the direction of the resultant pressure R (see ^(7. p. 259.) 
 upon the teeth is very nearly perpendicular to the line 01 
 centres BC, whatever may be the particular forms of the 
 teeth; provided only that they be of such forms as will 
 cause them to act truly with one another. In this case, 
 therefore, the angle BOK represented by d +9 is very nearly 
 
 if 
 equal to -, and cosec. (d + <p)=l. 
 
 2i 
 
 Since, moreover, RP is very nearly perpendicular to the 
 line of centres at A, and that the point of contact P of the 
 teeth deviates but little from that line, it is evident that the 
 line AP represented by X differs but little from an arc of 
 the pitch circle of the driven wheel, and that it differs the 
 less as the supposition made at the head of this article more 
 nearly obtains. Let us suppose -^ to represent the angle 
 subtended by this arc at the centre C of the pitch circle of 
 the driven wheel, then will the arc itself be represented by 
 TV!', and therefore ^=r^ very nearly. Substituting this 
 value of X in equation (250), observing that cosec. 
 
 and that (equation 227), and integrating, 
 
 , I'iPi , I'a 
 
2YO INVOLUTE TEETH. 
 
 ..... (251). 
 
 But the driven or working pressure P 2 being supposed tc 
 remain constant, whilst any two given teeth are in action, 
 P 3 ^ 2 4^ represents the work u 9 yielded by that pressure whilst 
 those teeth are in contact : also r^ represents the space S, 
 described by the circumference of the pitch circle of either 
 wheel whilst this angle is described. Now let 4/ be con- 
 ceived to represent the angle subtended by the pitch of one 
 of the teeth of the driven wheel, these teeth being supposed 
 
 2* 
 to act only behind the line of centres, then ^= , n y repre- 
 
 senting the number of teeth on the driven wheel, and J^ 
 
 :. U a = J 1+ *(-+-) sin. 9 + sin. 9, + sin. <p a 1 
 | \njnj ay, a,r 9 a ( 
 
 U.+ N. S ..... (252), 
 
 which relation between the work done at the moving and 
 working points, whilst any two given teeth are in contact, is 
 evidently also the relation between the work similarly done, 
 whilst any given number of teeth are in contact. It is there- 
 fore the MODULUS of any system of two toothed wheels, the 
 numbers of whose teeth are considerable. 
 
 221. THE MODULUS OF A SYSTEM OF TWO WHEELS WITH INVO- 
 LUTE TEETH OF ANY NUMBEES AND DIMENSIONS. 
 
 The locus of the points of contact of the teeth has been 
 shown (Art. 201.) to be in this case 
 a straight line DE, which passes 
 through the point of contact A of 
 the pitch circles, and touches the 
 circles (EF and DG) from which the 
 involutes are struck. Let P repre- 
 sent any position of this point of 
 contact, then is AP measured along 
 the given line DE the distance re- 
 presented by X in Art. 216., and the 
 angle CAD, which is in this case 
 constant, is that represented by 6. Since, moreover, the 
 point of contact of the teeth moves precisely as a point P 
 upon a flexible cord DE; unwinding from the circle EF and 
 winding upon DG, would (see note, p. 238.), it is evident 
 
INVOLUTE TEETH. 
 
 271 
 
 that the distance AP, being that which such a point would 
 traverse whilst the pitch circle AH revolved through a cer- 
 tain angle 4^ measured from the line of centres is precisely 
 equal to the length of string which would wind upon DU 
 whilst this angle is described by it; or to the arc of that 
 circle which subtends the angle 4>. If, therefore, we repre- 
 sent the angle ACD by *j, so that CD CA cos. ACD=r a 
 cos. *), then X=?" 2 4' cos - i- Substituting this value for X in 
 
 equation (249), and observing that 
 
 + <p = - *j -f <p = - 
 
 
 ) <p), and that = , we have 
 
 n 
 
 ..... (253); 
 
 from which equation we obtain by the same steps as in 
 Art. 219, observing that n is constant, 
 
 IL \ 
 ( 
 
 ) 
 
 n 
 
 cos. v\ sn. 
 
 sec. ^- 
 
 (254), 
 
 which is the modulus of a system of two wheels having any 
 given numbers of involute teeth. 
 
 222. THE ESTVOLTJTE TOOTH OF LEAST RESISTANCE. 
 
 It is evident that the value of Uj in equation (254), or of 
 the work which must be done 
 upon the driving wheel to cause 
 a given amount U 2 to be yielded 
 by the driven wheel is dependent 
 for its amount upon the value of 
 the co-efficient of U 2 in the 
 second member of that equation ; 
 and that this co-efficient, again, is 
 dependent for its value (other 
 things being the same) upon the 
 value of *j representing the angle 
 ACD, or its equal the angle DAI, 
 
272 THE INVOLUTE TOOTH OF LEAST RESISTANCE. 
 
 which the tangent DE to the circles from which the invo- 
 lutes are struck makes with a perpendicular AI to the line 
 of centres. Moreover, that the co-efficient N not involving 
 this factor ij (equation 247), the variation of the value of 
 [Ju so far as this angle is concerned, is wholly involved in 
 the corresponding variation of the co-efficient of U 2 and 
 becomes a minimum with it ; so that the value of i which 
 gives to the function of f\ represented by this co-efficient, its 
 minimum value, is the value of it which satisfies the condi- 
 tion of the greatest economy of power, and determines that 
 inclination DAI of the tangent DE to the perpendicular to 
 the line of centres, and those values, therefore, of the radii 
 CD and BE of the circles whence the involutes are struck, 
 which correspond to the tooth of least resistance. 
 
 To determine the value of *\ which corresponds to a mini- 
 mum value of this co-efficient, let the latter be represented 
 by u ; then, for the required value of *j, 
 
 du 
 -r=0, 
 
 -r 
 an 
 
 Let * (- + -}= A, ^ii sin. 9, + sin. 9 3 =B ; 
 Vf*j nj ' a^ a^ 
 
 cos. ^ sin. 9 + B) sec. (i 9); 
 sec. (19)4- A sin. 9 cos. ij sec. (19); 
 
 .*. -y- =B sec. (q 9) tan. (n 9) A sin. 9 { sin. v sec. (*) 9} 
 
 cos. y tan. (>i9) sec. (^9)} ; 
 . /. ^r =B sec. \v 9) sin. (*j 9) 
 A sin. 9 sec. X 1 ? 9) {sin. ^ cos. (q 9) cos. 3 sin. (>) -9)} ; 
 .'. j-=sec. '(*) 9){B sin. (>) <p) A sin. '9} ..... (255). 
 
 In order, therefore, that -r-may vanish for any value of 
 
 *i, one of the factors which compose the second member of 
 the above equation must vanish for that value of >j ; but 
 this can never be the case in respect to the first factor, for 
 the least value of the square of the secant of an arc is the 
 square of the radius. If, therefore, the function u admit of 
 
THE INVOLUTE TOOTH OF LEAST RESISTANCE. 273 
 
 a minimum value, the second factor of the above equation 
 vanishes when it attains that value ; and the corresponding 
 value of y is determined by the equation, 
 
 B sin. (ij 9) A sin. a 9=0 (256). 
 
 or by sin. (*] <p)= ^-sim "9 or byi=<p+sin. ( -g-sin. 2 9J; 
 or substituting the values of A and: B, 
 
 1 1 
 
 (257).. 
 
 yj=9+sn. 
 
 'sin. 9 r + sin. 9 2 
 
 Now the function u admits of a minimum, to> which this 
 value of v corresponds, provided that when, substituted in 
 
 -3-5 this value of *} gives to that second differential co-effi- 
 
 cient of u in respect to n & positive value.. 
 Differentiating equation (255),, we have- 
 
 5=2 sec. "(q 9). tan-, (i 9){Bsin. (>) 9)- 
 
 A sin., "9} +B;sec. a (*i ?) cos. (*j 9) 
 
 But the proposed value- of i (equation 256) has been 
 shown to be that which,, being substituted in the factor {B 
 sin. (>) 9) A sin. a 9}> will cause it to vanish, and therefore, 
 
 with it, the whole of the first term of the value of -y-j : it 
 
 Ojf\ 
 
 corresponds, therefore, to a minimum, if it gives to the 
 second term B sec. \y 9) cos. (11 9) a positive value ; or, 
 since sec. 3 (^--9) is essentially positive, and B does not 
 involve *), if it gives to cos. fy 9) a positive value, or if 
 
 if ~~ 1 /A \ if A 
 
 i 9 < or i f sm - sin - a <P < or if sin. 8 9<1 ; or if 
 
 A sin. 3 9 <B ; or if 
 
 sin - 
 
 This condition being satisfied, the value of tj. determined 
 
 18 
 
274 THE BEST DIVISION OF THE ANGLE OF CONTACT. 
 
 by equation (257), corresponds to a minimum, and deter- 
 mines the INVOLUTE TOOTH OF LEAST RESISTANCE.* 
 
 223. To DETERMINE IN WHAT PROPORTION THE ANGLE OF- 
 CONTACT OF EACH TOOTH SHOULD BE DIVIDED BY THE LINE 
 OF CENTRES ; OR THROUGH HOW MUCH OF ITS PITCH EACH 
 TOOTH SHOULD DRIVE BEFORE AND BEHIND THE LINE OF 
 CENTRES, THAT THE WORK EXPENDED UPON FRICTION MAY 
 BE THE LEAST POSSIBLE. 
 
 Let the proportion in which the angle of contact of each 
 tooth is divided by the line of centres be represented by a?, 
 
 2* 
 so that x may represent the angular distance from the line 
 
 n y 
 
 of centres of a line drawn from the centre of the driven 
 wheel to the point of contact of the teeth when they first 
 
 2* 
 come into action before the line of centres, and (1 a?) 
 
 n i 
 
 the corresponding angular distance behind the line of centres 
 when they pass out of contact ; and let it be observed that, 
 on this supposition, if U 2 represent as before the work 
 yielded by the driven wheel during the contact of any two 
 teeth, a?U t will represent the portion of that work done 
 before, and (1 c)(J 2 that done behind, the line of centres. 
 Then proceeding in respect to equation (253) by the same 
 method as was used in deducing from that equation the 
 modulus (Equation 254), but integrating first between the 
 
 limits and x , in order to determine the work u. done by 
 
 < 
 the driving pressure before the point of contact passes the 
 
 2* 
 line of centres, and then between the limits and (1 x) 
 
 n * 
 
 to determine the work u^ done after the point of contact has 
 passed the line of centres ; observing moreover, that in the 
 former case 9 is to be substituted in sec. (779) for 9 (Art. 
 217.), we have 
 
 * It may easily be shown by eliminating 77 between equations (254) and 
 (256) that the modulus corresponding to this condition of the greatest economy 
 of power, where involute teeth are used, is represented by the formula 
 
 U 1= | l-j-*A sin. 20-f-(B 2 -A 2 sin. I U 2 +NS. 
 
THE BEST DIVISION OF THE ANGLE OF CONTACT. 275 
 
 - + -)cos *)sin 4-^sm 
 n, nj ' a,r 
 
 --^sin. <p 2 }sec. (11+9) 
 
 Or assuming 
 
 *( + /cos. v sin. 9=0, and sin. <p a + - 2 - i sin. <p a =i 
 \, V a^ V 2 
 
 s 1 representing the space described by the pitch circle of 
 either wheel before the line of centres is passed ; similarly, 
 
 sec. 11- 
 
 Adding these equations together, and representing by Uj the 
 whole work t^H-if, done by the driving pressure during the 
 contact of the teeth, and by S the whole space described by 
 the circumference of either pitch circle, we have 
 
 sec. 
 
 . . (259) 
 
 by which equation is determined the modulus of two wheels 
 driven by involute teeth, when the contact takes place partly 
 before and partly behind the line of centres. 
 
 Let the portion of the work U^, which is expended upon 
 the friction of the teeth be represented by u. Then 
 
 u=. I (ax* -f- lx) sec. (q -f 9) + 
 
 the value of a?, which gives to this function its mini- 
 mum, and which therefore determines that division of the 
 driving arc which corresponds to the greatest economy of 
 power, is evidently the value which satisfies the condition 
 
 ^-0 
 
 dx~ dx* 
 
 But differentiating and reducing 
 
276 THE BEST DIVISION OF THE ANGLE OF CONTACT. 
 
 ~r= \ 2##{sec. (>j+9) + sec. (n 9)} -f 
 
 5 {sec. (*i + 9) sec. Oi <p)} 2# sec. (?) 9) j- U a 
 cPu 
 
 Whence it appears that the second condition is always satis- 
 fied, and that the first condition is satisfied by that value of 
 a?, which is determined by the equation 
 
 (q-f 9)+sec. (^9)} +5 {sec. (*]-f 9) sec. (^9)} 
 
 Whence we obtain by transposition and reduction 
 #=- j 1 (in I tan. if] tan. </> j- . 
 
 So that the condition of the greatest economy of power is 
 satisfied in respect to involute teeth, when the teeth first 
 come into contact before the line of centres at a point whose 
 angular distance from it is less than one half the angle sub- 
 tended by the pitch by that fractional part of the last-men- 
 
 tioned angle, which is represented by the formula -Jjl-j I 
 
 tan. TJ tan. 9, or substituting for J and a their values by the 
 formula 
 
 1 
 
 2 " 
 
 i , a^i ,^ 
 
 ~/l 1\~~ r tan ' ^ tan * 9 ' ^ 260 )- 
 
 tfi | icos.^ sin. 
 
 That division of the angle of contact of any two teeth by 
 the line of centres, which is consistent with the greatest 
 economy of power, is always, therefore, an unequal division, 
 the less portion being that which lies before the line of cen- 
 tres ; and its fractional defect from one half the angle of con- 
 tact, as also the fractional excess of the greater portion above 
 one half that angle, is in every case represented by the above 
 formula, and is therefore dependent upon the dimensions of 
 the wheels, the forms and numbers of the teeth, and the cir- 
 cumstances under which the driving and working pressures 
 are applied to them.* 
 
 * The division of the arc of contact which corresponds to the greatest eco- 
 nomy of power in epicycloidal teeth, may be determined by precisely the same 
 steps. 
 
THE MODULUS OF A SYSTEM OF TWO WHEELS. 277 
 
 224. THE MODULUS OF A SYSTEM OF TWO WHEELS DRIVEN BY 
 EPIOYCLOIDAL TEETH. 
 
 The locus of the point of contact P of any two such teeth 
 is evidently the generating circle APH of 
 the epicycloidal face of one of the teeth, and 
 the hypocycloidal flank of the other (Art. 
 202.) ; for it has been shown (Art. 199.), 
 that if the pitch circles of the wheel and the 
 generating circle APH of the teeth be con- 
 ceived to revolve about fixed centres B, C, 
 D by their mutual contact at A, then will a 
 point P in the circumference of the last-men- 
 tioned circle move at the same time upon 
 the surfaces of both the teeth which are in 
 contact, and therefore always coincide with their point of 
 contact, so that the distance AP of the point of contact P of 
 the teeth from A, which distance is represented in equation 
 (250) by X, is in this case the chord of the arc AP, which 
 the generating circle, if it revolved by its contact with 
 the pitch circles, would have described, whilst each of the 
 pitch circles revolved through a certain angle measured 
 from the line of centres. Let the angle which the driven 
 wheel (whose centre is C) describes between the period 
 when the point of contact P of the teeth passes the line of 
 centres, and that when it reaches the position shown in the 
 figure be represented as before by -^, the arc of the pitch 
 circle of that wheel which passes over the point A during that 
 period will then be represented by r^. ~N ow the generating 
 circle APH having revolved in contact with this pitch circle, 
 an equal arc of that circle will have passed over the point A ; 
 whence it follows that the arc AP=r 2 4' ; and that if the radius 
 of the generating circle be represented by r, then the angle 
 
 M 
 
 ADP subtended by the arc AP is represented by ^, or 
 
 7* 
 
 by 2^, if 20 be taken to represent the ratio of the radius 
 
 of the pi tcli circle of the driven wheel to the radius of the 
 generating circle. Now the chord AP=2AD sin. ADP; 
 
 /Y* 
 
 therefore \=2r sin. e-^ sin. e^. Substituting this value 
 
 C? 
 
 of X in equation (250) ; observing, moreover, that the angle 
 
278 THE MODULUS OF A SYSTEM OF TWO WHEELS 
 
 PAD represented by 6 in that equation is equal to -- -J 
 
 2 
 
 ADP, or to ^ e\, and that the whole angle 4- through 
 which the driven wheel is made to revolve by the contact of 
 each of its teeth is represented by , we have 
 
 sn - 
 
 sn.<p 2 }sec. <-< 
 tt-fi. 
 
 or, assuming L x and L 2 to remain constant during the con- 
 
 tact of any two teeth representing the constant 1 -f -^-sin.tpj -f 
 
 t 1 r l 
 
 ^sin. 9 a by A, and observing that = , 
 
 27T 2r 
 
 sn. e> sec, 
 
 11,=?^ j A / sec. (e-\> <p)^J/H (lH ^J sin. 9/ si 
 o ' o 
 
 Now the general integral, / sec. (^ 9)<^4 / 5 r 
 
 1 /* 
 
 - / sec. (e^ $)d(e\ 0) being represented* by the function 
 
 17 
 
 1 /^cos.(^-0)c?( 
 ~2e J 1+sin. (cV~ 
 
 rcos.Wtydieiit) 1. ( . . . , , .. ) 1. 
 
 I v y - 1/. - -^'rr: lo^ ^ 14- Sin. (ei/> 0) f log. 
 
 J 1 sin. (V-0) 2e e ' M 2e 6 
 
 , , v I 1, ( 1+sin. (eV-0) ) i_ 
 
 1 sin. (e^fy) \ -log. < - 
 
 M e b e ( 1 sin. (ei/> 0) ) 
 
HAVING EPICYOLOIDAL TEETH. ^ 279 
 
 -log. tan. \ 7+ 4(^0) r > its definite integral between the 
 
 6 e ( * ) 
 
 limits and has for its expression, 
 
 ff) 
 
 tan. (-- 
 
 27T 
 
 s* 74 y 2 
 
 Also/ sec.(^ 9) &in.e^d^==/ sec.(^ 9) sin. {(^9) +9} d\> 
 o o 
 
 27T 
 ^^Ti^ 
 
 =/sec. (e-^ 9) {sin. (6^9) cos. 9-f-cos. (e-^ 9) sin. 
 o 
 
 27T 
 
 /T2/2 
 jcos. 9 tan. (e-^ 9)+ sin. 9}^ 
 o 
 
 27T 
 
 "1 y^ ^*2 ' ^^ 
 
 = -cos. 9/ tan. (^9) ^ (^4 / 9) + sin. 9. 
 
 Now the general integral/ tan. (0^9) <f(e^ 9) has for 
 its expression log. g cos. (^9).* Taking its definite inte- 
 gral between the limits and , we have, therefore, 
 
 2 / 2^ \ 
 
 / 1 cos -\ - ~V) 2* 
 
 sec. (e^ 9) sin. #4'^= cos.9loff. _ x n * ' H -- sin.9. 
 e /^rka m OT. 
 
 COS. 9 
 
 1*1., f JTTi ^Lxl 
 
 2 sin. -j ~~rh (eipd) C cos. ) - h(ey<l> f j 2 
 * , i ^* ' ^4 
 
 -Ion. 
 
 ( 2 sin. | j_i(^- J cos. | J 
 
280 THE MODULUS OF A SYSTEM OF TWO WHEELS 
 
 Substituting these expressions in the modulus, representing 
 - - by 9', and observing that if U a represent the work 
 
 4: 2 
 
 yielded by the driven wheel during the action of each tooth, 
 then P a a a .?5=U a , so that P a a a =??2l?, we have 
 
 '* 
 
 cos. 9 
 
 . (261). 
 cos. 9 
 
 OOS/^m^ ( 20* ) 26* 
 
 JSTow log. __V?___/ = log. e ] l+tan.--tan.9 [ cos.--= 
 e cos. 9 a 2 
 
 log. cos. +log. jl + tan. tan.9[=log. cos. + 
 
 tan. -tan. 9 tan. a . tan. a 9 + &c. Substituting this 
 
 expression in the preceding equation, and neglecting terms 
 above the first dimension in tan. 9 and sin. 9, 
 
 cos. lJ a + NS (262). 
 
 26* 
 
 225. If the radius r of the generating circle be equal to 
 one half the radius r^ of the pitch circle of the driven wheel, 
 according to the method generally adopted by mechanics 
 
 (Art. 203.), then e=^ = J =1. 
 n * r r 
 
 In this case, therefore that is, where the flanks of the 
 driven wheel are straight (Art. 210.) the modulus becomes 
 
 . cos.- 
 
 ^ 9 
 
 . 
 2* ( 6 * tan. 9 \ n 
 
 (263). 
 
HAVING EPICYCLOIDAL TEETH. 281 
 
 226. Substituting (in equation 262.) for 9' its value - __ - 
 
 4: 2 
 
 lp g-. tn! ' =log '< 
 
 *** tan.U-i 
 
 If, therefore, we assume the teeth in the driven wheel to 
 be so numerous, or n^ to be so great a number, that the third 
 
 power and all higher powers of tan. I- -\ may be ne- 
 
 \n^ 2/ 
 
 glected as compared with its first power, and if we neglect 
 
 powers of tan. - above the second, 
 
 2 
 
 which expression becomes if we suppose the two arcs 
 
 which enter into it to be so small as to equal their respec- 
 tive tangents. 
 
 ! 
 
 Again, log.g cos. = %\ J very nearly.* 
 
 * For assume log.e cos. x=a l x*-}-a< i x*-\-a*x*-+- . . . . ; then differentiating, 
 
 tan. z=2a 1 aH-4a 2 o; 3 4-6a 8 z 6 -}- ..... ; 
 
 2 
 but (Miller, Diff. Cal. p. 95.) tan. x= x $x* -x 6 ---- ; equating, 
 
 therefore, the co-efficients of these identical series, we have 
 
 , , 3 
 
 x* x* 2x 
 ...l og . COS .* = _ T ______ 
 
282 THE MODULUS OF THE BACK AND PINION. 
 
 Substituting these values in equation (262), and perform- 
 ing actual multiplication by the factor -- 2 -, we have 
 
 -i + -)sin. 29 
 n l nj 
 
 or substituting for A its value ; and assuming -J- sin. 2<p= 
 sin. 9, since 9 is exceedingly small, 
 
 n.<p U,+NS ..... (264), 
 
 which is the modulus of a wheel and pinion having epicy- 
 cloidal teeth, the number of teeth 7i 2 in the driven wheel 
 being considerable (see equation 252). 
 
 It is evident that the value of Uj in the modulus (equa- 
 tion 261), admits of a minimum in respect to the value of / 
 there is, therefore, a given relation of the radius of the 
 generating circle of the driving, to that of the driven wheel, 
 which relation being observed in striking the epicycloidal 
 faces and the hypocycloidal flanks of the teeth of two wheels 
 destined to work with one another, those wheels will work 
 with a greater economy of power than they would under any 
 other epicycloidal forms of their teeth. This value of e may 
 be determined by assuming the differential co-efficient of the 
 co-efficient of U 2 in equation (261) equal to zero, and solving 
 the resulting transcendental equation by the method of 
 approximation. 
 
 227. THE MODULUS OF THE BACK AND PINION. 
 
 If the radius r 9 of the pitch circle of the driven wheel be 
 supposed infinite (Art. 213.), that wheel becomes a rack, and 
 the radius r l of the driving wheel remaining of finite dimen- 
 sions, the two constitute a rack and pinion. To determine 
 the modulus of the rack and pinion in the case of teeth of 
 any form, the number upon the pinion being great, or in 
 the case of involute teeth and epicycloidal teeth of any 
 number and dimensions, we have only to give to r^ an 
 infinite value in the moduli already determined in respect 
 
THE MODULUS OF THE RACK AND PINION. 283 
 
 to these several conditions. But it is to be observed in 
 respect to epicycloidal teeth, that n^ becomes infinite with 
 
 v 
 
 ?* whilst the ratio remains finite, and retains its equality 
 n i 
 
 to the ratio ^ (equation 227), so that^= -^ =*^.=^ ; 
 
 T 
 
 if we represent the ratio by 2^. Making n^ and r a infinite 
 in each of the equations (252), (254:), and (261), and sub- 
 
 P P 
 
 stituting for in equation (262) ; we have 
 
 1. For the modulus of the rack and pinion when the teeth 
 are very small, whatever may be their forms, provided that 
 they work truly. 
 
 'sin. 9x + -sm. 9 ! U a +NS ..... (265). 
 
 ^i^"i ^i 
 
 2. For the modulus of a rack and pinion, with involute 
 teeth of any dimensions (see^. 1. p. 255), 
 
 11!= ] 1+ I cos. ?? sin. <H sin. 0J sec. (rj <j>) [ U 3 + 
 
 (. \W' 1 Gbfl*^ I 
 
 JSTS . . (266). 
 
 3. For the modulus of the rack and pinion, with cycloidal 
 and epicycloidal teeth respectively (equation 261), 
 
 . 
 
 tan. 9' 
 
 20j cos. 9 ) 
 
 In each of which cases the value of N" is determined by 
 making r* infinite in equation (24:7). 
 
 2 )I (z + ^ (! +i)4 (1 + l) = i., becau8e 
 
 e is infinite. The friction of the rack upon its guides is not taken into account 
 in the above equations. 
 
284 
 
 CONICAL WHEELS. 
 
 CONICAL OB BEVIL WHEELS. 
 
 228. These wheels are used to communicate a motion of 
 rotation to any given axis from another, inclined to the first 
 at any angle. 
 
 Let AF be an axis to which a motion of rotation is to be 
 communicated from another axis AE 
 inclined to the first at any angle EAF, 
 by means of bevil wheels. 
 
 Divide the angle EAF by the straight 
 line AD, so that DO and DN, perpen- 
 diculars from any point D in AD upon 
 AE and AF respectively, may be to 
 one another as the numbers of teeth 
 which it is required to place upon the 
 two wheels.* 
 
 Suppose a cone to be generated by the revolution of the 
 line AD about AE, and another by the revolution of the 
 line AD about AF. Then if these cones were made to 
 revolve in contact about the fixed axes AE and AF, their 
 surfaces would roll upon one another along their whole line 
 of contact DA, so that no part of the surface of one would 
 slide upon that of the other, and thus the whole surface 
 of. the one cone, which passes in a given time over the line 
 of contact AD, be equal to the whole surface of the other, 
 which passes over that line in the same time. For it is 
 evident that if n v times the circumference of the circle DP 
 be equal to n^ times that of the circle DI and these circles 
 be conceived to revolve in contact carrying the cones with 
 them, whilst the cone DAP makes n^ revolutions, the cone 
 
 * This division of the angle EAF may be made as follows : Draw ST and 
 UW from any points S and U in the straight lines AE and AF at right angles 
 to those lines respectively, and having their 
 lengths in the ratio of the numbers of teeth 
 which it is required to place upon the two wheels ; 
 and through the extremities T and W of these 
 lines draw TD and WD parallel to AE and AF 
 respectively, and meeting in D. A straight line 
 drawn from A through D will then make the 
 required division of the angle ; for if DO and 
 DN be drawn perpendicular to AE and AF, they 
 will evidently be equal to UW and ST, and there- 
 fore in the required proportion of the numbers 
 of the teeth ; moreover, any other two lines 
 drawn perpendicular to AE and AF from any 
 
 other point in AD will manifestly be in the same proportion as F*0 and DN. 
 
CONICAL WHEELS. 285 
 
 DAI will make n^ revolutions; so that whilst any other 
 circle GH of the one cone makes n^ revolutions, the corre- 
 sponding circle HK of the other cone will make n^ revolu- 
 tions: but n^ times the circumference of the circle GH 
 is equal to n^ times that of the circle HK, for the diameters 
 of these circles, and therefore their circumferences, are to 
 one another (by similar triangles) in the same proportion as 
 the diameters and the circumferences of the circles DP and 
 DI. Since, then, whilst the cones make n v and n^ revolutions 
 respectively, the circles HG and HK are carried through n l 
 and n^ revolutions respectively, and that n^ times the circum- 
 ference of HG is equal to n^ times that of HK, therefore 
 the circles HG and HK roll in contact through the whole of 
 that space, nowhere sliding upon one another. And the 
 same is true of any other corresponding circles on the cones ; 
 whence it follows that their whole surfaces are made to roll 
 upon one another by their mutual contact, no two parts 
 being made to slide upon one another by the rolling of the 
 rest. 
 
 The rotation of the one axis might therefore be communi- 
 cated to the other by the rolling of two such cones in con- 
 tact, the surface of the one cone carrying with it the surface 
 of the other, along the line of contact AD, by reason of the 
 mutual friction of their surfaces, supposing that they could 
 be so pressed upon one another as to produce a friction equal 
 to the pressure under which the motion is communicated, or 
 the work transferred. In such a case, the angular velocities 
 of the two axes would evidently be to one another (equation 
 227) inversely, as the circumferences of any two correspond- 
 ing circles DP and DI upon the cones, or inversely as their 
 radii ND and OD, that is (by construction) inversely as the 
 numbers and teeth which it is supposed to cut upon the 
 wheels. 
 
 When, however, any considerable pressure accompanies 
 the motion to be communicated, the friction of two such 
 cones becomes insufficient, and it becomes necessary to 
 transfer it by the intervention of bevil teeth. It is the cha- 
 racteristic property of these teeth that they cause the motion 
 to be transferred by their successive contact, precisely as it 
 would by the continued contact of the surfaces of the 
 cones. 
 
286 
 
 CONICAL WIIICKLS. 
 
 229. To describe the teeth of bevil wheels.* 
 
 From D let FDE be wrawn at right angles to AD, inter- 
 secting the axes AE and AF of the two cones in E and F ; 
 suppose conical surfaces to be generated by the revolution 
 of the lines DE and DF about AE and AF respectively ; 
 
 and let these conical surfaces be truncated by planes LM 
 and XY respectively perpendicular to their axes AE and 
 AF, leaving the distances DL and DY about equal to the 
 depths which it is proposed to assign to the teeth. Let now 
 the conical surface LDPM be conceived to be developed 
 upon a plane perpendicular to AD, and passing through the 
 point D, and let the conical surface XIDY be in like 
 manner developed, and upon the same plane. When thus 
 developed, these conical surfaces will have be- 
 come the plane surfaces of two segmental annuli 
 l&Ppm and IXa^f, whose centres are in the 
 points E and F of the axes AE and AF, and 
 which touch one another in the point D of the 
 line of contact AD of the cones. 
 
 Let now plane or spur teeth be struck upon 
 the circles Pj? and K, such as would cause them 
 
 * The method here given appears first to have been published by Mr. Tred- 
 gold in his edition of Buchanan's Essay on Mill-work, 1823, p. 103. 
 
 f The lines MP and pm in the development, coincided upon the cone, as also 
 the lines IX and ix; the other letters upon the development in the above 
 
CONICAL WHEELS. 287 
 
 to drive one another as they would be driven by theii 
 mutual contact ; that is, let these circles Yp and Ii be taken 
 as the pitch circles of such teeth, and let the teeth be 
 described, by any of the methods before explained, so that 
 they may drive one another correctly. Let, moreover, their 
 pitches be such, that there may be placed as many such 
 teeth on the circumference P> as there are to be teeth 
 upon the bevil wheel HP, and as many on Ii as upon the 
 wheel III. 
 
 Having struck upon a flexible surface as many of the first 
 teeth as are necessary to constitute a pattern, apply it to 
 the conical surface DLMP, and trace off the teetli from it 
 upon tli at surface, and proceed in the same manner with the 
 surface DIXY. 
 
 Take DH equal to the proposed lengths of the teeth, draw 
 ef through H perpendicular to AD, and terminate the wheels 
 at their lesser extremities by concave surfaces HGmZ and 
 HKxy described in the same way as the convex surfaces 
 which form their greater extremities. Proceed, moreover, 
 in the construction of pattern teeth precisely in the same 
 way in respect to those surfaces as the other ; and trace out 
 the teeth from these patterns on the lesser extremities as on 
 the greater, taking care that any two similar points in the 
 teetli traced upon the greater and lesser extremities shall lie 
 in the same straight line passing through A. The pattern 
 teeth thus traced upon the two extremities of the wheels are 
 the extreme boundaries or edges of the teeth to be placed 
 upon them, and are a sufficient guide to the workman in 
 cutting them. 
 
 230. To prove that teeth thus constructed will work truly 
 with one another. 
 
 It is evident that if two exceedingly thin wheels had been 
 taken in a plane perpendicular to AD (fig. p. 286.) passing 
 
 figure represent points which are identical with those shown by the game let- 
 ters in the preceding figure. In that figure the conical surfaces are shown, 
 developed, not in a plane perpendicular to AD, but in the plane which contains 
 that line and the lines AE and AF, and which is perpendicular to the last-men- 
 tioned plane. It is evidently unnecessary, in the construction of the pattern 
 teeth, actually to develope the conical extremities of the wheels as above 
 described ; we have only to determine the lengths of the radii DE and DF by 
 construction, and with them to describe two arcs, Pjo, It, for the pitch circles 
 of the teeth, and to set off the pitches upon them of the same lengths as the 
 pitches upon the circles DP and DI, which last are determined by the numbeifl 
 of teeth required to be cut upon the wheels respectively. 
 
THE MODULUS OF A SYSTEM 
 
 through the point D, and having their centres in E and F, 
 and if teeth had been cut upon these wheels according to 
 the pattern above described, then would these wheels have 
 worked truly with one another, and the ratio of their angu- 
 lar velocities have been inversely tkat of ED to FD, or (by 
 similar triangles) inversely that of ND to OD ; which is the 
 ratio required to be given to the angular velocities of the 
 bevil wheels. 
 
 Now it is evident that that portion of each of the conical 
 surfaces DPML and DIXY which is at any instant passing 
 through the line LY is at that instant revolving in the plane 
 perpendicular to AD which passes through the point D, the 
 one surface revolving in that plane about the centre E, and 
 the other about the centre F ; those portions of the teeth of 
 the bevil wheels which lie in these two conical surfaces will 
 therefore drive one another truly, at the instant when they 
 are passing through the line LY, if they be cut of the forms 
 which they must have had to drive one another truly (and 
 with the required ratio of their angular velocities) had they 
 acted entirely in the above-mentioned plane perpendicular 
 to AD and round the centres E and F. Now this is pre- 
 cisely the form in which they have been cut. Those por- 
 tions of the bevil teeth which lie in the conical surfaces 
 DPML and DIXY will therefore drive one another truly at 
 the instant when they pass through the line LY ; and there- 
 fore they will drive one another truly through an exceedingly 
 small distance on either side of that line. Now it is only 
 through an exceedingly small distance on either side of that 
 line that any two given teeth remain in contact with one 
 another. Thus, then, it follows that those portions of the 
 teeth which lie in the conical surfaces DM and DX work 
 truly with one another. 
 
 Now conceive the faces of the teeth to be intersected by an 
 infinity of conical surfaces parallel and similar to DM and 
 DX ; precisely in the same way it may be shown that those 
 portions of the teeth which lie in each of this infinite num- 
 ber of conical surfaces work truly with one another; whence 
 it follows that the whole surfaces of the teeth, constructed as 
 above, work truly together. 
 
 231. THE MODULUS OF A SYSTEM OF TWO CONICAL OB 
 BEVIL WHEELS. 
 
 Let the pressure P, and P a be applied to the conical 
 
OF TWO CONICAL WHEELS. 289 
 
 wheels represented in the accompanying figure at perpen- 
 dicular distances a t and # 2 from their axes CB and CG ; let 
 the length AF of their teeth be represented by & ; let the 
 distance of any point in this line from F be represented by 
 a?, and conceive it to be divided into an exceedingly great 
 number of equal parts, each represented by A#. Through 
 each of these points of division imagine planes to be drawn 
 
 perpendicular to the axes CB and CG of the wheels, dividing 
 the whole of each wheel into elements or laminae of equal 
 thickness ; and let the pressures P, and P 2 be conceived to be 
 
 equally distributed to these laminae. The pressure thus dis- 
 
 p 
 tributed to each will then be represented bv A# on the 
 
 o 
 p 
 one wheel, and ?Aa? on the other. Let^ and^> 2 represent 
 
 the two pressures thus applied to the extreme laminae AH 
 and AK of the wheels, and let them be in equilibrium when 
 thus applied to those sections separately and independently 
 of the rest ; then if R, represent the pressure sustained along 
 that narrow portion of the surface of contact of the teeth of 
 the wheels which is included within these laminaa, and if R, 
 and R 2 represent the resolved parts of the pressure R in the 
 directions of the planes AH and AK of these laminae, the 
 pressures^ and 3^ applied to the circle AH are pressures 
 in equilibrium, as also the pressures p^ and R 2 applied to the 
 circle AK. If, therefore, we represent as before (Art. 216.) 
 by m l and m 2 , the perpendiculars from B and G upon the 
 
 19 
 
290 THE MODULUS OF A SYSTEM 
 
 directions of K, and R 2 , and by L t and L 2 , the distances be- 
 tween the feet of the perpendiculars & m t and , m a we 
 have (equation 236, 237), neglecting the weights of the 
 wheels, 
 
 1 f /Pi L i\ 1 
 
 l "&, + -- 1 sm. 0, Y R, 
 
 ' 
 
 p! and p a representing the radii of the axes of the two wheels, 
 and 9, and 9 2 the corresponding limiting angles of resistance. 
 Let 7j and y % represent the inclinations of the direction of E 
 to the planes of AH and AK respectively ; then 
 
 R!= R cos. 7 15 R 2 = R cos. y a . 
 
 Now it has been shown in the preceding article, that the 
 action of that part of the surface of contact of the teeth which 
 is included in each of the laminse AH, AK, is identical with 
 the action of teeth of the same form and pitch upon two 
 cylindrical wheels AD and AL of the same small thickness, 
 situated in a plane EAD perpendicular to AC, and having 
 their centres in the intersections, l> and </, with that plane of 
 the axes CB and CG produced. The reciprocal pressure R 
 of the teeth of the element has therefore its direction in the 
 plane EAD ; and if its direction coincided with the line of 
 centres DL of the two circles EA and AD, then would its 
 inclinations to the planes of AH and AK be represented by 
 D AH and LAK, or by ACB and ACG. 
 
 The direction of R is however, in overy case, inclined to 
 the line of centres at a certain angle, which has been shown 
 (Art. 216.) to be represented in every position of the teeth, 
 after the point of contact has passed the line of centres by 
 (4 + 9) j where & represents the inclination to AL of the line 
 X, which is drawn from the point of contact A of the pitch 
 circles to the point of contact of the teeth, and where 9 repre- 
 sents the limiting angle of resistance between the surfaces of 
 the teeth. To determine the inclination / t of RA to the 
 plane of the circle AH, its inclination RAD to the line of 
 centres being thus represented by (0 + <p), and the inclination 
 of the plane AD, in which it acts, to the plane AH being 
 DAH, which is equal to ACB, let this last angle be repre- 
 
OF TWO CONICAL WHEELS. 291 
 
 sented by i, ; and let Aa in the accom- 
 panying figure represent the intersection 
 of the planes AD and AH ; Aard repre- 
 senting a portion of the former plane and 
 Aacfi of the latter. Let moreover AT 
 represent the direction of the pressure R 
 in the former plane and let Ad and Ah be portions of the 
 lines AD and AH of the preceding figure. Draw re per- 
 pendicular to the plane Aach, and rd and ch parallel to A#, 
 and join dh ; then rAc represents the inclination 7, of the 
 direction of R to the plane AD, dAr represents the inclina- 
 tion (9 + A) of AR- to AD, and dAh represents the inclination 
 i x of the planes AD and AH to one another. Also, since Aa 
 is perpendicular to the plane AM, therefore dr is perpen- 
 dicular to that plane, 
 
 /. re = Ar sin. 7, = Ad sec. (6 4-9) sin. 7^ 
 Also hd = Ad sin. i iy but re = hd, 
 
 :. Ad sec. (d-f-9) sin. 7 1 = 
 /. sin. 7, cos. (d + <p) sin. 
 
 In like manner it may be shown that sin. 7 2 = cos. (d-f 9) 
 sin. i a , i 2 being taken to represent the inclination KAL of the 
 planes AE and AK, which angle is also equal to the angle 
 ACG. 
 
 From the above equations it follows that 
 
 !=R cos. 7,=E \/l cos. 2 (d+9) sin. 
 2 =R cos. 7 2 =R \/l cos. 2 (^ +9) sin. 
 
 From the centre b of the circle AD draw fim perpendicular 
 to RA, then is BM (the perpendicular let fall from the 
 centre of the circle AH upon the direction of R,) the projec- 
 tion of lym upon the plane of the circle AH. To determine 
 the inclination of bm to the plane AH, draw An parallel to 
 bm ; the sine of the inclination of An to the plane AH is 
 then determined to be cos. DArc/ . sin. precisely as the sine 
 of the inclination of Am to the same plane was before deter- 
 mined to be cos. DAm . sin. i e 
 
 Now T>An=Abm = 2 DAR = ^ (d + <p) ; therefore the 
 sine of the inclination of ATI, and therefore of bm, to the plane 
 
292 
 
 THE MODULUS OF A SYSTEM 
 
 Cv%:: 
 
 AH is represented by the formula sin. (0+9) sin. i,, and the 
 cosine of its inclination by 4/1 sin. 2 (0 + 9) sin. \ ; 
 
 . m 1 =BM=5m i/l sm. 2 (d+9)sin. a 1 . 
 
 Now it has been shown (Art. 216.) that the perpendicular 
 ~bm let fall from the centre of a spur wheel upon the direc- 
 tion of the pressure upon its teeth is, in any position of their 
 point of contact, represented (equation 239) by the formula, 
 
 r l sin. 
 
 sn. 9, 
 
 where 0, 9, X represent the same quantities which they have 
 been taken to represent in this article ; but r l represents the 
 radius 5 A of the circle AD, instead of the radius 13 A of the 
 circle AH; now 5A=rBA sec. DAH=7 | 1 sec. ^5 substituting 
 this value for r^ in the preceding formula, we have 
 
 m=7\ sn. +<p sec. 1,+ sn. 9 ; 
 /. m l = \T! sin. (6 +9) sec. ^4-^ sin. 9} 
 
 Similarly it may be shown that 
 
 ra a \r^ sin. (6 + 9) sec. 2 X sin. 9} 
 
 (269). 
 
 Substituting the values of in l and m^ above determined, 
 and also the values of R t and R 2 (equations 268) in equations 
 
OF TWO CONICAL WHEELS. 293 
 
 (267), and eliminating R between those equations, a relation 
 will be determined between^?, and p^ which is applicable to 
 any distance of the point of contact of the teeth from the 
 line of centres. 
 
 Let it now be assumed that the number of the teeth of 
 
 the driven wheel is considerable, so that the angle tra 
 
 n* 
 
 versed by the point of contact of each tooth may be small, 
 and the greatest value of the line X, the chord of an exceed- 
 ingly small arc of the pitch circle of the driven wheel. In 
 
 this case d+9 will very nearly equal -(Art. 220.); so that 
 
 cos. 8 (^+<p) will be an exceedingly small quantity and may 
 be neglected, and sin. (^ + 9) very nearly equal unity. Sub- 
 stituting these values in equations (268) and (269) we have 
 
 K 1 =E, R 2 =R, 
 
 m 1 =r 1 +X sin. 9 cos. i a , m 2 =/' 2 X sin. 9 cos. 2 . 
 
 Substituting these values in equations (26Y) and dividing 
 those equations by one another so as to eliminate R, 
 
 ^4-Xsin. 9 cos. ,+ I 1 sin. 9, 
 Pi _ <\ m _ > ^1 / _ . 
 
 1 r^ Xsin. 9 cos. < 3 1-^ M sin. 9 a 
 
 \ $ 2 / 
 
 x . /P,LA 
 
 1 H sm. 9 cos. j+ - sin. 9. 
 
 *i* 22 . 
 
 ** 1 -- sin. 9 cos. - - sin. 9, 
 
 r t \apj 
 
 "Whence performing actual division by the denominator of 
 the fraction, and neglecting terms involving dimensions 
 above the first in sin. 9, sin. 9,, sin. 9 2 , 
 
 Now if 4/ represent the angle described by the driven 
 wheel or circle ELA, whilst any two teeth are in contact, 
 since X is very nearly a chord of that circle subtending this 
 small angle 4/ (Art. 220.) ; /. X = r^. Let * represent the 
 
294: THE MODULUS OF A SYSTEM. 
 
 angle described by the conical wheel FK, whilst the circle 
 ELA describes the angle ^ ; then, since the pitch circle of 
 the thin wheel AK and the circle ELA revolve in contact at 
 A, they describe equal arcs whilst they thus revolve, respec- 
 tively, through the unequal angles 4< and *. Moreover, the 
 radius Ag of the circle AL=AG sec. GA.g=r t sec. * a , there- 
 fore 4^3 sec. i 3 =ir a ; 
 
 .-.,),=* cos. i f (270). 
 
 Substituting the above valves of 4' and X, and observing 
 
 ?"o 0*0 
 that = , 
 r, n. 
 
 L. <p + 
 
 Multiplying both sides of this equation by p^ a i n * y and ob- 
 serving that p.a. = p,a, 2 A T , and that A* is the exceed- 
 * n^ l T^A^' n^ 
 
 ingly small angle described by the driving wheel AN, whilst 
 the driven wheel describes the angle Ai', so that if A^ repre- 
 sent the work done by the pressure^ upon the lamina AH, 
 whilst the angle A* is described by the driven wheel, then 
 
 p d A^izrA'w we have 
 X 
 
 cos. ^ cos. < 2 \ 
 
 1 1 * cos. 3 sin. <p -f 
 
 n, n n I 
 
 or assuming Ai' infinitely small, and integrating between the 
 limits and (Art. 220.), 
 
 I. cos. 2 
 
 cos - 8ID - 
 
 sn. ^ + sn. 9. 
 
OF TWO CONICAL WHEELS. 295 
 
 Now the above relation between the work u l done by the 
 pressure p t upon the extreme element AH of the driving 
 wheel whilst any two teeth are in contact, and the pressure 
 j? 2 opposed to the motion of the corresponding element of 
 the driven wheel, is evidently applicable to any other two 
 corresponding elements ; the values of p^ r^ r Lj and L a 
 proper to those elements being substituted in the formula. 
 If, therefore, we represent by AU, that increment of the 
 whole work Uj done upon the driving wheel, which is due 
 to any one of the elements into which we have imagined 
 that wheel to be divided, and if we substitute for j? 2 its 
 
 p 
 value -T ?A a?, assign to L 15 L 2 , r^ r^ their values proper to that 
 
 element, and represent those values by L, L', r, /, 
 
 or assuming A$ infinitely small, and integrating between the 
 
 limits and 5, and observing that P 2 # 2 represents the 
 
 n, 
 
 whole work U 2 done upon the driven wheel under the con- 
 stant pressure P 2 during the contact of any two teeth, 
 
 iB in.<p + 
 
 Now a+x being taken to represent the distance of the 
 point of contact of any two such elements from C, and a to 
 represent the distance OF, the radii r and / of these ele- 
 ments are evidently (by similar triangles) represented by 
 
 a+x I so\ a+x / a?\ 
 
 r, or ^ 1 + -| / and j-r a or 1 1 + - j r M r, and r, repre- 
 senting the radii of the extreme elements NF and OF, or of 
 the pitch circles of the lesser extremities of the wheels. 
 Also assuming, as we have done, the pressures R 1 and R 9 
 
296 
 
 THE MODULUS OF A SYSTEM 
 
 to be perpendicular to the lines BA, 
 -ft GA joining the centre of each ele- 
 ment with their point of contact A, 
 so that the points M and N (see fig. 
 p. 292.) coincide with the point A 
 (see accompanying figure)* ; and re- 
 presenting the angles ABD and ACE 
 made by the perpendiculars DB and 
 CE with the line of centres by ^ and 
 
 ^respectively; observing also that AD 2 =BA a 2BA. BD 
 cos. ABD + BD 5 , so that (^-r\ = 1 2 (^\ cos. ABD + 
 
 ( - ) , we have, substituting, in the second number of this 
 \-t)A/ 
 
 equation, for BA or r its value r l { 1 -f- - \ 
 
 \ al 
 
 or expanding the binomials in this expression, observing 
 that - is an exceedingly small quantity, neglecting terms 
 
 Ob 
 
 involving powers of that quantity above the first, and 
 reducing, 
 
 <". 
 
 Now Lj representing the value of L when #=0, and 6 re 
 maining constant, 
 
 * The circles in this figure represent two of the corresponding laminae into 
 which wheels have been imagined to be divided ; they are not, therefore, in the 
 same planef Their planes intersect in AH. 
 
OF TWO CONICAL WHEELS. 297 
 
 Let now the angle ADB, made in respect to the first ele- 
 ment of the driving wheel between the perpendicular BD or 
 #! and the chord AD or L, be represented by 7/ 1? and let T/, 
 represent the corresponding angle in the driven wheel, then 
 
 , T , a 
 
 = r* ' - 
 
 , 
 
 L, - 2LA cos. 77, + a? = r*, :. ( ' 
 
 L.a. 
 
 -' cos. fl.+ r=; 
 
 O 1 * 1 I -1 1 I 1 1 O / _ 
 
 Substituting these values of I I and 2 1 ) I cos. d 1 
 in equation (273) ; 
 
 \rj "" {rj ~"" w \ y, 2 / \a) cos< ^- - \7r) 
 
 Extracting the square root of the binomial, and neglecting 
 
 terms involving powers of - above the first, 
 
 a 
 
 L L x / a, \ lx\ a, ( L, x \ 
 
 -. -- i^j i_i cos>7?= _^ ---- cos. 77. f ; 
 T r, \rj \a/ r l ( a t a 
 
 P. sn. 9. 
 
 . ... p a sin.9 2 fL' p a sin.9 a ( L a 5 ) 
 
 Similarly -^- / ? & = ^ \ - - $ - cos. r,, \ . 
 
 Substituting these values in the modulus (equation 272), 
 
 TT TT S -, / cos - 'i cos - ' 3 
 
 U, = U, ] 1+n - + - cos. i, sin. 9 + 
 
 I \ 7l l 7l a 
 
 P.sin.^/L, 5 
 
 ~ - - os - 
 
298 THE MODULUS OF A SYSTEM 
 
 Now let the angle BCG, or the inclination of the axes, 
 from one to the other of which motion is transferred by the 
 wheels, be represented by 2< ; therefore 1 -H a =2<. Also a 
 Bin. ^=7*! and a sin. < a =r a , 
 
 sin. i, T*! n t 
 sin. a 7", 7i a ' 
 
 sin. \ __ sin. \ ^ 1 cos. \ _ 1 cos. \ 
 1 1 cos.\^cos.%_ /cos. , cos. i a \ /cos.^ cos. i,\ 
 
 (COS.*. COS. i a \ /I COS. I. 1\ 
 1 J \ ^-j- rf] COS. i a 
 
 (COS. <. COS. l a \ W i ^a 
 ^H M cos. i 3 = ^ ^ . 
 n^ n^ 1 \_ cos. i t 
 
 W, COS. a " ^ a 
 
 ^ cos. i, cos. j'+iO, 01 1 tan. -|(..) tan. i 
 JNow 
 
 cos. ' 2 ~~ cos. i i(, 2 ) ~ l + tan.-J 3 )tan. J 
 
 y^ sin. >, sin. {t + K'i 0} _ tan - ' + tan.-j-^ < 2 ) 
 al80 n~ sin. i a ~ sin. j'-K',-',)} ~~ tan. i - tan. (,- a ) 
 
 . cos.< 
 
 cos, . yy-M, A 
 
 JL ~r~ - tan. i 
 
 1 cos. ',1 1 / cos. i, \ 
 ,". ---- a = : - 1 ^ -- n, ] = 
 
 w-, cos. a n n^ \ 2 cos. a V 
 
 1 (n*nf) + (V yQtan^ _ 
 
OF TWO CONICAL WHEELS. 299 
 
 - 8ec - 1 
 
 .+!)_ (_1) tan.', A + M cos ,,_(!_!\ gin V 
 n, nj \n, nj \V </ \n^ nj 
 
 I cos. ! cos. a /I 1 \ 
 
 . -- - + - -] cos. i a = -- -- cos. a 
 \ , n t I *'*t ***' 
 
 /I 1\ , /I 1\ 2sin. 9 i 
 
 I -- 1 sin. ' I + I -- . 
 VT^ nj \n 1 nj n^ 
 
 Substituting in the preceding relation, between Uj and U a , 
 
 which is the modulus of the conical or hevil wheel, neglecting 
 the influence of the weight of the wheel. 
 
 If for cos. ?, and cos. 17, we substitute their values (see 
 p. 297), we shall obtain by reduction 
 
 from which equation it is manifest that the most favourable 
 directions of the driving or working pressures are those 
 determined by the equations 
 
 232. It is evident, that if the plane of the revolution 
 of such a wheel be vertical, the influence of its weight must 
 be very nearly the same as that of a cylindrical or spur 
 
300 THE MODULUS OF A SYSTEM OF TWO CONICAL WHEELS. 
 
 wheel of the same weight, having a radius equal to the mean 
 radius of the conical wheel, and revolving also in a vertical 
 plane. If the axis of the wheel be not horizontal, its weight 
 must be resolved into two pressures, one acting in the plane 
 of the wheel, and the other at right angles to it; the latter is 
 effective only on the extremity of the axis, where it is borne 
 as by a pivot, so that the work expended by reason of it may 
 be determined by Art. 175, and will be found to present 
 itself under the form of N 2 . S, where N a is a constant and S 
 the space described by the pitch circle of the wheel, whilst 
 the work 'U l is done. The resolved weight in the plane 
 of the wheel must be substituted for the weight of the wheel 
 in equation (247), which determines the value of "N. Assum- 
 ing the value of K", this substitution being made, to be repre- 
 sented by Nj, the whole of the second term of the modulus 
 will thus present itself under the form (N t + IST 2 )S. 
 
 ,)S ..... (276). 
 
 233. Comparing the modulus of a system of two conical 
 wheels with that of a system of two cylindrical wheels 
 (equation 252), it will be seen that the fractional excess 
 of the work U 2 lost by the friction of the latter over that 
 lost by the friction of the former is represented by the 
 formula 
 
 2* sin. 2 < sin, <p ., 5 /p. p. \ 
 
 + - -j- -+ t- (-cos. ^ sin. 9 a + - cos.?? 2 sin.<p 2 j . . . 
 
 The first term of this expression is due to the friction of 
 the teeth of the wheels alone, as distinguished from the fric- 
 tion of their axes ; the latter is due exclusively to the friction 
 of the axes. Both terms are essentially positive, since ^ 
 
 and % are in every case less than -. 
 
 2 
 
 Thus, then, it appears that the loss of power due to the 
 friction of bevil wheels is (other things being the same) 
 essentially less than that due to the friction of spur wheels, 
 so that there is an economy of power in the substitution of 
 
THE MODULUS OF A TRAIN OF WHEELS. 301 
 
 a bevil for a spur wheel wherever such substitution is prac- 
 ticable. This result is entirely consistent with the experience 
 of engineers, to whom it is well known that bevil wheels run 
 lighter than spur wheels. 
 
 234. THE MODULUS OF A TRAIN OF WHEELS. 
 
 In a train of wheels such as that shown in the accompany- 
 ing figure, let the radii of their 
 pitch circles be represented in 
 order by r r r 3 . . . r t , begin- 
 ning from the driving wheel ; 
 and let a^ represent the perpen- 
 dicular distance of the driving 
 pressure from the centre of that 
 wheel, and a^ that of the driven 
 pressure or resistance from the centre of the last wheel of the 
 train ; U, the work done upon the first wheel, u y the work 
 yielded by the second wheel to the third, u s that yielded by 
 the fourth to the fifth, &c., and U 2 the work yielded by the 
 last or n th wheel upon the resistance, then is the relation be- 
 tween Uj and u t determined by the modulus (equation 252), 
 it being observed that the point of application of the resist- 
 ance on the second wheel is its point of contact 5 with the 
 third wheel, so that in this case a^=^r s . 
 
 These substitutions being made, and L 2 being taken to 
 represent the distance between the point Z> and the projection 
 of the point a upon the third wheel, we have 
 
 11,= j 1+W-+-W 9+ ^ sin. 9,+ 
 ( \n, nj a,r t 
 
 ^ Bin. ? j ,+*. 8,. 
 
 To determine, in like manner, the relation between u^ and 
 u^ or the modulus of the third and fourth wheels, let it be 
 observed that the work u^ which drives the third wheel has 
 been considered to be done upon it at its point of contact b 
 with the fourth ; so that in this case the distance between the 
 point of contact of the driving and driven wheels and the 
 foot of the perpendicular let fall upon the driving pressure 
 from the centre of the driving wheel vanishes, and the term 
 
 * See note p. 266. 
 
302 THE MODULUS OF A TRAIN OF WHEELS. 
 
 which involves the value of L representing that line disap- 
 pears from the modulus, whilst the perpendicular upon the 
 driving pressure from the centre of the driving wheel be- 
 comes r s . Let it also be observed, that the work of the 
 fourth wheel is done at the point of contact c of the fifth and 
 sixth wheels, so that the perpendicular upon the direction of 
 that work from the axis of the driven wheel is /,. We shall 
 thus obtain for the modulus of the third and fourth wheels, 
 
 In which expression L 3 represents the distance between the 
 point G and the projection of the point & upon the fifth 
 wheel. 
 
 In like manner it may be shown, that the modulus of the 
 fifth and sixth wheels, or the relation between u 3 and u# is 
 
 s > ; 
 
 and that of the seventh and eighth wheels, or the relation 
 between u 4 and u# 
 
 <= \ 
 
 and that, if the whole number of wheels be represented by 
 2p. or the number of pairs of wheels in the train by p, then 
 is the modulus of the last pair, 
 
 In which expressions the symbols !N",, !N",, N 8 . . . "N p , are 
 taken to represent, in respect to the successive pairs of wheels 
 of the train, the values of that function (equation 247), 
 which determines the friction due to the weights of those 
 wheels ; and each of the symbols L,, L s , L 4 . . . 1^ , the dis- 
 tance between the point of contact of a corresponding pair 
 of wheels and the projection upon its plane of the point of 
 contact of the next preceding pair in the train ; whilst the 
 symbols n^ n^ n s . . . n^ p , represent the numbers of teeth in 
 the wheels ; r^ r r^ . . . r^ P , the radii of their pitch circles ; 
 and S 15 S a , S 8 . . . S p , the spaces described by their points ot 
 
THE MJDULTIS OF A TRAIN OF WHEELS. 303 
 
 contact a, 5, c, &c. whilst the work TJj is done upon the first 
 wheel of the train. 
 
 Let us suppose the co-efficients of i* a , u^ u . . . U a , in these 
 moduli to be represented by (l+f\), (l+M- 2 )> (1+f*,) . . . . 
 (1-f fly) ; they will then become 
 
 Eliminating w a , w 8 , % 4 ...%,, between these equations, we 
 shall obtain an equation of the form 
 
 . S . . . (2TT), 
 where 
 
 p ..... (278). 
 
 Now let it be observed, that the space described by the first 
 wheel, at distance unity from its centre, whilst the space S x 
 
 Q 
 
 is described by its circumference, is represented by , and 
 
 r, 
 
 o 
 
 that this same space is represented by if S represent the 
 
 <*i 
 
 space described in the same time by the foot of the per- 
 pendicular 15 or the space through which the moving 
 pressure may be conceived to work during that time ; so 
 
 that = . Also let it be observed that the space de- 
 fi a i 
 
 scribed by the third wheel, at distance unity from its centre, 
 is the same with that described at the same distance from 
 
 S S 
 
 its centre by the second wheel, so that = ; in like 
 
 r s r, 
 
 manner that the spaces described at distances unity from 
 their centres by the fourth and fifth wheels are the same, so 
 
 Q C Q Q 
 
 that ; and similarly, that = , &c.=&c. ; and 
 ^ r, r , r t ' 
 
 finally, 
 
304 THE MODULUS OF A TRAIN OF WHEELS. 
 
 Multiplying the two first of these equations together, then 
 the three first, the four first, &c., and transposing, \ve have 
 
 , ., 
 
 a, 0, . r, \aj \n,l 
 
 , n, . m, 
 
 g r..n.n.A g /M (!'8, 
 
 0, . r a . r 4 . T-. \V \ 
 &c.=&c. 
 
 Substituting these values of S,, S a , &c. in equation (278), 
 and dividing by S, we have 
 
 or if we observe that the quantities i^,, ^ 3 , f* 3 , are composed 
 of terms all of which are of one dimension in sin. 9, sin. <p 1? 
 sin. 9 2 , &c. and that the quantities !N" 15 N 3 , N",, &c. (equation 
 247) are all likewise of one dimension in those exceedingly 
 small quantities ; and if we neglect terms above the first 
 dimension in those quantities, then 
 
 '-* }"> 
 
 If in like manner we neglect in equation (277) terms of 
 more than one dimension in M- a , f* a , ^ 8 , &c. we have 
 
 Now 1^= * ( j __ ) sin. 9 + ^ sin. 9, + 2^ si 
 
 \n, nj a,r, r.r, 
 
 a f = if I -- [_ ) sin. 9 + ^ sin. 9 
 
 \/i 8 nj r 4 r t 
 
 . S. 
 sin. 
 
THE MODULUS OF A TRAIN OF WHEELS. 305 
 
 /* = if ( + ) sin. 9 + ^ sin. <p 4 , 
 \n^ nj r r, 
 
 &c.=&c. 
 
 = r /-!_ + \ sin. 9 + 5^ si 
 W,P-I ' n, p ] r, p a, 
 
 Substituting these values of f^, f* a , &c. in the preceding 
 equation, 
 
 jL + .L + -L ---- L\ sin.9 + ^ sin.?! 
 n l n t n n, p J a.r., 
 
 which is a general expression for the modulus of a train of 
 any number of wheels. 
 
 235. The work IT, which must be done- upon the first 
 wheel of a train to yield a given amount U 3 .at the last wheel, 
 exceeds the work IJ 2 , or, in other words, the work done upon 
 the driving point exceeds- that yielded at the working point, 
 by a quantity which is represented by the expression 
 
 + -- h . ..- + - iwiM'.IL-fi-^ sm - 9 H sin. 9, 
 ^ r T 
 
 . . . (281). 
 
 In which expression the first term represents the expenditure 
 of work due to the friction of the teeth, and varies directly as 
 the work U a , which is done by the machine. The second 
 term represents the expenditure of work due to the friction 
 of the axes of the wheels, and varies in like manner directly 
 as the work done. Whilst the third term represents the 
 expenditure of work due to the weights of the wheels of the 
 train, and is wholly independent of the work done, but only 
 upon the space S, through which that work is done at the 
 point where the driving pressure is applied to the train. 
 
 20 
 
306 FRICTION OF THE AXES OF A TKAIN. 
 
 236. The expenditure of work due to the friction of the teeth. 
 
 The work expended upon the friction of the teeth is repre- 
 sented by the formula 
 
 r j( h - - + +...+ } sin. 9 (282), 
 
 whose value is evidently less as the factor sin. 9 is less, or as 
 the coefficient of friction between the common surfaces of the 
 teeth is less ; and as the numbers of the teeth in the different 
 wheels which compose the train are greater. The number 
 of teeth in any one wheel of the train may, in fact, be taken 
 so small, as to give this formula a considerable value as com- 
 pared with U 2 , or to cause the expenditure of work upon the 
 friction of the teeth to amount to a considerable fraction of 
 the work yielded by the train : and the numbers of teeth 
 of two or more wheels of such a train might even be taken 
 so small as to cause the work expended upon their friction to 
 equal or to surpass by any number of times the work yielded 
 by the train at its working point. This will become the 
 more apparent if we consider that the surfaces of contact of 
 the teeth of wheels are for the most part free from unguent 
 after they have remained any considerable time in action, so 
 that the limiting angle of resistance assumes in most cases 
 a much greater value at the surfaces of the teeth of the 
 wheels than at their axes. From this consideration the 
 importance of assigning the greatest possible number of 
 teeth to the wheels of a train individually and collectively 
 is apparent. 
 
 23V. The expenditure of work due to the friction of the axes. 
 This expenditure is represented by the formula 
 
 forming the second term of formula 280. Now, evidently, the 
 value of this formula is less as the quantities sin. 9,,. sin. 9,, 
 &c. are less, or ae the limiting angles of resistance between 
 the surfaces -of the axes and their bearings are less, or the 
 
FRICTION OF THE AXES OF A TRAIN. 307 
 
 lubrication of the axes more perfect ; and it is less as the 
 
 fractions^-', ^, ^, Ac. are less. 
 
 flyy y a 7y P 4 jy 
 
 Now, L a being the distance between the point of contact 1> 
 
 of the third and fourth wheels 
 and the projection of -the point of 
 contact a of the first and second 
 upon the plane of those wheels, 
 it follows that, generally, L 2 is 
 least when the projection of a 
 falls on the same side of the axis 
 as the point 5 ;* and that it 
 is least of all when this line falls on that side and in the line 
 joining the axis with the point 5; whilst it is greatest of all 
 when it falls in this line produced to the opposite side of the 
 axis. In the former case its value is represented by /,/ 
 and in the latter by ^ t +y a ; so that, generally, the maximum 
 and minimum values of L a are represented by the expression 
 
 y 3 y 2 , and the maximum and minimum values of ^- 2 - by 
 
 ** 
 
 JL | p 2 . And similarly it appears that the maximum and 
 
 r rj 
 
 minimum values of ^ are represented by( 1 ) P 3 ; and 
 
 TV \T T I 
 
 ' 4 ' 5 V/ 4 ' 6 ' 
 
 so of the rest. So that the maximum and minimum values of 
 the work lost by the friction of the axes are represented 
 by the expression 
 
 from which expression it is manifest, that in every case the 
 expenditure of work due to the friction of the axes is less as 
 the radii of the axes are less when compared with the radii 
 of the wheels ; being wholly independent of actual dimensions 
 of these radii, but only upon the ratio or proportion of the 
 radius of each axis to that of its corresponding wheel : more- 
 
 * Thte important condition is but a particular case of the general principle 
 established in Art. 168. ; from which principle it follows, that the driving 
 pressure on each wheel should be applied on the same side of the axis as the 
 driven pressure. 
 
308 THE WEIGHTS OF THE WHEELS. 
 
 over, that this expenditure of work is the least when the 
 wheels of the train are so arranged, that the projection of the 
 point of contact of any pair upon the plane of the next 
 following pair shall lie in the line of centres of this last pair, 
 between their point of contact and the axis of the driving 
 wheel of the pair ; whilst the expenditure is greatest when 
 this projection falls in that line but on the other side of the 
 axis. The difference of the expenditures of work on the 
 friction of the axes under these two different arrangements 
 of the train is represented by the formula 
 
 A- sin. <P! + sin. 9, + sin. <p 3 -f A S m.<p 4 + . . i U 
 r, r s r & r, ) 
 
 which, in a train of a great number of wheels, may amount 
 to a considerable fraction of U 2 ; that fraction of U a repre- 
 senting the amount of power which may be sacrificed by a 
 false arrangement of the points of contact of the wheels. 
 
 238. The expenditure of work due to the weights of the several 
 wheels of the train. 
 
 The third and last term !N" . S of the expression (280) repre- 
 sents the expenditure of work due to the weights of the 
 several wheels of the train; of this term the factor 1ST is 
 represented by an expression (equation -279), each of the 
 terms of which involves as a factor one of the quantities ~N 19 
 N 3 , N s , <fec., whose general type or form is that given in 
 equation (247), it being observed that the direction of the 
 driving pressure on any pair of the wheels being supposed 
 that of a tangent to their point of contact ; the case is that 
 discussed in the note to page 266. The other factor of each 
 term of the expression (equation 279) for !N", is a fraction 
 having the product n a n & . . . of the numbers of teeth in all 
 the preceding drivers of the train, except the first, for its 
 numerator, and the product n y . n 4 . n 6 . . , of the numbers of 
 teeth in the preceding followers of tlie train for its denomi- 
 nator ; so that if the train be one by which the motion is to 
 be accelerated, the numbers of teeth in the followers being 
 small as compared with those in the drivers, or if the multi- 
 plying power of the train be great, and if the qiiantities 
 Kj, N 9 , N 3 , &c., be all positive ; then is the expenditure of 
 work by reason of the weights of the wheels considerable, as 
 
MODULUS OF A TRAIN IN WHICH THE DRIVEKS ARE EQUAL. 309 
 
 compared with the whole expenditure. Since, moreover, the 
 coefficients of N 13 N a , !N",, &c., in the expression for N (equa- 
 tion 279) increase rapidly in value, this expenditure of work 
 is the greatest in respect to those wheels of the train which 
 are farthest removed from its first driving wheel : for which 
 reason, especially, it is advisable to diminish the weights 
 of the wheels as they recede from the driving point of the 
 train, which may readily be done, since the strain upon each 
 successive wheel is less, as the work is transferred to it under 
 a more rapid motion. 
 
 239. The modulus of a tram in which all the drivers cure 
 equal to one another and all the followers, and in which 
 the points of contact of the drivers and followers are all 
 similarly situated. 
 
 The numbers of teeth in the drivers of the train being in 
 this case supposed equal, and also the radii of these wheels, 
 n 1 = n s =n 6 =n 1 =&c.j r 1 =r 3 =r f> =?\=&c. The numbers of 
 teeth in the followers being also equal, and also the radii of 
 the followers n^n^n^&c., r^r^r^&c. 
 
 If, moreover, to simplify the investigation, the driving 
 
 work U 1 be supposed to be done upon the first wheel of the 
 
 9 train at a point situated in re- 
 
 spect to the point of contact a of 
 that wheel with its pinion pre- 
 cisely as that point of contact is 
 in respect to the point of contact 
 I of the next pair of wheels of 
 the train ; and if a similar sup- 
 position be made in respect to 
 the point at which the driven work U a is done upon the last- 
 pinion of the train, then, evidently, L,=L 2 =:L 3 . . . =L P , 
 and (see equation 247) 1^=^",= . . . =N P . 
 
 The modulus (equation 280) will become, these substitu- 
 tions being made in it, the axes being, moreover, supposed 
 all to be of the same dimensions and material, and equally 
 lubricated, and it being observed that the drivers and the 
 followers are each p in number, 
 
 ---- (284), 
 which is the modulus required. 
 
% 
 310 THE TRAIN OF LEAST RESISTANCE. 
 
 Moreover, the value of !N" (equation 27Y) will become by 
 the like substitutions, 
 
 
 THE TRAIN OF LEAST RESISTANCE. 
 
 240. A. train of equal driving wheels and equal followers 
 being required to yield at the last wheel of the train a 
 given amount of work U 2 , under a velocity m times greater 
 or less than that under which the work U, which drives the 
 train is done by the moving power upon the first wheel; it 
 is required to determine what should be the number p of 
 pairs of wheels in the train, so that the work TJ, expended 
 through a given space S, in driving it, may be a minimum. 
 
 Since the number of revolutions made by the last wheel 
 of the train is required to be a given multiple or part of the 
 number of revolutions made by the first wheel, which mul- 
 tiple or part is represented by m, therefore (equation 231), 
 
 _ 
 :. = , and = ; 
 
 ' * ' 
 
 Substituting these values in the modulus (equation 284); 
 substituting, moreover, for N its value from equation (285), 
 we have 
 
THE TKAIN OF LEAST RESISTANCE. 311 
 
 (286X 
 
 It is evident that the question is solved by that value of p 
 which renders this function a minimum, or which satisfies 
 
 the conditions -j- 2 = and -^ > 0. The first condition 
 dp dp ' 
 
 gives by the differentiation of equation (286), 
 Iog. E 
 
 (m 
 
 This equation may be solved in respect to p, for any given 
 values of the other quantities which enter into it, ly approxi- 
 mation. If, being differentiated a second time, the above 
 expression represents a positive quantity when the value of 
 p (before determined) is substituted in it, then does that 
 value satisfy both the conditions of a minimum, and sup- 
 plies, therefore, its solution to the problem. 
 
 If we suppose 9 1 =0 and'N^O, or, in other words, if we 
 neglect the influence of the friction of the axes and of the 
 weights of the wheels of the train upon the conditions of the 
 question, we shall obtain 
 
 , 
 
 gin. 9 H sin. 9=0 ; 
 p n^ 7i, 
 
 whence by reduction, 
 
 p = ^-' m i * ...... (288). 
 
 * This formula was given by the late Mr. Davis Gilbert, in his paper on the 
 " Progressive improvements made in the efficiency of steam engines in Corn- 
 wall," published in the Transactions of the Royal Society for 1830. Towards 
 the conclusion of that paper, Mr. Gilbert has treated of the methods best 
 adapted for imparting great angular velocities, and, in connection with that 
 subject, of the friction of toothed wheels ; having reference to the friction of 
 the surfaces of their teeth alone, and neglecting all consideration of the influ- 
 
THE INCLINED PLANE. 
 
 THE INCLINED PLANE. 
 
 241. Let AB represent the surface of an inclined plane on 
 which is supported a body whose centre of gravity is C, and 
 its weight W , by means of a pressure acting in any direction, 
 and which may be supposed to be supplied by the tension of 
 a cord passing over a pulley and carrying at its extremity 
 a weight. 
 
 Let OR represent the direction of the resultant of P and 
 W. If the direction of this line be inclined to the perpen- 
 dicular ST to the surface of the plane, at an angle OST 
 equal to the limiting angle of resistance, on that side of ST 
 which is farthest from the summit B of the plane (as in 
 Jig. 1), the body will be upon the point of slipping upwards; 
 and if it be inclined to the perpendicular at an angle OST, 
 
 ence due to the weights of the wheels and to the friction of their axes. The 
 author has in vain endeavoured to follow out the condensed reasoning by which 
 Mr. Gilbert has arrived at this remarkable result ; it supplies another example 
 of that rare sagacity which he was accustomed to bring to the discussion of 
 questions of practical science. Mr. Gilbert has given the following examples 
 of the solution of the formula by the method of approximation: If ra=120, 
 or if the velocity is to be increased by the train 120 times, then the value of p 
 given by the above formula, or the number of pairs of wheels which should 
 c impose the train, so that it may work with a minimum resistance, reference 
 being had only to the friction of the surfaces of the teeth, is 3-745 ; and the value 
 
 <f the factor p(m p -f-1) (equation 286), which being multiplied by sin. (f> U a 
 
 HI 
 
 i ^presents the work expended on the friction of the surfaces of the teeth, is in 
 this case 17'9 ; whereas its value would, according to Mr. Gilbert, be 121 if the 
 velocity were got up by a single pair of wheels. So that the work lost by the 
 friction of the teeth in the one case would only be one seventh part of that in 
 the other. In like manner Mr. Gilbert found, that if m=100, then p should 
 equal 3*6 ; in which case the loss by friction of the teeth would amount to the 
 sixth part only of the loss that would result from that cause if _p=l, or if the 
 required velocity were got up by one pair of wheels. 
 
 If m=40, then jt)=2'88, with a gain of one third over a single pair. 
 
 If 7tt=3-69, thenjt>=l. 
 
 If ra=12'85, then p=2. 
 
 If ra=46'3, then jt>=3. 
 
 If m=166'4, thenjo=4. 
 
 It is evident that when p, in any of the above examples, appears under the 
 form of a fraction, the nearest whole number to it, must be taken in practice. 
 The influence of the weights of the wheels of the train, and that of the friction 
 of the axes, so greatly however modify these results, that although they are 
 fully sufficient to show the existence in every case of a certain number of 
 wheels, which being assigned to a train destined to produce a given accelera- 
 tion of motion shall cause that train to produce the required effect with the 
 least expenditure of power, yet they do not in any case determine correctly 
 what that number of wheels should be. 
 
THE INCLINED PLANE. 
 
 313 
 
 (i.) 
 
 equal to the limiting angle of resistance, but on the side of 
 ST nearest to the summit B (as in fig. 2.), then the body will 
 be upon the point of slipping downwards (Art. 138.) ; the 
 former condition corresponds to the superior and the latter 
 to the inferior state bordering upon motion (Art. 140.). 
 
 Now the resistance of the plane is equal and opposite to 
 the resultant of P and "W ; let it be represented by K. 
 
 There are then three pressures P, W, and K in equili- 
 brium. 
 
 sn. 
 
 Let /BAC=i, ZOST=:lime. Z of resistance =9, let 
 represent the inclination PQB of the direction of P to the 
 surface of the plane, and draw OY perpendicular to AB ; 
 then, 
 
 mfig. 1, 
 and POK 
 
 in fig. 2., WOR=WOY-SOY=BAC-OST=i--9, 
 and 
 
 =PQB+-OST=^+*-9 ; 
 
 2 a 
 
 and 
 
 the upper or lower sign being taken according as the body 
 is upon the point of sliding up the plane, as in fig. 1, or 
 down the plane, as in fig. 2. Or if we suppose the angle 9 
 to be taken positively or negatively according as the body is 
 on the point of slipping upwards or downwards ; then gene- 
 
 rally WOK=+<p 
 
314 
 
 THE INCLINED PLANE. 
 
 P sin. (1 + 9) __sin. (+<p) t 
 
 "W"~~~ lie ~\~ cos. (09 ' 
 sin. (-+0-9) 
 
 sn - 
 
 * cos.(d 9) 
 
 (289). 
 
 If the direction of P be parallel to the plane, / PQB 01 
 1=0 ; and the above relation becomes 
 
 If i=0 the plane becomes horizontal (fig. 3)., and the re- 
 lation between P and W assumes the form 
 
 P=W 
 
 sin. 9 
 
 9) 
 
 (291). 
 
 If 0=0, P=W . tan. 9, as it ought (see Art. 138.). 
 
 If the angle PQB or & (fig. 1.) be increased so as to be- 
 come d, rQ will assume the direction shown in fig. 4, 
 and the relation (equation 289), between P and "W will be- 
 
 come 
 
 sn. 
 
 (292). 
 
 cos. (d + <p) ' 
 
 The negative sign showing that the direction of P must, 
 in order that the body may slip up the plane, be opposite to 
 that assumed in fig. 1. ; or that it must be a pushing pres- 
 sure in the direction PO instead of a pulling pressure in the 
 direction OP. 
 
 If, however, the body be upon the point of slipping down 
 the plane, so that 9 must be taken negatively ; and if, more- 
 over, 9 be greater than i, then sin. + 9), will become sin. 
 (19)^= sin. (91), so that P will in this case assume the 
 positive value 
 
 . 
 cos. (d 9) 
 
 (293), 
 
THE MOVEABLE INCLINED PLANE. 
 
 315 
 
 which determines the force just necessary under these cir- 
 cumstances to pull the body down the plane. 
 
 If i=<p, P 0, the body will therefore, in this case, be upon 
 the point of slipping down the plane without the application 
 of any pressure whatever to cause it to do so, other than its 
 own weight. The plane is under these circumstances, said 
 to be inclined at the angle of repose, which angle is there- 
 fore equal to the limiting angle of resistance. 
 
 242. The direction of least traction. 
 
 Of the infinite number of different directions in which the 
 pressure P may be applied, each requiring a different amount 
 to be given to that pressure, so as to cause the body to slide 
 up the plane, that direction will require the least value to be 
 assigned to P for this purpose, or will be the direction of 
 least traction, which gives to the denominator of the fraction 
 in equation (289) its greatest value, or 
 which makes 09=0 or d=<p. The di- 
 rection of P is therefore that of least 
 traction when the angle PQB is equal to 
 the limiting angle, a relation which ob- 
 tains in respect to each of the cases dis- 
 cussed in the preceding article. 
 
 243. THE MOVEABLE INCLINED PLANE. 
 
 Let ABC represent an inclined 
 plane, to the back AC of which 
 is applied a given pressure P,, 
 and which is moveable between 
 the two resisting surfaces GH and 
 KL, of which either remains fixed, 
 and the other is upon the point 
 of yielding to the pressure of the 
 plane. 
 
 If we suppose the resultants of the resistances upon the 
 different points of the two surfaces AB and BG of the plane 
 to be represented by T^ and R 3 respectively, it is evident 
 that the directions of these resistances and of the pressure P t 
 
316 THE MOVEABLE INCLINED PLANE. 
 
 will meet, when produced, in the same point O* ; and that, 
 since the plane is upon the point of slipping upon each of 
 the surfaces, the direction of each of these resistances is 
 inclined to the perpendicular to the surface of the plane, at 
 the point where it intersects it, at an angle equal to the cor- 
 responding limiting angle of resistance. 
 
 So that if ET and FS represent perpendiculars to the 
 , surfaces AB and BC of the plane at the points E and F and 
 9 1? 9 a , the limiting angles of resistance between these surfaces 
 of the plane and the resisting surfaces GH and KL re- 
 spectively, then K.ET^, K 2 FS=9 ? . 
 
 Now the pressures P 15 3^, R 2 being in equilibrium (Art. 
 W), 
 
 P, sin. EOF P^sin. EOF 
 
 K^sin.DOF' * K 3 -~sin.DOE' 
 
 But the four angles of the quadrilateral figure BEOF 
 being equal to four right angles (Euc. 1'32), EOF 2* 
 
 EBF-OEB-OFB; but EBF=, OEB+9, OFB= 
 
 Similarly, DOE=2rr-ADO-AEO-DAE; but ADO = 
 J AEO=~ ?l , BAC=^-: .-.DOE=J+.+ 9l . 
 
 Since, moreover, DO is parallel to BC, both being per- 
 pendicular to AC, /.DOF=tf OFC; but OFC=^ <p, : 
 
 P. _sin. {((+ 9, + 9,)} sin. Q + 9. + 9.) . 
 
 sn. 
 
 cos. 9 2 
 t _ sin, {if Q + 9i + <p 9 ){ _sin. 
 
 Since either is equal and opposite to the resultant of the other two. 
 
A SYSTEM OF TWO MOVEABLE INCLINED PLANTS. 317 
 
 cos. 
 
 In the case in which the surface GH yields to the pressure 
 of the plane, KL remaining fixed, we obtain (equation 121.) 
 for the modulus (see Art. 148.) observing that P 1 ()=K 1 sin. i 
 (equation 294), 
 
 u sin. 0+9,+yJ _ (29g) _ 
 
 sm. i . cos. <p a 
 
 In the case in which the surface KL yields, CH remaining 
 fixed, observing that P 1 ()=Tl a tan. i (equation 295), we have 
 
 sn. 
 
 (297) * 
 
 Equations (296) and (297) may be placed respectively un- 
 der the forms 
 
 and U =U CQS> (<p * +(p * J tan.i + tan.(9 1 +9.) 1 
 
 sin. j ( cot. 9 1 tan. ) tan. i ) * 
 
 The value of II, corresponding to a given value of U 3 is in 
 
 the former equation a minimum when =-, and in the latter 
 
 2t 
 
 when 
 
 tan. i=. \ A/-. C !' <P ; , x -l 1 tan. (9, +9.) .... (298). 
 
 f l sin. <p 1 sm.(<p 1 +<p,) ) 
 
 From the former of these equations it follows, that the work 
 lost by friction (when the driving surface of the plane is its 
 hypothenuse) is less as the inclination of the plane is greater, 
 or as its mechanical advantage is less. 
 
 244. A system of two moveable inclined planes. 
 Let A and B represent two inclined planes, of which A 
 
318 A SYSTEM OF TWO MOVEABLE INCLINED PLANES. 
 
 P rests upon a horizontal surface, and 
 
 receives a horizontal motion from 
 the action of the pressure P, ; com- 
 municating to B a motion which is 
 restricted to a vertical direction by 
 
 the resistance of the obstacle D, 
 
 ^ \ ~^ which vertical motion of the plane 
 
 ^ T3 i\. """ * s PP se d by the pressure P 2 ap- 
 * plied to its superior surface. It is 
 
 required to determine a relation between the pressures P x 
 and P 2 , in their state bordering upon motion ; and the mo- 
 dulus of the machine. 
 
 Let R 1 represent the pressure of the plane A upon the 
 plane B, or the resistance of the latter plane upon the former, 
 and R 3 the resistance of the obstacle D upon the back of the 
 plane B ; then is the relation between Rj and P, determined 
 by equation (294). And since R 1? R 3 , P 2 are pressures in 
 equilibrium, the relation between R 1 and P 2 is expressed 
 
 (Art. 14.) by the relation^- 1 = sm ' \ Now R 3 Q is 
 
 inclined to a perpendicular to the back of the plane B, at an 
 angle equal to the limiting angle of resistance between the 
 surface of that plane and the obstacle D on which it is upon 
 the point of sliding. Let this angle be represented by 9,, 
 then is the inclination of R 3 to the back of the plane or P 2 Q 
 
 represented by^ 9 3 ; so that P 2 QR 3 - 9,. 
 
 And if R 8 Q be produced so as to meet the surface of the 
 plane A in Y, and YS be drawn horizontally, 
 
 where i represents the inclination of the superior surface of 
 the plane A or the inferior surface of the plane B to the 
 horizon. Substituting these values of P a QR s and R^R,, we 
 obtain 
 
 cos.<p 8 
 
 Multiplying this equation by equation (294), and solving in 
 respect to Pj, 
 
A SYSTEM OF THREE INCLINED PLANES. 
 
 319 
 
 COS. (t + P. + 9.) COS. <f, 
 
 .-. (Art. 152.) U,=U, 
 
 sn. 
 
 ,) cos.? s 
 
 cos. 
 
 3 ) tan. t cos.<? 2 
 
 _ (goo) 
 
 A system of three inclined planes, two of which are movea- 
 lie, and the third fixed. 
 
 245. The inclined plane A, in the accompanying figure, is 
 fixed in position, the plane B is 
 moveable upon A, having its upper 
 surface inclined to the horizon at a 
 less angle than the lower ; and C is 
 an inclined plane resting upon B, 
 which is prevented from moving 
 horizontally by the obstacle D, but 
 may be made to slide along this 
 obstacle vertically. It is required 
 to determine a relation between 
 Pj and P a , applied, as shown in the figure, when the system 
 is in the state bordering upon motion. 
 
 Let Rj, R 2 , R 3 represent the resistances of the surfaces on 
 which motion takes place, <p a <P 2 9 3 their limiting angles of 
 resistance respectively, and i l9 2 the inclinations of the two 
 surfaces of contact of B to the horizon. Since P K 1? R 2 are 
 pressures in equilibrium, as also P 2 , E 2 , K 8 
 
 . P } __sin 1 R 2 OR 1 R 2 sin.P,QR 3 
 ' R 2 ~ sin. P.OIV P 2 - sin. R 2 QR 8 - 
 
 Multiplying these equations together, 
 
 P 1 _sin.E a OB 1 .8in.P il QB, 
 P 2 sin. P/JK, . sin. R 2 QR 3 
 
 Draw OS and OT parallel to the faces of the plane B ; then 
 
 -TOS; but R.OS^ --<?,, since OS is 
 
 parallel to the inferior face of the plane B, also QOT= <p a , 
 
 2 
 since OT is parallel to the superior face of the plane B ; and 
 
320 A SYSTEM OF THREE INCLINED PLANES. 
 
 TOS = the inclination of the faces of the plane B to one 
 another = i 1 a . 
 
 /.K^OK^ (2 -<P> 
 Also P 9 QK 3 = I -K 3 QM= | - 9.. 
 
 LetP.O be produced to Y; therefore P 1 OK l =*'-K 1 OY= 
 
 ^-(I^OS-SOV)^- j g ~9,) -i, | = I + i, + 9,- Lastly 
 K 2 QK 3 = OQM+MQK 3 '. Now, MQK 3 =9 3 ; also, OQM = 
 *-(QOT+TOY)=*- j (9.) +',} =J-',+9 
 
 .% E 9 QE 3 =r i, +9,+9 8 = (',9, 9.). 
 
 a sin. (s+'i + Pi) 8 ^ n - ] o ~( t *V*V*) f 
 , __p sin. j(9 1 + 9 a ) + Qi Q} cos. 9 8 
 
 Whence we obtain for the modulus (Art. 152.), observing; 
 that^<)= 8in - ( '~'' ) . 
 
 COS. , COS. I, 
 
 TT ^-rr sin. (9,+9,+^-Qcos.^cos. i cos.9, 
 
THE WEDGE DRIVEN BY PRESSURE. 
 
 321 
 
 THE WEDGE DRIVEN BY PRESSURE. 
 
 246. Let ACB represent an isosceles wedge, whose angle 
 ACB is represented by 2*, and which is 
 driven between the two resisting surfaces 
 DE and DF, by the pressure P,. Let R, 
 and R 2 represent the resistances of these 
 surfaces upon the acting surfaces CA and 
 CB of the wedge when it is upon the 
 point of moving forwards. Then are the. 
 directions of R, and R 2 inclined respec- 
 tively to the perpendicular Gs and; R' 
 to the faces C A and CB of the wedge^ at 
 angles each equal to the limiting angle of 
 resistance 9. The pressures R r and R a are 
 therefore equally inclined to the axis 
 of the wedge, and to the direction; of I\, whence it follows 
 that R^E,,, and therefore (Art. 13;) that P 1 =2R 1 cos. GOR. 
 Now, since CGOE is a quadrilateral figure, its four angles 
 are equal to four right angles ; therefore GOR=2* GCR 
 
 OGC ORC. But GCR=2i; OGC^ORC = +9 : 
 
 a 
 
 (303). 
 
 Whence it follows (equation 121) that the modulus of the 
 wedge is 
 
 . (304). 
 
 sn. i 
 
 This equation may be placed under the form 
 11!= U 3 jcot. 9 -f- cot. i\ sin. 9. 
 
 The work lost by reason of the friction of the wedge is 
 greater, therefore, as the angle of the wedge is less; and 
 infinite for a finite value of 9, and an infinitely small value 
 of*. 
 
 The angle of the wedge. 
 
 247. Let the pressure P instead of being that just suffi- 
 21 
 
322 
 
 THE WEDGE DRIVEN BY PRESSURE. 
 
 cient to drive tlie wedge, be now supposed 
 to be that which is only just sufficient tc 
 keep it in its place when driven. The two 
 surfaces of the wedge being, under these 
 circumstances, upon the point of sliding 
 u backwards upon those between which the 
 wedge is driven, at their points of contact 
 G and R, it is evident that the directions 
 of the resistances if* and i^R upon those 
 points, must be inclined to the normals 
 6-G and tH at angles, each equal to the 
 limiting angle of resistance, but measured 
 on the sides of those normals opposite to 
 
 those on which the resistances RjG and R 2 R are applied.* 
 In order to adapt equation (303) to this case, we have 
 
 only then to give to 9 a negative value in that equation. It 
 
 will then become 
 
 P l =2E 1 sin.( 9) 
 
 (305). 
 
 So long as i is greater than 9, or the angle C of the wedge 
 greater than twice the limiting angle of resistance, P l is 
 positive ; whence it follows that a certain pressure acting in 
 the direction in which the wedge is driven, and represented 
 in amount by the above formula, is, in this case, necessary 
 to keep the wedge from receding from any position into 
 which it has been driven. So that if, in this case, the pres- 
 sure Pj be wholly removed, or if its value become less than 
 that represented by the above formula, then the wedge will 
 recede from any position into which it has been driven, or 
 it will be started. If i be less than 9, or the angle C of the 
 wedge less than twice the limiting angle of resistance, P, 
 will become negative ; so that, in this case, a pressure, oppo- 
 site in direction to that by which the wedge has been driven, 
 will have become necessary to cause it to recede from the 
 position into w T hich it has been driven ; whence it follows, 
 that if the pressure P, be now wholly removed, the wedge 
 will remain, fixed in that position ; and, moreover, that it 
 will still remain fixed, although a certain pressure be applied 
 to cause it to recede, provided that pressure do not exceed 
 the negative value of P 1? determined by the formula. 
 
 * This will at once be apparent, if we consider that the direction of the 
 resultant pressure upon the wedge at G must, in the one case, be such, that if 
 it acted alone, it would cause the surface of the wedge to slip downwards on 
 the surface of the mass at that point, and in the other case upwards ; and that 
 the resistance of the mass is in each case opposite to this resultant pressure. 
 
THE WEDGE DRIVEN BY IMPACT. 323 
 
 It is this property of remaining fixed in any position into 
 which it is driven when the force which drives it is removed, 
 that characterises the wedge, and renders it superior to 
 every other implement driven by impact. 
 
 It is evidently, therefore, a principle in the formation of a 
 wedge to be thus used, that its angle should be less than 
 twice the limiting angle of resistance between the material 
 which forms its surface, and that of the mass into which it 
 is to be driven. 
 
 THE WEDGE DRIVEN BY IMPACT. 
 
 248. The wedge is usually driven by the impinging of a 
 heavy body with a greater or less velocity upon its back, in 
 the direction of its axis. Let W represent the weight of 
 such a body, and V its velocity, every element of it being 
 conceived to move with the same velocity. The work 
 accumulated in this body, when it strikes the wedge, will 
 
 then be represented (Art. 66.) by - V 2 . Now the whole of 
 
 $ 
 this work is done by it upon the wedge, and by the wedge 
 
 upon the resistances of the surfaces opposed to its motion ; 
 if the bodies are supposed to come to rest after the impact, 
 and if the influence of the elasticity and mutual compression 
 of the surfaces of the striking body and of the wedge are 
 neglected, and if no permanent compression of their surfaces 
 
 1 W V 3 
 
 follows the impact.* .*. Uj = - - . 
 
 2 9 
 
 * The influence of these elements on the result may be deduced from the 
 principles about to be laid down in the chapter upon impact. It results from 
 these, that if the surfaces of the impinging body and the back of the wedge, 
 by which the impact is given and received, be exceedingly hard, as compared 
 with the surfaces between which the wedge is driven, then the mutual pressure 
 of the impinging surfaces will be exceedingly great as compared with the 
 resistance opposed to the motion of the wedge. Now, this latter being 
 neglected, as compared with the former, the work received or gained by the 
 wedge from the impact of the hammer will be shown in the chapter upon 
 
 impact to be represented by * ~r~ e ' - 1 - - , where Wj represents the 
 
 " 
 
 weight of the hammer, W 2 the weight of the wedge, and e that measure of 
 'the elasticity whose value is unity when the elasticity is perfect. Equating 
 this expression with the value of Ui (equation 304), and neglecting the effect 
 of the elasticity and compression of the surfaces G and R, between which the 
 wedge is driven, we shall obtain the approximation 
 
 _ (l-f- g ) 3 W 1 a W a V a sin, i 
 sin. 
 
324 THE 'WEDGE DRIVEN BY IMPACT. 
 
 Substituting this value of U l in equation 303, and solving in 
 respect to U 2 , we have 
 
 sn.. ..... 
 
 2 g sin. (* 4- 9) 
 
 by which equation the work U 2 yielded upon the resistances 
 opposed to the motion of the wedge by the impact of a given 
 weight "W with a given velocity Y is determined ; or the 
 weight "W necessary to yield a given amount of work when 
 moving with a given velocity ; or, lastly, the velocity Y with 
 which a body of given weight must impinge to yield a given 
 amount of work. 
 
 If the wedge, instead of being isosceles, be of the form of 
 a right angled triangle, as shown 
 in the accompanying figure, the 
 relation between the work U, done 
 upon its back, and that yielded 
 upon the resistances opposed to 
 its motion at either of its faces, is 
 represented by equations (296) 
 and (297). Supposing therefore 
 this wedge, like the former, to be 
 
 driven by impact, substituting as before for Uj its value 
 
 1"W 
 
 - Y 2 , and solving in respect to U a , we have, in the c 
 
 which the face AB of the wedge is its driving surface 
 
 sn 
 
 2 g si 
 when the base BC of the wedge is its driving surface, 
 
 q. __1 WY 2 tan, i cos. 
 8 2 g ' 8in. 
 
 From this expression it follows, that the useful work is the greatest, other 
 things being the same, when the weight of the wedge is equal to the weight 
 of the hammer, and when the striking surfaces are hard metals, so that the 
 value of e may approach the nearest possible to unity. 
 
THE MEAN PBESStTEE OF IMPACT. 
 
 325 
 
 I 
 
 249. If the power of th6 wedge 
 be applied by the intervention of 
 an inclined plane moveable in a 
 direction at right angles to the di- 
 rection of the impact*, as shown in 
 the accompanying figure, then sub- 
 stituting for U, in equation (300) 
 
 respect to U 2 , we have 
 
 cos. (i -h 9 t +9,) tan. < cos. 9 a 
 cos 9 8 
 
 half the vis viva of the impinging 
 body, and solving, as befo 
 
 ore, in 
 
 sn. 
 
 (309). 
 
 If instead of the base of the 
 plane being parallel to the direc- 
 tion of impact, it be inclined to 
 it, as shown in the accompanying 
 figure, then, substituting as above 
 in equation 302, we have 
 
 a 9,) cos. 
 
 sn. (t 1 
 
 3 2 g sin. fo 
 
 i 2 cos. i, cos. * 2 eos.<p 8 
 
 ^^ 
 
 ' ' 
 
 THE MEAN PKESSUKE OF IMPACT. 
 
 250. It is evident from equations 306, 307, 308, that, since, 
 whatever may be the weight of the impinging body or the 
 velocity of the impact, a certain finite amount of work U 2 is 
 yielded upon the resistances opposed to the motion of the 
 wedge ; there is in every such case a certain mean resistance 
 H overcome through a certain space S, in the direction in 
 which that resistance acts, which resistance and space are 
 such, that 
 
 KS=U 2 , and therefore E=^ 
 
 b 
 
 If therefore the space S be exceedingly small as compared 
 
 * This is the form under which the power of the wedge is applied for the 
 expressing of oil. 
 
326 THE SCREW. 
 
 with U s , there will be an exceedingly great resistance E 
 overcome by the impact through that small space, however 
 slight the impact. From this fact the enormous amount of 
 the resistances which the wedge, when struck by the ham- 
 mer, is made to overcome, is accounted for. The power of 
 thus subduing enormous resistances by impact is not how- 
 ever peculiar to the wedge, it is common to all implements 
 of impact, and belongs to its nature ; its effects are rendered 
 permanent in the wedge by the property possessed by that 
 implement of retaining permanently any position into which 
 it is driven between two resisting surfaces, and thereby op- 
 posing itself effectually to the tendency of those surfaces, by 
 reason of their elasticity, to recover their original form and 
 position. It is equally true of any the slightest direct impact 
 of the hammer as of its impact applied through the wedge, 
 that it is sufficient to cause any finite resistance opposed to 
 it to yield through a certain finite space, however great that 
 resistance may be. The difference lies in this, that the sur- 
 face yielding through this exceedingly small but finite space 
 under the blow of the hammer, immediately recovers itself 
 after the blow if the limits of elasticity be not passed ; 
 whereas the space which the wedge is, by such an impact, 
 made to traverse, in the direction of its length, becomes a 
 permanent separation. 
 
 THE SCKEW. 
 
 251. Let the system of two moveable inclined planes re- 
 presented in fig. p. 318. be formed of ex- 
 ceedingly thin and pliable laminse, and con- 
 ceive one of them, A for instance, to be 
 wound upon a convex cylindrical surface, as 
 shown in the accompanying figure, and the 
 other, B, upon a concave cylindrical surface 
 having an equal diameter, and the same axis 
 with the other; then will the surfaces 
 EF and GH of these planes represent truly 
 the threads or helices of two screws, one of them of the form 
 called the male screw, and the other the female screw. Let 
 the helix EF be continued, so as to form more than one spire 
 or convolution of the thread ; if, then, the cylinder which 
 carries this helix be made to revolve upon its axis by the 
 action of a pressure Pj applied to its circumference, and the 
 cylinder which carries the helix GH be prevented from re- 
 
THE SCKEW. 327 
 
 volving upon its axis by the opposition of an obstacle D, 
 which leaves that cylinder nevertheless free to move in a 
 direction parallel to its axis, it is evident that the helix EF 
 will be made to slide beneath GH, and the cylinder which 
 carries the latter helix to traverse longitudinally ; moreover, 
 that the conditions of this mutual action of the helical sur- 
 faces EF and GrH will be precisely analogous to those of the 
 surfaces of contact of the two moveable inclined planes dis- 
 cussed in Art. 244. So that the conditions of the equili- 
 brium of the pressures P, and P, in the state bordering upon 
 motion, and the modulus of the system, will be the same in 
 the one case as in the other ; with this single exception, that 
 the resistance R 2 of the mass on which the plane A rests (see 
 fig. p. 318.) is not, in the case of the screw, applied only to 
 the thin edge of the base of the lamina A, but to the whole 
 extremity of the solid cylinder on which it is fixed, or to a 
 circular projection from that extremity serving it as a pivot. 
 Now if, in equation 299, we assume <? 2 =0, we shall obtain 
 that relation of the pressures P x and P 2 in their state border- 
 ing upon motion, which would obtain if there were no fric- 
 tion of the extremity of the cylinder on the mass on which it 
 rests ; and observing that the pressure P 2 is precisely that. 
 by which the pivot at the extremity of the cylinder is pressed 
 upon this mass, arid therefore the moment (see Art. 176, 
 equation 183) of the resistance to the rotation of the cylinder 
 
 2 
 produced by the friction of this pivot by -P 2 p tan. <p 2 , where 
 
 p represents the radius of the pivot ; observing, moreover, 
 that the pressure which must be applied at the circumfe- 
 rence of the cylinder to overcome this resistance, above that 
 which would be required to give motion to the screw if there 
 
 were no such friction, is represented by- P 2 -tan. <p 2 , r being 
 
 taken to represent the radius of the cylinder, we obtain for 
 the entire value of the pressure P x in the state bordering 
 upon motion 
 
 sin. (* + ?,) cos. 9. 2 p 
 
 +rt< 
 
 The pressure P x has here been supposed to be applied to 
 turn the screw at its circumference ; it is customary, however, 
 to apply it at some distance from its circumference by the 
 intervention of an arm. If a represent the length of such an 
 
328 
 
 THE SCREW. 
 
 arm, measuring from the axis of the cylinder, it is evident 
 that the pressure P! applied to the extremity of that arm, 
 would produce at the circumference of the cylinder a pressure 
 
 represented by P^, which expression being substituted for 
 
 P! "in the preceding equation, and that equation solved in 
 respect to P 1? we obtain finally for the relation between P, 
 and P 3 in their state bordering upon motion, 
 
 m j sin. (,+,.) COB , 8/j.V tan _ ) 
 
 \af ( cos. (i + Vi+Vi) 3\r/ 
 
 If in like manner we assume in the modulus (equation 300) 
 9 2 =0, and thus determine a relation between the work done 
 at the driving point and that yielded at the working point, 
 on the supposition that no work is expended on the friction 
 of the pivot ; and if to the value of U, thus obtained we add 
 the work expended upon the resistance of the pivot which is 
 shown (equation 184) to be represented at each revolution 
 4. 
 
 by ^pP 2 tan. 9,, and therefore during n revolutions by 
 o 
 
 4. 
 
 -*^pP a , we shall obtain the following general expression for 
 
 o 
 
 the modulus ; the whole expenditure of work due to the 
 prejudicial resistances being taken into account. 
 
 TT TT sn. i + 9 cos. 9, ^ 
 
 U '= U - ' + ^ tan ' 
 
 Representing by X the common distance between the threads 
 of the screw, i. e. the space which the nut B is made to 
 traverse at each revolution of the screw ; and observ- 
 
 4 4 IT 
 
 that n^P 2 =U 2 , so that ^wP 3 tan. 9,=-* 2 p tan. 9 3 = 
 
 O O A. 
 
 *r P TT , . , . * 
 
 - -.-.U 2 tan. 9, in which expression = cot. t, we 
 6 A. r ** 
 
 obtain finally for the modulus of the screw 
 
 i sin, fr + 9,) cos. 9 3 2 P_ J ^ ( 
 
 8 I cos. ^ + 9 1 -f-9 3 ) 3r } 
 
 It is evidently immaterial to the result at what distance 
 from the axis the obstacle D is opposed to the revolution of 
 
APPLICATIONS OF THE SCKEW. 
 
 329 
 
 that cylinder which carries the lamina B ; since the amount 
 of that resistance does not enter into the result as expressed 
 in the above formula, but only its direction determined by 
 the angle 9 8 , which angle depends upon the nature of the 
 resisting surfaces, and not upon the position of the resisting 
 point. 
 
 APPLICATIONS OF THE SCREW. 
 
 252. 'The accompanying figure represents an application 
 of the screw under the circumstances described in the pre- 
 ceding article, to the well known machine called the V ICE. 
 
 AB is a solid cylinder carrying on its surface the thread of a 
 male screw, and within the piece CD is a hollow cylindrical 
 surface, carrying the corresponding thread of a female 
 screw; this female screw is prevented from revolving with 
 the male screw by a groove in the piece CD, which carries 
 it, and which is received into a corresponding projection EF 
 of the solid frame of the machine, serving it as a guide ; 
 which guide nevertheless allows a longitudinal motion to 
 the piece CD. A projection from the frame of the instru- 
 ment at B, met by a pivot at the extremity of the male 
 screw, opposes itself to the tendency of that screw to tra- 
 verse in the direction of its length. The pressure P 2 to be 
 overcome is applied between the jaws H and K of the vice, 
 and the driving pressure Pj to an arm which carries round 
 with it the screw AB. 
 
 It is evident that, in the state bordering upon motion, the 
 resistance R upon the pivot at the extremity B of the screw 
 AB, resolved in a direction parallel to the length of that 
 screw, must be equal to the pressure P 2 (see Art. 16.) ; so 
 that if we imagine the piece CD to become fixed, and the 
 
330 
 
 APPLICATIONS OF THE 6CKEW. 
 
 piece BM to become moveable, being prevented from revolv 
 ing, as CD was, by the intervention of a groove and guide, 
 then might the instrument be applied to overcome any given 
 resistance E opposed to the motion of this piece CD by the 
 constant pressure of its pivot upon that piece. 
 
 The screw is applied under these circumstances in the 
 common screw press. The piece 
 A, fixed to the solid frame of the 
 machine, contains a female screw 
 whose thread corresponds to that 
 of the male screw ; this screw, 
 when made to turn by means of a 
 handle fixed across it, presses by 
 the intervention of a pivot B, at its 
 extremity, upon the surface of a 
 solid piece EF moveable verti- 
 cally, but prevented from turning 
 with the screw by grooves receiv- 
 ing two vertical pieces, which 
 serve it as guides, and form parts 
 of the frame of the machine. 
 
 The formulae determined in 
 Art. 251. for 'the preceding case 
 of the application of the screw, obtain ialso in this case, if 
 we assume 9 3 0. The loss of power due to the friction of 
 the piece EF upon its guides will, however, in this calcu- 
 lation, be neglected ; that expenditure is in all cases exceed- 
 ingly small, the pressure upon the guides, whence their 
 friction results, being itself but the result of the friction of 
 the pivot B upon its bearings ; and the former friction being 
 therefore, in all cases, a quantity of two dimensions in 
 respect to the coefficient of friction. 
 
 If, instead of the lamina A (p. 326.) being fixed upon the 
 convex surface of a solid cylinder, and B upon the concave 
 surface of a hollow cylinder, the order be reversed, A being 
 fixed upon the hollow and B on the solid cylinder, it is evi- 
 dent that the conditions of the equilibrium will remain the 
 same, the male instead of the female screw being in this case 
 made to progress in the direction of its length. If, however, 
 the longitudinal ' motion of the male screw B (p. 326.) be, 
 under these circumstances, arrested, and that screw thus 
 become fixed, whilst the obstacle opposed to the longitudinal 
 motion of the female screw A is removed, and that screw 
 thus becomes free to revolve upon the male screw, and also 
 to traverse it longitudinally, except in as far as the latter 
 
THE DIFFERENTIAL SCREW. 
 
 331 
 
 motion is opposed by a certain resistance 
 R, which the screw is intended, tinder 
 these circumstances, to overcome ; then 
 will the combination assume the well 
 known form of the screw and nut. 
 
 To adapt the formulae of Art. 251. to 
 
 this case, 9 3 must be made = 0, and 
 instead of assuming the friction upon the extremity, of the 
 screw (equation 311) to be that of a solid pivot, we must 
 consider it' as that of a hollow pivot, applying to it (by 
 exactly the same process as in Art. (251.), the formulas of 
 Art. (177.) instead of Art. (176.). 
 
 THE DIFFERENTIAL SCREW. 
 
 253. In the combination of three inclined planes discussed 
 in Art. 245., let the plane B be conceived of much greater 
 width than is given to it in the figure (p. 319.), and let it 
 then be conceived to be wrapped upon a convex cylindrical 
 surface. Its two edges ab and cd will thus become the 
 helices of two screws, having their threads of different incli- 
 nations wound round different portions of the same cylinder, 
 
 mm* 
 
 as represented in the accompanying figure, where the thread 
 of one screw is seen winding upon the surface of a solid 
 cylinder from A to C, and the thread of another, having a 
 different inclination, from D to B. 
 
 Let, moreover, the planes A and (p. 319.) be imagined 
 to be wrapped round two hollow cylindrical surfaces, of 
 equal diameters with the above-mentioned solid cylinder, 
 and contained within the solid pieces E and F, through 
 which hollow cylinders AB passes. Two female screws will 
 thus be generated within the pieces E and F, the helix of 
 
332 
 
 HUNTER'S SCREW. 
 
 the one adapting itself to that of the male screw extending 
 from A to 0, and the helix of the other to that upon the 
 male screw extending from D to B. If, then, the piece E 
 be conceived to be fixed, and the piece F moveable in the 
 direction of the length of the screw, bnt prevented from 
 turning with it by the intervention of a guide, and if a pres- 
 sure Pj be applied at A to turn the screw AB, the action of 
 this combination will be precisely analogous to that of the 
 system of inclined planes discussed in Art. 245.,. and the 
 conditions of the equilibrium precisely the same ; so that the 
 relation between the pressure P a applied to turn the screw 
 (when estimated at the circumference of the thread) and that 
 P 2 , which it may be made to overcome, are determined by 
 equation (301), and its modulus by equation (302). 
 
 The invention of the differential screw has been claimed 
 by M. Prony, and by Mr. White of Manchester. A com- 
 paratively small pressure may be made by means of it to 
 yield a pressure enormously greater in magnitude.* It 
 admits of numerous applications, and, among the rest, of 
 that suggested in the preceding engraving. 
 
 HUNTER'S SCREW. 
 
 254. If we conceive the plane B (p. 319.) to be divided 
 by a horizontal line, and the upper part 
 to be wrapped upon the inner or concave 
 surface of a hollow cylinder, whilst the 
 lower part is wrapped upon the outer or 
 convex circumference of the same cylin- 
 der, thus generating the thread of a fe- 
 male screw within the cylinder, and a 
 male screw without it ; and if the plane 
 C be then wrapped upon the convex sur- 
 face of a solid cylinder just fitting the in- 
 
 side or concave surface of the above-mentioned hollow cylin- 
 
 * It will be seen by reference to equation (301), that the working pressure 
 P 2 depends for its amount, not upon the actual inclinations ii i% of the threads, 
 but on the diiference of their inclinations ; so that its amount may be enor- 
 mously increased by making the threads nearly of the same inclination. Thus, 
 
 neglecting friction, we have, by equation (301), P 2 =Pi 77 r- ; which 
 
 sin. (ii h) 
 expression becomes exceedingly great when ^ nearly equals 3 . 
 
VARIABLE INCLINATION OF THE THREAD. 333 
 
 der, and the plane A upon a concave cylindrical surface just 
 capable of receiving and adapting itself to the outside or 
 convex surface of that cylinder, the male screw thus genera- 
 ted adapting itself to the thread of the screw within the hol- 
 low cylinder, and the female screw to the thread of that 
 without it ; if, moreover, the female screw last mentioned 
 be fixed, and the solid male screw be free to traverse in the 
 direction of its length, but be prevented turning upon its 
 axis by the intervention of a guide ; if, lastly, a moving pres- 
 sure or power be applied to turn the hollow screw, and a re- 
 sistance be opposed to the longitudinal motion of the solid 
 screw which is received into it ; then the combination will 
 be obtained, which is represented in the preceding engraving, 
 and which is well known as Mr. Hunter s screw, having been 
 first described by that gentleman in the seventeenth volume 
 of the Philosophical Transactions. 
 
 The theory of this screw is identical with that of the pre- 
 ceding, the relation of its driving and working pressures is 
 determined by equation (301), and its modulus by equation 
 (302). 
 
 THE THEORY OF THE SCREW WITH A SQUARE THREAD IN RE- 
 FERENCE TO THE VARIABLE INCLINATION OF THE THREAD AT 
 DIFFERENT DISTANCES FROM THE AxiS. 
 
 255. In the preceding investigation, the inclined plane 
 which, being wound upon the cylinder, generates the thread 
 of the screw, has been imagined to be an exceedingly thin 
 sheet, on which hypothesis every point in the thread may be 
 conceived to be situated at the same distance from the axis 
 of the screw ; and it is on this supposition that the relation 
 between the driving and working pressure in the screw and 
 its modulus have been determined. 
 
 Let us now consider the actual case in which the thread 
 
334: VARIABLE INCLINATION OF THE THREAD. 
 
 of the screw is of finite thickness, and different elements of 
 it situated at different distance from its axis. 
 
 Let mb represent a portion of the square thread of a screw, 
 in which form of thread a line be, drawn from any point 5 on 
 the outer edge of the thread perpendicular to the axis ef, 
 touches the thread throughout its whole depth ~bd. Let AC 
 represent a plane perpendicular to its axis, and of the pro- 
 jection of 1)6 upon this plane. Take^? any point in bd, and 
 let q be the projection of p. Let ep=r, mean radius of 
 thread =R, inclination of that helix of the thread whose 
 radius is E*=I, inclination of the helix passing through p=t, 
 whole depth of thread =2D, distance between threads (or 
 pitch) of screw =L. Now, since the helix passing through 
 p may be considered to be generated by the enwrapping of 
 an inclined plane whose inclination is i upon a cylinder 
 whose radius is r, the base of which inclined plane will then 
 become the arc tq, we h&VB<j?=zfo . tan. i. But, if the angle 
 Afa be increased to 2*, pq will become equal to the com- 
 mon distance L between the threads of the screw, and tq 
 will become a complete circle, whose radius is r ; therefore 
 L=2*r tan. t, and this being true for all values of r, there- 
 fore L= 2tfR tan. I. Equating the second members of these 
 equations, and solving in respect to tan. *, 
 
 R tan. I f ON 
 
 tan. i= (313). 
 
 From which expression it appears, that the inclination of the 
 thread of a square screw increases rapidly as we recede from 
 its edg^e and approach its axis, and would become a right 
 angle if the thread penetrated as far as the axis. Consider- 
 ing, therefore, the thread of the screw as made up of an in- 
 finite number of helices, the modulus of each one of which 
 is determined by equation (312), in terms of its correspond- 
 ing inclination , it becomes a question of much practical im- 
 portance to determine, if the screw act upon the resistance 
 at one point only of its thread, at what distance from its axis 
 that point should be situated, and if its pressure be. applied 
 at all the different points of the depth of its thread, as is 
 commonly the case, to determine how far the conditions of 
 its action are influenced by the different inclinations of the 
 thread at these different depths. 
 
 * This may be called the mean helix of the thread. The term helix is here 
 taken to represent any spiral line drawn upon the surface of the thread ; the 
 distance of every point in which, from the axis of the screw, is the same. 
 
VARIABLE INCLINATION OF THE THREAD. 335 
 
 We shall omit the discussion of the former case, and pro- 
 ceed to the latter. 
 
 Let P a represent the pressure parallel to its axis which is 
 to be overcome by the action of the screw. ~Now it is evi- 
 dent that the pressure thus produced upon the thread of the 
 screw is the same as though the whole central portion of it 
 within the thread were removed, or as though the whole 
 pressure P 2 were applied to a ring whose thickness is As or 
 2D. 'Now the area of this ring is represented by * |(R+D) a 
 (R D) 2 5 , or by 4<KD. So that the pressure of P a , upon 
 
 p 
 every square unit of it, is represented by -p* Let &r 
 
 represent the exceedingly small thickness of such a ring 
 whose radius is r, and which may therefore be conceived to 
 represent the termination of the exceedingly thin cylindrical 
 surface passing through the point p ; the area of this ring is 
 then represented by far^r, and therefore the pressure upon 
 
 it by V,-p-pv > or by ' .. Now this is evidently the 
 
 pressure sustained by that elementary portion of the thread 
 which passes through p, whose thickness is A/ 1 , and which 
 may be conceived to be generated by the enwrapping of a 
 thin plane, whose inclination is *, upon a cylinder whose ra- 
 dius is r ; whence it follows (by equation 311) that the ele- 
 mentary pressure AP 15 which must be applied to the arm of 
 the screw to overcome this portion of the resistance P 2 , thus 
 applied parallel to the axis upon an element of the thread, 
 is represented by 
 
 AP _ . r \ f sin. Q + 9Q cos. 9 3 P t \ . 
 
 r ~ ( cos..+9+9 ~~V a 9 * i ' 
 
 cos.(.+9 1 +9.) 
 
 whence, passing to the limit and integrating, we have 
 R+D 
 
 sn. 
 
 'co S 
 
 R-D 
 
 Now 
 
 sin. + 9,) cos. 9 3 tan. -f tan. 9, 
 
 cos. (' + 9,+9s) 1 tan. 9. tan. 9, tan. (tan. 9, 4- tan. $} 
 
 \ ' 1 T^ 
 
 tan, t+tan. 9 t ) ___ 
 
 ~(1 tan. 9, tan. 9,) {Itan. tan. (9^93) j " : an * Pi+ra 11 - ' 
 
836 THE SCREW WITH A TRIANGULAR THREAD. 
 
 +tan. fo-f <p,) tan. *t. Neglecting dimensions of tan. 9 1 and 
 tan. <p 8 above the first*, 
 
 R+D 
 tan. <pj + tan. i + tan. (<pj -j- 9 8 ) tan. "A/" 9 -}- 
 
 R D 
 
 fp^ tan. y^dr (314). 
 
 Substituting in this expression for tan. i its value (equation 
 313), it becomes 
 R+D 
 
 ' a tan. 9, + Rr tan. I + R a tan. 3 1 tan. (<pj + 9,) + 
 R-D 
 
 Integrating and reducing, 
 
 tan. 9 I tan. fo+cp,) j ..... (315); 
 whence we obtain by (equation 121) for the modulus, 
 
 ..... (316). 
 
 256. "Whence it follows that the best inclination of the 
 thread, in respect to the economv of power in the use of 
 the square screw, is that which satisfies the equation 
 
 The inclination of thread of a square screw rarely exceeds 
 7, so that the term tan. 2 I tan. (f t $$,) rarely exceeds *015 
 tan. (pi+pj,), and may therefore be neglected, as compared 
 
 * The integration is readily effected without this omission ; and it might be 
 desirable so to effect it where the theory of wooden screws is under discussion, 
 the limiting angle of resistance being, in respect to such screws, considerable. 
 The length and complication of the resulting expression has caused the omis- 
 sion of it in the text. 
 
THE BEAM OF THE STEAM ENGINE. 
 
 337 
 
 with the other terms of the expression; as also may the 
 
 term i( j tan. 9 15 since the depth 2D of a square screw 
 
 \Iv 
 
 being usually made equal to about th of the diameter, this 
 term does not commonly exceed T ^ tan. <p a . 
 
 Omitting these terms, observing that L=2tfR tan. I, and 
 eliminating tan. I, 
 
 ..... (317). 
 ..... (318). 
 
 THE BEAM OF THE STEAM 
 
 257. Let P,, P,, P 3 , P 4 represent the pressuo-es applied by 
 the piston rod, the crank rod, the air pump- rod, and the cold 
 
 water pump rod, to the beam of a steam engine ; and sup- 
 pose the directions of all these pressures to be vertical.* 
 
 Let the rods, by which the pressures P 15 P 2 , P 3 , P 4 are 
 applied to the beam, be moveable upon solid axes or gud- 
 geons, whose centres are #, d, ~b, 6, situated in the same 
 straight line passing through the centre C of the solid axis 
 of the beam. 
 
 Let pj, p 2 , p 8 , p 4 represent the radii of these gudgeons, p the 
 radius of the axis of the beam, and <p,, <p a , <p 9 , <p 4 , p the limit- 
 ing angles of resistance of these axes respectively. Then, if 
 the beam be supposed in the state bordering upon motion 
 
 
 
 * A supposition which in no case deviates greatly from the truth, and any 
 error in which may be neglected, inasmuch as it can only influence the results 
 about to be obtained in as far as they have reference to the friction of the 
 beam ; so that any error in the result must be of two dimensions, at least, in 
 respect to the coefficient of friction and the small angle by which any pressure 
 deviates from a vertical direction. 
 
 22 
 
338 THE BEAM OF THE STEAM ENGINE. 
 
 by the preponderance of P,, each gudgeon or axis being 
 upon the point of turning on its bearings, the directions of 
 the pressures P 15 P 2 , P 3 , P 4 , R, will not be through the cen- 
 tres of their corresponding axes, but separated from them by 
 perpendicular distances severally represented by p t sin. 9,, p 2 
 sin. 9 2 , p,, sin. <p a , p 4 sin. <p 4 , and p sin. 9, which distances, being 
 perpendicular to the directions of the pressures, are all 
 measured horizontally. 
 
 Moreover, it is evident that the direction of the pressure 
 Pj is on that side of the centre a of its axis which is nearest 
 to the centre of the beam, since the influence of the friction 
 of the axis a is to diminish the effect of that pressure to turn 
 the beam. And for a like reason it is evident that the 
 directions of the pressures P 2 , P 8 , P 4 are farther from the 
 centre of the beam than the centres of their several axes, 
 since the effect of the friction is, in respect to each of these 
 pressures, to increase the resistance which it opposes to the 
 rotation of the beam ; moreover, that the resistance R upon 
 the axis of the beam has its direction upon the same side of 
 the centre C as P 15 since it is equal and opposite to the 
 resultant pressure upon the beam, and that resultant would, 
 by itself, turn the beam in the same direction as l turns it. 
 Let now a l =Ga^ # 2 C<#, a a =Cb, a 4 =Ce. Draw the hori- 
 zontal line of/C<7, and let the angle aCf=&. Let, moreover, 
 W be taken to represent the weight of the beam, supposed 
 to act through the centre of its axis. Then since P 1? P 2 , P 3 , 
 P 4 , W, R are pressures in equilibrium, we have, by the 
 principle of the equality of moments, taking o as the point 
 from_which the moments are measured, P x . 0/'=P 2 . 
 P, .<^+P 4 .0+W.0C. 
 
 Now of=CfCo=a 1 cos. 8 p, sin. (p, p sin. 9, og 
 00=0,2 cos - ^ + p 2 sin. 9 2 + p sin. 9, oh=Ch Co=a s cos. d + 
 P 8 sin. 9 3 p sin. 9, ok=Ck + Co=a t cos. d + p 4 sin. 9 4 + p sin. 9. 
 
 .*. PJ&j cos. d (p^in. 9,-j-psin. 9)^ = 
 
 P 2 1& 2 cos. 6 -f (p 2 sin. 9 a + p sin. 9)} + 
 
 ) 3 sin. 9 3 P sin. 9) j + [ . . . (319). 
 
 i + p sin. 9) j + "Wp sin. 9 
 
 Multiplying this equation by ^, observing that af repre- 
 sents the space described by the point of application of P 15 
 so that Pj^ represents the work TJ 1 of Pj ; and similarly 
 that P 2 <^ 2 ^ represents the work U 2 of P 2 , P 3 &A that U 3 of P 8 , 
 and P 4 &/, that U 4 of P 4 , also that a represents the space S, 
 
THE BEAM OF THE STEAM ENGINE. 
 
 339 
 
 described by the extremity of the piston rod very nearly ; 
 we have 
 
 ,.(320), 
 
 which is the modulus of the beam. 
 
 Its form will be greatly simplified if we assume cos. d=l, 
 since 6 is small,* suppose the coefficient of friction at each 
 axis to be the same, so that 9=^=9, =<p 8 =9 4 , and divide by 
 the coefficient of U M omitting terms above the first dimen- 
 
 sion in sin, 9, &c. ; whence we obtain by reduction 
 
 -(321). 
 
 258. The best position of the axis of the foam. 
 
 Let a be taken to represent the length of the beam, and x 
 the distance aC of the centre of its axis from the extremity 
 to which the driving pressure is applied. 
 
 * In practice the angle 6 never exceeds 20, so that cos. 6 never differs from 
 unity by more than -060807. The error, resulting from which difference, in 
 the friction, estimated as above, must in all cases be inconsiderable. 
 
340 THE BEAM OF THE STEAM ENGINE. 
 
 Let the influence of the position of the axis on the 
 economy of the work necessary to open the valves, to work 
 the air-pump, and to overcome the friction produced by the 
 weight of the axis, be neglected ; and let it be assumed to 
 be that, by which a given amount of work U 2 may be 
 yielded per stroke upon the crank rod, by the least possible 
 amount u , of work done upon the piston rod. If, then, in 
 equation (321), we assume the three last terms of the second 
 member to be represented by A, and observe that a l in that 
 equation is represented by a?, and # a by a x, we shall 
 obtain 
 
 The best position of the axis is determined by that value 
 of x which renders this function a minimum ; which value 
 of x is represented by the equation 
 
 x= _ - _ . (322.) 
 
 If p 2 >p,, then! J >1 and a?<-|#; in this case, there- 
 
 fore, the axis is to be placed nearer to the driving than to 
 the working end of the beam. If p 2 <pj, the axis is to be 
 fixed nearer to the working than to the driving end of the 
 beam. 
 
 259. It has already been shown (Art. 168.), that a 
 machine working, like the beam of a steam engine, under 
 two given pressures about a fixed axis, is worked with the 
 greatest economy of power when both these pressures are 
 applied on the same side of the axis. This principle is 
 manifestly violated in the beam engine; it is observed in 
 the engine worked by Crowther's parallel motion,* and in 
 the marine engines recently introduced by Messrs. Seaward, 
 and known as the Gorgon engines. It is difficult indeed to 
 defend the use of the beam on any other legitimate ground 
 than this, that in some degree it aids the fly-wheel to 
 equalise the revolution of the crank arm,f an explanation 
 
 * As used in the mining districts of the north of England. 
 
 f The reader is referred to an admirable discussion of the equalising power 
 of the beam, by M. Coriolis, contained in the thirteenth volume of the Journal 
 de VEcole Poly technique. 
 
THE CKANK. 
 
 341 
 
 which does not extend to its use in pumping engines, 
 where, nevertheless, it retains its place; adding to the 
 expense of construction, and, by its weight, greatly increas- 
 ing the prejudicial resistances opposed to the motion of the 
 engine. 
 
 THE CKANK. 
 
 260. The modulus of the crank, the direction of the resist- 
 ance being parallel to that of the driving pressures. 
 
 Let CD represent the arm of the crank, and AD the con- 
 necting rod. And to simplify the 
 investigation, let the connecting 
 rod be supposed always to retain 
 its vertical position.* Suppose the 
 weight of the crank arm CD, act- 
 ing through its centre of gravity, 
 to be resolved into two other 
 weights (Art. 16), one of which W 2 
 is applied at the centre C of its axis 
 and the other at the centre c of 
 the axis which unites it with the 
 connecting rod. Let this latter 
 weight, when added to the weight 
 of the connecting rod, be repre- 
 sented by Wj. Let P 2 represent a 
 / pressure opposed to the revolution 
 x -;. ^S of the crank, which would at any 
 
 instant be just sufficient to balance 
 
 the driving pressure P a transmitted through the connecting 
 rod ; and to simplify the investigation, let us suppose the 
 direction of the pressure P 2 to be vertical and downwards. 
 
 Let Cc=a, CA 1 =a 1 , CA 2 =& 2 , <?CW 2 =d, radii of axes C 
 and c=p,, p 2 , lim. /s of resistance =<p,, <p a , W= whole weight 
 of crank arm and connecting rod=W 1 + W a . 
 
 Since the crank arm is in the state bordering upon 
 motion, the perpendicular distance of the direction of the 
 resistance upon its axis C from the centre of that axis, is 
 
 * Any error resulting from this hypothesis affecting the conditions of the 
 question only in as far as the friction is concerned, and being of two dimen- 
 sions at least in terms of the coefficient of friction and the small angular devi- 
 ation of the connecting rod from the vertical. 
 
342 
 
 THE CKANK. 
 
 represented by p x sin. <p t (Art. 153.). The resistance is also 
 equal to P a (P a + W) ; P x being supposed greater than P 2 + W ; 
 
 and the sign being taken 
 according as the direction of 
 P, is downwards or upwards, 
 or according as the crank arm 
 is describing its descending or 
 ascending arc. Whence it 
 follows, that the moment of 
 the resistance of the axis about 
 its centre is represented by 
 iP,(P, + W)j Pl sin. 9 , 
 Now the pressures P 15 P 2 , and 
 the resistance of the axis, are 
 pressures in equilibrium. 
 Therefore, by the principle of 
 the equality of moments, ob- 
 serving that the driving pressure is represented by PjSfcW,, 
 according as the arm is descending or ascending, 
 
 (P,W,) ,=P A + SP,(P, + W)} Pl sin. ?, 
 
 Since moreover the axis 0, which unites the connecting 
 rod and the crank arm, is upon the point of turning upon 
 its bearings, the direction of the pressure P x is not through 
 the centre of that axis, but distant from it by a quantity 
 represented by p 2 sin. <p a , which distance is to be measured 
 on that side of the centre o which is nearest to C, since the 
 friction diminishes the effect of P x to turn the crank arm. 
 
 != a sin. d p 3 sin. <p 2 
 
 (323). 
 
 Substituting this value of #, in the preceding equation, 
 O (a sin. 0-p 2 sin. 9,)=I>,+ { 
 
 (324). 
 
 Pl sn. <p a 
 Transposing and reducing 
 
 P,{a sin. & p a sin. ? 2 p, sin. 9,} =P 2 Kp 1 sin. 9,} 
 "Wp 1 sin. 9 1 =fW 1 (# sin. $ p a sin. 9,); 
 
 which is the relation between P t and P 2 in their state bor- 
 dering upon motion. JSTow if Ad represent an exceedingly 
 small angle described by the crank arm, a^b will represent 
 the space through which the resistance P 2 is overcome 
 whilst that angle is described, and P 2 # 2 A0 will represent the 
 
THE CRANK. 343 
 
 increment AtT 2 of the work yielded by the crank whilst that 
 small angle is described. Multiplying the above equation 
 by a^y we have 
 P^Ja sin. A p 2 sin. <p 2 p, sin. ^ l \A&={a 9 p l sin. 9 1 } A "D' 3 
 
 "Wa 2 p x sin. ^AdqpW^,, ( a sm - & P* sin - ^) A ^ ( 325 )- 
 whence passing to the limit, integrating from 0=6 to d= 
 tf 0, and dividing by # 2 
 
 P, |2acos.6 (* 28) 
 
 W <y_26) Pi s in. ^qp Wj {2a cos. Pa (V 26) sin. <p 2 } . . (326). 
 
 Now, let it be observed that 2a cos. 6 represents the pro- 
 jection of the path of the point c upon the vertical direction 
 of P,, whilst the arm revolves between the positions 6 and 
 tf _6 ; so that P^a cos. 6 represents (Art. 52.) the work 
 Uj done by P, upon the crank whilst the arm passes from 
 one of these positions to the other, or whilst the work U 2 is 
 
 TJ 
 yielded by the crank. Whence it follows that PI=^ J f sec. 6. 
 
 Substituting this value of P 1? and reducing we obtain 
 
 sin. 9, U 2 W (-rr-26) Pl sin. V^W, [2a cos. 
 P 2 (* 20) sin. 9 a { ..... (327). 
 
 By which equation is determined the modulus of the crank 
 in respect to the descending or ascending stroke, according 
 as we take the upper or lower signs of the ambiguous terms. 
 Adding these two values of the modulus together, and 
 representing by Uj the whole work of P,, and by TJ 2 the 
 whole work of P 2 , whilst the crank arm makes a complete 
 revolution, also by u t the work of P 2 in the down stroke, 
 and i 2 in the up stroke, we obtain 
 
 - - e sin - sin - ? > =v, 
 
 K-^ 2 )sin. 9, ..... (328), 
 which is the modulus of the crank in respect to a vertical 
 
344 THE CRANK. 
 
 direction of the driving pressure and of the resistance, the 
 arm being supposed in each half revolution, first, to receive 
 the action of the driving pressure when at an inclination of 
 to the vertical, and to yield it when it has again attained 
 the same inclination, so as to revolve under the action of 
 the driving pressure through the angle if 2. 
 
 In the double-acting engine, u^ u^=0 ; in the single-act- 
 ing engine ^=0. The work expended by reason of the 
 friction of the crank is therefore less in the latter engine than 
 in the former, when the resistance P 2 is applied, as shown 
 in the figure, on the side of the ascending arc. 
 
 If the arm sustain the action of the driving pressure con- 
 stantly, 9=0, and the modulus becomes, for the double-act- 
 ing engine, 
 
 or, dividing by the co-efficient of U, and neglecting dimen- 
 sions above the first in sin. 9 1? sin. <p 2 , 
 
 The modulus not involving the symbol W which repre- 
 sents the weight of the crank, it is evident that so long as P t 
 and P 2 are vertical and P, greater than P 2 + "W, the economy 
 of power in the use of the crank is not at all influenced by 
 its weight and that of the connecting rod, the friction being 
 upon the whole as much diminished by reason of that weight 
 in the ascending stroke as it is increased by it in the descend- 
 ing stroke. 
 
 It is evident, moreover, that if the friction produced by 
 the weight of the crank be neglected, the modulus above de- 
 duced, for the case in which the directions of the pressures 
 P, and P 3 are vertical, applies to every case in which the 
 directions of those pressures are parallel. 
 
 The condition P,>P 2 -f-W evidently obtains in every other 
 position of the crank arm, if it obtain in the horizontal position. 
 
 Now, in this position, P 2 = P 15 if we neglect friction. The 
 
 a i 
 
 required condition obtains, therefore, if ?,>?! -f-W. To 
 
 a * 
 
 satisfy this condition, # 2 must be greater than #, or the 
 
 resistance be applied at a perpendicular distance from the 
 
THE DEAD POINT IN THE CRANK. 345 
 
 axis greater than the length of crank arm, and so much 
 greater, that P l (1 -- j may exceed "W. These conditions 
 
 \ #2/ 
 
 commonly obtain in the practical application of the crank. 
 
 261. Should it, however, be required to determine the mo- 
 dulus in the case in which P x is not, in every position of the 
 arm, greater than P,-f W, let it be observed, that this condi- 
 tion does not affect the determination of the modulus (equa- 
 tion 327) in respect to the descending, but only the ascend- 
 ing stroke ; there being a certain position of the arm as it 
 ascends in which the resultant pressure upon the axis repre- 
 sented by the formula {P, (P 2 +W)j , passing through zero, 
 is afterwards represented by |(P 2 -f- W) P^ ; and when the 
 arm has still further ascended so as to be again inclined to the 
 vertical at the same angle, passes again through zero, and is 
 again represented by the same formula as before. The value 
 of this angle may be determined by substituting P a for 
 P a + W inequation (324), and solving that equation in re- 
 spect to A. Let it be represented by ^ ; let equation (325) 
 be integrated in respect to the ascending stroke from 0=0 
 to 0==^, the work of P 2 through this angle being represented 
 by u l ; let the signs of all the terms involving p x sin. <pj then 
 be changed, which is equivalent to changing the formula re- 
 presenting the pressure upon the axis from {P x (P 2 +W) 
 to {(P 2 + W) P a }; and let the equation then be integrated 
 
 from 6=6 l to 0=5, the work of P a through this angle being re- 
 
 presented by u^\ 2(1^ + w a ) will then represent the whole 
 work U 2 done by P 2 in the ascending arc. To determine 
 this sum, divide the first integral by the co-efficient of u l9 
 and the second by that of u 9 , add the resulting equations, 
 and multiply their sum by 2 ; the modulus in respect to the 
 ascending arc will then be determined ; and if it be added 
 to the modulus in respect to the descending arc, the modu- 
 lus in respect to an entire revolution will be known. 
 
 THE DEAD POINTS IN THE QRANK. 
 
 362. If equation (324) be solved in respect to P, it be- 
 comes 
 
346 THE DOUBLE CIJANK. 
 
 p _p ( ^PiSJn. 9, _ ) 
 
 2 1 a-am. 6 p 2 sin. <p a p 1 sin. ?, f 
 
 'Wp 1 sin.^ 'WXflBin.d p a sin. 9 8 ) 
 a sin. 4 p a sin. <p 2 p : sin. 9 t 
 
 In that position of the arm, therefore, in which 
 s in.O= f '' sin -^ + p - sin -' P - .... (330), 
 
 the driving pressure P, necessary to overcome any given re- 
 sistance P 2 opposed to the revolution of the crank, assumes 
 an infinite value. This position from which no finite pres- 
 sure acting in the direction of the length of the connecting 
 rod is sufficient to move the arm, when it is at rest in that 
 position, is called its dead point. 
 
 Since there are four values of 6, which satisfy equation 
 (330) two in the descending and two in the ascending semi- 
 revolution of the arm, there are, on the whole, four dead 
 points of the crank.* The value of P l being, however, in all 
 cases exceedingly great between the two highest and the two 
 lowest of these positions, every position between the two 
 former and the two latter, and for some distance on either 
 side of these limits, is practically a dead point. 
 
 THE DOUBLE CRANK. 
 
 263. To this crank, when applied to the steam engine, are 
 affixed upon the same solid shaft, two arms at right angles 
 to one another, each of which sustains the pressure of the 
 steam in a separate cylinder of the engine, which pressure is 
 transmitted to it, from the piston rod, by the intervention of 
 a beam and connecting rod as in the marine engine, or a 
 guide and connecting rod as in the locomotive engine. 
 
 * It has been customary to reckon theoretically only two dead points of the 
 crank, one in its highest and the other in its lowest position. Every practical 
 man is acquainted with the fallacy of this conclusion. 
 
THE DOUBLE CKANK. 
 
 347 
 
 Fig. 1. 
 
 In either case, the connecting rods 
 may be supposed to remain con- 
 stantly parallel to themselves, and 
 the pressures applied to them in 
 different planes to act in the samg 
 plane,* without materially affecting 
 the results about to be deduced.f 
 
 Let the two arms of the crank be 
 supposed to be of the same length a ; 
 let the same driving pressure P l be 
 supposed to be applied to each ; and 
 let the same notation be adopted in 
 other respects as was used in the 
 case of the crank with a single arm; 
 and, first, let us consider the case 
 represented in fig. 1, in which both 
 arms of the crank are upon the same 
 side of the centre C. 
 
 Let the angle W^B^d ; therefore W 1 CE=^+^ : whence 
 
 it follows by precisely the same reasoning as in Art. 260., 
 that the perpendicular upon the direction of the driving 
 pressure applied by the connecting rod AB is represented 
 (see equation 323) by a sin. 6 p 2 sin. <p a , and the per- 
 pendicular upon the pressure applied by the rod ED by 
 
 a 
 
 cos - & P 2 sin. 9,. Let now 
 
 be taken to represent the perpendicular distance from the 
 axis C, at which a single pressure, equal to 2P,, must be ap- 
 plied, so as to produce the same effect to turn the crank as 
 is produced by the two pressures actually applied to it by 
 the two connecting rods ; then, by the principle of the equa- 
 lity of moments, 
 
 2P 1 1 =P 1 ( sin. & p Q sin. 9 a ) + P 1 ( cos. & p a sin. 9,) ; 
 /. a l =^a (sin. 6 -f- cos. d) p a sin. 9, ; 
 
 * This principle will be more fully discussed by a reference to the theory of 
 statical couples. (See Pritchard on Statical Couples.) 
 
 f The relative dimensions of the crank arm and connecting rod are here sup- 
 posed to be those usually given to these parts of the engine ; the supposition 
 does not obtain in the case of a short connecting rod. 
 
34:8 THE DOUBLE CRANK. 
 
 a I . if *\ 
 
 /. #,= I sin. & cos. j+cos.d sin. -r I p a sin. <p a = 
 
 which expression becomes identical with the value of # n de- 
 termined by equation (323), if in the latter equation a be 
 
 replaced by -fL, and 6 by 6 +-. "Whence it follows that the 
 
 4/2 . * 
 
 conditions of the equilibrium of the double crank in the 
 state bordering upon motion, and therefore the form of the 
 modulus, are, whilst both arms are on the same side of the 
 centre, precisely the same as those of the single crank, the 
 direction of whose arm bisects the right angle BCE, and 
 the length of whose arm equals the length of either arm of 
 the double crank divided by |/2. 
 
 Now, if 6 l be taken to represent the inclination W X CF of 
 this imaginary arm to "W^C, both arms will be found on the 
 
 same side of the centre, from that position in which 6 t = _ 
 to that in which it equals ( * -I. If, therefore, we substi- 
 
 tute - for 6, in equations (326), and for &, , and add these 
 
 4/2 
 
 equations together, the symbol 2 U 2 in the resulting equa- 
 tion will represent the whole work yielded by the working 
 pressure, whilst both arms remain on the same side of the 
 centre, in the ascending and the descending arcs. We thus 
 obtain, representing the sum of the driving pressures upon 
 the two arms by P : , 
 
 2P>- (p a sin.<p 2 + Pl 8^.901=211, ..... (331).* 
 
 Throughout the remaining two quadrants of the revolution 
 of the crank, the directions of the two equal and parallel 
 pressures applied to it through the connecting rods being 
 opposite, the resultant pressure upon the axis is represented 
 by (P. + W), instead of |P, (P 2 + W) { ; whilst, in other 
 respects, the conditions of the equilibrium of the state bor 
 
 * Whewell's Mechanics, p. 25. 
 
THE DOUBLE CEANK. 
 
 849 
 
 dering upon motion remain the same as before ; that is, the 
 same as though the pressure P : were 
 applied to an imaginary arm, whose 
 
 length is - t and whose position co- 
 incides with OF. Now, referring to 
 equation (324), it is apparent that 
 this condition will be satisfied if, in 
 that equation, the ambiguous sign of 
 (P 2 + W) be suppressed, and the 
 value of P! in the second member, 
 w^hich is multiplied by p, sin. 9 a , be 
 assumed =0 ; by which assumption 
 the term p x sin. 9, will be made to 
 disappear from the left-hand member 
 of equation (325), and the ambiguous 
 signs which affect the first and second 
 terms of the right-hand member will become positive. Now, 
 these substitutions being made, and the equation being then 
 
 integrated, first, between the limits and -, and then be- 
 tween the limits and ir, the symbol U 3 in it will evidently 
 
 represent the work done during each of those portions of a 
 semi-revolutiou of the imaginary arm in which the two real 
 arms of the crank are not on the same side of the centre. 
 Moreover, the integral of that equation between the limits 
 
 and j? is evidently the same with its integral between the 
 
 limits -j- and if. Taking, therefore, twice the former inte- 
 gral, we have 
 
 P. sin. 9, 
 
 sn. 
 
 - sn. 
 
 Dividing this equation by (0 a + p x sin. 9,), or by # a 
 
 (1 H - sin. 9, )> and neglecting terms above the first dimen- 
 G> 9 ' 
 
 Bion in sin. 9! and sin. 9 2 , 
 
350 THE DOUBLE CRANK. 
 
 - sin. 9l )- 
 j p 2 sin. <p a I ; 
 
 in which equation 2U 2 represents the work done in the 
 descending or ascending arcs of the imaginary arm, accord- 
 ing as the ambiguous sign is taken positively or negatively. 
 Taking, therefore, the sum of the two values of the equation 
 given by the ambiguous sign, and representing by 4U 2 the 
 whole work done in the descending and ascending arcs, dur- 
 ing those portions of each complete revolution when both of 
 the arms are not on the same side of the centre, we have 
 
 or, observing that cos. 7 = ~~F> 
 
 2P, 0(|/2-l)-X 4/2-1) sin. 9l - Pa sin. 9, = 
 
 p 1 sin. <pj. 
 
 Adding this equation to equation (331), and representing by 
 U 2 the entire work yielded during a complete revolution of 
 the imaginary arm, 
 
 2P, a 4/2 - a(|/2 - 1) sin. 9, - (2 Pi sin.?, +^ 
 
 
 But if U 1 represent the whole work done by the driving 
 pressures at each revolution of the imaginary arm, then 
 
 4 P 1 =U 1 . Since 2 p is the projection of the space 
 
THE CRANK GUIDE. 351 
 
 described by the extremity of the arm during the ascending 
 and descending strokes respectively, therefore 2P l = j=. 
 Substituting this value for 2P X , 
 
 IT l-= 
 
 sin.^.. ..(332), 
 
 which is the modulus of the double crank, the directions of 
 the driving pressure and the resistance being both supposed 
 vertical; or if the friction resulting from the weight of the 
 crank be neglected, and W be therefore assumed =0, then 
 does the above equation represent the modulus of the 
 double crank, whatever may be the direction of the driving 
 pressure, provided that the direction of the resistance be 
 parallel to it. Dividing by the coefficient of Uj, and 
 neglecting terms of more than one dimension in sin. 9 1 and 
 sin. <p f , 
 
 ^ sin. ?,) | 
 
 (333). 
 
 THE CRANK GUIDE. 
 
 264. In some of the most important applications of the 
 steam engine, the crank is made to receive* its continuous 
 rotatory motion, from the alternating rectilinear motion of 
 the piston rod, directly through the connecting rod of the 
 crank, without the intervention of the beam or parallel 
 motion ; the connecting rod being in this case jointed at one 
 extremity, to the extremity of the piston rod, and the oblique 
 pressure upon it which results from this connexion being 
 sustained by the intervention of a cross piece fixed upon it, 
 and moving between lateral guides.* 
 
 * This contrivance is that well known as applied to the locomotive carriage. 
 
352 THE CEANK GUIDE. 
 
 Let the length CD of the connecting rod be represented 
 by J, and that BD of the crank arm by a, and let ft and P a 
 in the above figure be taken respectively, to represent the 
 pressure upon the piston rod of the engine and the connect- 
 ing rod of the crank, and RS to represent the direction of 
 the resistance of the guide in the state bordering upon 
 motion by the excess of the driving pressure P a . Then is 
 RS inclined to a perpendicular to the direction of the guides, 
 or of the motion of the piston rod, at an angle equal to the 
 limiting angle of resistance (Art. lil) of the surfaces of con- 
 tact of the guides. 
 
 Since, moreover, P 1? P 2 , R are pressures in equilibrium, 
 
 B P 3 sin. P,CS 
 
 Let /BCD =4 ; limiting angle of resistance of guide =9; 
 therefore, ^08=5-9, p a CS=^+9-4; 
 
 -FT t j-a^Fi) V* 
 
 sm.j- (4 <pU 
 
 Let BD = a, CD = &, and DBC = 4 n and assume P, to 
 remain constant, Pj being made to vary according to the 
 conditions of the state bordering upon motion ; 
 
 ^P, . AAC= P, . ABC= P, . A ( cos. &,+b cos. 4)= 
 
 P 2 sec. 9 cos. (49) (a sin. ^A^-j-5 sin. 
 AU a =-P a (ABC)cos. 4=P 2 (asin. 6^ 
 
 7T 
 
 ..U I =P,sec.9| / sin.4 l cos.(4 9)^^^ / sin. 4 cos. (4 9)$}. 
 
THE FLY-WHEEL* 353 
 
 <k 
 
 U,=P 3 {a J sin. ^ cos. &d& 1 + 1 J sin. & cos. 
 o o 
 
 The second integral in each of these equations vanishes 
 between the prescribed limits ; also sin. & = T sin. ^ ; there- 
 
 a? 
 fore cos. d = (1 sin. 3 ^)*; 
 
 7T 7T 
 
 .*. U,=P a # / sin. 6 l cos. ^ 1 =P 2 / (1 -TS sin. a ^,)* sin. ^^=s 
 
 TJ^Pa^sec. 9 / sin. 6 l cos. (^ 9)^ 1 =P a ^ / sin. ^ cos. ^, 
 o o 
 
 7T 7T 
 
 P,a tan. 9 / sin. 6 sin. ^ 1 ^ 1 =U a +P a ^-tan. 9 / sin. 9 ^ ^= 
 
 l 
 
 whence eliminating P 8 and reducing, we obtain 
 
 which is the modulus of the crank guide. 
 
 THE FLY-WHEEL. 
 265. TA0 angular velocity of the fly-wheel. 
 
 Let P x be taken to represent a constant pressure applied 
 through the connecting rod to the arm of the crank of a 
 
 * Church's Diff. and Int. Gal. Art. 199. 
 23 
 
854 
 
 THE FLY-WHEEL. 
 
 steam engine; suppose the direction of this pressure to 
 remain always parallel to itself, and let P 2 represent a con- 
 stant resistance opposed to the revolution of the axis which 
 carries the fly-wheel, by the useful work done and the pre- 
 judicial resistances interposed between the axis of the 
 fly-wheel and the working points of the machine. 
 
 Let the angle ACB=$, CB=&, CP 2 = 2 . 
 
 Now the projection, upon the direction of P 15 of the path 
 of its point of application B to the crank arm, whilst that 
 arm describes the angle ACB, is AM, therefore (Art. 52.), 
 the work done by Y l upon the crank, whilst this angle is 
 described, is represented by P, . AM, or by P, a vers. d. 
 And whilst the crank arm revolves through the angle $, the 
 resistance P 2 is overcome through the arc of a circle sub- 
 tended by the same angle 4, but whose radius is 2 , or 
 through a space represented by aj. So that, neglecting the 
 friction of the crank itself, the work expended upo'i the 
 resistances opposed to its motion is represented by P 2 # a 0, and 
 the excess of the work done upon it through the angle ACB 
 by the moving power, over that expended during the same 
 period upon the resistances, is represented by 
 
 Pavers. 4 P 2 ^ (336). 
 
 Now 2#Pj represents the work done by the moving pressure 
 P, during each effective stroke of the piston, and 2tf<2 2 P 2 the 
 work expended upon the resistance during each revolution 
 of the fly-wheel; so that if m represent the number of 
 strokes made by the piston whilst the fly-wheel makes one 
 
THE FLY-WHEEL. 355 
 
 revolution, and if the engine be conceived to have attained 
 its state of uniform or steady action (Art. 146.), then 
 
 ..J > .=*Pi ..... (337). 
 
 Eliminating from equation (336) the value of # 3 P 2 deter- 
 mined by this equation, we obtain for the excess of the work 
 done by the power (whilst the angle 6 is described by the 
 crank arm), over that expended upon the resistance, the 
 expression 
 
 P^jvers. *-^j ..... (338). 
 
 But this excess is equal to the whole work which has been 
 accumulating in the different moving parts of the machine, 
 whilst the angle 6 is described by the arm of the crank (Art. 
 145). Now, let the whole of this work be conceived to have 
 been accumulated in the fly-wheel, that wheel being pro- 
 posed to be constructed of such dimensions as sufficiently to 
 equalise the motion, even if no work accumulated at the 
 same time in other portions of the machinery (see Art. 150.), 
 or if the weights of the other moving elements, or their 
 velocities, were comparatively so small as to cause the work 
 accumulated in them to be exceedingly small as compared 
 with the work accumulated during the same period in the 
 fly-wheel. Now, if I f represent the moment of inertia of the 
 fly-wheel, ^ the weight of a cubic foot of its material, a 1 its 
 angular velocity when the crank arm was in the position 
 CA, and a its angular velocity when the crank arm has 
 
 passed into the position CB ; then will % (a 2 a^) represent 
 
 the work accumulated in it (Art. 75.) between these two 
 positions of the crank arm, so that 
 
 ..... (339). 
 
 266. The positions of greatest and least angular velocity of 
 the fly-wheel. 
 
 If we conceive the engine to have acquired its state of 
 steady or uniform motion, the aggregate work done by the 
 
356 
 
 THE FLY-WHEEL. 
 
 power being equal to that expended upon the resistances, 
 then will the angular velocity of the fly-wheel return to the 
 same value whenever the wheel returns to the same position ; 
 so that the value of OL I in equation (339) is a constant, and 
 the value of a a function of 6 ; a assumes, therefore, its mini- 
 mum and maximum values with this function of d, or it is a 
 
 minimum when -- = 0, and ->0> and a maximum when 
 
 da? A , d?a? _ 
 
 -^-=0, and w < 
 therefore -^-=0, when 
 
 -r, , da? . 
 
 But -=s 
 
 . m 
 sin. d 
 
 . m , (Pa? . 
 
 _ -, and w = cos. I, 
 
 (340.) 
 
 Now this equation is evidently satisfied by two values of 
 d, one of which is the supplement of the other, so that if ^ 
 represent the one, then will (*>]) represent the other; 
 which two values of 9 give opposite signs to the value cos. 
 6 of the second differential co-efficient of a 2 , the one being 
 positive or >0, and the latter negative or <0. The one 
 value corresponds, therefore, to a minimum and the other 
 to a maximum value of a. If, then, we take the angle ACB 
 
 Wit 
 
 in the preceding figure, such that its sine may equal 
 
 (equation 340), then will the position CB of the crank arm 
 be that which corresponds to the minimum angular velocity 
 

 
 THE FLY-WHEEL. 357 
 
 of the fly-wheel ; and if we make the angle ACE equal to 
 the supplement of ACB, then is CE the position of the 
 crank arm, which corresponds to the maximum angular 
 velocity of the fly-wheel. 
 
 267. The greatest variation of the angular velocity of the 
 fly-wheel. 
 
 Let a a be taken to represent the least angular velocity of 
 the fly-wheel, corresponding to the position CB of the crank 
 arm, and a a its greatest angular velocity, corresponding to 
 
 the position CE ; then does ^- (a 3 8 a 2 2 ) represent the work 
 
 accumulated in the fly-wheel between these positions, which 
 accumulated work is equal to the excess of that done by the 
 power over that expended upon the resistances whilst the 
 crank arm revolves from the one position into the other, 
 and is therefore represented by the difference of the values 
 given to the formula (338) when the two values K ^ and 
 i, determined by equation (340), are substituted in it for 0. 
 Now this difference is represented by the formula 
 
 _. ( m (TT 11 *}) ) 
 
 P,# -j vers. (if ?]) vers. q - r, 
 
 ( / 2>]\ ) 
 
 or by Pjfl | 2 cos. v m 1 1 J p 
 
 U.T ( / 2'i\ ) 
 . /_. 2 a\ ~p A l a pn o y, ^,1 i I y . 
 
 o V s a / ~~ i i ^v>to. ') //ti j. if) 
 
 ^0' \ \ it i } 
 
 /.a 3 2 a 2 2 = j^ i 2 cos. n--*n 7) | (341); 
 
 in which equation *j is taken (equation 340) to represent 
 that angle whose sine is . 
 
 268. The dimensions of the fly-wheel, such that its angular 
 velocity may at no period of a revolution deviate beyond 
 prescribed limits from the mean. 
 
 Let $N be taken to represent the mean number of revo- 
 
 u 
 
 lutions made by the fly-wheel per minute; then will -J 
 
 oU 
 
358 THE FLY-WHEEL. 
 
 represent the mean number of revolutions or parts of a 
 
 N NV 
 
 revolution made by it per second, and fcr^, or -^-, the 
 
 mean space described per second by a point in the fly-wheel 
 whose distance from the centre is unity, or the mean angular 
 velocity of the fly-wheel. Now, let the dimensions of the 
 fly wheel be supposed to be such as are sufficient to cause 
 its angular velocity to deviate at no period of its revolution 
 
 by more than -th from its mean value ; or such that the max- 
 J n 
 
 imum value a s of its angular velocity may equal -^r I 1 + - I 
 and that its minimum value a 2 may equal -^H 1 - I ; then 
 
 Substituting in equation (341), 
 
 2*] 
 
 Let H be taken to represent the horses' power of the 
 engine, estimated at its driving point or piston ; then will 
 33000H represent the number of units of work done per 
 minute, upon the piston. But this number of units of work 
 is also represented by %Nm . 2?^ ; since %Nm is the number 
 of strokes made by the piston per minute, and 2P^ is the 
 work done on the piston per stroke, 
 
 TT 
 
 :.&,a= 6600(%-. 
 
 Era 
 
 Substituting this value for 2P : a in the above equation, we 
 obtain, by reduction, 
 
 66000.30V) 
 - ^ f 
 
 Let & be taken to represent the radius of gyration of the 
 wheel, and M its volume; then (Art. 80.) MAf^I, therefore 
 M-M.^ 2 =^I. But M-M represents the weight of the wheel 
 in Ibs. ; let W represent its weight in tons ; therefore, 
 aM=2240W. Substituting this value, and solving in 
 respect to "W, 
 
THE FLY-WHEEL. 359 
 
 66000.30^) < /, to\\ H 
 
 Substituting their values for * and g, and determining the 
 numerical value of the co-efficient, 
 
 W = 86491 { 1 cos. ,- (l - 5) [ g- ..... (343). 
 
 If the influence of the work accumulated in the arms of 
 the wheel be given in, for an increase of the equalising 
 power beyond the prescribed limits, that accumulated in the 
 heavy rim or ring which forms its periphery being alone 
 taken into the account;* then (Art. 86.) H& 9 =I=2flrfaK 
 (R 2 4~Jc 2 ), where b represents the thickness, c the depth, and 
 R the mean radius of the rim. But by Guldiims's first 
 property (Art. 38.), 2^K=M; therefore & 2 =(R 2 + ic 2 ). 
 Substituting in equation (343) 
 
 W=86491 ) 1 co, ,- 
 
 If the depth c of the rim be (as it usually- is) small as 
 compared with the mean radius of the wheel, Jc j2 may be 
 neglected as compared with R 2 , the above equation then 
 becomes 
 
 i 2 / 2>)\ ) H?& 
 
 W=86491 { - cos. ,-(l - -] | ^ . . . . (345) ; 
 
 by which equation the weight "W in tons of a fly-wheel of a 
 given mean radius R is determined, so that being applied to 
 an engine of a given horse power H, making a given num- 
 ber of revolutions per minute JN", it shall cause the angular 
 
 velocity of that wheel not to vary by more than -th from its 
 
 mean value. It is to be observed that the weight of the 
 wheel varies inversely as the cube of the number of strokes 
 made by the engine per minute, so that an engine making 
 twice as many strokes as another of equal horse power, 
 
 * If the section of each arm be supposed uniform and represented by /c, and 
 the arms be p in number, it is easily shown from Arts. 79., 81., that the 
 momentum of inertia of each arm about its extremity is very nearly repre- 
 sented by i/c(R ic) 3 , where c represents the depth of the rim; so that the 
 
 whole momentum of inertia of the arms is represented by ^/c(R jc 1 ) 3 , which 
 
 o 
 
 expression must be added to the momentum of the rim to determine the whole 
 momentum I of the wheel. It appears, however, expedient to give the inertia 
 of the arms to the equalising power of the wheeL 
 
360 THE FLY-WHEEL. 
 
 would receive an equal steadiness of motion from a fly- 
 wheel of one eighth the weight; the mean radii of the 
 wheels being the same. 
 
 If, in equation (342), we substitute for I its value 2tffoR 3 , 
 or 2tfKH 3 (representing by K the section be of the rim), and 
 if we suppose the wheel to be formed of cast iron of mean 
 quality, the weight of each cubic foot of which may be 
 assumed to be 450 lb., we shall obtain by reduction 
 
 R^= 68521 i- cos. ^-(l--)l^... .(346); 
 ( m \ if I j N K 
 
 by which equation is determined the mean radius R of a fly- 
 wheel of cast iron of a given section K, which being applied 
 to an engine of given horse power H, making a given num- 
 ber of revolutions J-N" per minute, shall cause its angular 
 
 velocity not to deviate more than th from the mean ; or 
 
 conversely, the mean radius being given, the section K may 
 be determined according to these conditions. 
 
 269. In the above equations, m is taken to represent the 
 number of effective strokes made by the piston of the engine 
 whilst the fly-wheel makes one revolution, and ?i to represent 
 
 < 779 
 
 that angle whose sine is . 
 
 Let now the axis of the fly-wheel be supposed to be a 
 continuation of the axis of the crank, so that both turn with 
 the same angular velocity, as is usually the case ; and let its 
 application to the single-acting engine, the double-acting 
 engine, and to the double crank engine, be considered sepa- 
 rately. 
 
 1. In the single-acting engine, but one effective stroke of 
 the piston is made whilst the fly-wheel makes each revolution. 
 
 In this case, therefore, m=l, and sin. *)= =0-3183098 = 
 sin. 18 33'; therefore, cos. ?] = -9480460, also - = 
 
 it 
 
 103055 ; therefore, 1 = -793888. 
 
 co* ,-- = 
 
THE FLY-WHEEL. 361 
 
 Substituting in equations (345) and (346), 
 
 W= 95330-64 ^5, 
 rr~ 
 
 (347); 
 
 by which equations are determined, according to the pro- 
 posed conditions, the weight W in tons of a fly-wheel for a 
 single-acting engine, its mean radius in feet R being given, 
 and its material being any whatever; and also its mean 
 radius R in feet, its section (in square feet) K being given, 
 and its material being cast iron of mean quality ; and lastly, 
 the section K of its rim in square feet, its mean radius K 
 being given, and its material being, as before, cast iron. 
 
 2. In the double-acting engine, two effective strokes are 
 made by the piston, whilst the fly-wheel makes one 
 
 revolution. In this cases therefore, m = 2 and sin. ij=-= 
 
 if 
 
 0-636619 = sin. 39 32'; therefore, cos. *] = -7712549 j = 
 39 
 
 O 0' 
 
 / Oy, v 
 
 = -21963 ; therefore ( 1 - -^ ) = '56074 ; 
 
 - cos. *i - l - = -21051. 
 
 m 
 
 - (l - ) = 
 \ * i 
 
 Substituting in equations (345) and (346), 
 
 24-3593 'EM T E.n 
 
 R= S V -, =14424^^- ..... (348); 
 
 by which equations the weight of the fly-wheel in tons, the 
 mean radius in feet, and the section of the rim in square 
 feet are determined for the double-acting engine. 
 
 3. In the engine working with two cylinders and a double 
 crank, it has been shown (Art. 263.) that the conditions of 
 the working of the two arms of a double crank are precisely 
 the same as though the aggregate pressure 2P t upon their 
 extremities, were applied to the axis of the crank by the 
 intervention of a single arm and a single connecting rod; 
 
362 THE FKICTION OF THE FLY-WHEEL. 
 
 the length of this arm being represented by = instead of #, 
 
 1/2 
 
 ' and its direction equally dividing the inclination of the arms 
 of the double crank to one another. 
 
 Now, equations (345) and (346) show the proper dimen 
 sions of the fly-wheel to be wholly independent of the 
 length of the crank arm ; whence it follows that the dimen- 
 sion of a fly-wheel applicable to the double as well as a 
 single crank, are determined by those equations as applied 
 to the case of a double-acting engine, the pressure upon 
 whose piston rod is represented by 2Pj. But in assuming 
 Nm . ^P 1 =S3000H, we have assumed the pressure upon 
 the piston rod to be represented by Pj ; to correct this error, 
 and to adapt equations (345) and (346) to the case of a 
 double crank engine, we must therefore substitute -JH for H 
 in those equations. "We shall thus obtain 
 
 19-3339 
 
 
 by which equations the dimensions of a fly-wheel necessary 
 to give the required steadiness of motion to a double crank 
 engine are determined under the proposed conditions. 
 
 THE FKICTION OF THE FLY-WHEEL. 
 
 270. "W representing the weight of the wheel and 9 the 
 limiting angle of resistance between the surface of its axis 
 and that of its bearings, sin. <p will represent its coefficient 
 of friction (Art. 138.), and "W" sin. <p, the resistance opposed 
 to its revolution by friction at the surface of its axis. Now, 
 whilst the wheel makes one revolution, this resistance is 
 overcome through a space equal to the circumference of the 
 axis, and represented by 2tfp, if p be taken to represent the 
 radius of the axis. The work expended upon the friction of 
 the axis, during each complete revolution of the wheel, is 
 therefore represented by 2^pW sin. <p ; and if IS" represent 
 the number of strokes made by the engine per minute, and 
 
 vr 
 
 therefore the number of revolutions made by the fly-wheel 
 2 
 
MODULUS OF THE CRANK AND FLY-WHEEL. 363 
 
 per minute, then will the number of units of work expended 
 per minute, upon the friction of the axis be represented by 
 N*pW sin. 9 ; and the number of horses' power, or the frac- 
 tional part of a horse's power thus expended, by 
 
 . (350). 
 
 33000 
 
 If in this equation we substitute for W the weight in Ibs. 
 of the fly-wheel necessary to establish a given degree of 
 steadiness in the engine, as determined by equations (347), 
 (348), and (349), we shall obtain for the horse power lost by 
 friction of the fly-wheel, in the single-acting engine, the 
 double-acting engine, and the double crank engine, respec- 
 tively, the formulae 
 
 THE MODULUS OF THE CRANK AND FLY-WHEEL. 
 
 271. If Sj represent the space traversed by the piston of 
 the engine in any given time, and a the radius of the crank, 
 W the weight of the fly-wheel in Ibs., and p the radius of its 
 
 Q 
 
 axis, then will 2a represent the length of each stroke, the 
 
 2$ 
 
 Ql 
 
 number of strokes made in that time, and %fpW sin. <p . * 
 
 ' 
 
 or if WSj sin. <p the work expended upon the friction of the 
 a 
 
 fly-wheel during that time, which expression being added to 
 the equation (329) representing the work necessary to cause 
 the crank to yield a given amount of work U 2 to the ma- 
 chine driven by it (independently of the work expended on 
 the friction of the fly-wheel), will give the whole amount of 
 work which must be done upon the combination of the crank 
 and fly-wheel, to cause this given amount of work to be 
 yielded by it, on the machine which tiie crank drives. Let 
 this amount of work be represented by U^ then in the case 
 in which the directions of the driving pressure and the re- 
 sistance upon the crank are parallel (equation (329), and the 
 
364 
 
 THE GOVERNOR. 
 
 friction of the crane guide is neglected, we obtain for the 
 modulus of the crank and fly-wheel in the double-acting 
 engine 
 
 U = | 1 + fc sin. 9, + 1 sin. 9, ) } U, + *WS, f sin. 9 (352). 
 ( 2\# a / ) a 
 
 THE GOVERNOR. 
 
 272. This instrument is represented in the figure, under 
 that form in which it is most commonly applied tp the steam 
 engine. BD and CE are rods jointed 
 at A upon the vertical spindle AF, 
 and at D and E upon the rods DP 
 and EP, which last are again jointed 
 at their extremities to a collar fitted 
 accurately to the surface of the spin- 
 dle and moveable upon it. At their 
 extremities B and C, the rods DB 
 and EC carry two heavy balls, and 
 being swept round by the spindle 
 which receives a rapid rotation al- 
 ways proportional to the speed of the 
 machine, whose motion the governor 
 is intended to regulate these arms 
 by their own centrifugal force, and 
 that of the balls, are made to separate, and thereby to cause 
 the collar at P to descend upon the spindle, carrying with it, 
 by the intervention of the shoulder, the extremity of a lever, 
 whose motion controls the access of the moving power to 
 the driving point of the machine, closing the throttle valve 
 and shutting off the steam from the steam engine, or closing 
 the sluice and thus diminishing the supply of water to the 
 water-wheel. Let P be taken to represent the pressure of 
 the extremity of the lever upon the collar, Q the strain 
 thereby produced upon each of the rods DP and EP in the 
 direction of its length, W the weight of each of the balls, w 
 the weight of each of the rods BD and CE, AB=a, AD=&, 
 DP=c, FAB=d, APDr=^. Now upon either of these rods 
 as BD, the following pressures are applied : the weight of 
 the ball and the weight of the rod acting vertically, the 
 centrifugal force of the ball and the centrifugal force of the 
 rod acting horizontally, the strain Q of the rod DP, and 
 the resistance of the axis A. If a represent the angular 
 
THE GOVERNOR. 365 
 
 W "W 
 
 velocity of the spindle, a a . FB, or a 9 0sinJ, willrepre- 
 
 y & 
 
 sent the centrifugal force upon the ball (equation 102), 
 
 and V sin. 6 cos. & its moment about the point A ; also 
 
 9 
 
 the centrifugal force of the rod BD produces the same effect 
 as though its weight were collected in its centre of gravity 
 (Art. 124.), whose distance from A is represented by \(cib\ 
 so that the centrifugal force of the rod is represented by 
 
 ID 
 
 J ot?(a 5) sin. 6, and its moment about the point A by 
 
 1JO 
 
 J a* (a 5) 3 sin. & cos. 6. On the whole, therefore, the sum of 
 the moments of the centrifugal forces of the rod and ball are 
 represented by {~Wa?-\-%w(a 5) 3 j sin. 6 cos. 6. Now if f* 
 
 t/ 
 
 represent the weight of each unit in the length of the rod, 
 w = f*(+ &) ; therefore Wa* + %w(a - &) 3 = W a 9 
 (a b). Let this quantity be represented by " 
 
 - (-5) .... (353); 
 
 then will "W^* sin. 6 cos. 6 represent the sum of the moments 
 
 9 
 
 of the centrifugal forces of the rod and ball about A. More- 
 over, the sum of the moments of the weights of the rod and 
 ball, about the same point, is evidently represented by Wa 
 sin. b + w$(a V) sin. 6, or by \Wa+^(a? 5 2 )} sin. 6 ; let this 
 quantity be represented by W 2 # sin. 6, 
 
 Also the moment of Q about A=Q . AH=Q3 sin. 
 Therefore, by the principle of the equality of moments, ob- 
 serving that the centrifugal force of the rod and ball tend to 
 communicate motion in an opposite direction from their 
 weights and the pressure Q, 
 
 WX sin. 6 cos. t=Qb sin. (d+fiJ+Wji sin. 6. 
 
360 
 
 THE GOVERNOR. 
 
 Now P is the resultant of the pressures Q acting in the 
 directions of the rods PD and PE, and inclined to one 
 another at the angle 20, ; therefore (equation 13), 
 
 P=2Q cos. 6 l ; 
 
 .'. Q sin. (/) + &,) = JP -- = frp j sin . 6 + cos . 6 tan . ^ . 
 
 COS. fig 
 
 But since the sides 5 and <? of the triangle APD are oppo- 
 
 site to the angles 6 and 6, therefore sin * 1 = - ; therefore 
 
 sin. A c 
 
 cos ' 
 
 . 6 t = (l-l'sin. 2 *) V; 
 
 = sn. + - sn. cos. - 
 
 
 Substituting this value in the preceding equation, dividing 
 by sin. 0, and writing (1 cos. 2 d) for sin. 2 $, we obtain 
 
 a a. ..(355); 
 
 which equation, of four dimensions in terms of cos. 0, being 
 solved in respect to that variable, determines the inclination 
 of the arms under a given angular velocity of the spindle. 
 It is, however, more commonly the case that the inclination 
 of the arms is given, and that the lengths of the arms, 
 or the weights of the balls, are required to be determined, 
 so that this inclination may, under the proposed conditions, 
 be attained. In this case the values of W, and W 2 must be 
 substituted in the above equation from equations (353) and 
 (354), and that equation solved in respect to a or "W. 
 
 The values of b and c are determined by the position on 
 the spindle, to which it is proposed to make the collar 
 descend, at the given inclination of the arms or value of 0. 
 If the distance AP, of this position of the collar from A, be 
 represented by ^, we have h=b cos. b-\-e cos. ^, 
 
 -^$m.*4 V ---- (356); 
 
THE GOVERNOR. 
 
 367 
 
 from which equation and the preceding, the value of one 
 of the quantities 1) or c may be determined, according to the 
 proposed conditions, the value of the other being assumed to 
 be any whatever. 
 
 If N" represent the number of revolutions, or parts of a 
 revolution, made per second by the fly-wheel, and /N the 
 number of revolutions made in the same time by the spindle 
 of the governor, then will fyirylS represent the space a de- 
 scribed per second by a point, situated at distance unity from 
 the axis of the spindle. Substituting this value for a in 
 equation (355), and assuming & <?, we obtain 
 
 4ffV 2 N 8 
 
 -? WXcos. 0=PJ+W 2 a ____ (357) : 
 t/ 
 
 also by equation (356), 
 
 h=2b cos. A ..... (358). 
 
 Eliminating cos. d between these equations, and solving in 
 respect to A, 
 
 Let P (1-f) and P (l-~) represent the values of P 
 corresponding to the two states bordering upon motion 
 (Art. 140) and let N (l + ) and N (1^) be the correspond- 
 ing values of N ; so that the variation either way of ^th from 
 the mean number N" of revolutions, may be upon the point 
 of causing the valve to move. If these values be respectively 
 substituted for P and N in the above formula, it is evident 
 that the corresponding values of fi will be equal. Equating 
 those values of h and reducing, we obtain 
 
 By which equation there is established that relation between 
 the quantities W 2 , &, P, m which must obtain, in order that a 
 variation of the number of revolutions, ever so little greater 
 
368 
 
 THE CAKKIAGE-WHEEL. 
 
 than the ^th part, may cause the valve to move. Neglect- 
 ing \ as small when compared with n. 
 
 / w 
 
 I Jl 
 
 which expression, representing that fractional variation in the 
 number of revolutions which is sufficient to give motion to 
 the valve, is the true measure of the SENSIBILITY of the 
 governor. 
 
 273. The joints E and D are sometimes 
 fixed upon the arms AB and AC as in the 
 accompanying figure, instead of upon the 
 prolongations of those arms as in the pre- 
 ceding figure. All the formulae of the 
 last Article evidently adapt themselves 
 to this case, if b be assumed =0 (in equa- 
 tions 353, 354). The centrifugal force of 
 the rods EP and DP is neglected in this 
 computation. 
 
 THE CARRIAGE-WHEEL. 
 
 274:. Whatever be the nature of the resistance opposed to 
 the motion of a carriage- wheel, it is evidently equivalent to 
 that of an obstacle, real or imaginary, which the wheel may 
 be supposed, at every instant, to be in the act of surmount- 
 ing. Indeed it is certain, that, however yielding may be the 
 material of the road, yet by reason of its compression before 
 the wheel, such an immoveable obstacle, of exceedingly small 
 height, is continually in the act of being presented to it. 
 
 275. The two-wheeled carriage. 
 
 Let AB represent one of the wheels of a two-wheeled 
 carriage, EF an inclined plane, which it is in the act of as- 
 cending, O a solid elevation of the surface of the plane, or an 
 obstacle which it is at any instant in the act of passing over, 
 
THE CAKRIAGE-WHEEL. 
 
 369 
 
 P the corresponding trac- 
 tion, W the weight of the 
 wheel and of the load which 
 it supports. 
 
 Now the surface of the lox 
 of the wheel being in the 
 state bordering upon motion 
 on the surface of the axle, 
 the direction of the resist- 
 ance of the one upon the 
 other is inclined at the limit- 
 ing angle of resistance, to a 
 radius of the axle at their, 
 point of contact (Art. 14rl.)i. 
 This resistance has,, more- 
 over, its direction; through 
 the point of contact O of 
 the tire of the wheel with the obstacle on which it is in the 
 act of turning. If, therefore, OK be drawn intersecting the 
 circumference of the axis in a point <?, such that the angle 
 CcR may equal the limiting angle of resistance <p,, then will 
 its direction be that of the resistance of the obstacle upon 
 the wheel. 
 
 Draw the vertical GH representing the weight "W, and 
 through H draw HK parallel to OK, then will this line 
 represent (to the same scale) the resistance K, and GK the 
 traction P (Art. 14.) ; 
 
 , P _GK sin. GHK sin.. GHK 
 
 ' W~GH-sm. GKH-sin. (PGH-GHK)= 
 
 sin. WsO 
 sin. (PLW-W*Oy 
 
 Let K= radius of wheel, p= radius of axle, AGO =77, ACTW 
 i=inclination of the road to the horizon, ^inclination of 
 direction of the traction to the road. Now WsO=WCO-f- 
 
 COs, but 
 
 =* + iy, and 
 
 Let CO* be re- 
 
 presented by a, then WsO=i 
 
 sin. a^^sin. 9 .... (360). 
 
 Also PLW=^+*+d; therefore ~PLW-WsO=:--(r i +a-6)- 9 
 
 24: 
 
3TO THE CARRIAGE-WHEEL. 
 
 .-.P=W- ^r-^ (361); 
 
 COS. (*)+ a 6) 
 
 when the direction of traction is parallel to the road, d=0, 
 ..P=W{sin. 1 + cos. i tan. (*i+a)} .... (362). 
 
 If the road and the direction of traction be both horizontal 
 6=i=Q, and 
 
 P=W tan. (ij+a) (363). 
 
 In all cases of traction with wheels of the common dimen- 
 sions upon ordinary roads, AGO or v\ is an exceedingly small 
 angle ; a is also, in all cases, an exceedingly small angle 
 (equation 360); therefore tan. (77+ a) 77 + a very nearly. 
 Now if A be taken to represent the arc AO, whose length 
 is determined by the height of the obstacle and the radius 
 of the wheel, then 
 
 1=5 (364)- 
 
 Substituting the value of a from equation (360), 
 
 p w (A + p sin. 9) , 
 
 P=W.- -^ i (365). 
 
 276. It remains to determine the value of the arc A inter- 
 cepted between the lowest point to which the wheel sinks in 
 the road, and the summit of the obstacle, which it is at 
 every instant surmounting. Now, the experiments of Cou- 
 lomb, and the more recent experiments of M. Morin,* ap- 
 pear to have fully established the fact, that, on horizontal 
 roads of uniform quality and material, the traction P, when 
 its direction is horizontal, varies directly as the load W, and 
 inversely as the radius B, of the wheel ; whence it follows 
 (equation 365), that the arc A is constant, or that it is the 
 same for the same quality of road, whatever may be the 
 weight of the load, or the dimensions of the wheel.f The 
 
 * Experiences sur le Tirage des Voitures, faites en 1837 et 1888. (See AP- 
 PENDIX.) 
 
 f In explanation of this fact let it be observed, that although the wheel 
 sinks deeper beneath the surface of the road as the material is softer, yet the 
 obstacle yields, for the same reason, more under the pressure of the wheel, the 
 arc A being by the one cause increased, and by the other diminished. Also, 
 that although by increasing the diameter of the wheel the arc A would be ren- 
 dered greater if the wheel sank to the same depth as before, yet that it does 
 not sink to the same depth by reason of the corresponding increase of the sur- 
 face which sustains the pressure. 
 
THE CAKKI AGE-WHEEL. 371 
 
 constant A may therefore be taken as a measure of the re- 
 sisting quality of the road, and may be called the modulus 
 of its resistance. 
 
 The mean value of this modulus being determined in re- 
 spect to a road, whose surface is of any given quality, the 
 value of 7) will be known from equation (364), and the rela- 
 tion between the traction and the load upon that road, under 
 all circumstances ; it being observed, that, since the arc A 
 is the same on a horizontal road, whatever be the load, if the 
 traction be parallel, it is also the same under the same cir- 
 cumstances upon a sloping road ; the effect of the slope be- 
 ing equivalent to a variation of the load. The same substi- 
 tution may therefore be made for tan. (?)+a) in equation 
 (362), as was made in equation (363), 
 
 = sn. <+ cos. 
 
 277. The lest direction of traction in the two-wheeled 
 carriage. 
 
 This best direction of traction is evidently that which gives 
 to the denominator of equation (361) its greatest value ; it 
 is therefore determined by the equation 
 
 .... (367). 
 
 
 278. The four-wheeled carriage. 
 
 Let W 1? W a represent the loads borne by the fore and 
 hind wheels, together with their own weights, K K a their 
 radii, p,, p 2 the radii of their axles, and <p l5 <p a the limiting an- 
 gles of resistance. Suppose the direction of the traction P 
 parallel to the road, then, since this traction equals the sums 
 of the tractions upon the fore and hind wheels respectively, 
 we have by equation (366) 
 
372 THE CAKRIAGE-WHEEL. 
 
 or 
 
 3 sn. 
 
 ^) sin. 9l +W 2 (^)sin.9 2 jcos. . . . (368). 
 
 279. The work accumulated in the carriage-wheel.* 
 
 Let I represent the moment of inertia of the wheel about 
 its axis and M its volume; then will MR 2 +I represent its 
 moment of inertia (Art. 79.) about the point in its circum- 
 ferences about which it is, at every instant of its motion, in 
 the act of turning. If, therefore, a represent its angular 
 velocity about this point at any instant, U the work at that 
 instant accumulated in it, and p the weight of each cubical 
 
 unit of its mass, then (Art. 75.), U=Ja a -(MR a +I) = |-M 
 
 y y 
 
 M* 
 
 (aR) a -f-Ja a -I. Now if Y represent the velocity of the axis 
 of the wheel, aE=Y; 
 
 whence it follows, that the whole work accumulated in the 
 rolling wheel is equal to the sum obtained by adding the 
 work which would have been accumulated in it if it had 
 moved with its motion of translation only, to that which 
 would have been accumulated in it if it had moved with its 
 motion of rotation only. If we represent the radius of gyra- 
 tion (Art. 80.) by }, I=MK a ; whence substituting and 
 reducing, 
 
 ..... (369). 
 
 The accumulated work is therefore the same as though the 
 wheel had moved with a motion of translation only, but with 
 
 a greater velocity, represented by the expression 1 1 + -pa 1 Y. 
 
 * For a further discussion of the conditions of the rolling of a wheel, see a 
 paper in the Appendix on the Rolling Motion of a Cylinder. 
 
 f The angular velocity of the wheel would evidently be a, if its centre were 
 fixed, and its circumference made to revolve with the same velocity as now. 
 
ACCELERATED OE RETARDED MOTION. 373 
 
 280. ON THE STATE OF THE ACCELERATED OR THE RETARDED 
 MOTION OF A MACHINE. 
 
 Let the work U l done upon the driving point of a machine 
 be conceived to be in excess of that U 3 yielded upon the 
 working points of the machine and that expended upon its 
 prejudicial resistances. Then we have by equation (117) 
 
 11,= AU.+ BS, +^(V- V^tox' ; 
 
 where Y represents the velocity of the driving point of the 
 machine after the work 'U l has been done upon it, Y 4 that 
 when it began to be done, and 2i0X a the coefficient of equable 
 motion. ]Now let S x represent the space through which Uj 
 is done, and S 2 that through which U , is done ; and let the 
 above equation be differentiated in respect to S n 
 
 .ZU,_ dU. dB 1 dV 
 
 dS,- dS, ' 3 l + "*9 r < m ** * , 
 
 but TT" 1 PI (Art. 51.) if P, represent the driving pressure. 
 
 Also -SOT = P a , if P a represent the working pressure ; also 
 
 ' dV dt dV 1 dV 
 
 V ''-----=f (equation 72). 
 
 If, therefore, we represent by A the relation - 1 , between the 
 
 spaces described in the same exceedingly small time by the 
 driving and working points, we have 
 
 -2wX a ..... (370); 
 
 9 
 
 where/" (Art. 95.) represents the additional velocity actually 
 acquired per second by the driving point of the machine, if 
 Pj and P 2 be constant quantities, or, if not, the additional 
 velocity which would be acquired in any given second, if 
 these pressures retained, throughout that second, the values 
 which they had at its commencement. 
 
374 THE ACCELERATION OR RETARDATION 
 
 281. To determine the coefficient of equable motion. 
 
 a represents the sum of the weights of all the moving 
 elements of the machine, each being multiplied by the ratio 
 of its velocity to that of the driving point, which sum has 
 been called (Art. 151.) the coefficient of equable motion. If 
 the motion of each element of the machine takes place #bout 
 a fixed axis, and a 1} a, n &c., represent the perpendiculars 
 from their several axes upon the directions in which they 
 receive the driving pressures of the elements which precede 
 them in the series, and ^, J 2 , Z> 3 , &c., the similar perpen- 
 diculars upon the tangents to their common surfaces at the 
 points where they drive those that follow them ; then, 
 while the first driving point describes the small space AS 15 
 the point of contact of the p\h and p + 1th elements of the 
 series will be made (Art. 234.) to describe a space repre- 
 sented b 
 
 **M ^ 
 
 a^a^ ... ct>p 
 
 so that the angular velocity of the ^?th element will be 
 represented by 
 
 . 
 
 a p 
 
 and the space described by a particle situated at distance f 
 from the axis of that element by 
 
 and the ratio X of this space to that described by the driving 
 point of the machine will be represented by 
 
 a. t 
 
 The sum 2wX 2 w iH therefore be represented in respect to 
 this one element by 
 
 Or if Ip represent the moment of inertia of the element, and 
 PP the weight of each cubic unit of its mass, that portion of 
 the value of 2wX a which depends upon this element will be 
 represented by 
 
OF THE MOTION OF A MACHINE. 375 
 
 And the same being true of every other element of the 
 machine, we have 
 
 which is a general expression for the coefficient of equable 
 motion in the case supposed. The value of A in equation 
 (3T1) is evidently represented by 
 
 
 To determine the pressure upon the point of contact of 
 any two elements of a machine moving with an accelerated 
 or retarded motion. 
 
 Let j? 4 be taken to represent the resistance upon the point 
 of contact of the first element with the second, j9 2 that upon 
 the point of contact of the second element of the machine 
 with the third, and so on. Then by equation (3TO), observ- 
 ing that, P! and p l representing pressures applied to the 
 same element, ^w^ is to be taken in this case only in 
 respect to that element, so that it is represented by p.,!^ 
 
 whilst A is in this case represented by , we have, neglect- 
 ing friction, 
 
 Substituting the value of f from equation (371), and solving 
 in respect to j?,, 
 
 _! p _! / p _ A p \ j^L 
 
 b, 1 "!>,[ l V 2wK 
 
 where the value of A is determined by equation (373), and 
 that of 2i0X a by equation (37^). Proceeding similarly in 
 respect to the second element, and observing that the 
 impressed pressures upon that element are jp l and p we 
 have 
 
S76 ACCELERATED OR RETARDED MOTION. 
 
 fi representing the additional velocity per second of the 
 point of application of p which evidently equals /. 
 
 Substituting, therefore, the value of/ from equation (3T1) 
 as before, 
 
 Substituting the value of ^ from equation (374), and solv- 
 ing in respect tojp a , we have 
 
 And proceeding similarly in respect to the other points of 
 contact, the pressure upon each may be determined. It is 
 evident, that by assuming values of A and B in equations 
 (370) and (371) to represent the coefficients of the moduli in 
 respect to the several elements of the machine, and to the 
 whole machine, the influence of friction might, by similar 
 steps, have been included in the result. 
 
PART IV. 
 
 THEOEY OF THE STABILITY OF STKUCTUKES. 
 
 GENERAL CONDITIONS OF THE STABILITY OF A STRUCTURE OF 
 UNCEMETED STONES.* 
 
 A STRUCTURE may yield, under the pressures to which it is 
 subjected, either by the slipping of certain of its surfaces of 
 contact upon one another, or by their turning over upon the 
 edges of one another ; and these two conditions involve the 
 whole question of its stability. 
 
 THE LINE OF KESISTANCE. 
 
 283. Let a structure MNLK, composed of a single row of 
 uncemented stones of any forms, 
 s and placed under any given circum- 
 f stances of pressure, be conceived to 
 , be intersected by any geometrical 
 surface 1 2, and let the resultant a A 
 of all the pressures which act upon 
 one of the parts MN21, into which 
 this intersecting surface divides the 
 structure, be imagined to be taken. 
 Conceive, then, this intersecting 
 surface" to change its form and posi- 
 tion so as to coincide in succession 
 with all the common surfaces of 
 contact 8 4, 5 6, T 8, 9 10, of the 
 stones which compose the structure : 
 and let R, cO, dD, eE be the re- 
 
 * Extracted from a memoir on the Theory of the Arch by the author of this 
 work in the first volume of the " Theoretical and Practical Treatise on Bridges," 
 by Professor Hosking and Mr. Hann of King's College, published by Mr. Weale. 
 These general conditions of the equilibrium of a system of bodies in contact 
 were first discussed by the author in the fifth and sixth volumes of the " Cam- 
 bridge Philosophical Transactions." 
 
 871 
 
378 THE LINE OF RESISTANCE. 
 
 sultan ts, similarly taken with &A, which correspond to these 
 several planes of intersection. 
 
 In each such position of the intersecting surface, the result- 
 ant spoken of having its direction produced, will intersect 
 that surface either within the mass of the structure, or, when 
 that surface is imagined to be produced, without it. If it 
 intersect it without the mass of the structure, then the whole 
 pressure upon one of the parts, acting in the direction of 
 this resultant, will cause that part to turn over upon the 
 edge of its common surface of contact with the other part ; 
 if it intersect it within the mass of the structure, it will not. 
 
 Thus, for instance, if the direction of the resultant of the 
 forces acting upon the part NM 1 2 had been &'A', not inter- 
 secting the surface of contact 1 2 within the mass of the 
 structure, but when imagined to be produced beyond it to a' ; 
 then the whole pressure upon this part acting in a' A! would 
 have caused it to turn upon the edge 2 of the surface of con- 
 tact 1 2 ; and similarly if the resultant had been in a" A", 
 then it would have caused the mass to revolve upon the 
 edge 1. The resultant having the direction #A, the mass 
 will not be made to revolve on either edge of the surface of 
 contact 1 2. 
 
 Thus the condition that no two parts of the mass should be 
 made, by the insistent pressures, to turn over upon the edge 
 of their common surface of contact, is involved in this other, 
 that the direction of the resultant, taken in respect to every 
 position of the intersecting surface, shall intersect that sur- 
 face actually within the mass of the structure. 
 
 If the intersecting surface be imagined to take up an infi- 
 nite number of different positions, 1 2, 3 4, 5 6, &c., and the 
 intersections with it, #, J, <?, d, &c., of the directions of all 
 the corresponding resultants be found, then the curved line 
 cikcdef, joining these points of intersection, may with pro- 
 priety be called the LINE OF RESISTANCE, the resisting points 
 of the resultant pressures upon the contiguous surfaces lying 
 all in that line. f> . 
 
 This line can be completely determined by the methods of 
 analysis, in respect to a structure of any given geometrical 
 form, having its parts in contact by surfaces also of given 
 geometrical forms. And, conversely, the form of this line 
 being assumed, and the direction which it shall have through 
 any proposed structure, the geometrical form of that struc- 
 ture may be determined, subject to these conditions ; or 
 lastly, certain conditions being assumed, both as it regards 
 the form of the structure and its line of resistance, all that is 
 
THE LINE OF PRESSURE. 
 
 379 
 
 necessary to the existence of these assumed conditions may 
 be found. Let the structure ABCD have for its line of re- 
 sistance the line PQ. Now 
 it is clear that if this line 
 cut the surface MN of any 
 section of the mass in a point 
 n without the surface of the 
 mass, then the resultant of 
 the pressures upon the mass 
 CMN will act through n, 
 and cause this portion of the 
 mass to revolve about the 
 nearest point IS" of the in- 
 tersection of the surface of 
 secton K with the surface of the structure. 
 
 Thus, then, it is a condition of the equilibrium that the 
 line of resistance shall intersect the common surface of con- 
 tact of each two contiguous portions of the structure actually 
 within the mass of the structure / or, in other words, that it 
 shall actually go through each joint of the structure, avoid- 
 ing none : this condition being necessary, that no two po5> 
 tions of the structure may revolve on the edges of their 
 common surface of contact. 
 
 THE LINE OF PRESSURE. 
 
 
 
 284. But besides the condition that no two parts of the 
 structure should turn upon the edges of their common sur- 
 faces of contact, which condition is involved in the determi- 
 nation of the LINE OF RESISTANCE, there is a second condition 
 necessary to the stability of the structure. Its surfaces of 
 contact must no where slip upon one another. That this 
 condition may obtain, the resultant corresponding to each 
 surface of contact must have its direction within certain 
 limits. These limits are denned by the surface of a right 
 cone (Art. 1.39.), having the normal to the common surface 
 of contact v at the above-mentioned point of intersection of 
 the resultant) for its axis, and having for its vertical angle 
 twice that whose tangent is the co-emcient of friction of the 
 surfaces. If the direction of the resultant be within this 
 cone, the surfaces of contact will not slip upon one another ; 
 if it be without it, they will. 
 
 Thus, then, the directions of the consecutive resultants in 
 
380 * THE STABILITY OF A SOLID BODY. 
 
 respect to the normal to the point, where each intersects its 
 corresponding surface of contact, are to be considered as im- 
 portant elements of the theory. 
 
 Let then a line ABODE be taken, which is the locus of 
 the consecutive intersections of the 
 resultants # A, &B, cC, dD, &c. The 
 direction of the resultant pressure 
 upon every section is a tangent to 
 this line ; it may therefore with pro- 
 priety be called the LINE OF PRESSURE. 
 Its geometrical form may be deter- 
 mined under the same circumstances 
 as that of the line of resistance. A 
 straight line cC drawn from the point 
 0, where the LINE OF RESISTANCE abed 
 intersects any joint 5 6 of the struc- 
 ture, so as to touch the LINE OF PRES- 
 SURE ABCD, will determine the 
 direction of the resultant pressure 
 upon that joint: if it lie within the cone spoken of, the 
 structure will not slip upon that joint ; if it lie without it, 
 it will. 
 
 Thus the whole theory of the equilibrium of any structure 
 is involved in the determination with respect to that struc- 
 ture of these two lines the line of resistance, and the line 
 of pressure : owe of these lines, the line of resistance, de- 
 termining the point of application of the resultant of the 
 pressures upon each of the surfaces of contact of the system ; 
 and the other, the line of pressure, the direction of that 
 resultant. 
 
 The determination of both, under their most general forms, 
 lies within the resources of analysis ; and general equations 
 for their determination in that case, in which all the surfaces 
 of contact, or joints, are planes the only case which offers 
 itself as & practical case have been given by the author of 
 this work in the sixth volume of the " Cambridge Philo- 
 sophical Transactions." 
 
 THE STABILITY OF A SOLID BODY. 
 
 285. The stability of a solid body may be considered to be 
 greater or less, as a greater or less amount of work must be 
 aone upon it to overthrow it ; or according as the amount 
 
THE STABILITY OF A STRUCTURE. 381 
 
 of work which must be done upon it to bring it into 
 that position in which it will fall over of its own accord is 
 greater or less. Thus the stability of the solid represented 
 in fig. 1. resting on a horizontal Plg ^ Kjj 2 
 
 plane is greater or less, according 
 as the work which must be clone 
 upon it, to bring it into the position 
 represented in^g. 2., where its cen- 
 tre of gravity is in the vertical 
 passing through its point of sup- 
 port, is greater or less. Now this 
 work is equal (Art. 60.) to that 
 which would be necessary to raise its whole weight, verti- 
 cally, through that height by which its centre of gravity 
 is raised, in passing from the one position into the other. 
 Whence it follows that the stability of a solid body resting 
 upon a plane is greater or less, as the product of its weight 
 by the vertical height through which its centre of gravity is 
 raised, when the body is brought into a position in which it 
 will fall over of its own accord, is greater or less. 
 
 If the base of the body be a plane, and if the vertical 
 height of its centre of gravity when it rests upon a horizontal 
 plane be represented by A, and the distance of the point or 
 the edge, upon which it is to be overthrown, from the point 
 where its base is intersected by the vertical through its 
 centre of gravity, by & ; then is the height through which its 
 centre of gravity is raised, when the body is brought into a 
 position in which it will fall over, evidently represented by 
 (A 2 + #')* A; so that if "W represent its weight, and U the 
 work necessary to overthrow it, then 
 
 U=W \(V+lcJ-h\ .... (376). 
 U is a true measure of the stability of the body. 
 
 THE STABILITY OF A STRUCTURE. 
 
 286. It is evident that the degree of the stability of a 
 * structure, composed of any number of separate but contigu- 
 ous solid bodies, depends upon the less or greater degree of 
 approach which the line of resistance makes to the extrados 
 or external face of the structure ; for the structure cannot be 
 thrown over until the line of resistance is so del. ected as to 
 
332 rm<: WALL OK PIER. 
 
 intersect the extrados : the more remote is its direction frora 
 that surface, when free from any extraordinary pressure, the 
 less is therefore the probability that any such pressure will 
 overthrow it. The nearest distance to which the line of re- 
 sistance approaches the extrados will, in the following pages, 
 be represented by m, and will be called the MODULUS OF 
 STABILITY of the structure. 
 
 This shortest distance presents itself in the wall and but- 
 tress commonly at the lowest section of the structure. It is 
 evidently beneath that point where the line of resistance in- 
 tersects the lowest section of the structure that the greatest 
 resistance of the foundation should be opposed. If that point 
 be firmly supported, no settlement of the structure can take 
 place under the influence of the pressures to which it is ordi- 
 narily subjected.* 
 
 THE WALL OR PIER. 
 287. The stability of a wall. 
 
 If the pressure upon a wall be uniformly distributed along 
 its length,! and if we conceive it to be intersected by verti- 
 cal planes, equidistant from one another and perpendicular 
 to its face, dividing it into separate portions, then are the 
 conditions of its stability, in respect to the pressures applied 
 to its entire length, manifestly the same with the conditions 
 the stability of each of the individual portions into which it 
 is thus divided, in respect to the pressures sustained by that 
 portion of the wall ; so that if every such columnar portion 
 or pier into which the wall i,s thus divided be constructed so 
 as to stand under its insistent pressures with any degree of 
 firmness or stability, then will the whole structure stand with 
 the like degree of firmness or stability ; and conversely. 
 
 In the following discussion these equal divisions of the 
 length of a wall or pier will be conceived to be made one 
 foot apart ; so that in every case the question investigated 
 will be that of the stability of a column of uniform or varia- 
 
 * A practical rule of Vauban, generally adopted in fortifications, brings the 
 point where the line of resistance intersects the base of the wall, to a distance 
 from the vertical to its centre of gravity, of -|ths the distance from the latter 
 to the external edge of the base. (See Poncelet, Memoire sur la Stabilite des 
 Hevetemens, note, p. 8.) 
 
 f In the wall of a building the pressure of the rafters of the roof is thua 
 uniformly distributed by the intervention of the wall plates. 
 
THE LINE OF RESISTANCE IN A PIER. 
 
 383 
 
 ble thickness, whose width measured in the direction of the 
 length of the wall is one foot. 
 
 288. When a wall is supported by buttresses placed at 
 equal distances apart, the conditions of the stability will be 
 made to resolve themselves into those of a continuous wall, 
 if we conceive each buttress to be ex- 
 tended laterally until it meets the adja- 
 cent buttress, its material at the same 
 time so diminishing its specific gravity 
 that its weight when thus spread along 
 the face of the wall may remain the 
 same as before. There will thus be ob- 
 tained a compound wall whose external 
 and internal portions are of different 
 specific gravities ; the conditions of 
 whose equilibrium remain manifestly 
 unchanged by the hypothesis which has 
 been made in respect to it. 
 
 THE LINE OF KESISTANCE IN A PIEK. 
 
 289. Let ABEF be taken to repre- 
 /'' sent a column of uniform dimensions. 
 
 / Let PS be the direction of any pres- 
 
 ' & sure P sustained by it, intersecting its 
 
 axis in O. Draw any horizontal sec- 
 tion IK, and take ON to represent 
 the weight of the portion AKIB of 
 the column, and OS on the same scale 
 to represent the pressure P, and com- 
 plete the parallelogram ONES ; then 
 will OK- evidently represent, in mag- 
 nitude and direction, the resultant of 
 the pressures upon the portion AKIB 
 of the mass (Art. 3.), and its point of 
 intersection Q with IK will represent 
 a point in the line of resistance. 
 
 Let PS intersect BA (produced if necessary) in G, and let 
 ~ ~ " ~i\/r/^ "p/~\/"^ - * \-\4- P 
 
 each cubic foot of the material of the mass. Draw RL per- 
 pendicular to CD ; then, by similar triangles, 
 
 
 
 
 
 // 
 
 
 
 
 I 
 
 IT 
 
 
 lr n 
 
 
 
 1 
 
 : - 
 
 tt~ 
 
 1 
 
 
 
384: THE LINE OF KESISTANCE IN A PIEE. 
 
 QM_EL 
 OM""OL 
 
 But QM=y, OM=CM-CO=z-& cot._o, EL=EN 
 sin. ENL=P sin. , OL=ON+NL=ON+EN cos. ENL 
 =; +P cos. a; 
 
 y P sin. a 
 
 "xk cot. a~>#a?H-P cos. a ' 
 
 riB. .-* COB. . ..... 
 
 cos. a 
 
 which is the general equation of the line of resistance of a 
 pier or wall. 
 
 290. The conditions necessary that the stones of the pier may 
 not slip on one another. 
 
 Since in the construction of the parallelogram ONES, 
 whose diagonal OE determines the direction of the resultant 
 pressure upon any section IK, the side OS, representing the 
 pressure P in magnitude and direction, remains always the 
 same, whatever may be the position of IK ; whilst the side 
 ON, representing the weight of AKIB, increases as IK de- 
 scends : the angle EOM continually diminishes as IK de- 
 scends. Now, this angle is evidently equal to that made by 
 OE with the perpendicular to IK at Q ; if, therefore, this 
 angle be less than the limiting angle of resistance in the 
 highest position of IK, then will it be less in every subjacent 
 position. But in the highest position of IK, ON=0, so that 
 in this position EOM=a. Now, so long as the inclination 
 of OE to the perpendicular to IK is less than the limiting 
 angle of resistance, the two portions of the pier separated by 
 that section cannot slip upon one another (Art. 141.). It is 
 therefore necessary', and sufficient to the condition that no 
 two parts of the structure should slip upon their common 
 surface of contact, that the inclination a of P to the vertical 
 should be less than the limiting angle of resistance of the 
 common surfaces of the stones. All the resultant pressures 
 passing through the point O, it is evident that the line of 
 pressure (Art. 284.) resolves itself into that point. 
 
THE LINE OF RESISTANCE IN A PIER. 3S5 
 
 291. The greatest height of the pier. 
 
 At the point where the line of resistance intersects the 
 external face or extrados of the pier, y=%a\ if, therefore, H 
 represents the corresponding value of , it will manifestly 
 represent the greatest height to which the pier can be built, 
 so as to stand under the given insistent pressure P. Substi- 
 tuting these values for a? and y in equation (377), and solving 
 in respect to H, 
 
 Psin. a 
 
 
 If P sin. a -J-JX& 2 , H m/mYy y whence it follows that in 
 this case the pier will stand under the- given pressure P how- 
 ever great may be the height to which it is raised. 
 
 292. The line of resistance is an equilateral hyperbola. 
 
 Multiplying both sides of equation ($77) by the don >mi- 
 nator of the fraction in the second member, 
 
 y(pax-\-P cos. a)=Px sin. a P& cos. a ; 
 
 dividing by pa, transposing, and changing the signs of all the 
 terms, 
 
 Psin. a Pcos.a Pcos.a 
 
 / cos.a\ cos.a T 
 
 V I # H -- 1 = -- k : 
 M N / M-a 
 
 adding 
 
 Psin.a/ ' Pcos.a\ / Pcos.a PcOS.a/ Psin.a 
 
 / ' cos.a\ / cos.a\ PcOS.a/^ Psin.a\ 
 
 \ pa I y \ fx / M-a \ pa r 
 
 Psin. a Pcos.a Pcos.a Psin 
 
 en equa o 
 
 pa 
 
 VQ=y,, TV=a),, 
 
 Let Cil be taken equal to ifl nT= ; and let 
 
 pa pa 
 
 25 
 
386 - THE LINE OF RESISTANCE IN A PIER. 
 
 Pcos.a/ 7 rsm.a\ 
 /. x,y l = 1 k + 1 = a constant quantity. 
 
 This is the equation of a rectangular hyperbola, whose 
 asymtote is TX.* The line of resist- 
 'f / ance continually approaches TX 
 
 therefore, but never meets it ; whence 
 it follows, that if TX lie (as shown 
 in the figure) within the surface of 
 the mass, or if C H < C B or 
 
 Psin.ct ., 013 . 
 
 .. _ . <# or 2P sin. a<f*a. then 
 
 &'' ^ a 
 
 j , -'/ the line of resistance will no where 
 
 cut the extrados, and the structure 
 will retain its stability under the in- 
 sistent pressure P, however high it 
 may be built ; which agrees with 
 the result obtained in the preceding 
 article. 
 
 ii/frf 
 
 293. The thickness of the pier, so that when raised to a given 
 height it may have a given stability. 
 
 Let m be taken to represent the nearest distance to which 
 the line of resistance is intended to approach the extrados of 
 the pier, which distance determines the degree of its stability, 
 and has been called the modulus of stability (Art. 286.). It 
 is evident from the last article that this least distance will 
 present itself in the lowest section of the pier. At this 
 lowest section, therefore, y=^am. Substituting this value 
 for y in equation (377), and also the height h of the pier for 
 cc, and solving the resulting quadratic equation in respect to 
 a, we shall thus obtain 
 
 r P cos. a 
 
 2P 
 fr 
 
 Church's Analyt. Geom., Art. 161. 
 
 \ { 
 
 )cos.a[..(379). 
 
A WALL SUPPORTED BY SHORES. 
 
 387 
 
 294. To vary the point of application of the pressure P, so 
 that any required stability may he given to the pier. 
 
 It is evident, that if in equation (377) we substitute \a,m 
 for y, and k for a?, the modulus of stability m 
 may be made to assume any given value for 
 a given thickness a of the pier, by assigning 
 a corresponding value to k ; that is, by mov- 
 ing the point of application G- to a certain 
 distance from the axis of the pier, deter- 
 mined by the value of Jc in that equation. 
 This may be done by various expedients,- 
 and among others by that shown in the 
 figure. Solving equation (377) in respect to 
 & we have 
 
 . . . . (380), 
 
 It is necessary to the equilibrium of the pier, under these 
 circumstances, that the line of resistance should no where 
 intersect its intrados below the point D. 
 
 \ 
 
 THE STABILITY OF A WALL SUPPORTED BY SHORES. 
 
 295. Let the weight of the portion of the wall supported 
 by each shore or prop, and the 
 >T? pressure insistent upon it, be im- 
 / agined to be collected in a single 
 foot of the length of the wall ; tho 
 conditions of the stability of tho 
 wall evidently remain unchanged 
 by this hypothesis. Let ABCD 
 represent one of the columns or 
 piers into which the wall will thus 
 be divided, EF the corresponding 
 shore, P the pressure sustained upon 
 the summit of the wall, Q the 
 thrust upon the shore EF, 2w its 
 weight, x the point, where the line 
 of resistance intersects the base of 
 the wall, Csc=m, CJ?=b, FEC=: 
 and let the same notation be taken in other respects as in 
 
388 A WALL SUPPORTED BY SHORES. 
 
 the preceding articles. Then, since a? is a point in the direc 
 tion of the resultant of the resistances by which the base of 
 the column is sustained, the sum of the moments about that 
 point of the pressure P and half the weight of the shore, 
 supposed to be placed at E*, is equal to the sum of the 
 moments of the thrust Q, and the weight pah of the column; 
 or drawing soM. and %N perpendiculars upon the directions 
 of P and Q, 
 
 P. 
 
 Now #M = xs sin. a?*M=(HK Ht) sin. a ^ 
 cot. a} sin. a=A sin. a (k+^a mj cos. a, %~N=(b+m) cos. 
 
 m)cos. aj + 
 wm=Q(b+m) cos. P + pahQa m) 
 
 Solving this equation in respect to Q, and reducing, we 
 obtain. 
 
 +"* cos. 
 
 This expression may be placed under the form 
 Q=(P cos. a+pah+w) sec. (3 
 Pjfrcos. a A sin, a + (&+-|#) cos, a 
 
 If the numerator of the fraction in the second member of 
 this equation be a positive quantity (as in all practical cases 
 it will probably be found to be) the value of Q manifestly 
 diminishes with that of m. Now the least value of m, con- 
 sistent with the stability of the wall, is zero, since the line 
 of resistance no where intersects the extrados; the least 
 value of Q (the shore being supposed necessary to the sup- 
 port of the wall) corresponds, therefore, to the value zero of 
 m ; moreover this least value of the thrust upon the shore 
 consistent with the stability of the wall is manifestly that 
 which it sustains when the wall simply rests upon it, the 
 
 * The weight 2w of the shore may be conceived to be divided into two equal 
 parts and collected at its extremities. 
 
 } The expression (b-\-m) cos. ft may be placed under the form b cot ft sin. 
 3-if-m cos. 0=c sin. ft-\-m cos. 3, where c represents the height CE of the point 
 against which the prop rests. 
 
WALL SUPPORTED BY SHORES. 
 
 389 
 
 shore not being driven so as to increase the thrust sustained 
 by it Beyond that just necessary to support the wall.* 
 This least thrust is represented by the formula 
 
 rt __FjAsm.q 
 
 " 
 
 cos. a 
 
 The thrust which must be given to the prop in order that 
 there may be given to the wall any required stability, deter- 
 mined by the arbitrary constant m, is determined by equa- 
 tion (381). The stability will diminish as the value of m is 
 increased beyond <z, and the wall will be overthrown 
 inwards when it exceeds a. 
 
 296. The stability of a wall sustained lyy more than one 
 shore in the same plane. 
 
 Let EF, ef be shores in the same plane, sustaining the 
 wall ABCD, and both necessary to 
 its stability; so that if EF were re- 
 moved, the wall would turn over upon 
 /", and if ef were removed, upon some 
 point between F and C. 
 
 If the thrust of the shore EF be 
 only that just necessary to sustain 
 the tendency of the wall to overturn 
 upon /, it is evident that the line of 
 resistance must pass through that 
 point ; but if the thrust exceed that 
 just necessary to the equilibrium, or 
 if the shore be driven then the line 
 of resistance will intersect fg in some 
 points. ~Letfx=m; then represent- 
 ing the thrust upon EF by Q, the dis- 
 tances f D and fi by h and &, and the angle EFC by /3, the 
 value of Q is evidently determined by equation (381). 
 
 If z be taken in like manner to represent the point where 
 the line of resistance intersects the base of the wall, and 
 Cz=m l , CE b, ; Ce=b Cfe^P^ GD= A,, the thrust upon 
 the prop ef by Q t and its weight by 2w l ; then the sum of 
 the moments about the point z of Q and Q 1? and the weight 
 
 * This case presents an application of the principle of least resistance. 
 Theory of the Arch.) 
 
 (See 
 
390 
 
 THE STABILITY OF A GOTHIC STRUCTURE. 
 
 J of the wall, equals the sum of the moments of P, w, 
 and i0j ; or 
 
 QX&i+ff^) cos. ft + Q (#1+^1) cos. (3+pahi (%am^)= 
 P{A! sin. a (&-fia m^ cos. a}' + (^0+^) m, ...... (382.) 
 
 Substituting the value of Q in this equation, from equation 
 (381), and solving in respect to Q,, the thrust upon the prop 
 cf will be determined, so that the stability of the wall, upon 
 its section fg and upon its base CB, may be m and m t 
 respectively. 
 
 If m 1 =m t the portions of the wall above and below fg 
 are equally stable. 
 
 If m 1 =m=Q, the thrust upon each shore is only that 
 which is just necessary to support the wall, or which is pro- 
 duced by its actual tendency to overturn. In this case we 
 have 
 
 (P sin, a-^a 3 ) foft- 
 
 ft) 
 
 cos, a 
 
 ~ bb t cos. ft 
 
 the value of k being determined by equation (380). 
 
 Let 
 
 297. The stability of a structure having parallel walls, one 
 of which is supported l)y means of struts resting on the 
 summit of the other. 
 
 AB and CD be taken to represent the walls, and EF 
 one of the struts ; the thrust Q upon the 
 strut may be determined precisely as in 
 Art 295. So that the line of resistance 
 may intersect the base of the wall AB at 
 a given distance m from the extrados 
 (see note, p. 388.) 
 
 Let m, represent the distance Dx from 
 the extrados at which the line of resist- 
 ance intersects the base of the wall CD ; 
 then taking the moments of the pressures 
 applied to the wall CD about the point 
 a?, as in Art. 295, and observing that 
 besides the pressure Q the weight w of 
 one half the strut is applied at E, we 
 have 
 
 X sin. 
 
THE STABILITY OF A GOTHIC STKUOTITRE. 391 
 
 in which equation h^ and a t are taken to represent the 
 height and thickness of the wall CD, \ the distance of the 
 point E on which the strut rests from the axis of the wall, (3 
 the inclination of the strut to the vertical, and ^ the weight 
 of a cubic foot of the material of the wall. 
 
 Substituting for Q its value from equation (381), and 
 reducing, 
 
 P{Asin. (k+%a) cos. aj ^c?h+m(F cos, a -f 
 c sin. ft+m cos. ft 
 
 sin. ft 
 
 /OOON 
 
 By this equation is determined that relation between the 
 dimensions of the two walls and the amount of the insistent 
 pressure P, by which any required stability may be assigned 
 to each wall of the structure. If m=0, the pressure upon 
 the strut will be that only which is produced by the ten- 
 dency of AB to overturn ; and the value of m^ determined 
 from the above equation will give the stability of the exter- 
 nal wall on this supposition. 
 
 * If m=0 and m 1 =0, both walls will be upon the point of 
 overturning, and the above equation will express that rela- 
 tion between the dimensions of the wall and the amount of 
 the insistent pressure, which corresponds to the state of the 
 instability of the structure. 
 
 The conditions of the stability, when the wall AB is sup- 
 ported by two struts resting upon the summit of the wall 
 CD, may be determined by a method similar to the above 
 (see Art. 296). 
 
 The general conditions of the stability of the structure 
 discussed in this article evidently include those of a GOTHIC 
 BUILDING having a central nave, whose walls are supported, 
 under the thrust of its roof, by the rafters of the roof of its 
 side aisles. By a reference to the principles of the preceding 
 article, the discussion may readily be made to include the 
 case in which a further support is given to the walls of tho 
 nave by flying buttresses, which spring from the summits of 
 the walls of the aisles. The influence of the buttresses 
 which support the walls of the aisles upon the conditions of 
 the stabiiity of the structure forms the subject of a subse- 
 quent article. 
 
392 
 
 THE WALL OF A DWELLING. 
 
 298. The stability of a wall sustaining the floors of a 
 dwelling. 
 
 The 
 
 joists of the floors of a dwelling-house rest at their 
 extremities upon, and are sometimes 
 notched into, pieces of timber called 
 wall-plates, which are imbedded in 
 the masonry of the wall. They 
 serve thus to bind the opposite sides 
 of the house together ; and it is upon 
 the support which the thin walls of 
 modern houses receive from these 
 joists, that their stability is some- 
 times made to depend.* 
 
 Representing 
 that portion oi 
 
 rests upon the portion ABCD of the 
 wall, and the distance BE by <?, 
 taking a?, as before, to represent the 
 point where the line of resistance 
 intersects the base of the wall, and 
 measuring the moments from thig 
 
 by w the weight of 
 the flooring which 
 
 point, we have 
 
 whence, taking the same notation as in the preceding arti- 
 cles, and substituting, 
 
 <)Q + (%am)pah+(am)w= { Asm. a (k+^a m) cos. a} P ; 
 ,VQc = \h sin. a (k+$a) cos. a} P J^ 2 A 
 m (P cos. a+iiah+w) ..... (384) ; 
 
 from which expression it appears that Q is less as m is less. 
 "When, therefore, the strain upon the joints is that only 
 which is just necessary to preserve the stability of the wall, 
 or which it produces by its tendency to overturn, then 
 m=0. In this case, therefore, 
 
 * A house thus constructed evidently becomes unsafe when its wall-plates or 
 the extremities of its joists begin to decay. 
 
A WALL SUPPORTED BY BUTTRESSES. 393 
 
 \h Sill, a (&-J--J-&) COS. a} P \\L(ji?h~W(L /oosrs 
 
 2= - G ~ (doo). 
 
 If 13 be assumed a right angle, and if (am)w be substi- 
 tuted for mw, the case discussed in Art. 295. will 
 evidently pass into that which is the subject of 
 the present article, and the preceding equation 
 may thus be deduced from equation (381) (see 
 note, p. 388.). 
 
 In like manner, if the wall sustain the pres- 
 sure of two floors, and A be taken to represent 
 the distance from its summit to the lower floor, 
 and AJ its whole height ; then, representing by m 
 arid ra x the distances from the extrados at which 
 the line of resistance intersects the sections EG 
 and eg, and substituting (w + w^) (a m^ for 
 (w +w 1 )m 1 , the value of the strain Q on the 
 joists of the lower floor may be determined by 
 equation (382), it being observed that for the 
 coefficient of Q t in that equation must be substi- 
 tuted (as was shown above) the height (h l h] of 
 the lower floor from the bottom of the wall. If the strain 
 be only that produced by the tendency of the wall 
 turn at g and C, then 
 
 W Cd 
 
 . 0,+wa 
 
 P sin. a 
 
 The value of Q is determined by equation (385), G being 
 taken to represent the distance E0 between the floors. If 
 the joists be not notched into the wall-plates, the friction of 
 their extremities upon them, produced per foot of the length 
 by the weight which they support, must at least equal Q and 
 Qi respectively. 
 
 299. The stability of a wall supported by piers or buttresses 
 of uniform thickness. 
 
 Let the piers be imagined to extend along the whole 
 
394: WALL SUPPORTED BY BUTTRESSES. 
 
 length of the wall, as explained in Art. 288. ; 
 and let ABCD represent a section of the com- 
 pound wall thus produced. Let the weight of 
 each cubic foot of the material of the portion 
 ABFE be represented by f* 1? and that of each 
 cubic foot of GFCD by f* f , EA= 1 , GD = a 
 BC=a, AB = A 15 CD = A 2 , distance from CD, 
 produced, of the point where P intersects 
 AE=Z, x the intersection of the line of resist- 
 ance with CB, Cx= m. By the principle of the 
 equality of moments, the moment of P about 
 the point x is equal to the sum of the moments 
 of the weights of GO and AF about that point. 
 But (Art. 295.) moment of P=P \h, sin. a 
 (lm) cos. aj ; also moment of weight of AF 
 (a z m +-J# 1 )A 1 & 1 M- 1 ; moment of weight of GC= 
 
 P {A t sin. a (lm) cos. *} =(, 
 
 ..... (387) 
 
 If the material of the pier be the same with that of the 
 wall ; then, taking b to represent the breadth of each pier, 
 and c the common distance of the piers from centre to 
 centre (Art. 288.), ca^=ba^ therefore c^=b^ l . Repre- 
 
 si 
 
 senting =- by n, eliminating the value of ^ between this 
 equation and equation (387), writing ^ for i^, and reducing, 
 ^ sin. a I cos. )=if* (afh^ + 20,0^ + - 2 2 A 3 1 
 
 \ 7l> I 
 
 ra-JPcos. a+M< \a l h l +-aji t ] \. . . .(388); 
 / \ n V \ 
 
 by which equation a relation is determined between the 
 dimensions of a wall supported by piers, having a given 
 stability m, and its insistent pressure P. Solving it in 
 respect to & 2 , the thickness of the pier necessary to give any 
 required stability to the wall will be determined. (See 
 APPENDIX.) 
 
 If # 2 be assumed to represent that width of the pier by 
 which the wall would just be made to sustain the given 
 pressure P without being overthrown; then taking ra=0, 
 and solving in respect to # a , 
 
WALL SUPPORTED BY BUTTRESSES. 395 
 
 sin. -Z cos. )+ ( -1)) *.... (389). 
 A a \ A a / 
 
 300. .7^0 stability of a pier or buttress swr* 
 mounted by a pinnacle. 
 
 Let "W represent the weight of the pinnacle, 
 and e the distance of a vertical through its cen- 
 tre of gravity from the edge of the pier : then 
 assuming x to be the point where the line of 
 resistance intersects the base of the pier, and tak- 
 ing the same notation as before, equation (387) 
 will evidently become 
 
 P {k, sin. (lm) cos. *} = \a % 
 
 + (em)W. 
 
 Substituting for /x a its value-^ or , writing p for f^, and 
 reducing, 
 
 P(A! sin. a I cos. *)=if*|a 1 > A 1 -f2fl 1 o f A 1 -ht-a l 'A t j + 
 
 If a 9 represent the thickness of that pier by which the wall 
 will just be sustained under the pressure, taking m=0, and 
 
 solving in respect to a a , a^=na l -^- + 
 
 fP(A, Bin. .-I cos. .)_-W| + '-l ,' (391). 
 
396 WATT. SUPPORTED BY GOTHIC BUTTRESSES. 
 
 THE GOTHIC BUTTRESS. 
 
 301. In Gothic buildings the thickness of 
 a buttress is not unfrequently made to vary 
 at two or three different heights above its 
 base. Such buttress is represented in the 
 accompanying figure. 
 
 The conditions by which any required sta- 
 bility may be assigned to that portion of it 
 whose base is ~be may evidently be determined 
 by equation (390). To determine the condi- 
 tions of the stability of the whole buttress 
 upon CD, let the heights of the points Q, a, 
 and b above CD be represented by A,, A 2 and 
 7* 3 ; let DE=^, DY=a FC=a 3 , Cx=m 1 ; 
 then adopting, in other respects, the same 
 notations as in Arts. 299 and 300. Since the 
 distances from x of the verticals through the 
 centres of gravity of those portions of the 
 buttress whose bases are DE, DF, and FC 
 respectively, are (a^ + a^^a.m^ (0,-fi0 a 
 _ m ^ and ( : ka s 'm 1 ) we have, by the equality 
 of moments, 
 
 P {AJ sin. a (I m,) cos. a} =(^ + ^4-^ m,) AAM- + 
 
 (^-m 1 ) (392). 
 
 This equation establishes a relation between the dimen- 
 sions of the buttress and its stability, by which any one of 
 those dimensions which enter into it may be so determined 
 as to give to m t any required value, and to the structure any 
 required degree of stability. (See APPENDIX.) 
 
 It is evident that, with a view to the greatest economy of 
 the material consistent with the given stability of the but- 
 tress, the stability of the portion which rests upon the base 
 be should equal that of the whole buttress upon CE ; the 
 value of m^ in the preceding equation should therefore equal 
 that of m in equation (390) If m be eliminated between 
 these two equations, it being observed that h l and A 2 in equa> 
 tion (390) are represented by A, A 3 and A 2 A 3 in equation 
 (392), a relation will be established between a 1? # a 3 A 1? A,, 
 A 8 , which relation is necessary to the greatest economy of 
 
THE STABILITY OF WALLS SUSTAINING KOOFS. 397 
 
 material ; and therefore to the greatest stability of the struc- 
 ture with a given quantity of material. 
 
 THE STABILITY OF WALLS SUSTAINING ROOFS. 
 
 302. Thrust upon the feet of the rafters of a roof, the tie- 
 beam not being suspended from the, ridge. 
 
 If f^ be taken to represent the weight of each square foot 
 of the roofing, 2L the span, i the 
 inclination BAG of the rafters to 
 the horizon, q the distance between 
 each two principal rafters, and a 
 
 h 1 the inclination to the vertical of 
 the resultant pressure P on the 
 
 foot of each rafter ; then will L sec. i represent the length of 
 each rafter, and f^L^ sec. i the weight of roofing borne by 
 each rafter. Let the weights thus borne by each of the 
 rafters AB and BC be imagined to be collected in two equal 
 weights at its extremities ; the conditions of the equilibrium 
 will remain unchanged, and there will be collected at B the 
 weight supported by one rafter and represented by f^L^ 
 sec. t, and at A and C weights, each of which is represented 
 by -J^L^ sec. i. Now, if Q be taken to represent the thrust 
 produced in the direction of the length of either of the 
 rafters AB and BC, then (Art. 13.) ^Lq sec. * = 2Q cos. 
 |ABC: but ABC^tf 2t; therefore cos. JABC = sin. *; 
 therefore 2Q sin. i=p 1 ~Lo i sec. i ; 
 
 . o- T . 8ec '* M^Lg f^Lg 
 
 ' ""^V^! sin. L ~~ 2 sin. i cos. i ~~ sin. 2*' 
 
 The pressures applied to the foot A of the rafter are the 
 thrust Q and the weight -J^L^ sec. i ; and the required pres- 
 sure P is the resultant of these two pressures. Resolving Q 
 vertically and horizontally, we obtain Q sin. i and Q cos. , 
 or ^^Lq sec. L and j^L^ cosec. i. The whole pressure applied 
 vertically at A is therefore represented by p^Lq sec. t, and 
 the whole horizontal pressure by ^^Lq cosec. i ; whence it 
 follows (Art. 11.) that 
 
 P = V' sec. 1 *+Jf* I 'L cosec. '<= 
 
 i t/l+icot. 2 * ..... (393). 
 
398 RAFTERS OF A ROOF. 
 
 (394) " 
 
 If the inclination i of the roof be made to vary, the span 
 remaining the same, P will attain a minimum value when 
 
 tan. i = -, or when 
 
 4/2 
 
 =35 16' ..... (395). 
 
 It is therefore at this inclination of the roof of a given 
 span, whose trusses are of the simple form shown in the 
 figure, that the least pressure will be produced upon the feet 
 of the rafters. If <p represent the limiting angle of resistance 
 between the feet of the rafters and the surface of the tie, the 
 feet of the rafters would not slip even if there were no mor- 
 tice or notch, provided that a were not greater than <p (Art. 
 141.), or \ cot. i not greater than tan. 9, or 
 
 i not less than cot.- 1 (2 tan. ?>)* ..... (396). 
 
 303. The thrust upon the feet of the rafters of a roof in 
 which the tie-beam is suspended from the ridge l>y a 
 king-post. 
 
 It will be shown in a subsequent portion of this work 
 (see equation 558) that, in this case, 
 the strain upon the king-post BD is 
 equal to fths of the weight of the 
 tie-beam with its load. Represent- 
 ing, therefore, the weight of each 
 foot in the length of the tie-beam 
 by f* a , and proceeding exactly as in 
 the last article, we shall obtain for the pressure P upon the 
 feet of the rafters, and its inclination to the vertical, the 
 expressions 
 
 + 5 M' 2 ) 2 cot. 2 ^^ ---- (397). 
 
 ' . .(398). 
 
 - 
 
 * If the surfaces of contact be oak, and thin slips of oak plank be fixed 
 under the feet of the rafters, so that the surfaces of contact may present par- 
 allel fibres of the wood to one another (by which arrangement the friction will 
 be greatly increased), tan. <j>=-48 (see p. 133.); whence it follows that the 
 rafters will not slip, provided that their inclination exceed cot." 1 -96, or 
 46 10'. 
 
WALL SUSTAINING THE THRUST OF A KOOF. 
 
 399 
 
 304. The stability of a wall sustaining the thrust of a roof, 
 having no tie-beam. 
 
 Let 
 
 it be observed, that in the equation to the line of 
 resistance of a wall (equation 
 377), the terms P sin. a and P 
 cos. a represent the horizontal 
 and vertical pressures on each 
 foot of the length of the summit 
 of the wall ; arid that the former 
 of these pressures is represented 
 in the case of a roof (Art. 302.) 
 by i^L cosec. , and the latter 
 by M-jL sec. i ; whence, substitu- 
 ting these values in equation 
 (377), we obtain for the equation 
 to the line of resistance in a wall 
 sustaining the pressure of a roof, 
 without a tie-beam 
 
 
 I I 
 
 H*!/ 
 
 _ 
 
 in which expression a represents the thickness of the wall, 
 k the distance of the feet of the rafters from the centre of 
 the summit of the wall, L the span of the roof, ^ the weight 
 of a cubic foot of the wall, and /*, the weight of each square 
 foot of the roofing. The thickness a of the wall, so that, 
 being of a given height A, it may sustain the thrust of a 
 roof of given dimensions with any given degree of stability, 
 may be determined precisely, as in Art. 293, by substituting 
 h for x in the above equation, and \a m for y, and solving 
 the resulting quadratic equation in respect to a. 
 
 If, on the other hand, it be required to determine what 
 must be the inclination i of the rafters of the roof, so that 
 being of a given span L it may be supported with a given 
 degree of stability by w^alls of a given height h and thick- 
 ness a/ then the same substitutions being made as before, 
 the resulting equation must be solved in respect to i instead 
 of a. 
 
 The value of a admits of a minimum in respect to the 
 variable i. The value of t, which determines such a mini- 
 mum value of a, is that inclination of the rafters which is 
 
4:00 STABILITY OF A WALL. 
 
 consistent with the greatest economy in the material of the 
 wall, its stability being given. 
 
 305. The stability of a wall supported ~by buttresses, and 
 sustaining the pressure of a roof without a tie-beam. 
 
 The conditions of the stability of such a wall, when sup- 
 ported by buttresses of uniform thickness, will evidently be 
 determined, if in equation (388) we substitute for P cos. a 
 and P sin. a their values f^L sec. i and -J^L consec. i ; we 
 shall thus obtain 
 
 cosec. i I sec. *)=Jf* (0 1 i A 1 +2a l o 8 M a?h^m 
 
 n 
 
 fsL sec. I+P (#A + - <*A f (400). 
 
 From which equation the thickness a z of the buttresses 
 necessary to give any required stability m to the wall may 
 b# determined. 
 
 If the thickness of the buttresses be different at different 
 heights, and they be surmounted by pinnacles, the con- 
 ditions of the stability are similarly determined by substi- 
 tuting for P sin. a and P cos. a the same values in equations 
 (390) and (392). 
 
 To determine the conditions of the stability of a Gothic 
 building, whose nave, having a roof without a tie-beam, is 
 supported by the rafters of its two aisles, or by flying but- 
 tresses, which rest upon the summits of the walls of its 
 aisles, a similar substitution must be made in equation (383). 
 
 If the walls of the aisles be supported by buttresses, 
 equation (383) must be replaced by a similar relation 
 obtained by the methods laid down in Arts. 299 and 301 ; 
 the same substitution for P sin. a and P cos. a must then be 
 made. 
 
 306. The conditions of the stability of a wall supporting a 
 
 shed roof. 
 
 Let AB represent one of the rafters of such a roof, one ex- 
 
STABILITY OF A WALL. 
 
 401 
 
 tremity A resting against the face of 
 the wall of a building contiguous to 
 the shed, and the other B upon the 
 r summit of the wall of the shed. 
 
 It is evident that when the wall 
 BH is upon the point of being over- 
 thrown, the extremity A will be upon 
 the point of slipping on the face of 
 the wall DC ; so that in this state of 
 the stability of the wall BH, the direc- 
 tion of the resistance K of the wall 
 DC on the extremity A of the rafter, 
 will be inclined to the perpendicular AE to its surface at.ani 
 angle equal to the limiting angle of resistance. Moreover;, 
 this direction of the resistance R which corresponds to the- 
 state bordering upon motion is common to every other state ; 
 for by the principle of least resistance (see Theory of the 
 Arch) of all the pressures which might be supplied by the 
 resistance of the wall so as to support the extremity of the 
 rafter, its actual resistance is the least.. Now this least re- 
 sistance is evidently that whose direction is most nearly ver- 
 tical ; for the pressure upon the rafter is wholly a vertical 
 pressure. But the surface of the wall supplies no resistance 
 whose direction is inclined farther from the horizontal line 
 AE than AR ; AR is therefore the direction of the resist- 
 ance. 
 
 Resolving R vertically and horizontally, it becomes R sin. 
 9 and R cos. 9. Representing the span BF by L v the incli- 
 nation ABE by i, the distance of the rafters by , and the 
 weight of each square foot of roofing by f\ (Art. 10.), R sin. 
 9 + P cos. a,=t*< l Lq sec. i and Reos. 9 P sin. a=0 ; also the 
 perpendiculars let fall from A on P and upon the vertical 
 through the centre of AB, are represented by 
 
 L cos. (a, + i) sec. i and -JL ; therefore (Art. 7). 
 PL cos. (a + *)sec. i=^L . L^ q sec. t, and hence 
 P cos. (a + i)=^'L^ l q. Eliminating between these equa- 
 tions, we obtain 
 
 cot. a=tan. 9 + 2 tan. t 
 
 (401); 
 
 sin.(9 + 0' 
 
 cos. t( tan. 9 + tan. i) 
 26 
 
402 
 
 THE PLATE BANDE. 
 
 If the rafter, instead of resting at A 
 against the face of the wall, be received 
 into an aperture, as shown in the figure, 
 so that the resistance of the wall may be 
 applied upon its inferior suface instead of 
 at its extremity: then drawing AE per- 
 pendicular to the surface of the rafter, 
 the direction AR of the resistance is evi- 
 dently inclined to that line at the given 
 limiting angle 9. Its inclination to the hori- 
 zon is therefore represented by -^ 
 
 Substituting this angle for 9 in equations (401) and (402), 
 
 cot. a=cot. (* 9) + 2 tan. i (403). 
 
 ,2 sec. i 
 
 cos. (i 
 
 . (i 9)tan. i ' 
 
 cos.t{cot.(t-9)+tan.i 
 
 . 
 
 Substituting in equations (3 77) and (379) for Psin. a, P cos. a, 
 their values determined above, all the conditions of the sta- 
 bility of a wall supporting such a roof will be determined. 
 
 307. THE PLATE BANDE OR STRAIGHT ARCH. 
 
 Let MN represent any joint of 
 the plate bande ABCD, whose 
 points of support are A and B ; 
 PA the direction of the resistance 
 at A, WQ a vertical through the 
 centre of gravity of AMND, TR 
 the direction of the resultant pres- 
 sure upon M.'N ; the directions of 
 
 TR, WQ, and PA intersect, therefore, in the same point O. 
 Let OAD= a , AM=aj, MR=y, AD=H, AB=2L, weight 
 
 of cubic foot of material of apca==^,, Draw ~Rm a perpen- 
 
 dicular upon PA produced; then by the principle of the 
 
 equality of moments, 
 
 Em . P=MQ . (weight of DM). 
 
THE PLATE BANDE. 403 
 
 But Rm = x cos. a y sin. a, MQ = -Ja?, weight of DM = 
 Hf^ ; also resolving P vertically, 
 
 PcoB.*=LHf* I (405). 
 
 Whence we obtain, by substitution in the preceding equa- 
 tion, and reduction, 
 
 L(x y tan. a) =' (406), 
 
 which is the equation to the line of resistance, showing it to 
 be a parabola. If, in this equation, L be substituted for a?, 
 and the corresponding value of y be represented by Y, there 
 will be obtained the equation Y tan. a = |L, whence it 
 appears that a is less as Y is greater ; but by equation (405), 
 P is less as a is less. P, therefore, is less as Y is greater ; 
 but Y can never exceed H, since the line of resistance can- 
 not intersect the extrados. The least value of P, consistent 
 with the stability of the plate bande, is therefore that by 
 which Y is made equal to H, and the line of resistance 
 made to touch the upper surface of the plate bande in F. 
 
 Now this least value of P is, by the principle of least 
 resistance (see Theory of the Arch\ the actual value of the 
 resistance at A, 
 
 /.tan.a^ijj (40Y). 
 
 Eliminating a between equations (405) and (407), 
 
 (408). 
 
 Multiplying equations (405) and (407) together, 
 P sin. a^-JLX (409). 
 
 Now P sin. a represents the horizontal thrust on the point 
 of support A. From this equation it appears, therefore, that 
 the horizontal thrust upon the abutments of a straight arch 
 is wholly independent of the depth H of the arch, and that 
 it varies as the square of the length L of the arch ; so that 
 the stability of the abutments of such an arch is not at all 
 diminished, but, on the contrary, increased, by increasing 
 the depth of the arch. This increase of the stability of the 
 abutment being the necessary result of an increase of the 
 vertical pressure on the points of support, accompanied by 
 no increase of the horizontal thrust upon them. 
 
404 
 
 THE PLATE BANDE. 
 
 308. The loaded plate lande. 
 
 It is evident that the effect of a loading, distributed 
 uniformly over the extrados of the 
 plate bande, upon its stability, is in 
 every respect the same as would be 
 produced if the load were removed, 
 and the weight of the material of 
 the bande increased so as to leave 
 the entire weight of the structure 
 unchanged. Let ^ 3 represent the 
 weight of each cubic foot when thus 
 increased, M- 2 the weight of each 
 cubic foot of the load, and H x the height of the load ; then 
 
 H 
 
 (410). 
 
 The conditions of the stability of the loaded plate bande 
 are determined by the substitution of this value of ^ for ^ 1 
 in the preceding article. 
 
 309. Conditions necessary that the voussoirs of a plate lande 
 may not slip upon one another. 
 
 It is evident that the inclination of every other resultant 
 pressure to the perpendicular to the surface of its corres- 
 ponding joint, is less than the inclination of the resultant 
 
 pressure or resistance P, to the 
 perpendicular to the joint AD. 
 If, therefore, the inclination be 
 not greater than this limiting an- 
 gle of resistance, then will every 
 other corresponding inclination 
 be less than it, and no voussoir 
 will therefore slip upon the sur- 
 face of its adjacent voussoir. Now the tangent of the incli- 
 nation P to the perpendicular to AD is represented by cot. a 
 
 2H 
 
 or by -j- (equation 40 7) ; the required condition is therefore 
 
 determined by the inequality, 
 
 ?5<tan.9 (411). 
 
THE SLOPING BUTTRESS. 405 
 
 It is evident that the liability of the arch to failure by the 
 slipping of its voussoirs, is less as its depth is less as com- 
 pared to its length. In order the more effectually to pro- 
 tect the arch against it, the voussoirs are sometimes cut of 
 the forms shown by the dotted lines in the preceding figure, 
 their joints converging to a point. The pressures upon the 
 points A and B are dependent upon the form of that portion 
 of the arch which lies between those points, and indepen- 
 dent of the forms of the voussoirs which compose it ; these 
 pressures, and the conditions of the equilibrium of the piers 
 which support the arch, remain therefore unchanged by this 
 change in the forms of the voussoirs. 
 
 310. To determine the conditions of the equilibrium of 
 the upright piers or columns of masonry which form the 
 abutments of a straight arch, supposing them to be termi- 
 nated, as shown in the figure, on a different level from the 
 extrados CD of the arch, let b be taken t*o represent the 
 elevation of the top of the pier above the point A ; then will 
 
 Z> tan. a, or |^j (equation 407), represent the distance AG 
 
 (p. 383), or the value of Ic \a). Substituting for k in equa- 
 tion (377) and also the values of P sin. a, P cos. a, from 
 equations (409) and (405), we have 
 
 (412); 
 
 which is the equation to the line of resistance of the pier, a 
 representing its thickness, 5 the height of its summit above 
 the springing A of the arch, L the length of the arch, f* the 
 weight of a cubic foot of the material of the arch or abut- 
 ment (supposed the same). 
 
 The conditions of the stability may be determined from 
 this equation as in the preceding articles. If the arch be 
 uniformly loaded, the value of ^ 3 given by equation (410) 
 must be substituted for f. 
 
 311. THE CENTRE OF GRAVITY OF A BUTTRESS WHOSE FACES 
 ARE INCLINED AT ANY ANGLE TO THE VERTICAL. 
 
 Let the width AB of the buttress at its summit be repre- 
 
406 
 
 THE SLOPING BUTTRESS. 
 
 sented by #, its width CD at the base by lj 
 its vertical height AF by <?, the inclination 
 of its outer face or extrados BC to the 
 vertical by a l9 that of its intrados AD by 
 a a . 
 
 Let II represent the centre of gravity of 
 the parallelogram ADEB, and K that of 
 the triangle BCE, and G that of the but- 
 tress ; draw HM, GL, ICN", perpendiculars 
 upon AF. Then representing GL by \ 
 and observing that the area ADEB is represented by 
 the area EBC by %(b a}c, and the area ADCB by f 
 
 Now HM= 
 
 n. a 2 ) ; 
 Substituting these values and reducing, 
 
 a+b 
 
 tan. a^^a + c tan. a a ), 
 tan. a,= 
 
 J=CD=CF DF=c tan. a^a c tan. a 2 ; also (a 3 + ab + 6 2 ) 
 
 (ba)* + Sab = <? 2 (tan.a, tan.a 2 ) 2 + 3ac(tan.a 1 tan.a 2 ) + 3^ 2 , 
 
 (a + 25)<?tan. a a = |2c(tan. a, 1 tan.a 2 ) + 3( c tan. a a 
 
 c tan. a a 
 =2c 2 (tan. OL I tan. a 2 ) tan. a 2 + 3ac tan. a 2 ; 
 
 + 2) e tan. a 2 =c 2 (tan.*^ tan. 3 a 2 ) 
 + 3ac tan. aj + 3^ 2 . 
 ' 8a i~tan. 2 a 2 ) + ac tan. a x 
 
 a! tan. a 2 ) 
 
 . . . (414). 
 
 312. THE LINE OF RESISTANCE IN A BUTTRESS. 
 
 Let LM represent any horizontal section of the buttress, 
 TK a vertical line through the centre of 
 .Ji gravity of that portion AMLB of the but- 
 tress which rests upon this section. Pro- 
 duce LM to meet the vertical AE in V, 
 and let KY=^ and AV=a? ; then is the 
 value of X determined by substituting x 
 for c in equation (414). Let PO be the 
 direction in which a single pressure P is 
 applied to overturn the buttress. Take 
 
 * This equation is, of course, to be adapted to the case in which the inclina- 
 tion of AD is on the other side of the vertical, as shown by the dotted line 
 Ac? by making 2 , and therefore tan. a 2 negative. 
 
THE SLOPING BUTTRESS. 407 
 
 OS to represent P in magnitude and direction, and ON to 
 represent the weight of the portion AMLB of the buttress ; 
 complete the parallelogram SN, and produce its diagonal 
 OR to Q ; then will OR evidently be the direction 01 the 
 resultant pressure upon AMLB 5 and Q a point in the line of 
 resistance. 
 
 Let YQr=y, AG=&, /GOT i, (*:= weight of each cubic 
 foot of material ; and let the same notation be adopted in 
 other respects as in the last article. By similar triangles, 
 
 QK_RI 
 OK" 01 
 
 OK=TK TO=TK TG cot. GOT=ra; (X+&) cot. i,. 
 HI EN sin. ROT=P sin. i, 
 
 OI=OISr+OT=iaAV(AB+LM)-f RN cos. RNI= 
 Jf*aj{2a4-iB (tan. c^ tan. a 2 )J +P cos. ; 
 
 y X P sin, t 
 
 "x (X + #) cot. i ^px{%a-}-x(t&n. a x tan. a a )}-f-P cos. i 
 
 Transposing and reducing, 
 
 _jXfxa? \%a+x (tan, otj tan. a 2 )^ +P (a? sin, i 7c cos. 1 1 
 J^{2-fa?(tan. , tan. a. 2 )\ +P cos. t ~' 
 
 but substituting x for <? in equation (414), and multiplying 
 both sides of that equation by the denominator of the frac- 
 tion in the second member, and by the factor 4 a # 5 we have 
 
 (tan. a, tan. a 2 )} =^x 3 (tan.Vj tan. 2 
 tan. 
 
 n.^! tan.'q^-j-^atan. x -f-//a;a 2 4-2P(a; sin, f ^ cos, i) 
 Hx I2a4-actan.d-tan.aa)! -f-2P cos. i 
 
 which is the equation to the line of resistance in a buttress. 
 If the in trades AD be vertical, tan. a 2 is to be assumed =0. 
 If AD be inclined on the opposite side of the vertical to that 
 shown in the figure, tan. 2 is to be taken negatively. The 
 line of resistance being of three dimensions in x, it follows 
 that, for certain values of y, there are three possible values 
 of x ; the curve has therefore a point of contrary flexure. 
 The conditions of the equilibrium of the buttress are deter- 
 
408 WALL SUSTAINING THE 
 
 mined from its Hne of resistance precisely as those of the 
 wall. 
 
 Thus the thickness a of the buttress at its summit being 
 given, and its height c, and it being observed that the dis- 
 tance CE is represented by a+c tan. a 1? the inclination o^ of 
 its extrados to the vertical may be determined, so that its 
 line of resistance may intersect its foundation at a given dis- 
 tance m from its extrados, by solving equation (415) in re- 
 spect to tan. 1? having first substituted c for x and a + c tan. 
 ^m for y\ and any other of the elements determining the 
 conditions of the stability of the buttress may in like manner 
 be determined by solving the equation (the same substitu- 
 tions being made in it) in respect to that element. 
 
 313. A WALL OF UNIFORM THICKNESS SUSTAINING THE FEES- 
 SURE OF A FLUID. 
 
 If E be taken to represent the surface of the fluid, IK any 
 section of the wall, and EP two thirds 
 Isss^ ~=r= t ^ le d e P tn EK ; then will P be the cen- 
 tre of pressure* of EK, the tendency 
 of the fluid to overturn the portion 
 AKIB of the wall being the same as 
 would be produced by a single pressure 
 applied perpendicular to its surface at 
 P, and being equal in amount to the 
 weight of a mass of water whose base 
 is equal to EK, and its height to the 
 depth of the centre of gravity of EK, or 
 
 to ^EK. Let AK=#, AE=e, weight of each cubic foot of 
 
 the fluid f* 
 
 Let the direction of P intersect the axis of the wall in O ; 
 let it be represented in magnitude by OS ; take ON to 
 represent the weight of the portion AKIB of the wall ; com- 
 plete the parallelogram SIN", and produce its diagonal to 
 meet IK in Q ; then will Q be a point in the line of resist- 
 ance. Let QM=y, AB=#, weight of each cubic foot of 
 
 material of wall=fA. By similar triangles, r = :. Now 
 
 * Treatise on " Hydrostatics and Hydrodynamics," by the author of this 
 work, Art. 38. p. 26. 
 
PRESSURE OF A FLUID. 4()9 
 
 e)', NO = weight 
 
 Dividing numerator and denominator of this equation by 
 
 M, 
 
 jx l5 and observing that the fraction represents the ratio o 
 
 *i 
 
 of the specific gravities of the material of the wall and the 
 fluid, we have 
 
 which is the equation to the line of resistance in a wall of 
 uniform thickness, sustaining the pressure of a fluid. 
 
 314. To determine the thickness, a, of the wall, so that its 
 height, h, being given, the line of resistance may intersect 
 its foundation at a given distance, m, within the extrados. 
 
 Substituting, in equation (416), h for a?, and \a m for y, 
 and solving the resulting equation in respect to a, we obtain 
 
 |/~ i ^~ 6 ) 3 
 
 Equation (416) may be put under the form y= 
 
 1 / e\ 3 
 
 ~ a? 2 1 1 -1 ; whence it is apparent that y increases con- 
 tinually with x ; so that the nearest approach is made by 
 the line of resistance, to the extrados of the pier, at its 
 lowest section, m therefore represents, in the above expres- 
 sion, the modulus of stability (Art. 286). 
 
 315. The conditions necessary that the wall should not be 
 overthrown by the slipping of the courses of stones on one 
 another. 
 
 The angle SEO represents the inclination of the resultant 
 pressure upon the section IK to the perpendicular ; the pro- 
 posed condition is therefore satisfied, so long as SRO is less 
 than the limiting angle of resistance 9. 
 
410 
 
 Now, tan. 
 
 WALL SUSTAINING THE 
 
 OS 
 
 ; the proposed con* 
 
 (x tf) 2 
 dition is therefore satisfied, so long as ^ < tan. <p ; or, 
 
 reducing this inequality, so long as 
 
 316. THE STABILITY OF A WALL OF (VARIABLE THICKNESS 
 
 SUSTAINING THE PRESSURE OF A FLUID. 
 
 Let us firsteuppose the internal face AB of the wall to be 
 vertical ; let XY be any section of it, 
 P the centre of pressure of EX, and 
 SM a vertical through the centre of 
 gravity of the portion AXYD of the 
 wall. Produce the horizontal direc- 
 tion of the pressure P of the fluid, 
 supposed to be collected in its centre 
 of pressure, to meet MS in S, and let 
 SK be taken to represent it in mag- 
 nitude, and ST to represent the weight 
 of the portion AXYD of the wall, 
 and complete the parallelogram STRK ; then will its 
 diagonal SR represent the direction and amount of the 
 resultant pressure upon the mass AXYD, and if it be pro- 
 duced to intersect XY in Q, Q will be a point in the line of 
 resistance. 
 
 Let AX=#, XQ=2/, MX=X, AE=0, AD=a, inclination 
 of DC to vertical a, f*= weight of cubical foot of wall, 
 fji l = weight of cubical foot of fluid. By similar triangles, 
 QM_RT 
 
 ~* ' W 
 
 RT=pressure of fluid on EX=iEX.f* 1 EX=i* I ( 
 ST=weight of mass AY=-|-{2#+a? tan. a\x\*>. 
 
 * The centre of pressure of a rectangular plane surface sustaining the 
 pressure of a fluid is situated at two thirds the depth of its immersion. 
 Hydrostatics, p. 26. 
 
 f The pressure of a heavy fluid on any plane surface is equal to the weight 
 of a prism of the fluid whose base is equal in area to the surface pressed, and 
 its height to the depth of the centre of gravity of the surface pressed. 
 Hydrostatics, Art. 31. 
 
PRESSURE OF A FLUID. 411 
 
 M, 
 
 Let =0; then, if the fluid be water, a represents the 
 
 specific gravity of the material of the wall ; and if not, it 
 represents the ratio of the specific gravities of the fluid and 
 wall. 
 
 (x-ef 
 
 ._ 
 
 - 
 
 tan. a* 
 
 K"ow making a a =0 in equation (414), and substituting a 
 for a a and x for <?, 
 
 ^a? 2 tan. 2 a 4- ax tan. a + a* _-J#> 8 tan. 2 a + aw* tan. a + # 2 # 
 
 x tan. a + 2# %OjX + a? 2 tan. a 
 
 Adding this equation to the preceding, 
 
 o-(aj e) 3 -}-^x s tan.V + ^a? 2 tan. a+ofte 
 
 which is the equation to the line of resistance to the wall, 
 the conditions of whose stability may be determined from it 
 as before (see Arts. 291. 293.). 
 
 31T. The conditions necessary iJiat no course of stones com- 
 posing the wall may slip upon the subjacent course. 
 
 This condition is satisfied when the inclination of SQ to 
 the perpendicular to the surface of contact at Q is less than 
 the limiting angle of resistance 9 ; that is, when QSM <<p, 
 or when 
 
 ET 
 tan. 9 >tan. QSM, or >, or > 
 
 or tan. 9 > I- 
 
 ISTo course of stones will be made by the pressure of the 
 fluid to slip upon the subjacent course so long as this condi- 
 tion is satisfied. 
 
 It is easily shown that the expression forming the second 
 member of the above inequality increases continually with 
 
4:12 THE NATURAL SLOPE OF EARTH. 
 
 a?, so that the obliquity of the resultant pressure upon each 
 course, and the probability of its being made to slip upon 
 the next subjacent course, is greater in respect to the lower 
 than the upper courses, increasing with the depth of each 
 course beneath the surface of the fluid. 
 
 EARTH WORKS. 
 318. The natural slope of earth. 
 
 It has been explained (Art. 241.) that a mass, placed upon 
 an inclined plane and acted upon by no other forces than its 
 weight and the resistance of the plane, will just be supported 
 when the inclination of the plane to the horizon equals the 
 limiting angle of resistance between the surface of the plane 
 and that of the mass which it supports ; so that the limiting 
 angle of resistance between the surfaces of the component 
 parts of any mass of earth might be determined by varying 
 continually the slope of its surface until a slope or inclination 
 was attained, at which particular slope small masses of the 
 same earth would only just be supported on its surface, or 
 would just be upon the point of slipping down it. Now this 
 process of experiment is very exactly imitated in the case of 
 embankments, cuttings, and other earth-works, by natural 
 causes. If a slope of earth be artificially constructed at an 
 inclination greater than the particular inclination here 
 spoken of, although, at first, the cohesion of the material 
 may so bind its parts together as to prevent them from slid- 
 ing upon one another, and its surface from assuming its 
 natural slope, yet by the operation of moisture, penetrating 
 its mass and afterwards drying, or under the influence of 
 frost, congealing, and in the act of congelation expanding 
 itself, this cohesion of the particles of the mass is continually 
 in the process of being destroyed ; and thus the particles, so 
 long as the slope exceeds the limiting angle of resistance, 
 are continually in the act of sliding down, until, when that 
 angle is at length reached, this descent ceases (except in so 
 far as the particles continue to be washed down by the rain), 
 and the surface retains permanently its natural slope. 
 
 The limiting angle of resistance 9 is thus'detennined by 
 observing what is the natural slope of each description of 
 earth. 
 
THE PRESSURE OF EARTH. 
 
 4:13 
 
 The following table contains the results of some such 
 observations * : 
 
 NATURAL SLOPES OF DIFFERENT KINDS OF EARTH. 
 
 Nature of Earth. 
 
 Natural Slope. 
 
 Authority. 
 
 Fine dry sand (a single experiment) 
 Ditto 
 Ditto 
 Common earth pulverised and dry - 
 Common earth slightly damp - 
 Earth the most dense and compact - 
 Loose shingle perfectly dry 
 
 21 
 34 29' 
 39 
 46 50' 
 54 
 55 
 39 
 
 Gadroy. 
 Ron dele t. 
 Barlow. 
 Rondelet. 
 Rondelet. 
 Barlow. 
 Pasley. 
 
 SPECIFIC GRAVITIES OF DIFFERENT KINDS OF EARTH. 
 
 Nature of Earth. 
 
 Specific Gravity. 
 
 
 I ,A 
 
 
 1 .( 
 
 Marl 
 
 1-9 
 1*7 
 
 Rubble masonry of calcareous earth or siliceous stones 
 Rubble masonry of granite ..... 
 Rubble masonry of basaltic stones .... 
 
 1-7 to 2-3 
 2-3 
 2-5 
 
 319. THE PRESSURE OF EARTH. 
 
 Let BD represent the surface of a wall sustaining the 
 pressure of a mass of earth whose surface AE is horizontal. 
 
 Let P represent the resultant of the pressures sustained 
 by any portion AX of the wall ; a*nd let the cohesion of the 
 particles of the earth to one another be neglected, as also 
 their friction on the surface of the wall. It is evident that 
 
 * It is taken from the treatise of M. Navier, entitled Resume (Fun Cours ck 
 Construction, p. 160. \ 
 
414 
 
 THE PRESSURE OF EARTH. 
 
 any results deduced in respect to 
 the dimensions of the wall, these 
 ._.* elements of the calculation being 
 .*; neglected, will be in excess^ and 
 
 err on the safe side. 
 ? Now the mass of earth which 
 presses upon AX may yield in the 
 direction of any oblique section 
 XY, made from X to the surface 
 AE of the mass. Suppose YX to 
 be the particular direction in which 
 it actually tends to yield ; so that 
 if AX were removed, rupture 
 would first take place along this 
 section, and AXY be the portion of the mass which would 
 first fall. Then is the weight of the mass AYX supported 
 by the resistances of the different elements of the surface AX 
 of the wall, whose resultant is P, and by the resistance of 
 the surface XY on which it tends to slide. Suppose, now, 
 that the mass is upon the point of sliding down the plane 
 XY, the pressure P being that only which is just sufficient 
 to support it ; the resultant SE of the resistances of the 
 different points of XY is therefore inclined (Art. 241.) to the 
 normal ST, at an angle RST equal to the limiting angle of 
 resistance 9 between any two contiguous surfaces of the 
 earth. 
 
 Now the pressure P, the weight "W of the mass AXY, and 
 the resistance E, being pressures in equilibrium, any two of 
 them are to one another inversely as the sines of their incli- 
 nations to the third (Art. 14.). 
 
 **w 
 
 sin. WSE 
 : sin. PSE 
 
 sin. WSE 
 sin. PSE ' 
 
 But WSE=WST-EST=AYX-EST=^-<-<p T 
 
 =<; PSE=PST+EST==AXY+KST=*+<p. 
 
 .... (42.1). 
 
 if 
 
 Also W=-J^AX . AY^J-^a? 3 tan. i ; if ^ l= weight of each 
 cubic foot of earth, and AX=a? ; 
 
 ..P^z-Jfjo^ 8 tan. i cot. (* + <?) . . . . (422). 
 Now it is evident that this expression, which represents 
 
THE PRESSURE OF EARTH. 4:15 
 
 the resistance of the wall necessary to sustain the pressure of 
 the wedge-shaped mass of earth AXY, being dependent for 
 its amount upon the value of * (so that different sections, 
 such as XY, being taken, each different mass cut off by such 
 section will require a different resistance of the wall to sup- 
 port it), may admit of a maximum value in respect to that 
 variable. * And if the wall be made strong enough to supply 
 a resistance sufficient to support that wedge-shaped mass of 
 earth whose inclination i corresponds to the maximum value 
 of P, and which thus requires the greatest resistance to sup- 
 port it ; then will the earth evidently be prevented by it from 
 slipping at any inclination whatever, for it will evidently not 
 slip at that angle, the resistance necessary to support it at 
 that angle being supplied ; and it will not slip at any other 
 angle, because more than the resistance necessary to prevent 
 it slipping at any other angle is supplied. 
 
 If, then, the wall supplies a resistance equal to the maxi- 
 mum value of P in respect to the variable i>, it will not be 
 overthrown by the pressure of the earth on AX. Moreover, 
 if it supply any less resistance, it will be overthrown ; there 
 not being a sufficient resistance supplied by it to prevent the 
 earth from slipping at that inclination i which corresponds 
 to the maximum value of P. 
 
 To determine the actual pressure of the earth on AX, we 
 have then only to determine the maximum value of P in re- 
 spect to i. 
 
 This maximum value is that which satisfies the conditions 
 
 dP 
 
 But differentiating equation (422) in respect to t, we obtain 
 by reduction 
 
 , . 
 
 U cos. i sin. (1+9) 
 
 Let the numerator and denominator of the fraction in the 
 
 * The existence of this maximum will subsequently be shown : it is, how- 
 ever, sufficiently evident, that, as the angle i is greater, the wedge-shaped mass 
 to be supported is heavier; for which cause, if it operated alone, P would be- 
 come greater as L increased. But as i increases, the plane XY becomes less 
 inclined; for which cause, if it operated alone, P would become less as L in 
 creased. These two causes thus operating to counteract one another, deter- 
 mine a certain inclination in respect to which their neutralising influence is the 
 least, and P therefore the greatest. 
 
 f Church's Diff. and Int. Cal., Art. 41. 
 
416 REVETEMENTS. 
 
 second member of this equation be represented respectively 
 by p and q ; therefore -^r=i^ 1 ar' . -, i^-q ~~-/p\ ; but when 
 
 =- =0, p=0 ; in this case, therefore, -=-5-= J(* a? 8 --^ . Whence 
 w> cLi di 
 
 it follows, by substitution, that for every value of i by which 
 the first condition of a maximum is satisfied, the second dif- 
 ferential co-efficient becomes 
 
 Now it is evident from equation (423) that the condition 
 -y-=0 is satisfied by that value of t which makes 2(*-j-(p)=: 
 
 (il 
 
 * 2, or 
 
 <=H w 
 
 And if this value be substituted for i in equation (424), it 
 becomes 
 
 J 9\ ./* <P 
 
 cos - (4-2) ^ b+i 
 
 which expression is essentially negative, so that the second 
 condition is also satisfied by this value of L. It is that, there- 
 fore, which corresponds to the maximum value of P ; and 
 substituting in equation (422), and reducing, we obtain for 
 this maximum value of P the expression 
 
 which expression represents the actual pressure of the earth 
 on a surface AX of the wall, whose width is one foot and its 
 depth x. 
 
 REVETEMENT WALLS. 
 320. If, instead of a revetement wall sustaining the pres- 
 
KEVETEMENTS. 
 
 417 
 
 sure of a mass of earth, the weight 
 Y K of each cubic foot of which is re- 
 tt:.r?t?r^wp:$jg:*jl presented by f^, it had sustained 
 $1 the pressure of a fluid, the weight 
 / |) of each cubic foot of which was re. 
 
 $*/ F presented by f* r tan. 2 1 - - 1 , then 
 
 \ ! / \4r 2if 
 
 "fi? |i would the pressure of that fluid 
 
 ' | V $ upon the surface AX have been 
 
 S! represented* by J|mX tan - a ( - -1 
 
 >**. \ 4: 4'A 
 
 .";?=g?.; ftv,ugjs^ go ijj^ ^g pressure of a mass o^ 
 earth upon a revetement wall (equation 427), when its, sur- 
 face is horizontal (and when its horizontal surface extends,, 
 as shown in the figure, to the very surface' of the wall)^is 
 identical with that of an imaginary fluid whose specific gra 
 vity is such as to cause each culic foot of it to have a weight 
 M-j, represented in pounds by the formula 
 
 Substituting this value far ^ in equations (416) and (419), 
 we determine therefore, at once, the lines of resistance in 
 revetement walls of uniform and variable thickness, under 
 the conditions supposed, to be respectively 
 
 tan. a I -_ | J(x e) a -h-^'tan.V + aa? a tan.a -f cfx 
 11-=. 
 
 & O/v/vj I /vj2 4-^.-n 
 
 ..(430); 
 
 where a represents the ratio of the specific gravity of the 
 material of the wall to that of the earth. The conditions of 
 the equilibrium of the revetement wall may be determined 
 from the equation to its line of resistance, as explained in 
 the case of the ordinary wall. 
 
 27 
 
 Hydrostatics, Art. 31. 
 
418 
 
 REVETEMENTS. 
 
 321. The conditions necessary that a revetement wall may 
 not le overthrown ~by the slipping of the stones of any 
 course upon those of the subjacent course. 
 
 These are evidently determined from the inequality (420) 
 by substituting /* (equation 428) for ^ in that inequality ; 
 we thus obtain, representing the limiting angle of resistance 
 of the stones composing the wall by <PI to distinguish it from 
 that 9 of the earth, 
 
 where a represents the ratio of the specific gravity of the 
 material of the wall to that of the earth. 
 
 As before, it may be shown from this expression that the 
 tendency of the courses to slip upon one another is greater 
 in the lower courses than the higher. 
 
 322. The pressure of earth whose surface is inclined to the 
 
 horizon. 
 
 Let AB represent the surface of such a mass of earth, YX 
 
 the plane along which the 
 rupture of the mass in 
 contact with the surface 
 AX of a revetement wall 
 tends to take place, AX= 
 a, AXY=*, XAB=/3. 
 Then if W be taken to 
 represent the weight of 
 the mass AXY, it may be 
 shown, as in Art, 319, 
 equation (421), that P= 
 W cot, 
 
 tTT 
 fore W= 
 
 =fr l AX. AY. sin. 0, 
 # a sin. i sin. /3 
 
 ; there- 
 
 cot. 
 
 Now the value of i in this function is that which renders 
 it a maximum (Art, 31^). Expanding cot, (1 + 9), and dif- 
 
REVETEMENTS. 
 
 419 
 
 ferentiating in respect to tan. t, this value of i is readily 
 determined to be that which satisfies the equation 
 
 cot. t=tan. (p + sec. 9 4/r+cot7/3 cot. 9 .... (433). 
 Substituting in equation (432), and reducing, 
 
 {COS (p I 
 
 l + sin.^l + cot.gcot.M ..... (43 *> 
 
 From which equation it is apparent, that the pressure of the 
 earth is, in this case, identical with that of a fluid, of such a 
 density that the weight j* a , of each cubic foot of it, is repre- 
 sented by the formula 
 
 
 1 + sin. 9 v 1+cot. <f> cot. /3 
 
 (435). 
 
 The conditions of the equilibrium of a revetement wall 
 sustaining the pressure of such a mass of earth are therefore 
 determined by the same conditions as those of the river wall 
 (Arts. 313 and 316). 
 
 323. THE RESISTANCE or EAKTH. 
 
 Let the wall BDEF be supported by the resistance of a 
 
 mass of earth upon its sur- 
 face AD, a pressure P, ap- 
 plied to its opposite face, 
 tending to overthrow it. Let 
 the surface AH of the earth 
 be horizontal ; and let Q 
 represent the pressure which, 
 being applied to AX, would 
 just be sufficient to cause the 
 mass of earth in contact 
 with that portion of the wall 
 to yield ; the prism AXY 
 slipping backwards upon the 
 surface XY. Adopting the same notation as in Art. 319, 
 and proceeding in the same manner, but observing that US 
 is to be measured here on the opposite side of TS (Art. 241), 
 since the mass of earth is supposed to be upon the point of 
 slipping upwards instead of downwards, we shall obtain 
 
 ^-l^aj 8 tan. i cot. (49) (436). 
 
420 
 
 WALLS BACKED BY EARTH. 
 
 Now it is evident that XY is that plane along which rup- 
 ture may be made to take place by the least value of Q ; / 
 in the above expression is therefore that angle which gives 
 to that expression its minimum value. Hence, observing 
 that equation (436) differs from equation (422) only in the 
 sign of <p, and that the second differential (equation 426) is 
 rendered essentially positive by changing the sign of 9, it is 
 apparent (equation 427) that the value of Q necessary to 
 overcome the pressure of the earth upon AX is represented 
 
 324. It is evident that a fluid would oppose the same 
 resistance to the overthrow of the wall as the resistance of 
 the earth does, provided that the weight f* 4 of each cubic 
 foot of the fluid were such that 
 
 (V=Man.' + ....(438); 
 
 so that the point in AX at which the pressure Q may be 
 conceived to be applied, is situated at fds the distance AX. 
 
 325. The stability of a wall of uniform thickness which a 
 given pressure P tends to overthrow, and which is sus- 
 tained by the resistance of earth. 
 
 Let y be the point in whidi any section XZ of the wall 
 
 would be intersected by the 
 resultant of the pressures 
 upon the wall above that sec- 
 tion, if the whole resistance 
 Q, which the earth in con- 
 tact with AX is capable of 
 supplying, were called into 
 action. Let BX=a?, ~Ky=y, 
 BA = e, BE=0, Bp = jfc, 
 weight of cubic feet of ma- 
 terial of wall fA, inclination 
 of P to vertical^. Taking 
 the moments about the point 
 y of the pressures applied to BXZE, we have, by the prin- 
 ciple of the equality of moments, observing that "'~ " 
 
WALLS BACKED BY EARTH. 
 
 421 
 
 (a? e), and that the perpendicular from y, upon P is repre- 
 sented by x sin. &(ky) cos. d, 
 
 r \x sin. d (fc y) cos. e\ =$(x( 
 
 or substituting for Q its value (equation 437), and solving in 
 respect to y, 
 
 irM' 4 (aJ e}*+^a?x P(a? sin. 6 Jc, cos. 6) 
 
 y=. =r (4o9). 
 
 JL cos. o -j- \*<ax 
 
 Now it is evident that the wall will not be overthrown 
 upon any section XZ, so long as the greatest resistance Q, 
 which the superincumbent earth is capable of supplying, is 
 sufficient to cause the resultant pressure upon EX to inter- 
 sect that section, or so long as y in the above equation has 
 a positive value ; moreover, that the stability of the wall is 
 determined by the minimum value of y in respect to x in 
 that equation, and the greatest height to which the wall can 
 be buut, so as to stand, by that value of x which makes y=0. 
 
 326. The stability of a wall which a given pressure tends to 
 overthrow, and which is supported lyy a mass of earth 
 whose surface is not horizoni " 
 
 Let (3 represent the inclination of the surface AB of earth 
 
 to the horizon. By reasoning 
 similar to that of Art. 322., it is 
 apparent that the resistance Q 
 of the earth in contact with any 
 given portion AX of the wall to 
 displacement, is determined by 
 assigning to 9 a negative value 
 in equation (434). Whence it 
 follows, that this resistance is 
 equivalent to that which would 
 be produced by the pressure of 
 a fluid upon the wall, the weight 
 ^ 6 of each cubic foot of which 
 was represented by the formula 
 
 % 
 
 A i/ 
 
 icos. 9 ) 2 
 T Y 
 1 sin. 9 1/1 cot. 9 cot. j3 ) 
 
 (440). 
 
 The conditions of the stability of an upright wall sub- 
 jected to any given pressure P tending to overthrow it, and 
 
422 
 
 REVETEMENTS. 
 
 sustained by the pressure of such a mass of earth, are there- 
 fore precisely the same as those discussed in the last article ; 
 the symbol ^ 4 (equation 439) being replaced by |u- 5 (equation 
 440). 
 
 327. The stability of a revetement wall whose interior face 
 is inclined to the vertical at any angle ; taking into account 
 the friction of the earth upon the face of the wall. 
 
 Let 2 represent the inclination of the face BD of such a 
 wall to the vertical, <p 2 the limiting angle of resistance 
 between the mass of earth and the surface of the wall ; and 
 let the same notation be adopted as in the last article in 
 respect to the other elements of the 
 question, and the same construction 
 made. Draw PQ perpendicular to BD ; 
 then is the direction PS of the resist- 
 ance of the wall upon the mass of earth, 
 evidently inclined to QP at an angle 
 QPS equal to the limiting angle of 
 resistance <? 2 , in the state bordering 
 upon motion by the overthrow of the 
 wall* (Art. 241.). 
 
 Draw Pn horizontally and X$ verti- 
 cally, produce TS and ES to meet it in 
 m and n, and let 
 
 P 
 W 
 
 sin. WSE sin. (WST-TSE) 
 sin. PSK sin. (EmP + SPm). 
 
 But 
 
 = AYX= - 
 
 = - -t, TSE=<p, 
 2 
 
 Also W= 
 
 sn. t + 9 + 9 a + a 2 sn. ^ 
 
 ^^X (tan. i + tan. a a ) ; if X=a?, 
 
 * It is not only in the state of the wall bordering upon motion that thia 
 direction of the resistance obtains, but in every state in which the stability of 
 the wall is maintained. (See the Principle of 'Least Resistance.) 
 
REVETEMENT8. 4:23 
 
 a a ) ; ; 
 
 Assuming a a -f-9+9 a =/3, then differentiating in respect to t, 
 
 7T> 
 
 and assuming -^- = 0, we -obtain by reduction 
 
 (tan. i + tan. a a ) cos. ( 9) -f 
 
 cos. (t-f (p)sin. (*+/3)sec. a *=0; or, 
 
 (tan. i + tan. a a ) (1 + tan. (3 tan. (p) -f- 
 
 (1 tan. i tan. (p) (tan. i+ tan. /3)=0 ; 
 
 .'. tan. 3 1 + 2 tan. i tan. /3 tan. (3 cot. 9 + 
 
 (cot. 9 + tan. /3) tan. a a = 0. 
 
 Solving this quadratic in respect to tan. , neglecting the 
 negative root, since tan. i is essentially positive, and reducing, 
 
 tan. i (tan. /3 tan. a,)*(tan. + cot. 9)* tan. /3 . . . (442.) 
 
 Now the value of t determined by this equation, when 
 substituted in the second differential coefficient of P in 
 respect to , gives to that coefficient a negative value ; it 
 therefore corresponds to a maximum value of P, which 
 maximum determines (Art. 319.) the thrust of the earth 
 upon the portion AX of the wall. To obtain this maximum 
 value of P by substitution in equation (441), let it be 
 observed that 
 
 cos. (* + <p)_l tan. i tan. 9 /cos. <p \ 
 sin. (i + $)~~ ( tan - i+tan. /3) \cos. j8'/ 
 
 1 tan. i tan. 9=1 + tan. /3 tan. 9 tan. 9 (tan. /3 
 
 tan. a 2 )*(tan. /3-fcot. 9)*, 
 
 =tan. 9 (tan. /3 + cot. 9)M( tan - 1 s + cot. 9)* (tan. /3 tan. 
 tan. i+tan. ^=(tan. /3 + cot. 9) f (tan. /3 tan. a a )*; 
 
 cos. (t. + 9) sin. 9 j / tan. /3 -f cot. 9 \ * ) 
 '"sin. (t + /3 = cos. /3 ( \tan. /3 tan. a) " ) ' 
 
 Also tan. i + tan. a 2 =(tan. /3-fcot. 9)*(tan. /3 
 
 tan. a a ) } (tan. /3 tan. a a ) 
 (tan. /3 tan. a,)* {(tan. ^ + cot. 9)* (tan. p tan. 
 
 ^y-Ktan. /3 + cot. 9)*-(tan. -tan. ,)*{ a ; 
 
 COS. /3 
 
424 
 
 RE VETE3IENTS . 
 
 which expression may be placed under the following form, 
 better adapted to logarithmic calculation, 
 
 P_i sin - 9 \ /cos. (ff-<p)\ *_ /sin. (/3-q,)\ * ) 8 . 
 ~ tM/1 cos. 2 /3 ( \ sin. 9 / \ cos. a a / f 
 
 or substituting for (3 its value a 3 +<P+<p 2? 
 
 P=l ^ sin - <P J /COS.(a a + 9,)\*_ 
 ~ 2 \ sin. 9 / 
 
 - >( 
 
 By a comparison of this equation with equation (427) it 
 is apparent, that the pressure of a mass of earth upon a 
 revetement wall, under the supposed conditions, is identical 
 with that which it would produce if it were perfectly fluid, 
 provided that the weight of each cubic foot of that fluid had 
 a value represented by the coefficient of -Jar 2 in the above 
 equation ; so that the conditions of the stability of such a 
 revetement wall are identical (this value being supposed) 
 with the conditions of the stability of a wall sustaining the 
 pressure of a fluid, except that the pressure of the earth is 
 not exerted upon the wall in a direction perpendicular to its 
 surface, as that of a fluid is, but in a direction inclined to 
 the perpendicular at a given angle, namely, the limiting 
 angle of resistance. 
 
 328. THE PRESSURE OF EARTH WHICH SURMOUNTS A REVETE- 
 
 MENT WALL AND SLOPES TO ITS SUMMIT. 
 
 Hitherto we have supposed the surface of the earth whose 
 
 ippose 
 
 vated above the summit of the wall, 
 and to descend to it by the natural 
 slope ; the wall is then said to be 
 surcharged, or to carry a parapet. 
 Let EF represent the natural slope 
 of the earth, FY its horizontal sur- 
 face, BX any portion of the internal 
 face or intrados of the wall, P the 
 horizontal pressure just necessary 
 
REVETEMENTS. 425 
 
 f 
 
 to support the mass of earth HXYF, whose weight is W, 
 upon the inclined plane XY. Produce XB and YF to meet 
 in A, and let AX^a?, AH=, AXY=u, ^= weight of each 
 cubic foot of the earth, 9 the natural slope of its surface 
 FE. Now it may be shown, precisely by the same reason- 
 ing as before, that the actual pressure of the earth upon the 
 portion BX of the wall is represented by that value of P 
 which is a maximuni in respect to the variable i ; moreover, 
 that the relation of P and i is expressed by the function P 
 = W cot. 0+9); where W^f^area HXYF)=^(AXY 
 |-c a cot. 9); 
 
 /.P ^(a? 8 tan. i <? cot. 9) cot. (4+9) ..... (4M). 
 Expanding cot. (^+9), 
 
 P__JL (# a tan. i c* cot. 9) (1 tan, i tan. 9) 
 ~"*V tan. i+ tan. 9 
 
 To facilitate the differentiation of this function, let 
 tan. *-f tan. 9 be represented by z, and let it be observed 
 that whatever conditions determine the maximum value of P 
 in respect to z determine also its maximum value in respect 
 to i.* Then tan. i=z tan. 9 ; therefore 1 tan. i tan. 9= 
 1z tan. 9 + tan. 2 9= z tan. 9 + sec. "9. Also, a? a tan. < 
 c* cot. q>=a?z (a? tan. 9+c a cot. 9). 
 
 Substituting these values in the preceding expression for 
 P, and reducing, 
 
 j (?" tan. 9 + c 2 cot. 9) sec. '9 
 
 P=i^i \ z ^n. 9 - - - - l - -+ 
 
 ..... (445). 
 
 dl* ( (a; 8 tan. 9 + c a cot. 9) sec. *9 ) 
 
 a -p- - \ , 
 
 ^_ dP dPdz , d*P rfTP/&\ rfP<f dz , t 
 
 * For - r =- T - -j-. and =_ _( -f - ; now - =sec. *f. there- 
 dt dz dt dt 1 dz*\dt ) dz dt* dt 
 
 7T> /JT> 
 
 fore -j- = -=- sec. 3 i; and for all values of i less than -, sec. 2 i has a finite 
 dt dz 2' 
 
 value, so that -3- = when =0. 
 <w efe 
 
 </P <i a P d*P idz\* d*P 
 
 When, moreover, -T-=O, -TT=-rr ( -7- ) 5 so tnat when -r-r- is negative. 
 az at dz \di / dz 
 
 -; is also negative. 
 
426 BEVETEMENTS. 
 
 <? P_ (j* tan, 9 + c 9 cot. 9) sec. '9 
 
 ~~ 
 
 The first condition of a maximum is therefore satisfied by 
 the equation 
 
 (a; 2 tan. 9 +<? 8 cot. 9) sec. "9 A ,.,, 
 
 a? a tan.9 + - - - 9 - -=0 . . . . (446); 
 
 or, solving this equation in respect to z, and reducing, it is 
 satisfied by the equation 
 
 / <? 2 \* 
 
 '<= I sec. 9 + a cosec. 9 1 . 
 
 E"ow the second condition of a maximum is evidently 
 satisfied by any positive value of z, and therefore by the 
 positive root of this equation. Taking, therefore, the posi- 
 tive sign, substituting for z its value, and transposing, 
 
 / c 2 \* 
 
 tan. i= I sec. \ -f cosec. 2 <p 1 tan. 9 ..... (447) ; 
 
 which equation determines the tangent of the inclination 
 AXY to the vertical, of the base XY of that wedge-like 
 mass of earth HXYF, whose pressure is borne by the sur- 
 face BX of the wall. To determine the actual pressure 
 upon the wall, this value of tan. i must be substituted in the 
 expression for P (equation 445). Now the two first terms 
 of the expression within the brackets in the second member 
 of that equation may be placed under the form 
 
 ( 
 z I 
 
 (a? 2 tan. 9+c 2 cot. 9) sec. 3 <p 
 a? 2 tan. 9 + ^ - -- " - 
 
 But it appears by equation (446) that the two terms which 
 compose this expression are equal, so that the expression is 
 equivalent to 2^ 2 tan. 9 ; or, substituting for the value of 
 
 Z) to 2ft 2 tan. 9 (sec. a 9+-j cosec. *<p)*, or to 2a? sec. 9 
 (a? 2 tan. 2 9 + c 2 ) * . Substituting in equation (445), 
 
 P ^{ 2a?sec. 9(# 3 tan. a 9 + c a )* +(a? a tan. 2 9 + c a )+ar l sec. '9} 
 ;.P=lfi. l {jB sec. 9 (# 2 tan. 2 9 + <? 2 )^ 2 ..... (448); 
 
 by which expression is determined the actual pressure upon 
 a portion of the wall, the distance of whose lowest point 
 from A is represented by x. 
 
427 
 
 329. The conditions necessary that a revetement wall carry- 
 ing a parapet may not be overthrown by the slipping of 
 any course of stones on the subjacent course. 
 
 Let <pj represent the limiting angle of the resistance of the 
 stones of the wall upon one another ; and let OQ represent 
 the direction of the resultant pressure 
 on the course XZ. The proposed 
 conditions are then involved (Art. 
 141.) in the inequality 9 1 >QOM, or 
 tan. 9j > tan. QOM, or tan. 9, > 
 
 T^> ^tnr-^ov*; or substituting 
 OS weight of BZ* 
 
 for P its value (equation 448), and 
 fx(2#a?-fa? 2 tan. a) for the weight of 
 BZ, it appears that the proposed 
 conditions are determined by the 
 inequality 
 
 tan. 
 
 tan. a 
 
 330. 
 
 of resistance in a revetement wall carrying a 
 parapet. 
 
 Let OT be taken to represent the pressure P, and OS the 
 weight of BZ. Complete the parallelogram ST, and pro- 
 duce its diagonal OR to Q ; then will Q be a point in the 
 line of resistance. Let AX=#, QX=/, AB=5, AP=X, 
 
 =X, W = weight of BZf. By similar triangles, r?= 
 
 g ; but QM=(y-X), OM=aj-X, KS=P, OS= 
 y-X P Wx+Paj-PX 
 
 ~ 
 
 Now the value of X is determined from equation (414), by 
 
 * The influence, upon the equilibrium of the wall, of 'the small portion of 
 earth BHE is neglected in this and the subsequent computation. 
 
 f The influence of the weight of the small mass of earth BEH which rests 
 on the summit of the wall is here again neglected. 
 
428 
 
 REVETEMENTS. 
 
 substituting in that equation (x 1) for c: whence we obtain, 
 observing that tan. a a =0, and substituting a for a 1? 
 
 . _i(# &) a tan. a a + #(#) tan. a 
 (a? 5)tan.aH-2# 
 
 Also W=JKa-5){(a-&)tan.a + 2a{ ..... (451); 
 /. WX= Jf*( &) $(o? &) a tan. 2 a + a(aj 5) tan. a 
 
 It remains, therefore, only to determine the value of the 
 term P . X. Now it is evident (Art. 16.) that the product 
 P . X is equal to the sum of the moments of the pressures 
 upon the elementary surfaces which compose the whole sur- 
 face BX. But the pressure upon any such elementary sur- 
 face, whose distance from A is a?, is evidently represented 
 
 by -J-AX* ; its moment is therefore represented by -r-ajAo?, 
 and the sum of the moments of all such elementary pressures 
 by 2 -ccAaj, or when AX is infinitely small, by 
 
 / -T-xdx ; therefore P . X= / 
 
 b b 
 
 But differentiating equation (448), 
 
 2 <p 
 
 Performing the actual multiplication of the factors in the 
 second member of this equation, observing that -^ -- ~ -^-yr 
 
 3 , , c a 
 
 =( x tan.V + ^y / rr - a , av , and re- 
 y 2a aJ 
 
 ducing we obtain 
 
 * P being a function of #, let it be represented by f(x); then will f(x) repre- 
 Bent the pressure upon a portion of the surface BX terminated at the distance 
 x from A, and /(a-f-Az) that upon a portion terminated at the distance x-\-Ax ; 
 therefore /( x -f-Aic) fx will represent the pressure upon the small element Aa? 
 of the surface included between these two distances. But by Taylor's theorem, 
 
 /(ar-f-Aa;) fx= -- As -f- ^-~- -f, &c. ; therefore, neglecting terms ii> 
 ux dx 1*2 
 
 volving powers of A# above the first, pressure on element = -j-A*. 
 
THE ABOH. 429 
 
 dx~ 
 
 <? sec. 9 
 
 Multiplying this equation by a?, and integrating between the 
 limits 6 and a?, 
 
 'i(sec. a 9+tan. 2 9)(a? 8 5 8 ) | sec. 9 cot. '9 |(a? a tan. a 9 
 P.X^^-I +c a )f (V 2 tan. a 9 + <? a )f} +c 2 sec. 9 cot. 2 9 
 
 .... (452). 
 
 This value of P . X being substituted in equation (450), 
 and the values of W\ W, P, from equations (448) and 
 (451), the line of resistance to the revetement wall will be 
 determined, and thence all the conditions of its stability 
 may be found as before.* 
 
 THE ARCH. 
 
 331. Each of the structures, the conditions of whose sta- 
 bility (considered as a system of bodies in contact), have 
 hitherto been discussed, whatever may have been the pres- 
 sures supposed to be insistent upon it, has been supposed to 
 rest ultimately upon a single resisting surface, the resultant 
 of the resistances on the different elements of which was at 
 once determined in magnitude and direction by the resultant 
 of the given insistent pressures! being equal and opposite 
 to that resultant. 
 
 The arch is a system of bodies in contact which reposes 
 ultimately upon two resisting surfaces called its abutments. 
 The resistances of these surfaces are in equilibrium with the 
 
 * The limits which the author has in this work imposed upon himself do not 
 leave him space to enter further upon the discussion of this case of the 
 revetement wall, the application of which to the theory of fortification is so 
 direct and obvious. The reader desirous of further information is referred to 
 the treatise of M. Poncelet, entitled " Memoire sur la Stabilite des Revete- 
 ments et do lours Fondations." He will there find the subject developed in all 
 ith practical relations, and treated with the accustomed originality and power 
 of that illustrious author. The above method of investigation has nothing in 
 common with the method adopted by M. Poncelet except Coulomb's principle 
 of the wedge of maximum pressure. 
 
 f The weight of the structure itself is supposed to be included among these 
 pressures. . 
 
430 THE PRINCIPLE OF LEAST RESISTANCE. 
 
 given pressures insistent upon the arch (inclusive of its 
 weight), but the direction and amount of the resultant pres- 
 sure upon each surface is dependent upon the unknown 
 resistance of the opposite surface ; and thus the general 
 method applicable to the determination of the fine of 
 resistance, and thence of the conditions of stability, in that 
 large class of structures which repose on a single resisting 
 surface, fails in the case of the arch. 
 
 332. THE PRINCIPLE OF LEAST RESISTANCE. 
 
 If there ~be a system of pressures in equilibrium amona which 
 are a given number of resistances, then is each of these a 
 minimum, subject to the conditions imposed by the equili- 
 brium of the whole* 
 
 Let the pressures of the system, which are not resistances, 
 be represented by A, and the resistances by B ; also let any 
 other system of pressures which may be made to replace the 
 pressures B and sustain A, be represented by C. 
 
 Suppose the system B to be replaced by C ; then it is 
 apparent that each pressure of the system C is equal to the 
 pressure propagated to its point of application from the 
 pressures of the system A ; or it is equal to that pressure, 
 together with the pressure so propagated to it from the 
 other pressures of the system C. In the former case it is 
 identical with one of the resistances of the system B ; in the 
 latter case it is greater than it. Hence, therefore, it appears 
 that each pressure of the system B is a minimum, subject 
 to the conditions imposed by the equilibrium of the whole. 
 
 If the resultant of the pressures applied to a body, other 
 than the resistances, be taken, it is evident from the above 
 that these resistances are the least possible so as to sustain 
 that resultant ; and therefore that if each resisting point be 
 capable of supplying its resistance in any direction, then are 
 all the resistances parallel to one another and to the result- 
 ant of the other pressures applied to the body. 
 
 * The principle of least resistance was first published by the author of thia 
 work in the Philosophical Magazine for October, 1833. 
 
THE ARCH. 431 
 
 333. Of oil the pressures which can le applied to the highest 
 voussoir of a semi-arch, different in their amounts and 
 points of application, hut all consistent with the equili- 
 orium of the semi-arch, that which it would sustain from 
 the pressure of an opposite and equal semi-arch is the least. 
 
 Let EB represent the surface by which an arch rests upon 
 
 either of its abutments ; then are the resistances upon the 
 different points of that surface (Art. 331.) the least pressures, 
 which, being applied to those points, are consistent with the 
 equilibrium of the arch. They are, moreover, parallel to one 
 another : their resultant is therefore the least single pressure, 
 which, being applied to the surface EB, would be sufficient 
 to maintain the equilibrium of the arch, if the abutment were 
 removed. 
 
 Now, if this resultant be resolved vertically and horizon- 
 tally, its component in a vertical direction will evidently be 
 equal to the weight of the semi-arch : it is therefore given in 
 amount. In order that the resultant may be a minimum, its 
 vertical component being thus given, it is therefore necessary 
 that its horizontal component should be a minimum ; but 
 this horizontal component of the resistance upon the abut- 
 ment is evidently equal to the pressure P of the opposite 
 semi- arch upon its key-stone : that pressure is therefore a 
 minimum; or, if the semi-arches be equal in every respect, 
 it is the least pressure which, being applied to the side of the 
 key-stone, would be sufficient to support either semi-arch ; 
 which was to be proved. 
 
 The following proof of this property may be more intelli- 
 gible to some readers than the preceding. It is independent 
 of the more general demonstration of the principle of least 
 resistance.* 
 
 * See Memoir by the author of this work in Mr. Harm's " Treatise on the 
 Theory of Bridges," p. 10. 
 
432 
 
 THE AKCH. 
 
 The pressure which an opposite semi-arch would produce 
 upon the side AD of the key-stone, is equal to the tendency 
 of that semi-arch to revolve forwards upon the inferior edges 
 of one or more of its voussoirs. Now this tendency to motion 
 is evidently equal to the least force which would support the 
 opposite semi-arch. If the arches be equal and equally 
 loaded, it is therefore equal to the least force which would 
 support the semi-arch ABED. 
 
 334:. GENERAL CONDITIONS OF THE STABILITY OF AN 
 
 Suppose the mass ABDO to be acted upon by any number 
 
 of pressures, among which 
 is the pressure Q, being the 
 resultant of certain resist- 
 ances, supplied by different 
 points in a surface BD ; 
 common to the mass arid to 
 an immoveable obstacle 
 BE. 
 
 Now it is clear that un- 
 der these circumstances we 
 may vary the pressure P, 
 both as to its amount, di- 
 rection, and point of appli- 
 cation in AC, without disturbing the equilibrium, provided 
 only the form and direction of the line of resistance continue 
 to satisfy the conditions imposed by the equilibrium of the 
 system. 
 
 These have been shown (Art. 283) to be the following : 
 that it no where cut the surface of the mass, except at P, 
 and within the space BD ; and that the resultant pressure 
 upon no section MN of the mass, or the common surface BD 
 of the mass and obstacle, be inclined to the perpendicular to 
 that surface, at an angle greater than the limiting angle of 
 resistance. 
 
 Thus, varying the pressure P, we may destroy the equi- 
 librium, either, first, by causing the resultant pressure to 
 take a direction without the limits prescribed by the resist- 
 ance of any section MN through which it passes, that is, 
 without the cone of resistance at the point where it inter- 
 
 * Theoretical and Practical Treatise on Bridges, vol. i. ; Memoir by the au- 
 thor of this work, p. 11. 
 
THE ARCH. 433 
 
 sects that surface ; or, secondly, by causing the point Q to 
 fall without the surface BD, in which case no resistance can 
 be opposed to the resultant force acting in that point ; or, 
 thirdly, the point Q lying within the surface BD, we may 
 destroy the equilibrium by causing the line of resistance to 
 cut the surface of the mass somewhere between that point 
 and P. 
 
 Let us suppose the limits of the variation of P, within 
 which the first two conditions are satisfied, to be known ; and 
 varying it, within those limits, let us consider what may be 
 its least and greatest values so- as- to satisfy the third condition,, 
 
 Let P act at a given point in AC, and in a given direc- 
 tion. It is evident that by diminishing it under these 
 circumstances the line of resistance will be- made continually 
 to assume more nearly that direction which it would have 
 if P were entirely removed. 
 
 Provided, then, that if P were thus removed, the line of 
 resistance would cut the surface, that is, provided the 
 force P be necessary to the equilibrium, it follows that by 
 diminishing it we may vary the direction and curvature of 
 the line of resistance, until we at length make it touch some 
 point or other in the surface of the mass. 
 
 And this is the limit ; for if the diminution be carried 
 further, it will cut the surface, and the equilibrium will be 
 destroyed. It appears, then, that under the circumstances 
 supposed, when P, acting at a given point and in a given 
 direction, is the least possible, the line of resistance touches 
 the interior surface or intrados of the. mass. 
 
 In the same manner it may be shown that when it is the 
 greatest possible, the line of resistance touches the exterior 
 surface or extrados of the mass. 
 
 The direction and point of application of P in AC have 
 here been supposed to be given ; but by varying this direc- 
 tion and point of application, the contact of the line of 
 resistance with the intrados of the arch may be made to 
 take place in an infinite variety of different points, and each 
 such variation supplies a new value of P. Among these, 
 therefore, it remains to seek the absolute maximum and 
 minimum values of that pressure. 
 
 In respect to the direction of the pressure P, or its incli- 
 nation to AC, it is at once apparent that the least value of 
 that pressure is obtained, whatever be its point of applica- 
 tion, when it is horizontal. 
 
 There remain, then, two conditions to which P is to be 
 subjected, and which involve its condition of a minimum. 
 
 28 
 
434 THE ARCH. 
 
 The first is, that its amount shall ~be such as will give to the 
 line of resistance a point of contact with the intrados ; the 
 second, that its point of application in the key-stone AC 
 shall be such as to give it the least value which it can receive, 
 subject to the first condition. 
 
 335. PRACTICAL CONDITIONS OF THE STABILITY or AN AECH 
 
 OF TJNCEMENTED STONES. 
 
 The condition, however, that the resultant pressure upon 
 the key-stone is subject, in respect to the position of its 
 point o'f application on the key-stone, to the condition of a 
 minimum, is dependent upon hypothetical qualities of the 
 masonry. It supposes an unyielding material for the arch- 
 stones, and a mathematical adjustment of their surfaces. 
 These have no existence in the uncemented arch. On the 
 striking of the centres the arch invariably sinks at the 
 crown, its voussoirs there slightly opening at their lower 
 edges, and pressing upon one another exclusively by their 
 upper edges. Practically, the line of resistance then, in an 
 arch of uncemented stones, touches the extrados at the crown ; 
 so that only the iirst of the two conditions of the minimum 
 stated above actually obtains : that, namely, which gives to 
 the line of resistance a contact with the intrados of the 
 arch. This condition being assumed, all consideration of 
 the yielding quality of the material of the arch and its 
 abutments is eliminated. 
 
 The form of the solid has hitherto been assumed to be 
 given, together with the positions of the different sections 
 made through it ; and the forms of its lines of resistance and 
 pressure, and their directions through its mass have thence 
 been determined. 
 
 It is manifest that the converse of this operation is pos- 
 sible. 
 
 Having given the form and position of the line of resist- 
 ance or of pressure, and the positions of the different sections 
 to be made through the mass, it may, for instance, be 
 inquired what form these conditions impose upon the surface 
 which bounds it. 
 
 Or the direction of the line of resistance or pressure and 
 the form of the bounding surface may be subjected to certain 
 conditions not absolutely determining either. 
 
 If, for instance, the form of the intrados of an arch be 
 given, and the direction of the intersecting plane be always 
 
THE ARCH. 435 
 
 perpendicular to it, and if the line of pressure be supposed 
 to intersect this plane always at the same given angle with 
 the perpendicular to it, so that the tendency of the pressure 
 to thrust each from its place may be the same, we may 
 determine what, under these circumstances, must be the 
 extrados of the arch. 
 
 If this angle equal constantly the limiting angle of resist- 
 ance, the arch is in a state bordering upon motion, each 
 voussoir being upon the point of slipping downwards, or up- 
 wards, according as the constant angle is measured above or 
 below the perpendicular to the surface of the voussoir. 
 
 The systems of voussoirs which satisfy these two con- 
 ditions are the greatest and least possible. 
 
 If the constant angle be zero, the line of pressure being 
 every where perpendicular to the joints of the voussoirs, the 
 arch would stand even if there were no friction of their sur- 
 faces. It is then technically said to be equilibriated ; and 
 the equilibrium of the arch, according to this single con- 
 dition, constituted the theory of the arch so long in vogue, 
 and so well known from the works of Emerson, Hutton, and 
 "Whewell. It is impossible to conceive any arrangement of 
 the parts of an arch by which its stability can be more 
 effectually secured, so far as the tendency of its voussoirs to 
 slide upon one another is concerned: there is, however, 
 probably, no practical case in which this tendency really 
 affects the equilibrium. So great is the limiting angle of 
 resistance in respect to all the kinds of stone used in the 
 construction of arches, that it would perhaps be difficult to 
 construct an arch, the resultant pressure upon any of the 
 joints of which above the springing should lie without this 
 angle, or which should yield by the slipping of any of its 
 voussoirs. 
 
 Traced to the abutment of the arch, the line of resistance 
 ascertains the point where the direction of the resultant 
 pressure intersects it, and the line of pressure determines the 
 inclination to the vertical of that resultant ;* these elements 
 determine all the conditions of the equilibrium of the abut- 
 ments, and therefore of the whole structure ; they associate 
 themselves directly with the conditions of the loading of the 
 arch, and enable us so to distribute it as to throw the points 
 of rupture into any given position on the intrados, and give 
 to the line of resistance any direction which shall best con- 
 
 * The inclination of the resultant pressure at the springing to the vertical 
 may be determined independently of the line of pressure, as will hereafter be 
 shown 
 
4:36 THE LINE OF RESISTANCE IN THE AKCH. 
 
 duce to the stability of the structure ; from known ^dimen- 
 sions, and a known loading of the arch, they determine the 
 dimensions of piers which will support it ; or conversely, 
 from known dimensions of the piers they ascertain the 
 dimensions and loading of the arch, which may safely be 
 made to span the space between them. 
 
 336. To DETERMINE THE LINE OF RESISTANCE IN AN ARCH 
 WHOSE INTRADOS 18 A CIRCLE, AND WHOSE LOAD IS COL- 
 LECTED OVER TWO POINTS OF ITS EXTRADOS SYMMETRICALLY 
 PLACED IN RESPECT TO THE CROWN OF THE ARCH. 
 
 Let ADBF represent any portion of such an arch, P a 
 
 pressure applied at its extreme 
 voussoir, and X and Y the ho- 
 rizontal and vertical compo- 
 nents of any pressure borne 
 upon the portion DT of its ex- 
 trados, or of the resultant of 
 any number of such pressures ; 
 let, moreover, the co-ordinates, 
 from the centre C, of the point 
 of application of this pressure, 
 or of this resultant pressure, be 
 x and y. 
 
 Let the horizontal force P 
 be applied in AD at a vertical distance p from C ; also let 
 CT represent any plane which, passing through 0, intersects 
 the arch in a direction parallel to the joints of its voussoirs. 
 Let this plane be intersected by the resultant of the pres- 
 sures applied to the mass ASTD in R. These pressures are 
 the weight of the mass ASTD, the load X and Y, and the 
 pressure P. Now if pressures equal and parallel to these, 
 but in opposite directions, were applied at K, they would of 
 themselves support the mass, and the whole of the subjacent 
 mass TSB might be removed without affecting the equili- 
 brium. (Art. 8.) Imagine this to be done ; call M the weight 
 of the mass ASTD, and k the horizontal distance of its cen- 
 tre of gravity from C, and let CR be represented by p, and 
 the angle ECS by 0, then the perpendicular distances from 
 C of the pressures M+ Y and P X, imagined to be applied 
 to R, are p sin. 6 and p cos. 6 ; therefore by the condition of 
 the equality of moments, 
 
THE ANGLE OF KUPTUKE IN THE ARCH. 
 
 437 
 
 (M+Y) p sin. d + (P X) p cos.a=MA+ Y-Xy+Pp ; 
 
 P ~~ 
 
 (M+Y) sin.d-f (P-X) cos. 6 
 
 . . . (453), 
 
 which is the equation to the line of resistance. 
 
 M and h are given functions of & ; as also are X and Y, if 
 
 the pressure of the load extend continuously over the surface 
 
 of the extrados from D to T. 
 
 It remains from this equation 
 to determine the pressure P, be- 
 ing that supplied by the opposite 
 semi-arch. As the simplest case, 
 let all the voussoirs of the arch 
 be of the same depth, and let the 
 inclination ECP of the first joint 
 of the semi-arch to the vertical be 
 represented by 0, and the radii 
 of the extrados and intrados by 
 R and r. Then, by the known 
 principles of statics.* 
 
 R e 
 = / J r 3 sin. &d&dr= i(R s ^(cos. 6 cos. 0) ; 
 
 r 6 
 
 also, M=-KR f r>)(a e) ; 
 
 /. p $(R 8 -r a X0-0) sin. 6+ Y sin. d-X cos. d+P cos. A] = 
 
 which is the general equation to the line of resistance. 
 
 THE ANGLE OF RUPTURE. 
 
 337. At the points of rupture the line of resistance meets 
 the intrados, so that there p=r : if then Y be the correspond- 
 ing value of d, 
 
 r $(R 2 r*)(v ) sin. Y + Y sin. Y X cos. Y + P cos. Y} =- 
 3 /)(cos. cos.)+Y# Xy + P^? (455). 
 
 See Note 1 at end of PART IV. ED. 
 
438 THE ANGLE OF KUPTUKE 
 
 Also at the points of rupture the line of resistance toucJws 
 the intrados, so that there -1=-^= 5 assuming then, tu 
 
 OJO U/a 
 
 simplify the results, that the pressure of the load is wholly 
 in a vertical direction, so that X=0, and that it is collected 
 
 /7~V 
 
 over a single point of the extrados, so that =0, and dif- 
 
 ferentiating equation (454), and assuming -^=0, when d=Y 
 and p=r, we obtain 
 
 r {i(K 2 -r 2 ) (v 0) cos. Y + (E a r a ) sin. Y + 
 
 Y cos. Y P sin. ^J =4(K 3 -O sin. Y ; 
 hence, assuming R^T 1 (1+a), 
 
 (456). 
 
 Eliminating (Y 0) between equations (455) and (456), we 
 have 
 
 (457). 
 
 Eliminating P between equations (455) and (456), and 
 reducing. 
 
 
 sin. Y {(+'+$') cos. (Ja'+a) cos.Yj gin.Y 
 
 ..... (458). 
 
 * This equation might have been obtained by differentiating equation (454) 
 
 7T> 
 
 in respect to P and 6, and assuming - = when r and are substituted for 
 p and 0; for if that equation be represented by w 0, u being a function of 
 
 is 
 
 therefore obtained, whether we assume- = 0, or- =0, which last supposi- 
 
 dv da 
 
 tion is that made in equation (456), whence equation (458) has resulted. The 
 
IN THE AKCH. 
 
 439 
 
 Let 
 
 (1+X) cos. . 
 value of , 
 
 T 
 
 ; therefore -= 
 Substituting this 
 
 cos. 
 
 jl-(l+X) cos. cos. }(-- 
 
 -1 =(ia a + a) 
 . cos.)sin.Y I 
 
 by which equation the angle of rupture Y is determined. 
 
 If the arch be a continuous segment the joint AD is ver- 
 tically above the centre, and CD coinciding with CE, =0; 
 if it be a broken segment, as in the Gothic arch, has a 
 given value determined by the character of the arch. In the 
 pure or equilateral Gothic arch, = 30. Assuming 0=0, 
 and reducing, 
 
 (460.) 
 
 It may easily be shown that as Y increases in this equa- 
 tion, Y increases, and conversely ; so that as the load is 
 increased, the points of rupture descend. When Y=0, or 
 there is no load upon the extrados, 
 
 ..... (461). 
 
 7T> 
 
 hypotheses -=0, p = r, determine the minimum of the pressures P, which 
 
 do 
 
 being applied to a given point of the key-stone will prevent the semi-arch from 
 turning on any of the successive joints of its voussoirs. 
 
440 
 
 THE LINE OF RESISTANCE. 
 
 When 05=0, or the load is placed on the crown of the 
 arch, 
 
 Y/ 1 2 
 \^-a -f 
 
 ? r ~" 
 
 Y 
 
 l ' 
 
 x 
 "When - 
 
 / 
 ( tan. -^ 
 
 cot. 
 
 \ Y 
 
 I = 0,-^- becomes infinite ; 
 
 an infinite load is therefore required to give that value to the 
 angle of rupture which is determined by this equation. 
 
 Y 
 Solved in respect to tan. , it gives, 
 
 2 
 
 tan 
 
 2 
 
 . . (463). 
 
 No loading placed upon the arch can cause the angle of rup- 
 ture to exceed that determined by this equation. 
 
 THE LINE OF RESISTANCE IN A CIRCULAR ARCH WHOSE 
 VOUSSOIRS ARE EQUAL, AND WHOSE LOAD IS DISTRIBUTED 
 OVER DIFFERENT POINTS OF ITS EXTRADOS. 
 
 338. Let it be supposed that the pressure of the load is 
 wholly vertical, and such that any 
 portion FT of the extrados sustains 
 the weight of a mass GFTY imme- 
 diately superincumbent to it, and 
 bounded by the straight line GY 
 inclined to the horizon at the an- 
 gle t; let, moreover, the weight 
 of each cubical unit of the load be 
 equal to that of the same unit of 
 the material of the arch, multiplied 
 by the constant factor ^ ; then, re- 
 presenting AD by K/3, ACF by @, 
 ACT by 0, and DZ by z, we have, 
 
 area GFTY 
 
 = /w. 
 
 e 
 
 dz: 
 
SEGMENTAL ARCH. 
 
 441 
 
 but TY= MZ-(MT+ YZ), and MZ = CD = E-hE/3, MT= 
 E cos. 0, YZ = DZ tan. * = E sin. 6 tan. t. Therefore MT+ 
 YZ E cos. 0+E sin. 6 tan. i = E {cos. 6 cos. *+sin. sin. 1} 
 sec. t = E cos. (0i) sec. * ; 
 
 /. TY= E{1 + /3-cos. (0-*) sec. 1} ; 
 
 also, s=DZ=E sin. 0; 
 
 
 
 .-. areaGFTY=y*TY . ^de=wJ*{l+(3- 
 
 e 
 
 cos. (0*) sec. *} cos. 0d0 ; 
 
 e 
 
 .-. Y=weight of mass GFTY=f*E a C{1 +/3 see.* cos.(0)f 
 
 e 
 
 cos. 0^0 = fxE 3 1 (1 +)8) (sin. sin. 0) i sec. * {sin. (2 Bi) 
 
 sin.(20-t)} J(0-0) ... (464).* 
 
 e 
 = moment of GFTY= ^E 3 /* {(1 +13) sec. t cos. (0 
 
 
 
 sin. cos. 0^0=fxE 3 {i(l+/3)(cos. 3 0-cos. 2 0) J(cos. '0 
 cos. '0)- 1 tan. i (sin. 8 sin. 8 0) . . . 465).* 
 
 A SEGMENTAL ARCH WHOSE EXTRADOS IS HORIZONTAL. 
 
 -v a 339. As the simplest case, let us first 
 
 1 i || 1 1^| | ' i-H suppose DY horizontal, the material of 
 '"' 'i ! I 'J^TfTTl A the loading similar to that of the arch, 
 and the crown of the arch at A, so 
 that i=0, f*=l, and 0=0. Substi- 
 tuting the values of Y and Ya? (equa- 
 tions 464, 465) which result from these 
 suppositions, in equation (455), solving 
 
 that equation in respect to _, and re- 
 
 ducing, we have, = 
 
 r. 
 
 * See Note 2, at end of PART IY. ED. 
 
442 
 
 THE GOTHIC A ECU. 
 
 Assuming -= =0 (see note, page 438.), and X = a , and 
 reducing, 
 
 1(1 -2a) cos. 3 T _ {(l_ a l + /3 + l+ a i_2 a cos. 
 
 COS. 
 
 In the case in which the line of resistance passes through 
 the bottom of the key-stone, so that X 0, equation (466) 
 becomes 
 
 (l + cos. Y) cos. T ^Ycot. JY+J=0 ____ (468); 
 whence assuming _4_ 0, we have 
 
 Sill, i 
 
 -a COS. 
 =0. . . .(469.) 
 
 GOTHIC AECH, THE EXTRADOS OF EACH SEMI-ARCH BEING- 
 A STRAIGHT LINE INCLINED AT ANY GIVEN ANGLE TO THE 
 HORIZON, AND THE MATERIAL OF THE LOADING DIFFERENT 
 FROM THAT OF THE ARCH. 
 
 340. Proceeding in respect to this general case of the 
 stability of the circular arch, by precisely the same steps as 
 in the preceding simpler case, we obtain from equation 
 (455), 
 
 |cos.i--(l-|-A)co8.e} ...(470) 
 
THE GOTHIC ARCH. 44:3 
 
 in which equation the values of Y and Ya? are those deter- 
 mined by substituting Y for 6 in equations (464) and (465). 
 
 Differentiating it in respect to Y, assuming-^- =0 (note, 
 p. 438.), and X a , we obtain 
 
 '-ia 3 K) cos. sin. T_( a 2 + a ) sin. Y cos. * 
 
 -(l+a) COS. COS. {(* 0) 
 
 Y ~Yx 
 }1 (1 + a) cos. ^ cos. }-^ + a sin. ^ + {cos. Y 
 
 . j 1 d(Yx) sin. Y <ZY) 
 (l + ) cos. 0} I - 3 -^--^-^=0 . . . (471). 
 
 Y YOJ 
 Substituting in this equation the values of -5 and 5 , de 
 
 termined by equations (464) and (465) the following equa 
 tion will be obtained after a laborious reduction : it deter 
 mines the value of Y : 
 
 A+B cos. T C cos. 3 Y + D cos. 3 T+E sin. Y 
 F sin. Y cos. T G sin. 3 Y H cot. 
 
 I(1-K cos. T) +a =0 .... (472) 
 where 
 
 ") tan f sin 8 0-(l+/3) J2-(l+) 
 
 |(l+a) COS. 8 i +(2a + a 2 a 8 fa 4 ) COS. 
 
 COS. 0-(l~f*)} +1. 
 
 l-2 a COS. . 
 
 a ) 2 (l 2a) tan. i=fD tan. i. 
 
 a ) 3 (l 2a) tan. i COS. =E(l+a) COS. 0. 
 
 a ) 2 (l 2a) tan. i=~D tan. i. 
 
 /3-sec. t cos. -* sin. 20. 
 
 K=(l+a) COS. . 
 
 L=M-(l + a ) 2 {2(H-/3) sec. i cos. ( )}sin. . 
 
 Tables might readily be constructed from this or any of 
 
44:4 AN ARCH SUSTAINING THE PRESSURE OF WATER. 
 
 the preceding equations by assuming a series of values of , 
 and calculating the corresponding values of (3 for each given 
 value of a, *, M-, . The tabulated results of such a series of 
 calculations would show the values of Y corresponding to 
 given values of a, /3, t, (*, . These values of Y being sub- 
 stituted in equation (470), the corresponding values of the 
 horizontal thrust- would, be determined, and thence the polar 
 equation to the line of resistance (equation 454). 
 
 A CIRCULAR ARCH HAVING EQUAL VOUSSOIRS AND SUSTAINING 
 THE PRESSURE OF 'WATER. 
 
 341. Let us next take a case of oblique pressure on the 
 extrados, and let us suppose it to be 
 the pressure of water, whose surface 
 stands at a height /3R above the sum- 
 mit of the key-stone. The pressure of 
 this water being perpendicular to the 
 extrados will everywhere have its di- 
 rection through the centre C, so that its 
 motion about that point will vanish, 
 and Y# Xy=0; moreover, by the 
 principles of hydrostatics,* the vertical 
 component Y of the pressure of the water, superincumbent 
 to the portion AT of the extrados, will equal the weight of 
 that mass of water, and will be represented by the formula 
 (464), if we assume i=0. The horizontal component Xf of 
 the pressure of this mass of water is represented by the 
 formula 
 
 -cos . sn. =^ + + / cos. - 
 
 
 
 cos. 0) J(cos. 2 cos. 2 4)J ..... (473). 
 
 Assuming then =0, we have (equation 464), in respect 
 to that portion of the extrados which lies between the crown 
 and the points of rupture, 
 
 Y 
 
 sin. T- sin. 2 Y- 
 
 and (equation 473) =^(1 + a ) 9 {(1+/3) vers. Y -J sin. 
 
 * See Hydrostatics and Hydrodynamics, p. 30, 31. 
 | See Note 3, end of PART IV. ED. 
 
AN AIICR SUSTAINING THE PRESSURE OF WATER. 445 
 
 ... sin. y_ cos. Y=Kl+ a )l+/ vers. Y_T sn. v 
 
 ...... (474). 
 
 Substituting this value in equation (455), making Yx Xy=0, 
 
 P #> 
 
 solving that equation in respect to and making =1+X, 
 
 we have 
 
 a_|_vers. * 
 
 If, instead of supposing the pressure of the water to be 
 borne by the extrados, we suppose it to 
 take effect upon the intrados, tending to 
 blow up the arch, and if /3 represent the 
 __^^_ __ height of the water above the crown of 
 the intrados, we shall obtain precisely 
 the same expressions for X and Y as 
 M before, except that r must be substituted 
 m for (1 + aV, and X and Y must be taken 
 
 Y X 
 
 negatively ; in this case, therefore, -5- sin. Y ,- cos. = 
 
 f*}(l + /3) vers. |T sin. Y} ; whence, by substitution in 
 equation (455), and reduction, 
 
 P (-ia' + a-f-^Ysin.Y j a _f- a 3 -f i a s -f fA(l_f-/3)}vers.Y , 
 
 > . (4:70) 
 
 Now by note, page 438, -^ =0 ; differentiating equa- 
 tions (475) and (476), therefore, and reducing, we have, 
 
 tan. Xcot.^ vers. Y+Ax=0 ..... (477); 
 
 which equation applies to both the cases of the pressure of a 
 fluid upon an arch with equal voussoirs ; that in which its 
 pressure is borne by the extrados, and that in which it is 
 borne by the intrados ; the constant A representing in the 
 
 , ,., 
 
 first case the quantity --- 1 3 ~7 v~7T~7 \* j an d in the 
 
 20. +a tjf^l-l-a) 
 
 K + X + KH-/ 3 ) .1 r 
 second case --* . If the line of resistance 
 
 Ja +-f"t* 
 
 pass through the summit of the key-stone, X must be taken =a, 
 
446 EQUILIBRIUM OF AN ARCH. 
 
 If it pass along the inferior edge of the key-stone, 
 
 Y 
 
 > ^=0. In this second case, tan. {Y sin. "} 0, therefore, 
 
 1=0; so that the point of rupture is at the crown of the 
 arch. For this value of Y equations (475) and (476) become 
 vanishing fractions, whose values are determined by known 
 methods of the differential calculus to be, when the pressure 
 is on the extrados, 
 
 <*)'.... (4T8); 
 when the pressure is on the intrados, 
 
 =-f,'-ft ..... (479). 
 
 It is evident that the line of resistance thus passes through 
 the inferior edge of the key-stone, in that state of its equili- 
 brium which precedes its rupture, by the ascent of its crown. 
 The corresponding equation to the line of resistance is deter- 
 
 , P 
 mined by substituting the above values of in equation 
 
 (454). In the case in which the pressure of the water is 
 sustained by the intrados, we thus obtain, observing that 
 
 X X 
 
 sinJ -- 3 cos. &= (*|(1 +/3) vers. 6^6 sin. dj; 
 
 _ a a + 2 a -/fy-( a 8 + a a + a )cos. 6 
 
 =r2 ----^ - '- * ' * ( '* 
 
 If for any value of 6 in this equation, less than the angle 
 of the semi-arch, the corresponding value of p exceed 
 (l+ay, the line of resistance will intersect the extrados, c 
 the arch will blow up. 
 
 THE EQUILIBRIUM OF AN ARCH, THE CONTACT OF WHOSE 
 VOUSSOIRS IS GEOMETRICALLY ACCURATE. 
 
 342. The equations (459) and (456) completely determine 
 
EQUILIBRIUM OF AN ARCH. 
 
 447 
 
 N| 
 
 the value of P, subject to the first 
 of the two conditions stated in 
 Art. 333., viz. that the line of re- 
 sistance passing through a given 
 point in the key-stone, determined 
 by a given value of \ shall have a 
 point of geometrical contact with 
 the intrados. It remains now to 
 determine it subject to the second 
 condition, viz. that its point of ap- 
 plication P on the key-stone shall 
 be such as to give it the least va- 
 lue which it can receive subject to 
 the first condition. It is evident 
 that, subject to this first condition, every different value of 
 X will give a different value of "V ; and that of these values of 
 "Y that which gives the least value of P, and which corres- 
 ponds to a positive value of X not greater than a, will be the 
 true angle of rupture, on the hypothesis of a mathematical 
 adjustment of the surfaces of the voussoirs to one another. 
 To determine this minimum value of P, in respect to the va- 
 riation of Y dependent on the variation of X or of p, let it be 
 observed that X does not enter into equation (456) ; let that 
 equation, therefore, be differentiated in respect to P and Y, 
 
 and let -=- be assumed =0, and Y constant, we shall thence 
 du * 
 
 obtain the equation 
 
 sec. * = 
 
 
 whence, observing that 
 
 sec. 
 
 we obtain by elimination in equation (456) 
 
 4T 
 
 sin. 2Y 2Y= 
 
 : 20 
 
 (482), 
 
 from which equation Y may be determined. Also by equa- 
 tion (481) 
 
448 APPLICATIONS OF THE THEORY OF THE ARCH. 
 
 ---- (483); 
 
 and by eliminating sec. Y between equations (457) and (481), 
 and reducing, 
 
 cos. e= ) |/ a ( a +2) { 
 
 Yi ) 
 
 .--prl >. . . (484). 
 
 The value of X given b y this equation determines the actual 
 direction of the line of resistance through the key-stone, on 
 the hypothesis made, only in the case in which it is a positive 
 quantity, and not greater than a ; if it be negative, the line 
 of resistance passes through the bottom of the key-stone, or 
 if it be greater than a, it passes through the top. 
 
 Such a mathematical adjustment of the surfaces of contact 
 of the voussoirs as is supposed in this article is, in fact, sup- 
 plied by the cement of an arch. It may therefore be con- 
 sidered to involve the theory of the cemented arch, the influ- 
 ence on the conditions of its stability of the adhesion of its 
 voussoirs to one another being neglected. In this settlement, 
 an arch is liable to disruption in some of those directions in 
 which this adhesion might be necessary to its stability. That 
 old principle, then, which assigns to it such proportions as 
 would cause it to stand firmly did no such adhesion exist, 
 will always retain its authority with the judicious engineer. 
 
 APPLICATIONS OF THE THEORY OF THE ARCH. 
 
 343. It will be observed that equation (459) or (472) 
 determines the angle "f of rupture in terms of the load Y, 
 and the horizontal distance x of its centre of gravity from 
 the centre C of the arch, its radius r, and the depth &r of its 
 voussoirs ; moreover, that this determination is wholly inde- 
 pendent of the angle of the arch, and is the same whether 
 its arc be the half or the third of a circle ; also, that if the 
 angle of the semi-arch be less than that given by the above 
 equation as the value of Y, there are no points of rupture, 
 such as they have been defined, the line of resistance passing 
 through the springing of the arch and cutting the iiitradoa 
 there. 
 
THEORY OF THE ARCH. 449 
 
 The value of Y being known from this equation, P is 
 determined from equation (456), and this value of P being 
 substituted in equation (454), tne line of resistance is com- 
 pletely determined ; and assigning to d the value ACB 
 (p. 437.), the corresponding value of p gives us the position 
 of the point Q, where the line of resistance intersects the 
 lowest voussoir of the arch, or the summit of the pier. 
 Moreover, P is evidently equal to the horizontal thrust on 
 the top of the pier, and the vertical pressure upon it is the 
 weight of the arch and load: thus all the elements are 
 known, which determine the conditions of the stability of a 
 pier or buttress (Arts. 293. and 312.) of given dimensions 
 sustaining the proposed arch and its loading. 
 
 Every element of the theory of the arch and its abutments. 
 is involved, ultimately, in the solution in respect to- V of 
 equation (459) or equation (472). Unfortunately this solu- 
 tion presents great analytical difficulties.. In the- failure of 
 any direct means of solution, there are, however, various 
 methods by which the numerical 1 relation of Y and Y may 
 be arrived at indirectly. Among thei% one of the simplest 
 is this : 
 
 Let it be observed that that equation is readily soluble in 
 respect to Y ; instead, then, of determining the value of Y 
 for an assumed value of Y, determine conversely the value 
 of Y for a series of assumed values of Y. Knowing the dis- 
 tribution of the load Y, the values of as will be known in 
 respect to these values of Y, and thus the values of Y may 
 be numerically determined, and may be tabulated. From 
 such tables may be found, by inspection, values of Y corres- 
 ponding to given values of i . 
 
 The values of Y, P, and r are completely determined by 
 equations (482, 483, 484), and all the circumstances of the 
 equilibrium of the circular arch are thence known, on the 
 hypothesis, there made, of a true mathematical adjustment 
 of the surfaces of the voussoirs to one another ; and although 
 this adjustment can have no existence in practice when, 
 the voussoirs are put together without cement, yet may it 
 obtain in the cemented arch. The cement, by reason of 
 its yielding qualities when fresh, is made to enter into so 
 intimate a contact with the surfaces of the stones between 
 which it is interposed that it takes, when dry, in respect 
 to each joint (abstraction being made of its adhesive proper- 
 ties), the character of an exceedingly thin voussoir, having 
 its surfaces mathematically adjusted to those of the adjacent 
 v^ussoirs ; so that if we imagine, not the adhesive properties 
 
 29 
 
450 APPLICATIONS OF THE 
 
 of the cement of an arch, but only those which tend to tho 
 more uniform diffusion of the pressures through its mass, to 
 enter into the conditions of its equilibrium, these equations 
 embrace the entire theory of the cemented arch. The hypo- 
 thesis here made probably includes all that can be relied 
 upon in the properties of cement as applied to large struc- 
 tures. 
 
 An arch may FALL either by the sinking or the rising of 
 its crown. In the former case, the line of resistance passing 
 through the top of the key-stone is made to cut the extrados 
 beneath the points of rupture ; in the latter, passing through 
 the bottom of the key-stone, it is made to cut the extrados 
 between the points of rupture and the crown. 
 
 In the first case the values of X, Y, and P, being deter- 
 mined as before and substituted in equation (454), and p 
 being assumed = (!+)?, the value of d, which corresponds 
 to p=(l-f-a)r, will indicate the point at which the line of 
 resistance cuts the extrados. If this value of d be less than 
 the angle of the semi-arch, the intersection of the line of 
 resistance with the extrados will take place above the 
 springing, and the arch will fall. 
 
 In the second case, in which the crown ascends, let the 
 maximum value of p be determined from equation (454), p 
 being assumed =r ; if this value of p be greater than R, and 
 the corresponding value of & less than the angle of rupture, 
 the line of resistance will cut the extrados, the arch will 
 open at the intrados, and it will fall by the descent of the 
 crown. 
 
 If the load be collected over a single point of the arch, 
 the intersection of the line of resistance with the extrados 
 will take place between this point and the crown ; it is that 
 portion only of the line of resistance which lies between these 
 points which enters therefore into the discussion. Now if 
 we refer to Art. 336., it will be apparent that in respect to 
 this portion of the line, the values of X and Y in equations 
 (453) and (454) are to be neglected ; the only influence of 
 these quantities being found in the value of P. 
 
THEOKY OF THE ARCH. 
 
 451 
 
 Example 1. Let a circular arch of equal voussoirs have 
 the depth of each voussoir equal to 
 
 I T Vth the diameter of its intrados, so 
 r that a=*2, and let the load rest upon 
 _^ it by three points A, B, D of its 
 
 J A extrados, of which A is at the crown 
 and B D are each distant from it 45 ; 
 and let it be so distributed that -fths 
 of it may rest upon each of the points 
 B and D, and the remaining J upon 
 A ; or let it be so distributed within 
 60 on either side of the crown as to 
 produce the same effect as though it 
 rested upon these points. 
 
 Then assigning one half of the load 
 upon the crown to each semi-arch, 
 and calling x the horizontal distance 
 of the centre of gravity of the load upon either semi-arch 
 
 from C, it may easily be calculated that - = sin. 45 = 
 
 5303301. Hence it appears from equation (463) that no 
 loading can cause the angle of rupture to exceed 65. 
 Assume it to equal 60; the amount of the load necessary to 
 produce this angle of rupture, when distributed as above, 
 will then be determined by assuming in equation (460), 
 
 ^=60, and substituting a for X, -2 for a, and -5303301 for?. 
 
 Y Y 
 
 "We thus obtain -^=-0138. Substituting this value of -,, and 
 
 also the given values of a and T in equation (457), and 
 
 observing that in this equation - is to be taken =1+ a and 
 
 r 
 
 P P 
 
 0=0, we find -5 = -11832. Substituting this value of in 
 
 the equation (454), we have for the final equation to the line 
 of resistance beneath the point B 
 
 2426 vers.^H- -1493 
 
 0138 sin. & + -1183 cos. & + -22 & sin. 0* 
 
452 
 
 APPLICATIONS OF THE 
 
 If the arc of the -arch be a com- 
 plete semicircle, the value of p in this 
 
 equation corresponding to d = - will 
 
 a 
 
 determine the point Q, where the 
 line of resistance intersects the abut- 
 ment; this value is p=~L'Q9r. 
 
 If the arc of the arch be the third 
 o of a circle, the value of p at the 
 abutment is that corresponding to 
 
 & = - ; this will be found to be r, as 
 o 
 
 it manifestly ought to be, since the 
 points of rupture are in this case at 
 the springing. 
 
 In the first case the volume of the semi-arch and load is 
 represented by the formula 
 
 
 1 , ' 
 
 l^[ 
 
 /I 
 
 
 !/ 
 
 I 
 
 / 1 
 
 
 ;1 
 
 i 
 
 / 1 
 
 J- 
 
 /I 
 
 r 1 
 
 
 1 
 
 ! 
 
 1 
 
 and in the second case by 
 
 Thus, supposing the pier to be of the same material as the 
 arch, the volume of its material, which would have a weight 
 equal to the vertical pressure upon its summit, would in the 
 first case be *3594r 2 , and in the second case -2442^ 2 , whilst 
 the horizontal pressures P would in both cases be the same, 
 viz. -11832?' 2 ; substituting these values of the vertical and 
 horizontal pressures on the summit of the pier, in equation 
 (377), and for Ic writing -J- a (p r\ we have in the first 
 case 
 
 n [ _ 
 
 _ -3594(0 
 
 and in the second case, 
 H= 
 
 -2442 ar* 
 
THEORY OF THE AECH. 453 
 
 where H is the greatest height to which a pier, whose width 
 is a, can be built so as to support the arch. 
 
 If \tf -11832^=0, or =-4864r, then in either case the 
 pier may be built to any height whatever, without being 
 overthrown. In this case the breadth of the pier will be 
 nearly equal to Jth of the span. 
 
 The height of the pier being given (as is commonly the 
 case), its breadth, so that the arch may just stand firmly 
 upon it, may readily be determined. As an example, let us 
 suppose the height of the pier to equal the radius of the 
 arch. Solving the above equations in respect to a, we shall 
 then obtain in the first case en = 'SOTS/ 1 , and in the second 
 a='3r. 
 
 If the span of each arch be the same, and r l and r 2 repre- 
 sent their radii respectively, then r l =r 9 sin. 60* ; supposing 
 then the height of the pier in the second arch to be the same 
 as that in the first, viz. r^ then in the second equation we 
 must write for H, r^ sin. 60. We shall thus obtain for a the 
 value "28/v 
 
 The piers shown by the dark lines in the preceding 
 figures are of such dimensions as just to be sufficient to 
 sustain the arches which rest upon them, and their loads, 
 both being of a height equal to the radius of the semicircular 
 arch. It will be observed, that in both cases the load 
 Y='0138r 2 , being that which corresponds to the supposed 
 angle of rupture 60, is exceedingly small. 
 
 Example 2. Let us next take the example of a Gothic 
 arch, and let us suppose, as in the last examples, that the 
 angle of rupture is 60, and that a='2; but let the load in 
 this case be imagined to be collected wholly over the 
 
 crown of the arch, so that - = sin. 30. Substituting in equa- 
 tion (459), 30 for 0, and 60 for T, and -2 for a, and sin. 30 
 
 for -, we shall obtain the value *21015 for - ; whence bv 
 
 T r* ' J 
 
 p 
 
 equation (457)-^- = -2405, and this value being substituted, 
 
454 
 
 APPLICATIONS OF THE 
 
 equation (454) gives 1*1457' foi 
 
 the value of p when = _. "We 
 
 2 
 
 have thus all the data for deter- 
 mining the width of a pier of 
 given height which will just 
 support the arch. Let the 
 height of the pier be supposed, 
 as before, to equal the radius 
 of the intrados ; then, since the 
 weight of the semi-arch and its 
 load is 5556r 2 , and the horizon- 
 tal thrust -24057 12 , the width a 
 of the pier is found by equation 
 (379) to be 4195r. 
 The preceding figure represents this arch ; the square, 
 formed by dotted lines over the crown, shows the dimensions 
 of the load of the same materials as the arch which will cause 
 the angle of the rupture to become 60 ; the piers are of the 
 required width ^l&S/ 1 , such that when their height is equal 
 to AB, as shown in the figure, and the arch bears this insist- 
 ent pressure, they may be on the point of overturning. 
 
 TABLES OF THE THRUST OF ARCHES. 
 
 344. It is not possible, within the limits necessarily 
 assigned to a work like this, to enter further upon the dis- 
 cussion of those questions whose solution is involved in the 
 equations which have been given ; these can, after all, be- 
 come accessible to the general reader, only when tables shall 
 be formed from them. 
 
 Such tables have been calculated with great accuracy by 
 M. Garidel in respect to that case of a segmental arch* whose 
 loading is of the same material as the voussoirs, and the ex- 
 trados of each semi-arch a straight line inclined at any given 
 angle to the horizon. These tables are printed in the Ap- 
 pendix (Tables 2, 3). 
 
 * The term segmental arch is used, here and elsewhere, to distinguish that 
 form^of the circular arch in which the intrados is a contiguous segment from 
 that in which it is composed of two segments struck from different centres, aa 
 in the Gothic arch. 
 
THEORY OF THE ARCH. 455 
 
 Adopting the theory of Coulomb*, M. Garidel has arrived 
 at an equationf which becomes identical with equation (472) 
 in respect to that particular case of the more general condi- 
 tions embraced by that equation, in which ^=1 and 0=0. 
 
 By an ingenious method of approximation, for the details 
 of which the reader is referred to his work, M. Garidel has 
 determined the values of the angle of rupture Y, and the 
 
 p 
 quantity , in respect to a series of different values of a and 
 
 (3. The results are contained in the tables which will be 
 found at the end of this volume. 
 
 p 
 The value of -^ being known from the tables, and the 
 
 values of Y and Ya? from eouations (464), (465), the line of 
 resistance is determined by the substitution of these values 
 in equation (454). The line of resistance determines the 
 point of intersection of the resultant pressure with the sum- 
 mit of pier ; the vertical and horizontal components of this 
 resultant pressure are moreover known, the former being the 
 weight of the semi-arch, and the other the horizontal thrust 
 on the key. All the elements necessary to the determina- 
 tion of the stability of the piers (Arts. 289 and 312) are 
 therefore known. 
 
 It will be observed that the amount of the horizontal 
 thrust for each foot of the width of the soffit is determined 
 
 p 
 by multiplying the value of a , shown by the tables, by the 
 
 square of the radius of the intrados in feet, and by the 
 weight of a cubic foot of the material 
 
 * See Mr. Hann's Theory of Bridges, Art. 16. ; also p. 24. of the Memoir on 
 the Arch by the author of this work, contained in the same volume, 
 f Tables des Poussees des Voutes, p. 44. Paris, 1837. Bachelier. 
 
456 NOTES TO PAET IV. 
 
 NOTE 1. PART IY. 
 
 The length of an elementary arc ds of the intrados AS subtending the angle 
 dd is expressed by rdd ; an elementary volume of the arch will therefore be 
 expressed by rdftdr ; the perpendicular distance of the centre of gravity of 
 this volume from the vertical line CE is r sin. ; the moment of this volume, 
 with regard to CE, is therefore rdddr^r sin.0=rVr sin. 6d6; then from (Art. 
 31.) equation (20) there obtains 
 
 R 
 Wi=fr*drjlm. Odd. 
 
 NOTE 2. PART IV. General integrals of equations 464, 465. 
 ^Phe general integral, (equation 464) 
 
 y{l-|__co8. (0-fr) sec. i } cos. 6dd=f(l+(3) cos. 6dff 
 
 /sec. i (cos. cos. i+sin. 6 sin. i) cos. W0= /(l-f-0) cos. Odd 
 
 j sec. i cos. i cos. 2 0oR9 / see. i sin. i sin. 6 cos. 0d0. 
 But^y (l-f-/3) cos. 0o?0=(l-f-/?) sin. 0; /sec. t cos. t cos. *6dd= 
 
 . t cos. i^y (- | OS 2 )c?0=sec. t cos.t(-0-f- sin. 20); /sec. t sin t 
 
 /j j 
 
 -sin. 20cf(20) = sec. i sin. t cos. 20; 
 4 4 
 
 ,-.J | 1+/3-COS. (0-t) sec. i j-cos. 0c?0=(l-j-/?) sin. 0-i sec. z 
 (sin. 20 cos. t sin.t cos. 20)-^0=:(l+)8) sin. 0-^sec. tsin. (20- 4 )-^ 
 
 The general integral, / {(1 -}-/?) sec. t cos. (0 t)} sin. cos. 0<f0, (equation 
 465), =y(l -f/3) sin. cos. Odd i* sec. i {cos.Pcos. f-f-sin.08in. i}sin.0 cos.0^0. 
 
 sec 
 
 cos. 20= 
 
NOTES TO PART IV. 457 
 
 /%ec. tj cos. 6 cos. t-fsin. 0sin. 1 j sin. 6 cos. fid . 0==y co8. 2 0sin. 
 /tan. i sin. U 0cos. 6dd . /cos. 2 0dcos. 0-f-/tan. * sin - 2 ^ sin - 0= 
 
 _L COS . "0_f_I sin. "0 
 3 o 
 
 ... f | (14.5) -see. i cos. (0-0 j- sin. cos. 0rf . -|-(f0= -^(1-f/?) cos. 3 
 L (l-|-/3)4- J- cos. 3 0-i- tan. t sin. 8 . tan t. 
 
 4 o o 
 
 NOTE 3. PART IV. 
 
 In equation (427), (Art. 319), by making 0=0, we obtain P= Hi x* ; since 
 tan. r=l, and this answers to the case of the horizontal pressure of a perfect 
 
 fluid like water. From this expression there obtains dl*=/j.ixdx j to express 
 the elementary pressure at any depth x below the surface. This depth in 
 (Art. 341), equation (473), is TV=AD+AB=AD-f AC-BC=/3R-f R-Rcos.0, 
 
 .-. dP=iiR(l4-{3 cos. 6)Rd (1-f-/? cos. 0)=//R 2 J1+/3 cos. 0f sin. 
 
 * * ) 
 
 
 .-. P=X= y uR 2 > / > Jl-f /3-cos. } sin. 0d0. 
 
 e 
 
T V. 
 
 THE STRENGTH OF MATERIALS. 
 
 ELASTICITY. 
 
 345. From numerous experiments which have been made 
 upon the elongation, flexure, and torsion of solid bodies 
 under the action of given pressures, it appears that the 
 displacement of their particles is subject to the following 
 laws. 
 
 1st. That when this displacement does not extend beyond 
 a certain distance, each particle tends to return to the place 
 which it before occupied in the mass, with a force exactly 
 proportional to the distance through which it has been 
 displaced. 
 
 2dly. That if this displacement be carried beyond a 
 certain distance, the particle remains passively in the new 
 position which it has been made to take up, or passes finally 
 into some other position different from that from which it 
 was originally moved. 
 
 The effect of the first of these laws, when exhibited in 
 the joint tendency of the particles which compose any finite 
 mass to return to any position in respect to the rest of the 
 mass, or in respect to one another, from which they have 
 been displaced, is called elasticity. There is every reason to 
 believe that it exists in all bodies within the limits, more or 
 less extensive, which are imposed by the second law stated 
 above. 
 
 The force with which each separate particle of a body 
 tends to return to the position from which it has been 
 displaced varying as the displacement, it follows that the 
 force with which any aggregation of such particles, consti- 
 tuting a finite portion of the body, when extended or 
 compressed within the limits of elasticity, tends to recover 
 its form, that is the force necessary to keep it extended or 
 
ELONGATION. 459 
 
 compressed, is proportional to the amount of the extension 
 or compression ; so that each equal increment of the extend- 
 ing or compressing force produces an equal increment of its 
 extension or compression. This law, which constitutes 
 perfect elasticity, and which obtains in respect to fluid and 
 gaseous bodies as well as solids, appears first to have been 
 established by the direct experiments of S. Gravesande on 
 the elongation of thin wires.* 
 
 It is, however, by its influence on the conditions of 
 deflexion and torsion that it is most easily recognized as 
 characterizing the elasticity of matter, under all its solid 
 forms, f within certain limits of the displacement of its 
 particles or elements, called its elastic limits. 
 
 ELONGATION. 
 
 346. To determine tTie elongation or compression of a bar of 
 a given section under a given strain. 
 
 Let K be taken to represent the section of the bar in 
 square inches, L its length in feet, I its elongation or com- 
 pression in feet under a strain of P pounds, and E the strain 
 or thrust in pounds which would be required to extend a 
 bar of the same material to double its length, or to compress 
 
 * For a description of the apparatus of S. Gravesande, see Illustrations of 
 Mechanics, by the Author of this work, 2d edition, p. 30. In one of his 
 experiments, Mr. Barlow subjected a bar of wrought iron, one square inch in 
 section, to strains increasing successively from four to nine tons, and found the 
 elongations corresponding to the successive additional strains, each of one ton, 
 to be, in millionths of the whole length of the bar, 120, 110, 120, 120, 120. 
 In a second experiment, made with a bar two square inches in section, under 
 strains increasing from 10 tons to 30 tons, he found the additional elongations, 
 produced by successive additional strains, each of two tons, to be, in millionths 
 of the whole length, 110, 110, 110, 110, 100, 100, 100, 100, 95, 90. From an 
 extensive series of similar results, obtained from iron of different qualities, he 
 deduced the conclusion that a bar of iron of mean quality might be assumed 
 to elongate by 100 millionth parts, or the 10,000th part, of its whole length, 
 under every additional ton strain per square inch of its section. (Report to 
 Directors of London and Birmingham Railway. Fellowes, 1835.) 
 
 The French engineers of the Pont des Invalides assigned 82 millionth parts 
 to this elongation, their experiments having probably been made upon iron of 
 inferior quality. M. Vicat has assigned 91 millionth parts to the elongation 
 of cables of iron wire (No. 18.) under the same circumstances, MM. Minard 
 and Desormes, 1,176 millionth parts to the elongation of bars of oak. 
 (lllust. Mech., p. 393.) 
 
 f The experiments of Prof. Robison on torsion show the existence of this 
 property in substances where it might little be expected; in pipe-clay, for 
 instance. 
 
460 THE WOEK EXPENDED ON ELONGATION. 
 
 it to one half its length, if the elastic limit of the material 
 were such as to allow it to be so far elongated or compressed. 
 the law of elasticity remaining the same.* 
 
 Now, suppose the bar, whose section is K square inches, 
 to be made up of others of the same length L, each one inch 
 in section ; these will evidently be K in number, and th'3 
 
 p 
 strain or the thrust upon each will be represented by ^. 
 
 Moreover, each bar will be elongated or compressed, by this 
 strain or thrust, by I feet ; so that each foot of the length of 
 it (being elongated or compressed by the same quantity as 
 each other foot of its length) will be elongated or compressed 
 
 by a quantity represented, in feet, by y. But to elongate 
 
 or compress a foot of the length of one of these bars, by one 
 foot, requires (by supposition) E pounds strain or thrust ; to 
 
 elongate or compress it by - feet requires, therefore, 
 
 j 
 
 pounds. But the strain or thrust which actually produces 
 
 P P I 
 
 this elongation is =^ pounds. Therefore,^ = E^-. 
 
 PT 
 
 347. To find the number of units of work expended upon the 
 elongation ly a given quantity (I) of a bar whose section is 
 K and its length L. 
 
 If x represent any elongation of the bar (x being a part 
 of l\ then is the strain P corresponding to that elongation 
 
 KE 
 
 represented (equation 485) by -y-#; therefore the work 
 
 done in elongating the bar through the small additional 
 
 KE 
 space A#, is represented by -y-a?A# (considering the strain 
 
 to remain the same through the small space Aa?) ; and the 
 
 * The value of E in respect to any material is called the modulus of its elas- 
 ticity. The value of the moduli of elasticity of the principal materials of con- 
 struction have been determined by experiment, and will be found in a table at 
 the end of the volume. 
 
THE WORK EXPENDED ON ELONGATION. 461 
 
 whole work U done is, on this supposition, represented by 
 -= 2&A&, or (supposing A#J to be infinitely small) by 
 
 KE / . KE 7 , 
 
 -f-J d% or by ITT^ 
 
 (486). 
 
 TTT? 
 
 348. By equation (485) P=-pZ, therefore U = 
 
 whence it follows that the work of elongating the bar is one 
 half that which would have been required to elongate it by 
 the same quantity, if the resistance opposed to its elongation 
 had been, throughout, the same as its e'xtreme elongation I. 
 If, therefore, the whole strain P corresponding to the 
 elongation I had been put on at once, then, when the elonga- 
 tion I had been attained, twice as much work would have 
 been done upon the bar as had been expended upon its 
 elasticity. This work would therefore have been accumu- 
 lated in the bar, and in the body producing the strain under 
 which it yields ; and if both had been free to move on (as, 
 for instance, when the strain of the bar is produced by a 
 weight suspended freely from its extremity), then would 
 this accumulated work have been just sufficient yet further 
 to elongate the bar by the same distance Z,* which whole 
 elongation of 2Z coulcl not have remained; because the 
 strain upon the bar is only that necessary to keep it 
 elongated by I. The extremity of the bar would therefore, 
 under these circumstances, have oscillated on either side of 
 that point which corresponds to the elongation I. 
 
 
 
 * The mechanical principle involved in this result has numerous applica- 
 tions ; one of these is to the effect of a sudden variation of the pressure on a 
 mercurial column. The pressure of such a column varying directly with its 
 elevation or depression, follows the same law as the elasticity of a bar; 
 whence it follows that if any pressure be thrown at once or instantaneously 
 upon the surface of the mercury, the variation of the height of the column 
 will be twice that which it would receive from an equal pressure gradually 
 accumulated. Some singular errors appear to have resulted from a neglect of 
 this principle in the discussion of experiments upon the pressure of steam, 
 made with the mercurial column. No such pressure can of course be made to 
 operate, in the mathematical sense of the term, instantaneously ; and the term 
 gradually has a relative meaning. All that is meant is, that a certain relation 
 must obtain between the rate of the increase of the pressure and the amplitude 
 of the motion, so that when the pressure no longer increases the motion may 
 cease. 
 
462 RESILIENCE AND FRAGLITY. 
 
 349. Eliminating I between equations (485) and (486), we 
 obtain 
 
 U=ig (487); 
 
 whence it appears that the work expended upon the elonga- 
 tion of a bar under any strain varies directly as the square 
 of the strain and the length of the bar, and inversely as the 
 area of its section.* 
 
 THE MODULI OF RESILIENCE AND FRAGILITY. 
 
 (7 v a 
 y 1 KL (equation 486), it is evident 
 
 that the different amounts of work which must be done upon 
 different bars of the same material to elongate them by equal 
 
 fractional parts I y), are to one another as the product KL. 
 
 Let now two such bars be supposed to have sustained that 
 fractional elongation which corresponds to their elastic limit; 
 let U represent the work which must have been done upon 
 the one to bring it to this elongation, and M a that upon the 
 other : and let the section of the latter bar be one square 
 inch and its length one foot ; then evidently 
 
 U e =M,KL ..... (488). 
 
 M is in this case called the modulus of longitudinal resili- 
 ence.^ 
 
 It is evidently a measure of that resistance which the 
 material of the bar opposes to a strain in the nature of an 
 impact, tending to elongate it beyond its elastic limits. 
 
 If M/be taken to represent the work which must be simi- 
 larly done upon a bar one foot long and one square inch, in 
 section to produce fracture, it will be a measure of that 
 resistance which the bar opposes to fracture under the like 
 circumstances, and which resistance is opposed to its fra- 
 
 * From this formula may be determined the amount of work expended pre- 
 judicially upon the elasticity of rods used for transmitting work in machinery, 
 under a reciprocating motion pump rods, for instance. A midden effort of the 
 pressure transmitted in the nature of an impact may make the loss of work 
 double that represented by the formula ; the one limit being the minimum, and 
 the other the maximum, of the possible loss. 
 
 f The term "modulus of resilience" appears first to have been used by 
 Mr. Tredgold in his work on " the Strength of Cast Iron," Art. 304. 
 
A BAR SUSPENDED VERTICALLY. 4:63 
 
 gility ; it may therefore be distinguished from the last men- 
 tioned as the modulus of fragility. If TJ/ represent, the work 
 which must be done upon a bar whose section is K square 
 inches and its length L fee* to produce fracture; then, as 
 before, 
 
 U / =M / KL ..... (489). 
 
 If P e and P/ represent respectively the strains which 
 would elongate a bar, whose length is L feet and section^ K 
 inches, to its elastic limits and to rupture ; then, equation 
 (487), 
 
 " M<=1 Similarly M r =i ..... (490). 
 
 These equations serve to determine the values of the 
 moduli M e and M/by experiment.* 
 
 351. The elongation of a lor suspended^ vertically, and sus- 
 taining a given strain in the direction of its length, the 
 influence of its own weight being taken into the account. 
 
 Let x represent any length of the bar before its elonga- 
 tion, &x an element of that length, L the whole length of the 
 bar before elongation, w the weight of each foot of its 
 length, and K its section. Also let the length x have become 
 a? x when the bar is elongated, under the strain P and its own 
 weight. The length of the bar, below the point whose dis- 
 tance from the point of suspension was x before the elonga- 
 tion, having then been L a?, and the weight of that portion 
 of the bar remaining unchanged by its elongation, it is still 
 represented by (L a?) w. Now this weight, increased by P, 
 constitutes the strain upon the element AOJ; its elongation 
 under this strain is therefore represented (equation 485) by 
 
 -rr-n A #> and the length ^x l of the element when thus 
 
 * The experiments required to this determination, in respect to the princi- 
 pal materials of construction, have been made, and are to be found in the 
 published papers of Mr. Hodgkinson and Mr. Barlow. A table of the moduli 
 of resilience and fragility, collected from these valuable data, is a desideratunc 
 in practical science. 
 
464 THE VERTICAL OSCILLATIONS OF 
 
 I 
 
 elongated, by &x-\ T^- ^^5 whence dividing by A#, 
 
 and passing to the limit, we obtain 
 
 db 1 _ 1 P+(L-aQM 
 dx~ KE 
 
 Integrating between the limits and L, and representing 
 by L x the length of the elongated rod, 
 
 If the strain be converted into a thrust, P must be made 
 to assume the negative sign; and if this thrust equal one 
 half the weight of the bar, there will be no elongation at all. 
 
 352. THE VERTICAL OSCILLATIONS OF AN ELASTIC ROD OK 
 COED SUSTAINING A GIVEN WEIGHT SUSPENDED FROM ITS 
 
 EXTREMITY. 
 
 Let A represent the point of suspension of the rod (fig. 1. 
 on the next page), L its length AB before its elongation, and 
 \l the elongation produced in it by a given weight "W sus- 
 pended from its extremity, and C the corresponding position 
 of the extremity of the rod. 
 
 Let the rod be conceived to be elongated through an 
 additional distance CD=e by the application of any other 
 given strain, and then allowed to oscillate freely, carrying 
 with it the weight "W; and let P be any position of its 
 extremity during any one of the oscillations which it will 
 thus be made to perform. If, then, CP be represented by a?, 
 the corresponding elongation BP of the rod will be repre- 
 sented by iZ-ho?, and the strain which would retain it perma- 
 
 KE 
 nently at this elongation (equation 485) by -(i^+a?); the 
 
 unbalanced pressure or moving force (Art. 92.) upon the 
 weight W, at the period of this elongation, will therefore be 
 
 Trnn TT-p 
 
 represented by -^-(%l+x) W, or by -y-; since W, being 
 the strain which would retain the rod at the elongation -J-Z, is 
 
 ~ 
 
 represented by -y~JZ (equation 485). 
 
 * Whewell's Analytical Statics, p. 113. 
 
A LOADED BAR. 
 
 4:65 
 
 The unbalanced pressure, or moving force, upon the mass 
 "W varies, therefore, as the distance x of the point P from the 
 given point ; whence it follows by the general principle 
 established in Art 97., that the oscillations of the point P 
 extend to equal distances on either side of the point C, as a 
 centre, and are performed isochronously, the time T of each 
 oscillation being represented by the formula 
 
 T-/WLU 
 
 \JBSt 
 
 . (493). 
 
 The distance from A of the centre C, about which th$> 
 oscillations of the point P' take place, is represented b$- 
 L+-JZ; so that, representing this distance by L and substi- 
 tuting for %l its value, we have 
 
 f 
 i 
 
 353'. Let us now suppose that when in 
 making its first oscillation about C 
 (fig. 2.) the weight W has attained its 
 highest position d^ and is therefore, for 
 an instant, at rest in that position, a 
 second weight w is added to it ; a second 
 series of oscillations will then be com- 
 menced about a new centre C 15 whose 
 distance L 2 from A is evidently repre- 
 sented by the formula 
 
 So that the distance CO, of the two centres is ^ ; and the 
 
 ixiij 
 
 greatest distance CJ)^ beneath the centre 0,, attained in the 
 second oscillation, equal to the distance, C^ at which the 
 oscillation commenced above that point. Now C 1 D J = 
 
 the second oscillation is therefore %c -f- 
 
 30 
 
 1 =o+ ; the amplitude df), of 
 
4:66 THE OSCILLATIONS OF A LOADED BAR. 
 
 Let the weight w be conceived to be removed when the 
 lowest point ~D t of the second oscillation is attained, a third 
 series of oscillations will then be commenced, the position of 
 whose centre being determined by equation (494), is identical 
 with that of the centre C, about which the first oscillation 
 was performed. In its third oscillation the extremity of the 
 rod will therefore ascend to a point <# 2 as far above the point 
 C as D : is below it ; so that the amplitude of this third oscil- 
 
 lation is represented by 2TCD,, or by 2CJD~T r CC ! , or by 
 ^J). When the highest point d z of this third oscil- 
 
 lation is attained, let the weight w be again added ; a fourth 
 oscillation will then be commenced, the position of whose 
 centre will be determined by equation (495,) and will there- 
 fore be identical with the centre C,, about which the second 
 oscillation was performed ; so that the greatest distance C,D ? 
 beneath that point attained in this fourth oscillation will be 
 equal to CA or to CC^ -f CDj ; and its amplitude will be 
 
 represented by 2 I <?+__). And if the weight w be thus 
 \ K.Jii / 
 
 conceived to be added continually, when the highest point 
 of each oscillation is attained, and taken off at the lowest 
 point, it is evident that the amplitudes of these oscillations 
 will thus continually increase in an arithmetical series ; so 
 that the amplitude A^ of the n ili oscillation will be repre- 
 sented by the formula 
 
 The ascending oscillations of the series being made about 
 the centre C, and the descending oscillations about C 1? 
 if n be an even number, the centre of the n ih oscillation is 
 Cj ; the elongation c n of the rod corresponding to the lowest 
 point of this oscillation is therefore equal to BQ+^A^ or 
 substituting for BCj its value given by equation (495), and 
 for A n its value from equation (496), 
 
 Thus it is apparent that by the long continued and 
 
DEFLEXION. 467 
 
 periodical addition and subtraction of a weight w, so small 
 as to produce but a slight elongation or contraction of the 
 rod when first added or removed from it, an elongation c n 
 may eventually be produced, BO great as to pass limits of its 
 elasticity, or even to break it. Numerous observations have 
 verified this fact: the chains of suspension bridges have 
 been broken by the measured tread of soldiers ;* and M. 
 Savart has shown, that by fixing an elastic rod at its centre, 
 and drawing the wetted finger along it at measured inter- 
 vals, it may, by the strain resulting from the slight friction 
 received thus periodically upon its surface, be made with 
 great ease to receive an oscillatory movement of sufficient 
 amplitude to be measured, f M. Poncelet has compared the 
 measurement of M. Savart with theoretical deductions 
 analogous to those of the preceding article, and has shown 
 their accordance with it. 
 
 DEFLEXION. 
 354. The neutral surface of a deflected beam. 
 
 One surface of a beam becoming, when deflected, convex, 
 and the other concave, it is evident that the material form- 
 ing that side of the beam which is bounded by the one 
 surface is, in the act of flexure, extended, and that of the 
 other compressed. The surface which separates these two 
 portions of the material being that where its extension ter- 
 minates and its compression begins, and which sustains, 
 therefore, neither extension nor compression, is called the 
 NEUTRAL SURFACE. 
 
 355. THE POSITION OF THE NEUTRAL SURFACE OF A BEAM. 
 Let ABCD be taken to represent any thin lamina of the 
 
 * Such was the fate of the suspension bridge at Broughton near Manchester, 
 the circumstances of which have been ably detailed by Mr. E. Hodgkinson in 
 the fourth volume of the Manchester Philosophical Transactions. M. Navier 
 has shown, in his treatise on the theory of suspension bridges (Sur les Fonts 
 Suspendus, Paris, 1823), that the duration of the oscillations of the chains of 
 a suspension bridge may in certain cases extend to nearly six seconds ; there 
 might easily, in such cases, arise that isochronism at each interval, or after 
 any number of intervals, between the marching step of the troops and the 
 oscillations of the bridge, whence would result a continually increasing elon- 
 gation of the suspending chains. 
 
 f Mecanique Industrielle, p. 437, Art. 331. ED. 
 
468 THE NEUTRAL SURFACE 
 
 beam contained by planes 
 parallel to the plane of its 
 deflexion, and P 1? P,, P 8 the 
 resultants of all the pres- 
 sures applied, to it ; acb that 
 portion of the neutral sur- 
 iace of the beam which is 
 contained within this la- 
 mina, and may be called its 
 neutral line; PT and QY 
 planes exceedingly near to 
 one another, and perpen- 
 dicular to the neutral line at the points where they intersect 
 it ; and O the intersection of PT and QY when produced. 
 
 Now let it be observed that the portion ArTD of the 
 beam is held in equilibrium by the resultant pressure P 1? 
 and by the elastic forces called into operation upon the sur- 
 face PT ; of which elastic forces those acting in PR (where 
 the material of the beam is extended) tend to bring the 
 points to which they are severally applied nearer to the 
 plane SQ, and those acting in RT (where the material is 
 compressed), to carry their several points of application 
 farther from the plane SY. 
 
 Let aR=x, SR=A#, and imagine the lamina PQYT to be 
 made up of fibres parallel to SR ; then will Aa? represent 
 the length of each of these fibres before the deflexion of the 
 beam, since the length of the neutral fibre SR has remained 
 unaltered by the deflexion. Let dx represent the quantity 
 by which the fibre pq has been elongated by the deflexion 
 of the beam, then is the actual length of that fibre repre- 
 sented by AaH-cSa?. Whence it follows (equation 485), that 
 the pressure which must have operated to produce this 
 
 dx 
 
 elongation is represented by E A&, A& being taken to repre- 
 sent the section of the fibre, or an exceedingly small element 
 of the section PT of the lamina. Now PT and QY being 
 normals to SR, the point O in which they meet, when 
 produced, is the centre of curvature to the neutral line in 
 R. Let the radius of curvature OR be represented by R, 
 
 Op 
 and the distance ~Rp by p. By similar triangles, Q^= 
 
 pq R-fp Aa?+da? p <fo Al 
 
 ^jL.j or __/== 9 or !+:-= 1 + ; therefore. 
 
 p 
 
OF A BEAM. 469 
 
 . Substituting this value of in the expression for the 
 
 pressure which must have operated to produce the elonga- 
 tion of the fibre pq, and representing that pressure by AP, 
 we have 
 
 AP=*|A& ____ (498). 
 
 If, therefore, KP be represented by ^ and ET by &,, then 
 the sum of the elastic forces developed by the extension of 
 
 E kl 
 the fibres in RPQS is represented by ^-2 pA&; and, similarly, 
 
 the sum of those developed by the compression of the fibres 
 
 E * 2 
 in RTVS is represented by =-2 pA&. Now let it be observed 
 
 that (since the pressures applied to APTD, and in equili- 
 brium, are the forces of extension and compression acting 
 in KP and RT respectively, and the pressure PJ, if the 
 pressure P x be resolved in a direction perpendicular to the 
 plane PT, or parallel to the tangent to the neutral line in R, 
 this resolved pressure will be equal (Art. 16.) to the differ- 
 ence of the sums of the forces of extension and compression 
 applied (in directions perpendicular to that plane, but oppo- 
 site to one another) to the portions RP and RT of it respec- 
 tively. Representing, therefore, by 6 the inclination R0P, 
 of the direction of Pj to the normal to the neutral line in R, 
 we have 
 
 ~p* k l 
 
 P a sin. 4=-2 pA&_ 
 
 JK. K 
 
 But if Jc be taken to represent the whole section PT, and h 
 the distance of the point R from its centre of gravity, then 
 (Art. 18.) 
 
 .-. P, sin. 4= 
 
 RP 
 ...^-^sin.4 ..... (499); 
 
 which expression represents the distance of the neutral line 
 from the centre of gravity of any section PT of the lamina, 
 that distance being measured towards the extended or the 
 compressed side of the lamina according as d is positive or 
 
470 
 
 RADIUS OF CURVATURE. 
 
 negative; so that the neutral line passes from one side to 
 the other of the line joining the centres of gravity of the 
 cross sections of the lamina, at the point where d 0, or at 
 the point where the normal to the neutral line is parallel to 
 the direction of P. 
 
 356. Case of a rectangular beam. 
 
 If the form of the beam be such that it may be divided 
 into laminae parallel to ABCD of similar forms and equal 
 dimensions, and if the pressure P, applied to each lamina 
 may be conceived to be the same ; or if its section be a rec- 
 tangle, and the pressures applied to it be applied (as they 
 usually are) uniformly across its width, then will the distance 
 h of the neutral line of each lamina from the centre of gra- 
 vity of any cross section of that lamina, such as PT, be the 
 same, in respect to corresponding points of all the laminae, 
 whatever may be the deflection of the beam ; so that in thig 
 case the neutral surface is always a cylindrical surface. 
 
 357. Case in which the deflecting pressure P 1 is nearly per- 
 pendicular to the length of the learn. 
 
 In this case 0, and therefore sin. 0, is exceeding small, so 
 long as the deflexion is small at every point R of the neutral 
 line ; so that h is exceedingly small, and the neutral line 
 of the lamina passes very nearly, or accurately, through the 
 centre of gravity of its section PT. 
 
 358. THE RADIUS OF CURVATURE OF THE NEUTRAL SURFACE 
 
 OF A BEAM. 
 
 Since the pressures applied to the portion APTD of the 
 
 lamina ABCD are in equilibrium, the principle of the equality 
 of moments must obtain in respect to them ; taking, there- 
 
RADIUS OF CURVATURE. 471 
 
 fore, the point R, where the neutral axis of the lamina inter- 
 sects PT, as the point from which the moments are measured, 
 and observing that the elastic pressures developed by the 
 extension of the material in RP and its compression in RT 
 both tend to turn the mass APTD in the same direction 
 about the point R, and that each such pressure upon an 
 
 element &k of the section PT is represented (equation 498) by 
 
 
 _ pA& 5 and therefore the moment of that pressure about the 
 R 
 
 E 
 
 point R by p a A&, it follows that the sum of the moments 
 
 R 
 about the point R of all these elastic pressures upon PT is 
 
 represented by = -2p 2 A& 5 or by --, if I be taken to represent 
 
 R R 
 
 the moment of inertia of PT ^bout R. Observing, moreover, 
 that if p represent the length of the perpendicular let fall 
 from R upon the direction of any pressure P applied to the 
 portion APTD of the beam, fp will represent its moment, 
 and SPp will represent the sum of the moments of all the 
 similar pressures applied to that portion of the beam ; 
 we have by the principle of the equality of moments, 
 
 359. The neutral surface of the beam is a cylindrical sur- 
 face, whatever may be its deflection or the direction of its 
 deflecting pressure, provided that its section is a rectangle 
 (Art. 356.) ; or whatever may be its section, provided that its 
 deflection be small, the direction of the deflecting pressure 
 nearly perpendicular to its length, and its form before de- 
 flexion symmetrical in respect to a plane perpendicular to the 
 plane of deflexion. In every such case, therefore, the neutral 
 lines of all the laminae similar to ABCD, into which the 
 beam may be divided, will have equal radii of curvature at 
 points similar to R lying in the same right line perpendicular 
 to the plane of deflection ; taking, therefore, equations simi- 
 lar to the above in respect to all the laminae, multiplying 
 both sides of each by I, adding them together, and observ- 
 ing that R and E are the same in all, we have _ _ P m 
 
 R E 
 In this case, therefore, I may be taken in equation (COO) to 
 
472 MOMENT OF INERTIA. 
 
 represent the moment of inertia of the whole section of the 
 beam, and P the pressure applied across its whole width. 
 
 360. The radius of curvature of a beam whose deflexion is 
 small, and the direction of me deflecting pressures nearly 
 perpendicular to the length of the beam. 
 
 In this case the neutral line is very nearly a straight line, 
 perpendicular to the directions of the deflecting pressures ; 
 so that, representing its length by a?, we have, in this case, 
 jp=x ; and equation (500) becomes 
 
 which relation obtains, whatever may be the form of the 
 transverse section of the beam, I representing its moment of 
 inertia in respect to an axis passing through its centre of 
 gravity and perpendicular to the plane of deflexion. 
 
 361. The moment of inertia I of the transverse section of a 
 beam about the centre of gravity of the section. 
 
 In treating of the moments of inertia of bodies of different 
 geometrical forms in a preceding part of this work (Art. 82, 
 &c.), we have considered them as solids ; whereas the mo- 
 ment of inertia I of the section of a beam which enters into 
 equation (500) and determines the curvature of the beam 
 when deflected, is that of the geometrical area of the section. 
 Knowing, however, the moment of inertia of a solid about 
 any axis, whose section perpendicular to that axis is of a 
 given geometrical form, we can evidently determine the 
 moment of the area of that section about the same axis, by 
 supposing the solid in the first place to become an exceed- 
 ingly thin lamina (i. e. by making that dimension of the 
 solid which is parallel to the axis exceedingly small in the 
 expression for the moment of inertia), and then dividing 
 the resulting expression by the exceedingly small thickness 
 of this lamina. We shall thus obtain the following values 
 of I:- 
 
MOMENT OF INERTIA. 473 
 
 362. For a beam with a rectangular section, ) 
 whose breadth is represented by b and its depth V&a 
 by G (equation 61), ) 
 
 363. For a beam with a triangular ) i c 
 section, whose base is b and its height c > I = ~^r 
 (equation 63), ) 
 
 364. For a beam or column with a circular ) 
 section, whose radius is c (equation 66), j 
 
 365. To determine the moment of inertia I in respect to a 
 A B beam whose transverse section is of the 
 
 1 n f - ' form represented in the accompanying 
 figure, about an axis ab passing through 
 its centre of gravity ; let the breadth of 
 the rectangle AB be represented by b l and 
 its depth by d^ and let & 2 and d z be simi- 
 larly taken in respect to the rectangle EF, 
 and b a and d 3 in respect to CD ; also let I t 
 represent the moment of inertia of the section about the axis 
 cd passing through the centre of CD, A 1? A,, A,, the areas 
 of the rectangles respectively, and A the area of the whole 
 section. 
 
 Now the moments of inertia of the several rectangles, 
 about axes passing through their centres of gravity, are 
 represented by yV^A'i A^A 3 ? rV^AN an( ^ tne distances of 
 these axes from the axis cd are respectively 
 0. Therefore (equation 58), 
 
 but A,=M A,=M A S = 
 
 Also if h represent the distance between the axes ab and cd, 
 then (Art. 18) AA=i(d a + rf,)A a i^ + ^^o and (equation 
 
 58) Irnl.-A'A. 
 
 If d l and d^ be exceedingly small as compared with <2 8 , 
 
4T4 
 
 DEFLEXION OF A BEAM. 
 
 neglecting their values in the two last terms of the equation 
 and reducing, we obtain 
 
 (503). 
 
 If the areas AB and EF be equal in every respect, 
 1=1 {d* + *& + %)*} At + JsAj; ..... (504). 
 
 366. THE WORK EXPENDED UPON THE DEFLEXION OF A BEAM 
 TO WHICH GIVEN PRESSURES ARE APPLIED. 
 
 If AP represent the pressure which must have operated 
 
 to produce the elongation or 
 compression which the ele- 
 mentary fibre pq receives, 
 by reason of the deflexion 
 of the beam, AOJ the length 
 of that fibre before the de- 
 flexion of the beam, and &k 
 its section; then the work 
 which must have been done 
 upon it, thus to elongate 
 or compress it, is repre- 
 sented, equation (487) by 
 
 498) Ap=&4jfc 
 
 E Aft' 
 
 pended upon the extension or compression of pq is there- 
 fore represented by 
 
 And the same being true of the work expended on the 
 compression or extension of every other fibre composing the 
 elementary solid VTPQ, it follows that the whole work 
 expended upon the deflexion of that element of the beam 
 
 is represented by J-^j- 2p 3 A&, or by i A # 5 for 2p 2 A& repre- 
 sents the moment of inertia I of the section PT, about an 
 axis perpendicular to the plane of ABCD, and passing through 
 the point K. If, therefore, t be taken to represent the 
 length of that portion of the beam which lies between D 
 
DEFLEXION OF A BEAM. 475 
 
 and M before its deflexion, and therefore the length of the 
 portion ac of its neutral line after deflexion, then the whole 
 work expended upon the deflexion of the part AM of the 
 
 01 1 1 
 
 beam is represented lyy % E2 ^- 2 Aa?. But (equation 500) ^= 
 
 o-tv J*> 
 
 yara" j whence, by substitution, the above expression 
 
 P 2 2i# a 
 
 becomes j~4r o y Aa? - Passing to the limit, and represent- 
 ing the work expended upon the deflexion of the part AM 
 of the beam by u^ 
 
 P, a 
 
 o 
 
 367. TA0 work expended upon the deflexion of a learn of 
 uniform dimensions, when the deflecting pressures are 
 nearly perpendicular to the surface of the beam. 
 
 In this case I is constant, and ^=8?; whence we obtain 
 by integrating (equation 505) be- 
 tween the limits and a 1? 
 
 u >=^if (506)> 
 
 where u v represents the work ex- 
 pended upon the deflexion of the 
 portion AM of the beam. Simi- 
 larly, if bc=a^ the work expended 
 
 upon the deflexion of the portion BM of the beam is repre- 
 
 sented by 
 
 so that the whole work U s expended upon the deflexion of 
 the beam is represented by 
 
 
 P 2 // 8 
 \ a \ 
 
 6EI 
 
 But by the principle of the equality of moments, if a 
 represent the whole length of the beam, 
 
476 
 
 DEFLEXION OF A BEAM. 
 
 Eliminating P 1 and P a between these equations and the pre- 
 ceding, we obtain by reduction 
 
 (507). 
 
 If the pressure P 3 be applied in the centre of the beam, 
 
 368. THE LINEAR DEFLEXION OF A BEAM WHEN THE DIRECTION 
 
 OF THE DEFLECTING PRESSURE IS PERPENDICULAR TO ITS 
 SURFACE. 
 
 Let 
 
 the section MK remain fixed, the deflexion taking 
 place on either side of that section ; 
 then u^ representing the work ex- 
 pended upon the deflexion of the 
 portion AM of the beam, and D 1 
 the deflexion of the point to which 
 P! is applied, measured in a direc- 
 tion perpendicular to the surface, we 
 
 ( e( l uatio11 40 )> ^i 
 
 du, du, dP* 
 ' 
 
 , 
 therefore P. = -~r- 
 
 But by equation (506), = l ; therefore P, - 
 
 -jf^ 5 therefore -^- = ^ Wy ; whence we obtain by integration 
 
 3EI 
 
 (509). 
 
 If the whole work of deflecting the beam be done by the 
 pressure P 3 , the points of application of P and P 2 having no 
 motions in the directions of these pressures (Art. 52.), then 
 proceeding in respect to equation (507) precisely as before in 
 respect to equation (506), and representing the deflexion 
 
 * Church's Diff. Cal. Art. 1Y. 
 
DEFLEXION OF A BEAM. 477 
 
 perpendicular to the surface of the beam at the point of 
 application of P 3 by D 3 , we shall obtain 
 
 If the pressure P 8 be applied at the centre of the beam 
 
 Eliminating P, between equations (506) and (509), and P, 
 between equations (507) and (510), we obtain 
 
 by which equations the work expended upon the deflexion 
 of a beam is determined in terms of the deflexion itself, as 
 by equations (506) and (507) it was determined in terms of 
 the deflecting pressures. 
 
 369. CONDITIONS OF THE DEFLEXION OF A BEAM TO WHICH ARE 
 APPLIED THREE PRESSURES, WHOSE DIRECTIONS ARE NEARLY 
 
 PERPENDICULAR TO ITS SURFACE. 
 
 Let AB represent any lamina of the beam parallel to its 
 
 H 
 
 plane of deflexion, and acb the neutral line of that lamina 
 intersected by the direction of P 8 in the point c. 
 
 Draw xx 1 parallel to the length of the beam before its 
 deflexion, and take this line as the axis of the abscissae, and 
 the point o as the origin ; then, representing by x and y the 
 
 * This result is identical with that obtained by a different method of inves- 
 tigation by M. Navier (Resume de Lemons de Construction, Art. 359.). 
 
4:78 EQUATION TO THE NEUTRAL LINE. 
 
 co-ordinates of any point in ac, and by ~R the radius of curva- 
 ture of that point, we have * 
 
 Now the deflexion of the beam being supposed exceed- 
 ingly small, the inclination to ex of the tangent to the 
 
 neutral line is, at all points, exceedingly small, so that I -^J 
 
 may be neglected as compared with unity ; therefore ^ -^. 
 
 Substituting this value in equation (501), and observing that 
 in this casejp is represented by (a, a?) instead of #, 
 
 (513). 
 
 the direction of the pressure P a being supposed nearly per 
 pendicular to the surface of the beam, and I constant. Let 
 the above equation be integrated between the limits and 
 a?, /3 being taken to represent the inclination of the tangent 
 
 at c to B, so that the value of f at c may be represented by 
 
 GuX 
 
 tan. 13, 
 
 -tan./SXo,-^ ...(514). 
 
 Integrating a second time between the limits and a?, and 
 observing that when a?=0, y=0, 
 
 y=^ l \fatf-ia?}+a>teai.P .... (515). 
 
 Proceeding similarly in respect to the portion ~bc of the neu- 
 tral line, but observing that in respect to this curve the value 
 
 of - at the point c is represented by tan. /3, we have 
 
 dx*~~ El 
 ./3 == ^M^-^} . . (516). 
 
 jL 
 
 Church's Diff. Cal. Art. 105. 
 
EQUATION TO THE NEUTRAL LINE. 479 
 
 If D l and D 2 be taken to represent the deflexions at the 
 points a and , and ca and cb be assumed respectively equal 
 to cd and ce, 
 
 by equation (515), D 1 = l-A +g t tan. /3, 
 
 P # 3 
 by equation (517), D 5 = ^~ 2 tan. . 
 
 If the pressures P 1 and P 2 be supplied by the resistances 
 of fixed surfaces, then T> l =T>, l . Subtracting the above equa- 
 tion we obtain, on this supposition, 
 
 tan. 
 
 Now F A '-P A '= ''-~'=P 8 a.^( gl - a .) ; ob- 
 serving that P 1 a=P,<*,, ?,=?, and ,+,:= a, 
 
 If /3 15 /3 2 represent the inclinations of the neutral line to 
 xx l at the points a and J, then by equations (514) and (516) 
 
 tan./3,_tan./3=, tan. ^ 
 
 Substituting for tan. /3 its value from equation (518), elimi- 
 nating and reducing, 
 
 _P i a.g,(g I + 2g.) . _P.g, 
 
 1D - ^ 1- a ' ^ 2 " 
 
 To determine the point m where the tangent to the neutral 
 line is parallel to cxx^ or to the undeflected position of the 
 
 fii i 
 
 beam, we must assume -/-=0 in equation (516)* ; if we 
 
 fftx 
 
 then substitute for tan. (3 its value from equation (518), 
 substitute for P 2 its value in terms of P 3 , and solve the 
 
 * Church's Diff. Cal. Art. 78. 
 
480 
 
 LENGTH OF THE NEUTRAL LINE. 
 
 resulting equation in respect to a?, we shall obtain for the 
 distance of the point m from c the expression 
 
 370. THE LENGTH OF THE NEUTRAL LINE, THE BEAM BEING 
 LOADED IN THE CENTRE. 
 
 Let the directions of the resistances upon the extremities 
 
 of the beam be supposed nearly perpendicular to its surface ; 
 then if x and y be the co-ordinates of the neutral line from 
 the point #, we have (equation 501), representing the hori- 
 zontal distance AB by 20, and observing that in this case 
 
 -=-5-5, and that the resistance at A or B = JP, 
 
 Integrating between the limits x and 0, and observing that 
 
 d/u 
 at the latter limit - = O r 
 
 Now if * represent the length of the curve ac, 
 
 Church's Int. Cal. Art. 197. 
 
THE DEFLEXION OF A BEAM. 
 
 481 
 
 the deflexion being small, -T-> is exceedingly small at every 
 point of the neutral line. 
 
 :.a = f | 
 
 '.* = + 60ET-... (521). 
 
 Eliminating P between this equation and equation (511), and 
 representing the deflexion by D, 
 
 D a * 
 
 -. 
 
 371. A BEAM, ONE PORTION OF WHICH IS FIRMLY INSERTED IN 
 MASONRY, AND WHICH SUSTAINS A LOAD UNIFORMLY DISTRI- 
 BUTED OVER ITS REMAINING PORTION. 
 
 Let the co-ordinates of the neutral line be measured from 
 
 * The following experiments were made by Mr. Hatcher, superintendant of 
 the work-shop at King's College, to-verify this result, which is identical with 
 that obtained by M. Navier (Resume des Lemons, Art. 86.). Wrought iron 
 rollers -7 inch in diameter were placed loosely on wrought iron bars, the sur- 
 faces of contact being smoothed with the file and well oiled. The bar to be 
 tested had a square section, whose side was '7 inch, and was supported on the 
 two rollers, which were adjusted to 10 feet apart (centre to centre) when the 
 deflecting weight had been put on the bar. On removing the weights-care- 
 fully, the distance to which the rollers receded as the bar recovered its hori- 
 zontal position was noted. 
 
 Deflecting Weight 
 
 Deflection in Inches. 
 
 Distance through .which 
 each Roller receded 
 in inches. 
 
 Distance through which 
 each Roller would have 
 receded by Formula. . 
 
 56 
 84 
 
 3-7 
 5-45 
 
 1 
 
 2 
 
 13 
 29 
 
 31 
 
THE DEFLEXION OF A BEAM 
 
 the point B where the beam 
 is inserted in the masonry, 
 and let the length of the 
 portion AD which sustains 
 the load be represented by 
 #, and the load upon each 
 unit of its length by p ; 
 then, representing by x and 
 y the co-ordinates of any 
 point P of the neutral line, 
 and observing that the pres- 
 sures applied to AP, and in 
 
 equilibrium, are the load ^(ax) and the elastic forces 
 developed upon the transverse section at P, we have by the 
 principle of the equality of moments, taking P as the point 
 from which the moments are measured, and observing that 
 since the load p(ax) is uniformly distributed over AP it 
 produces the same effect as though it were collected over the 
 centre of that line, or at distance \(a x) from P ; observing, 
 moreover, that the sum of the moments of the elastic forces 
 upon the section at P, about that point, is represented (Art. 
 
 358.) by 5 or by El J (Art. 369.) ; 
 
 Integrating twice between the limits and a?, and observing 
 
 CM i V/ 
 
 that when x=Q,-^-=Q and y=0, since the portion BC of the 
 beam is rigid, we obtain 
 
 which is the equation to the neutral line. 
 
 Let, now, a be substituted for x in the above equation ; 
 and let it be observed that the corresponding value of y 
 represents the deflexion D at the extremity A of the beam ; 
 we shall thus obtain by reduction 
 
LOADED UNIFORMLY. 
 
 483 
 
 Representing by P the inclination to the horizon of the tan- 
 gent to the neutral line at A, substituting a for x in equation 
 
 (523), and observing that when x=a, -/^= tan. /8, we obtain 
 tan. /3=|^ . . . . (526). 
 
 372. A BEAM SUPPORTED AT ITS EXTREMITIES AND SUSTAINING 
 A LOAD UNIFORMLY DISTRIBUTED OVER ITS LENGTH. 
 
 Let the length of the beam be represented by 20, the load 
 upon each unit of length by p ; take 
 x and y as the co-ordinate of any 
 point P of the neutral line, from the 
 origin A; and let it be observed 
 that the forces applied to AP, and in 
 equilibrium, are the load px upon that 
 portion of the beam, which may be 
 supposed collected over its middle 
 point, the resistance upon the point A, which is represented 
 by pa, and the elastic forces developed upon the section 
 atP; then by Art. 360., 
 
 Integrating this equation between the limits x and #, and 
 
 observing that at the latter limit -^ = 0, since y evidently 
 
 dx 
 attains its maximum value at the middle C of the beam, 
 
 = ifx(^- 8 )-4K a - 9 ) ____ (528). 
 
 Integrating a second time between the limits and a?, and 
 observing that when oj*=0, y=0, 
 
 3 -a'x) ____ (529), 
 which is the equation to the neutral line. Substituting a for 
 
THE DEFLEXION OF A BEAM 
 
 x in this equation, and observing that the corresponding 
 value of y represents the deflexion D in the centre of the 
 beam, we have by reduction 
 
 Representing by /3 the inclination to the horizon of the tan- 
 gent to the neutral line at A or B, and observing that when 
 
 a?=0 in equation (528),^= tan. /3, 
 ax 
 
 = L ..... (531). 
 
 Let it be observed that the length of the beam, which in 
 equation (511) is represented by #, is here represented by 
 2w, and that equation (530) may be placed under the form 
 
 D=j- . AQ-FT ? whence it is apparent that ihe deflexion 
 
 of a beam, when uniformly loaded throughout, is the same 
 as though -fths of that load (2^) were suspended from its 
 middle point. 
 
 373. A BEAM IS SUPPORTED BY TWO STRUTS PLACED SYM- 
 METRICALLY, AND IT IS LOADED UNIFORMLY THROUGHOUT 
 ITS WHOLE LENGTH; TO DETERMINE ITS DEFLEXION. 
 
 Let CD=2&, CA = a 1 , load upon each foot of the length 
 of the beam^fx ; then load on 
 each point of support =v-a. Take 
 C as the origin of the co-ordinates ; 
 then, observing that the forces 
 impressed upon any portion CP 
 of the beam, terminating between 
 C and A, are the elastic forces 
 upon the transverse section of the 
 beam at P, and the weight of the 
 load upon CP ; and observing that the weight M-CP of the 
 load upon CP, produces the same effect as though it were 
 collected over the centre of that portion of the beam, so that 
 its moment about the point P is represented by p. CP. 
 
LOADED UNIFORMLY. 485 
 
 or by -|^CP a ; we obtain for the equation to the neutral line 
 in respect to the part CA of the beam (Art. 360) 
 
 (532). 
 
 Since, moreover, the forces impressed upon any portion CQ 
 of the beam, terminating between A and E, are the elastic 
 forces developed upon the transverse section at Q, the 
 resistance pa of the support at A, and the load upon CQ, 
 whose moment about Q is represented by J-^CQ 2 , we have 
 (equation 501), representing CQ by a?, 
 
 EI =f(e-Ka'-0,) ..... (533). 
 
 Representing the inclination to the horizon of the tangent to 
 the neutral line at A by /3, dividing equation (532) by v*, 
 integrating it between the limits x and a 15 and observing 
 
 ffll 
 
 that at the latter limit -^=tan. /3, we have, in respect to the 
 portion CA of the beam, 
 
 Integrating equation (533) between the limits x and , and 
 observing that at the latter limit ^=sO, since the neutral 
 line at E is parallel to the horizon, 
 
 -7-= |# s \a(x a^ a*+%a(a a$ (535); 
 
 |X (JUvu 
 
 which equation having reference to the portion AE of the 
 
 beam, it is evident that when #=#., -y-=tan. P. 
 
 IJ dx 
 
 8-1 
 
 .... (536). 
 
 Substituting, therefore, for tan. P in equation (534), and 
 reducing, that equation becomes 
 
486 THE DEFLEXION OF A BEAM 
 
 Integrating equation (535) between the limits a, and a?, and 
 equation (537) between the limits and a?, and representing 
 the deflexion at C, and therefore the value of y at A, by D 
 
 El 
 
 (y-DO 
 
 (538); 
 
 the former of which equations determines the neutral line 
 of the portion AE, and the latter that of the portion CA of 
 the beam. Substituting a t for x in the latter, and observing 
 that y then becomes D : ; then substituting this value of D 1 
 in the former equation, and reducing, 
 
 '-&-ad\ (539); 
 
 El 
 
 ay-c?}x ---- (540). 
 
 f** 
 
 The latter equation being that to the neutral line of the por- 
 tion AE of the beam, if we substitute a in it for a?, and 
 represent the ordinate of the neutral line at E by y^ we 
 shall obtain by reduction 
 
 <*-,)-3 s 5 .... (541). 
 
 If a^O, or if the loading commence at the point A of the 
 beam, the value of y^ will be found to be that already deter- 
 mined for the deflexion in this case (equation 530). 
 
 Now, representing the deflexion at E by D 2 , we have evi- 
 dently D 2 D x y r 
 
 (542). 
 
 374. THE CONDITIONS OF THE DEFLEXION OF A BEAM LOADED 
 UNIFORMLY THROUGHOUT ITS LENGTH, AND SUPPORTED AT 
 
 ITS EXTREMITIES A AND D, AND AT TWO POINTS B AND C 
 SITUATED AT EQUAL DISTANCES FROM THEM, AND IN THE 
 SAME HORIZONTAL STRAIGHT LINE. 
 
 Let AB=^, AD =20. 
 
 Let A be taken as the origin of the co-ordinates ; let the 
 
LOADED UNIFORMLY. 
 
 487 
 
 pressure upon that point be 
 represented by P 15 and the 
 pressure upon B by P 2 ; also 
 the load upon each unit of 
 the length of the beam by p. 
 If F be any point in the 
 neutral line to the portion AB 
 of the beam, whose co-ordi- 
 nates are x and y, the pres- 
 
 sures applied to AP, and in equilibrium, are the pressure 
 P 1 at A, the load px supported by AP, and producing the 
 same effect as though it were collected over the centre of 
 that portion of the beam, and the elastic forces developed 
 upon the transverse section of the beam at P ; whence it 
 follows (Art. 360.) by the principle of the equality of 
 moments, taking P as the point from which the moments 
 are measured, that 
 
 =W-I> .... (6*8). 
 
 Integrating this equation between the limits a^ and #, and 
 representing the inclination to the horizon of the tangent to 
 the neutral line at B by /3 2 , 
 
 El(-tan. /3,) = 
 
 -<) .... (544). 
 
 Integrating again between the limits and a?, 
 
 . (545), 
 
 Whence observing that when x=a l , y= 
 El tan. pa P 2 
 
 (546). 
 
 Similarly observing, that if x and y be taken to represent 
 the co-ordinates of a point Q in the beam between B and C, 
 the pressures applied to AQ are the elastic forces upon the 
 section at Q, the pressures P, and P a and the load ^x } we 
 have 
 
 Integrating this equation between the limits a l and a?, and 
 
 fl'tl 
 
 observing that at the former limit the value of -/- is repre- 
 sented by tan. /3 a , we have 
 
4:88 THE DEFLEXION OF A BEAM 
 
 Big-tan. 0.) =Hrf-O-ff.('*-< 
 
 .... (548). 
 
 Now it is evident that, since the props B and C are placed 
 symmetrically, the lowest point of the beam, and therefore 
 of the neutral line, is in the middle, between B and C ; so 
 
 /m/M 
 
 that -/ = 0, when x=a. Making this substitution in equa- 
 
 tion (548), 
 
 -El tan. p^ipW-afi-tPW-a^-lY^a-aJ . . (549). 
 
 Since, moreover, the resistances at C and D are equal to 
 those at B and A, and that the whole load upon the beam is 
 sustained by these four resistances, we have 
 
 P.+P^f** ..... (550). 
 
 Assuming a i =na j and eliminating P 1? P a , tan. ft, between 
 the equations (546), (549), and (550), we obtain 
 
 -D pa ( 3 -f-127i a 24/1 + 8 
 
 8 
 
 24EI 
 
 (5^ a 16^+8) 
 ' 1 2*-8 "f 
 
 Making a?=0 in equation (544); and observing that the cor- 
 responding value of -y- is represented by tan. /3 1? we have 
 
 El (tan. ft tan. ft)== |i*a 1 $ +iP 1 a 1 i . 
 
 Substituting for tan. ^ 2 and P x their values from equations 
 (553) and (551), and reducing, 
 
 ) 
 [....(554). 
 
 Representing the greatest deflexions of the portions AB and 
 
LOADED UNIFORMLY. 
 
 489 
 
 BC of the beam, respectively, by D l and D 2 , and by a?, the 
 distance from A at which the deflexion Dj is attained, we 
 have, by equations (544) and (545), 
 
 -El tan. /3 i= = jf* (x*-a*)-<, 
 x a^tan. /3 2 )=- 
 
 The value of Dj is determined by eliminating a?, between 
 these equations, and substituting the values of P a and tan. J3 t 
 from equations (551) and (553). 
 
 Integrating equation (548) between the limits a t and 0, 
 and observing that at the latter limit y=D 2 , we have 
 
 ^EI^-^) tan. 
 
 Substituting in this equation for the values of tan. )8 S , P,, P a , 
 and reducing, we obtain 
 
 (556). 
 
 Representing BC by 2# t , and observing that ^ 2 = AE 
 AB=a na=(l ri)a, 
 
 ^ 
 - Ua ~48EI ' 3-2ril-n 
 
 375. A BEAM, HAVING A UNIFORM LOAD, SUPPORTED AT EACH 
 EXTREMITY, AND BY A SINGLE STRUT IN THE MIDDLE. 
 
 If, in the preceding article, a^ be assumed equal to a, or 
 
 w=l, the two props B and 
 C will coincide in the centre ; 
 and the pressure P 2 upon 
 the single prop, resulting 
 from their coincidence, win 
 be represented by twice the 
 corresponding value of P a in 
 equation (552) ; we thus ob- 
 tain 
 
490 
 
 THE DEFLEXION OF A BEAM 
 
 The distance x l of the point of greatest deflexion of either 
 portion of the beam from its extremities A or D, and the 
 amount D, of that greatest deflexion, are determined from 
 equations. (555). Making tan. /3 2 =0 in those equations, 
 substituting for P, its value, solving the former in respect to 
 0J, and the latter in respect to D we obtain 
 
 16 
 
 #=4215350 (559). 
 
 48EI 
 
 48EI 
 
 (560). 
 
 376. A BEAM WHICH SUSTAINS A UNIFORM LOAD THROUGHOUT 
 ITS WHOLE LENGTH. AND WHOSE EXTREMITIES ARE SO FIRMLY 
 
 IMBEDDED 
 RIGID. 
 
 IN A SOLID MASS OF MASONRY AS TO BECOME 
 
 Let the ratio of the lengths of the two portions AB and 
 AE of a beam, supported by two props (p. 487), be assumed 
 to be such as will satisfy the condition 5^ 2 16?i + 8=0 ; or, 
 solving this equation, let 
 
 n=- 
 
 (561). 
 
 The value of tan. 
 
 (equation 553) will then become 
 zero ; so that when this re- 
 lation obtains, the neutral 
 line will, at the point B, be 
 parallel to the axis of the 
 abscissae ; or, in other words, 
 the tangent to the neutral 
 line at the point B will retain, 
 after the deflexion of the beam, the position which it had' 
 before; i. e., its position will be that which it would have 
 retained if the beam had been, at that point, rigid. Now 
 this condition of rigidity is precisely that which results from 
 the insertion of the beam at its extremities in a mass of 
 masonry, as shown in the accompanying figure ; whence it 
 follows that the deflexion in the middle of the beam is the 
 same in the two cases. Taking, therefore, the negative sign 
 in equation (561), and substituting for n its value f (4 1/6) 
 or -6202041 in equation (557), and observing that, in that 
 
SUPPORTED AT ANY NUMBER OF POINTS. 
 
 491 
 
 equation, 2# a represents the distance BC in the accompany- 
 ing figure, we obtain 
 
 2EI 
 
 (562). 
 
 By a comparison of this equation with equation (530), it 
 appears that the deflexion of a learn sustaining a pressure 
 uniformly distributed over its whole length, and having its 
 extremities prolonged and firmly imbedded, is only one-fifth 
 of that which it would exhibit if its extremities were free.* 
 
 If the masonry which rests upon each inch of the portion 
 AB of the beam be of the same weight as that which rests 
 upon each inch of BC, the depth AB of the insertion of each 
 end should equal '62 of AE, or about three ten.hs of the 
 whole length of the beam. 
 
 1 
 
 , I 1 
 
 j 
 
 f 
 
 
 1 
 
 , J. 
 
 4 
 
 a D 
 
 4 
 
 A 
 
 J 
 
 1 
 
 i 
 
 1 
 
 I 
 
 1 
 
 
 1 
 
 [ 
 
 1 
 
 r~ 
 
 
 377. Conditions of the equilibrium of a beam supported at 
 any number of points and deflected by given pressures. 
 
 To simplify the investigation, let the points of support 
 
 ABC be supposed to be three 
 in number, and let the direc- 
 tions of the pressures bisect 
 the distances between them ; 
 the same analysis which de- 
 termines the conditions of the 
 equilibrium in this case will 
 be found applicable in the more general case. 'Let P 15 P 3 , 
 P B , be taken to represent the resistances of the several points 
 of support, a l and # 2 the distances between them, P Q P 4 the 
 deflecting pressures, and x y the co-ordinates of any point in 
 the neutral line from the origin B. Substituting in equation 
 
 (500) for ^ its value -y^, and observing that in respect to the 
 
 portion BD of the beam ^Pp=P 9 (^a l x) 'P 1 (a l x\ and 
 that in respect to the portion DA of the beam, 2Pj9= 
 ]?,(#, x\ we have for the differential equation to the 
 neutral line between B and D 
 
 * The following experiment was made by Mr. Hatcher to verify this result. 
 A strip of deal y 3 ^- in. byy 7 g- in. was supported with its extremities resting 
 loosely on rollers six feet apart, and was observed to deflect 1-2 inch in the 
 middle by its own weight. The extremities were then made rigid by confining 
 them between straight edges, and, the distance between the points of support 
 remaining the same, the deflexion was observed to be '22 inch. The theory 
 would have given it '24. 
 
492 BEAM SUPPORTED AT AlfY NUMBER OF POINTS. 
 
 =P,(K- a! )-P,( 1 - ! B) ---- (563), 
 between D and A 
 
 ^-P^-*) . . . (584). 
 
 .Representing by the inclination of the tangent at B to the 
 axis of the abscissae, and integrating the former of these 
 equations twice between the limits and #, 
 
 ii. ft .... (565); 
 
 tan - - 
 
 Substituting ^a l for x in these equations, and representing 
 by D! the value of y, and by 7 the inclination to the horizon 
 of the tangent at the point D, we obtain 
 
 El tan. y=P 2 a 1 2 -fI ) A 2 +EItan./3 .... (567), 
 Ero^APA'-APA'+iEL*, tan. . . . . (568). 
 Integrating equation (564) between the limits -^ and x 
 
 tan. 
 
 Eliminating tan. y between this equation and equation (567) 
 and reducing, 
 
 EI^^-PX^-^+EI^-^+iPA 2 (569). 
 
 Integrating again between the limits - and a?, and elimi- 
 nating the value of D, from equation (568), 
 
 ]S"ow it is evident that the equation to the neutral line in 
 respect to the portion CE of the beam, will be determined 
 by writing in the above equation P 5 and P 4 for P x and P, 
 respectively. 
 
 Making this substitution in equation (570), and writing 
 -tan. /3 for +tan. (3 in the resulting equation ; then assum- 
 
BEAM SUPPORTED AT ANY NUMBER OF POINTS. 493 
 
 ing x=a 1 in equation (570), and a?=# a in the equation thus 
 derived from it, and observing that y then becomes zero in 
 both, we obtain 
 
 0=-iPA' + s PA'+EI^ tan. ft 
 A 8 -EIa 2 tan. (3. 
 
 Also, by the general conditions of the equilibrium of parallel 
 pressures (Art. 15.), 
 
 Eliminating between these equations and the preceding, as- 
 suming a 1 + a. t =a^ and reducing, we obtain . 
 
 1600, 
 
 By equation (568), 
 
 D .=7 
 
 Similarly, 
 
 -9I > A 1 ..... (5T5); 
 
 'By equation (567), 
 
 ..... (577). 
 
 If a, be substituted for x in equation (569), and for P, and 
 tan. /3 their values from equations (571) and (576) ; and if 
 the inclination of the tangent at A to the axis of x be repre- 
 sented by p,j we shall obtain by reduction 
 
494: BEAM SUPPORTED AT ANT NUMBER OF POINTS. 
 
 ..... (578). 
 
 Similarly, if /3 2 represent the inclination of the tangent at 
 C to the axis of a?, 
 
 ..... (579). 
 
 378. If the pressures P 2 and P 4 , and also the distances 0, 
 tnd a a , be equal, 
 
 P,=P 5 =AP 5 , P,= VPtan. 0=0, tan. ft=tan. 0.= 
 
 379. If the distances ^ and a 2 be equal, and P 4 =3P a , 
 P,=iP,, P.=VP P.=tP tan. /3=-^fi, tan./3,=0.* 
 
 380. If ,=, and 3P,=13P a , P,=0, P ^ ', P,,.P 5 =fP,. 
 
 * The following experiments were made by Mr. Hatcher to verify this result. 
 The bar ACB, on which the experiment was to be tried, was supported on 
 knife edges of wrought iron at A, C, and B, whose distances AC and CB were 
 each five feet. The angles of the knife edges were 90, and the edges were 
 oled previous to the experiments. The weights were suspended at points D 
 
 and E intermediate between the points of support. In measuring the angles 
 of deflexion the instrument (which was a common weighted index-hand turn- 
 ing on a centre in front of a graduated arc) was placed so that the angle c 
 of vhe parallelogram of wood carrying the arc was just over the knife-edge B, 
 the side cd of the parallelogram resting on the deflected bar. This position 
 gave the angle at the point of support. 
 
 1st Experiment. A bar of wrought iron half an inch square, being loaded 
 at E with a weight of 18 Ib. 13 oz., and at D with 52 Ib. 3 oz., assumed a per- 
 fectly horizontal position at B, as shown by the needle. The proportion of 
 these weights is 2 - 77 : 1. 
 
 2d Experiment. A bar -7 inches square, being loaded at E with a weight of 
 87 '3 Ib., and at D with a weight of 112 Ib., assumed a perfectly horizontal 
 position at B. The weights were in this experiment accurately in the propor- 
 tion 3 : 1. 
 
 3d Experiment. A round bar, '75 inch in diameter, being loaded at E with 
 37 '3 Ib., and at D with 112 Ib., showed a deviation from the horizontal position 
 at B amounting to not more than 20'. The weights were in the proportion of 
 8:1. 
 
 The influence of the weight of the bar is not taken into account. 
 
A BEAM DEFLECTED BY PRESSURES. 
 
 495 
 
 381. CURVATURE OF A RECTANGULAR BEAM, THE DIRECTION OF 
 THE DEFLECTING PRESSURE AND THE AMOUNT OF THE DE- 
 FLEXION BEING ANY WHATEVER. 
 
 The moment of inertia I (Art. 358.) is to be taken, about 
 an axis perpendicular to the plane of deflexion, and passing 
 through the neutral line, the distance h of which neutral 
 line from the centre of gravity of the section is determined 
 by equation (499). 
 
 Now y^fo 3 representing (Art. 362.) the moment of inertia 
 of the rectangular section of the beam about an axis pass- 
 ing through its centre of gravity, it follows (Art. 79.) that 
 the moment I about an axis parallel to this passing through 
 a point at distance h from it is represented by 
 
 Substituting, therefore, the value of h from equation 
 (499), 
 
 Substituting this value in equation (500), and reducing, 
 1_ 12P,E^ 
 R~~12R f P 1 1 sin. 1 d + EW ' 
 
 Draw ax parallel to the 
 position of the beam be- 
 fore deflexion; take this 
 line as the axis of the 
 abscissae and a as the 
 origin ; then p l =Jlm=It,n 
 
 +nm=MJR cos. 
 
 aM. sin. 
 
 -\-x sin. M&m. 
 
 Let, now, the inclination 
 DaP l of the direction of 
 P! to the normal at a be 
 represented by d,, and the inclination Mat of the tangent to 
 
 the neutral line at a to a%, by f3 1 ; then 
 
 ^ cos - 
 
 :.p,=y sn. 1 +/ 1 -f cos. 
 Substituting this value of p l in the preceding equation, 
 
496 A BEAM DEFLECTED BY PRESSURES. 
 
 . in. ft +ft)+a cos. (*.+,){ ._.. 
 
 K~ 12RT 2 sin. 2 ^+E^V 
 
 where represents (Art. 355.) the inclination Rqa of the 
 normal at the point K to the direction of P,. 
 
 382. Case in which the deflexion of the beam is small. 
 
 If the deflexion be small, and the inclination d,, of the 
 direction of P t to the normal at its point of application, be 
 
 not greater than j ; then y sin. (^+/3 a ) is exceedingly small, 
 
 and may be neglected as compared with x cos. (^ 1 +/3 1 ); in 
 this case, moreover, 6 is, for all positions of R, very nearly 
 equal to 6 im Neglecting, therefore, , as exceedingly small, 
 we have 
 
 !._ .. 
 
 R~12RT 1 2 sin. f ^+E'JV ' 
 
 Solving this equation, of two dimensions, in respect to =p , and 
 taking the greater root, 
 1 6P t 
 
 . a ^ ic a sin. 2 ^} ---- (584). 
 
 383. THE WOKE: EXPENDED UPON THE DEFLEXION OF A UNI- 
 
 FORM RECTANGULAR BEAM, WHEN THE DEFLECTING PRES- 
 SURES ARE INCLINED AT ANY ANGLE GREATER THAN HALF A 
 RIGHT ANGLE TO THE SURFACE OF THE BEAM. 
 
 If u^ represent work expended on the deflexion of the 
 portion AM of the beam, then (equation 505) 
 
 but by equation (500)%s=p . -g 
 
A: BEAM DEFLECTED BY PRESSURES. 497 
 
 p, 6P 
 
 ~ 
 
 by equation (584), observing that the deflexion being small, 
 p^x cos. 6 l very nearly. Now the value of ^ (equation 
 584) becomes impossible at the point where a? cos. ^becomes 
 
 less than c sin. &. ; the curvature of the neutral line com- 
 
 1/3 
 
 mences therefore at that point, according to the hypotheses; 
 on which that equation is founded. Assuming, then, the 
 
 corresponding value 7=0 tan. d, of a? to be represented by x t> 
 
 the integral (equation 585) must be taken between the limits 
 a?, and a instead of and a t ; 
 
 SP/cosX/! -7-i *z i i ** ) j 
 
 /. uj= ^j 9 / \x cos. O.+x Vx cos. L wr sin. ",J o J 
 J^y(? */ 
 
 i 
 
 P a cos 2 <M 1 3 1 
 
 /,^= x ' H^ 3 - C 3 tan. 3 t). + (^ a ^ a tan. a ^) 2 h*(586). 
 
 xLOC l ,-i >4/o i/ j \ / 
 
 And a similar expression being evidently obtained for the 
 work expended in the deflexion of the portion BM of the 
 beam, it follows, neglecting the term involving o 3 as exceed- 
 ingly small when compared with &, 3 , that the whole work TJ^ 
 expended upon the deflexion is represented by the equation 
 
 C08 ' '* i - tan. + 
 
 But if ^ 3 be taken to represent the inclination of P 3 to the- 
 normal to the surface of the beam, as 6 l and d a represent the 
 similar inclinations of P l and P 2 , then, the deflexion being 
 small, 
 
 * Church's Int. Cal. Art. 149. 
 32 
 
498 A BEAM DEFLECTED BY PRESSURES. 
 
 ^ COS. =,a a COS. ,, 2 & COS. 2 = 3 1 COS. ,. 
 
 Eliminating P t and P 2 between these equations and the 
 preceding, 
 
 (587). 
 
 If the pressure P s be applied perpendicularly in the centre 
 of the beam, and the pressures P x and P 2 be applied at its 
 extremities in directions equally inclined to its surface ; then 
 ^=(1^=^^ 6 l =6 t =6^ and d g =0. Substituting these values 
 in the preceding equations, and reducing, 
 
 .... (588). 
 
 384. THE LINEAR DEFLEXION OF A RECTANGULAR BEAM. 
 
 Dj being taken as before (Art. 368.) to represent the de- 
 flexion of the extremity A measured in a direction perpen- 
 dicular to the surface of the beam, we have (Art. 52.) 
 
 ^I cos. 
 
 But by equation (586), neglecting the term involving c 8 , 
 
 CC (/ Cv 1 -^ j_ , QAf ^./ Q 10, O*\"Oi 
 
 a2 
 
 Dividing both sides by P 15 reducing, and integrating, 
 
 9P 2. 
 
 D.^^cos^.j^ + K-ic'tan. 'O,) 2 } ---- (589) 
 
 Proceeding similarly in respect to the deflection D 8 perpen 
 
INCLINED AT ANY ANGLE TO ITS SURFACE. 499 
 
 dicnlar to the surface of the beam at the point of application 
 of P 3 , we obtain from equation (587) 
 
 . . . . (590) 
 
 In the case in which P, and P 3 are equally inclined to the 
 extremities of the beam and the direction of P 3 bisects it, 
 this equation becomes 
 
 385. The work expended upon the deflexion of a beam sub- 
 jected p the action of pressures applied to its extremities, 
 and to a single intervening point, and also to the action 
 of a system of parallel pressures uniformly distributed 
 over its length. 
 
 Let - represent the aggregate amount of the parallel 
 
 pressures distributed over each unit of the length of the 
 beam, and a their common inclination to the perpendicular 
 to the surface ; then will px represent the aggregate of those 
 distributed uniformly over the surface DT, and these will 
 manifestly produce the same effect as though they were 
 collected in the centre of DT. Their moment about the 
 point R is therefore represented by M-OJJOJ cos. a, or by ^x 3 
 cos. a ; and the sum of the moments of the pressures applied 
 to AT is represented by (P^ cos. d a Jjxaj* cos. a). Substi- 
 tuting this value of the sum of the moments for Pj? a in 
 equation (505), we obtain 
 
 _ 1 /*(?,# COS. tjjwi? COS. a) a 
 
 ' ~~ ~ 
 
500 
 
 DEFLEXION OF A BEAM BY PRESSURES. 
 
 386. If the pressures le all perpendicular to the surface of 
 the ~beam, ^=0, a=0, and I is constant (equation 499); 
 whence we obtain, by integration and reduction, 
 
 (592). 
 
 If the pressure P 3 be applied in the centre of the beam, 
 P^-JP.+i^fl, and a 1 =^a j also the whole work U 3 of 
 deflecting the beam is equal to 2^; whence, substituting 
 and reducing, 
 
 ( 593 )- 
 
 387. A RECTANGULAR BEAM IS SUPPORTED AT ITS EXTREMITIES 
 BY TWO FIXED SURFACES, AND LOADED IN THE MIDDLE I IT 
 IS REQUIRED TO DETERMINE THE DEFLEXION, THE FRICTION 
 OF THE SURFACES ON WHICH THE EXTREMITD3S REST BEING 
 TAKEN INTO ACCOUNT. 
 
 It is evident that the work which produces the deflexion 
 
 of the beam is done upon it partly by the deflecting pressure 
 P, and partly by the friction of the surface of the beam 
 upon the fixed points A and B, over which it moves whilst 
 in the act of deflecting. Representing by 9 the limiting 
 angle of resistance between the surface of the beam and 
 either of the surfaces upon which its extremity rests, the 
 friction Q, or Q 2 upon either extremity will be represented 
 by fP tan. 9 ;. and representing by s the length of the 
 curve ca or cb, and by %a the horizontal distance between 
 the points of support ; the space through which the surface 
 of the beam would have moved over each of its points of 
 support, if the point of support had been in the neutral line, 
 is represented by s a, and therefore the whole work done 
 upon the beam by the friction of each point of support by 
 
 i tan. <p/P<&. Moreover,. I> representing the deflexion of 
 
THE SOLID OF THE STRONGEST FORM. 501 
 
 the beam under any pressure P, the whole work done by P 
 is represented by /P^D. Substituting, therefore, for the 
 
 work expended upon the elastic forces opposed to the 
 deflexion of the beam its value from equation (588), and ob- 
 serving that the directions of the resistances at A and B are 
 inclined to the normals at those points at angles equal to 
 the limiting angle of resistance, we have 
 
 / 
 
 But 
 
 e( l uati011 
 
 Substituting these values in the above equation, and dif- 
 ferentiating in respect to P, we have 
 
 T>dD_P K + 2 |c a tan. a <p)f| PV 
 
 " ~30ET tan ' 9 ' 
 
 Dividing by P, and integrating in respect to P, 
 
 PK + (^-^tan.^ PV 
 
 3 "60ET 9 ' 
 
 388. THE SOLID OF THE STRONGEST FORM WITH A GIVEN 
 QUANTITY OF MATERIAL. 
 
 The strongest form which can be given to a solid body in 
 the formation of which a given quantity of material is to be 
 used, and to which the strain is to be applied under given 
 circumstances, is that form which renders it equally liable to 
 rupture at every point. So that when, by increasing the 
 strain to its utmost limit, the solid is brought into the state 
 bordering upon rupture at one point, it may be in the state 
 bordering upon rupture at every other point. For let it be 
 supposed to be constructed of any other form, so that its 
 rupture may be about to take place at one point when it is 
 not about to take place at another point, then may a portion 
 of the material evidently be removed from the second point 
 without placing the solid there in the state bordering upon 
 
502 THE RUPTURE OF A BAR. 
 
 rupture, and added at the first point, so as to take it out 
 of the state bordering upon rupture at that point ; and thus 
 the solid being no longer in the state bordering upon 
 rupture at any point, may be made to bear a strain greater 
 than that which was before upon the point of breaking it, 
 and will have been rendered stronger than it was before. 
 The first form was not therefore the strongest form of which 
 it could have been constructed with the given quantity of 
 material; nor is any form the strongest which does not 
 satisfy the condition of an equal liability to rupture at every 
 point. 
 
 The solid, constructed of the strongest form, with a given 
 quantity of a given material, so as to be of a given strength 
 under a given strain, is evidently that which can be con- 
 structed, of the same strength, with the least material ; so 
 that the strongest form is also the form of the greatest 
 economy of material. 
 
 RUPTURE. 
 
 389. The rupture of a bar of wood or metal may take 
 place either by a strain or tension in the direction of its 
 length, to which is opposed its TENACITY ; or by a thrust or 
 compressing force in the direction of its length, to which is 
 opposed its resistance to COMPRESSION ; or each of these 
 forces of resistance may oppose themselves to its rupture 
 transversely, the one being called into operation on one side 
 of it, and the other on the other side, as in the case of 
 a TRANSVERSE STRAIN. 
 
 TENACITY. 
 
 390. The tenacities of different materials as they have 
 been determined by the best authorities, and by the mean 
 results of numerous experiments, will be found stated in a 
 table at the end of this volume. The unit of tenacity is that 
 opposed to the tearing asunder of a bar one square inch in 
 section, and is estimated in pounds. It is evident that the 
 tenacity of a fascile of n such bars placed side by side, or 
 of a single bar n square inches in section, would be equal 
 to n such units, or to n times the tenacity of one bar. 
 
 To find, therefore, the tenacity of a bar of any material 
 in pounds, multiply the number 'of square inches in its sec- 
 
RUPTURE OF A BAR SUSPENDED VERTICALLY. 503 
 
 tion by its tenacity per square inch, as shown by the 
 table. ' 
 
 391. A BAR, CORD, OR CHAIN IS SUSPENDED VERTICALLY, CAR- 
 RYING A WEIGHT AT ITS EXTREMITY I TO DETERMINE THE 
 
 'CONDITIONS OF ITS RUPTURE. 
 
 First. Let the bar be conceived to have a uniform section 
 represented in square inches by K ; let its length in inches 
 be L, the weight of each cubic inch |u<, the weight suspended 
 from its extremity W, the tenacity of its material per square 
 inch r ; and let it be supposed capable of bearing m times 
 the strain to which it is subjected. The weight of the bar 
 
 will then be represented by juJLK, and the strain upon its 
 highest section by ^LK-h W. Now the strain on this section 
 is evidently greater than that on any other ; it is therefore at 
 this section that the rupture will take place. But the resist- 
 ance opposed to its rupture is represented by Kr ; whence it 
 follows (since this resistance is m times the strain) that 
 
 (595). 
 
 By which equation is determined the uniform section K of a 
 bar, cord, or chain, so that being of a given length it may be 
 capable of bearing a strain m times greater than that to 
 which it is actually subjected when suspended vertically. 
 The weight Wj. of the bar is represented by the formula 
 
 392. Secondly. Let the section of the rod be variable ; and 
 let this variation of the section be such that its strength, at 
 every point, may be that which would cause it to bear, 
 without breaking, m times as great a strain as that which it 
 actually bears there. Let K represent this section at a point 
 whose distance from the extremity which carries the weight 
 W is x ; then will the weight of the rod beneath that point 
 
 be represented by I &~Kdx ; or, supposing the specific gravity 
 
504 RUPTURE OF A BAR SUSPENDED VERTICAL! ,T. 
 
 of the material to be every where the same, by f/Kdfo? : also 
 the resistance of this section to rupture is Kr. 
 
 Differentiating this expression in respect to a?, observing that 
 K is a function of a?, and dividing by Kr, we obtain 
 
 1 dK _ mp 
 K ^~ = : T ; 
 
 Integrating this expression between the limits and #, and 
 representing by Ko the area of the lowest section of the rod, 
 
 : But the strain sustained by the section KO is W, therefore 
 
 (597). 
 
 The whole weight W 2 of the rod, cord, or chain, is repre- 
 sented by the formula 
 
 -l . (598). 
 
 A rope or chain, constructed according to these conditions, 
 is evidently as strong as the rope or chain of uniform section 
 whose weight TV\ is determined by equation (596), the value 
 of 77i being taken the same in both cases. The saving of ma- 
 terial effected by giving to the cord or chain a section vary- 
 ing according to the law determined by equation (598) is 
 represented by "Wj W 2 , or by the formula 
 
 T^-T i^rriL v 
 
 ^_W ~~1 (599). 
 
 rm^L V* / 
 
 * Church's Int. Cal. Art. 159. 
 
THE SUSPENSION BKIDGE. 
 
 505 
 
 THE SUSPENSION BRIDGE. 
 
 393. General conditions of the equilibrium of a loaded 
 
 chain. 
 
 Let AEH represent a chain or cord hanging freely from 
 
 two fixed points A and H, 
 and having certain weights 
 w. 
 
 \ 
 
 f I T 7 .- * t 
 
 rods 
 
 M w n &c., suspended by 
 or cords from given 
 points B, C, D, &c., in its 
 length. Through the lowest 
 point E of the chain draw 
 the vertical E#, containing 
 as many equal parts as there 
 are units in the weight of 
 the chain between E and any 
 point of suspension B, to- 
 gether with the suspending 
 rods attached to it, and the weights which they severally 
 carry ; draw aP parallel to the direction of a tangent to the 
 curve at B, and produce the tangent at E to meet aP in P ; 
 then will aP and EP contain as many equal parts as there 
 are units in the tensions at B and E respectively ; and if E& 
 and EC be taken to represent the whole weights sustained by 
 EC and ED, and P5 and PC be joined, these lines will in 
 like manner represent the tensions upon the points C and D. 
 For the pressures applied to EB, and in equilibrium, being 
 the weight of the chain, the weights of the suspending rods, 
 the weights attached to the rods, and the tensions upon B 
 and E, the principle of the polygon of pressures (Art. 9.) 
 obtains in respect to these pressures. Now the lines drawn 
 to complete this polygon, parallel to the weights, form 
 together the vertical line E&, and the polygon (resolving 
 itself into a triangle) is completed by the lines aP and EP 
 drawn parallel to the tensions upon B and E. Each line 
 contains, therefore, as many equal parts (A^. 9.) as there 
 are units in the corresponding tension. Also, the pressures 
 applied to the portion EC of the curve, being the weights 
 whose aggregate is represented by E&, and the tensions upon 
 E and C, of which the former is represented in direction 
 and amount by EP, it follows (Art. 9.) that the latter is 
 represented also in direction and amount by the line P5, 
 
506 
 
 THE CATENARY. 
 
 which completes the triangle aPb ; so that 5P is parallel to 
 the tangent at C. 
 
 In like manner it is evident that the tension upon D is 
 represented in magnitude and direction by cP ; so that cP is 
 parallel to the tangent to the curve at D. 
 
 THE CATENARY. 
 
 394. If a chain of uniform section be suspended freely 
 between two fixed points A and B, being acted upon by no 
 other pressures than the weights of its parts, then it will 
 assume the geometrical form of a curve called the 
 
 Let PT be a tangent to any point P of the curve inter- 
 secting the vertical CD passing through its lowest point D 
 
 in T ; draw the horizontal line DM intersecting PT in Q ; 
 take this line as the axis of the abscissae ; and let DM =a?, 
 MP=y, DP =5, weight of each unit in the length of the 
 chain =f*, tension at D <?. Now DT being taken to repre- 
 sent the weight w of DP, it has been shown (Art. 393.) 
 that DQ will represent the tension c at D, and TQ that 
 at P. 
 
 /7v/ DT . 
 
 Also, jg= tan. PQM = tan. DQT = j)Q=y , 
 
 " dx 
 
 (600). 
 
 dx 
 
 be- 
 
THE CATENARY. 507 
 
 tween the limits and s,* and observing that when $=0, 
 
 By addition and reduction, 
 
 i*x 
 
 ..... (602> 
 
 / ^ i*x\ 
 
 =*?(/_.") 
 
 Substituting this value for s in equation (600), and inte- 
 grating between the limits and a?, 
 
 ~~ 
 
 +s _ 2 f* s _ s 
 
 which is the equation to the catenary. 
 
 395. The tension (c) on the lowest point of the catenary. 
 
 Let 2S represent the whole length of the chain, and 2a 
 the horizontal distance between the points of attachment. 
 Now when x=a, s=S ; therefore (equation 602), 
 
 / f*a f*a\ 
 
 s=| T ~l (so*); 
 
 for which expression the value of c may be determined by 
 approximation. 
 
 396. The tension at any point of the chain. 
 The tension T at P is represented by TQ= 
 
 Church's Int. Cal. Art. 144. 
 
508 THE CATENARY. 
 
 (605). 
 
 Now the value of c has heen determined in the preceding 
 article ; the tension upon any point of the chain whose dis- 
 tance from its lowest point is s is therefore known. 
 
 397. The inclination of the curve to the vertical at any 
 
 point. 
 
 Cl ?/ 
 
 Let i represent this inclination, then cot. i=-jr 
 
 ( ~^ x \ 
 c c ] (606). 
 - / 
 
 The inclination may be determined without having first 
 determined the value of <?, by substituting cot. i for in 
 
 equation (601) ; we thus obtain, writing also a and S for x 
 and #, 
 
 a 
 
 q=tan. i log. (cot. i + cosec. *)=tan. i log. cot. Jt; 
 
 a 
 8 
 
 /. tan. i log. tan. j^s== (607). 
 
 This equation may readily be solved by approximation ; a 
 the value of c may then be determined by the equati 
 
 and 
 equation 
 
 =^ tan. i. 
 
 398. A. chain of given length being suspended between two 
 given points in the same horizontal line : to determine the 
 depth of the lowest point beneath the points of attachment y 
 and, conversely, to determine the length of the chain whose 
 lowest point shall hang at a given depth below its points 
 of attachment. 
 
 The same notation being taken as before, 
 
THE CATENARY. 509 
 
 Integrating between the limits and s, and observing that 
 y=0 when s=0, 
 
 Solving this equation in respect to *, 
 
 + j ..... (609). 
 
 If H represent the depth of the lowest point, or the versed 
 sine of the curve, then y=H when =S. 
 
 (610). 
 
 - (611). 
 
 399. The centre of gravity of the catena/ry. 
 
 If Gr represent the height of the centre of gravity above 
 the lowest point, we have (Art. 32.) 
 
 S.G=fyds=fy~dx. 
 
 erefore, for y am 
 tions (602) and (603), we have : 
 
 Substituting, therefore, for y and - their values from equa- 
 
 cux 
 
 L. 
 
 fix 
 
 +s +2 2 ^ 
 
510 
 
 THE SUSPENSION BRIDGE 
 
 But by equation (604) B=^-l c c ) and by equation 
 
 V 'g S I 
 
 603), 
 
 -p 
 
 s 4' 
 
 + ' 
 
 .-. 0=iH-l- (613). 
 
 400. THE SUSPENSION BEIDGE OF GREATEST STRENGTH, THE 
 WEIGHT OF THE SUSPENDING RODS BEING NEGLECTED. 
 
 Let ADB represent the chain, EF the road-way ; and let 
 
 the weight of a bar of the material of the chain, one square 
 inch in section and one foot long, be represented by f*,, the 
 weight of each foot in the length of the road-way by f* a , the 
 aggregate section of the chains at any point P (in square 
 inches) by K, the co-ordinates DM and MP of P by a? and y, 
 and the length of the portion DP of the chain by s. Then 
 
 will the weight of DP be represented by ^ I K<&, and the 
 
 weight of the portion CM of the roadway by ^x ; so that 
 the whole load (u) borne by the portion DP of the chain 
 will be represented (neglecting the weight of the suspending 
 rods) by 
 
 (614). 
 
OF GREATEST STRENGTH. 511 
 
 Let this load (w), supported by the portion DP of the 
 chain, be represented by the line D#, and draw Dp in the 
 direction of a tangent at D, representing on the same scale 
 the tension c at that point ; then will ap be parallel to a 
 tangent to the chain at P (Art. 393). 
 
 .. 
 
 dx 
 
 (615). 
 
 Now let it be assumed that the aggregate section of the 
 chains is made so to vary its dimensions, that their strength 
 may at every point be equal to m times the strain which 
 they have there to sustain. But this strain is represented in 
 magnitude by the line ap (Art. 393.), or by (c a -fV)*; if, 
 therefore, r be taken to represent the tenacity of the mate- 
 rial of the chain, per square inch of the section, then 
 
 (616). 
 Therefore Kr=mc(l + ~j = me (l + ^J* (equation 615) 
 
 =mc-j- ; therefore-/ 1 ". Also / ~Kds= /K-^- 
 dx dx mo J J dx 
 
 (equation 616) ; 
 /.(equation 6I4 : )u=- 2 
 
 Differentiating in respect to a?, and observing that -r- 
 du dy u du . 
 
 T y a*r- T y ( e i uatlon 615 ). 
 
 du u du 
 
 u u 
 
 re r du r T udu 
 
512 THE SUSPENSION BRIDGE 
 
 Integrating these expressions,* we obtain 
 
 + 
 
 Substituting in this equation the value of u given by the 
 preceding equation, and reducing, 
 
 which is the equation to the suspension chain of uniform 
 strength, and therefore OF THE GREATEST STRENGTH WITH A 
 
 GIVEN QUANTITY OF MATERIAL. 
 
 401. To determine the variation of the section K of the 
 chain of the suspension bridge of the greatest strength. 
 
 Let the value of u determined by equation (617) be sub- 
 stituted in equation (616) ; we shall thus obtain by reduction 
 
 (l + ^-)* }* . (819).t 
 r \ cmpj 
 
 It is evident from this expression that the area of the sec- 
 tion of the chains, of the suspension bridge of uniform 
 strength, and therefore of the greatest economy of material^ 
 increases from the lowest point towards the points of suspen- 
 sion, where it is greatest. 
 
 * Church's Int. Cal. Art. 133, Case IV. 
 
 f = ; .-. s= /Kdx. Now the function K (equation 619) may be 
 1 dx me' mcj 
 
 integrated in respect to x by known rules of the integral calculus , the value 
 of s may therefore be determined in terms of #, and thence the length in 
 terms of the span. The formula is omitted by reason of its length. 
 Church's Int. Cal. Art. 1 29, Case II. 
 
OF GREATEST STBKNGTH. 
 
 513 
 
 402. To determine the weight "W of the chain of the suspen- 
 sion bridge of the greatest strength. 
 
 Let it be observed that W=f* 1 / ~K.ds=u ^x (equation 
 
 614) ; substituting the value of u from equation (617), we 
 have 
 
 ^-'-)*tan. I ^(l+- T A\^ I -w . . (620). 
 mwj ( r \ cmpj ( v ' 
 
 403. To determine the tension c upon the lowest point D of' 
 the chain of uniform strength. 
 
 Let H be taken to represent the depth of the lowest point 
 D, beneath the points of suspension, and 2& the horizontal 
 distance of those points : and let it be observed that H and 
 a are corresponding, values of y and x (equation 618) ; 
 
 TT T 1 ( W 1 , 
 
 ,VH= log. sec. \ - 
 
 m^ ( r 
 
 Solving this equation in respect to <?, 
 
 c=- 
 
 -1 
 
 sec. 
 
 -1 
 
 ..(621). 
 
 404. THE SUSPENSION BRIDGE OF GREATEST STRENGTH, THE 
 WEIGHT OF THE SUSPENDING RODS BEING TAKEN INTO AC- 
 
 COUNT. 
 
 Conceive the suspending rods to be replaced by a con- 
 
514 THE SUSPENSION BRIDGE 
 
 tinuous flexible lamina or plate connecting the roadway with 
 the chain, and of such a uniform thickness that the material 
 contained in it may be precisely equal in weight to the ma- 
 terial of the suspending rods. It is evident that the condi- 
 tions of the equilibrium will, on this hypothesis, be very 
 nearly the same as in the actual case. Let f/< 8 represent the 
 
 weight of each square foot of this plate, then will j* 8 / yd% 
 
 represent the weight of that portion of it which is suspended 
 from the portion DP of the chain, and the whole load u upon 
 that portion of the chain will be represented by 
 
 .... (622). 
 It may be shown, as before (Art. 400.), that 
 
 (623). 
 
 fKds= f(c*+u*)d%. Substituting in equation (622), 
 
 differentiating in respect to a?, and observing that =-=- , 
 
 ax c a/y 
 
 du udu m 
 
 tereTy^ 
 
 Transposing, reducing, and assuming, 
 
 = 
 
 -j 
 
 ay 
 
 A linear equation in u*, the integration of which by a well 
 known method gives 
 
 2 ay 
 
 Assuming the length of the shortest connecting rod DC to 
 be represented by &, integrating between the limits b and y, 
 and observing that when y=&, ^=0, 
 
 * Church's Int. Cal. Art. 176. 
 
OF GREATEST STRENGTH. 515 
 
 -2a& -2ayv 
 
 -y) + +*H-ls ( -1) (626). 
 
 Substituting this value of u* in equation (623), and 
 reducing, 
 
 by which expression the variation of the section of the chain 
 of uniform strengtli is determined. 
 
 Differentiating the equation -^=- in respect to a?, and 
 
 ci ?/ 
 substituting for -r- its value from equation (624:). 
 
 Substituting for w a its value from equation (626), 
 
 //7/ 
 
 Multiplying both sides of this equation by jag, and integrat- 
 ing between the limits 5 and y, observing that when y=5, 
 
 Now let it be observed, that the value of T, being in all 
 practical cases exceedingly great as compared with the 
 values of f*, and m, the value of a (equation 625) is exceed- 
 ingly small ; so that we may, without sensible error, assume 
 those terms of the series s 2a (y~ & ) which involve powers of 
 2a(y b) above the first, to vanish as compared with unity, 
 
 * Church's Int. Cal. Art. 140. 
 
516 THE SUSPENSION BRIDGE. 
 
 This supposition being made, we have s 2a (y-&)_ l=2a(y 5), 
 whence, by substitution and reduction, 
 
 \WMf 
 
 Extracting the square root of both sides, transposing, and 
 integrating. 
 
 integrating 
 
 the equation to a parabola whose vertex is in D, and its 
 axis vertical.* 
 
 The values a and H of a? and y at the points of suspension 
 being substituted in this equation, and it being solved in 
 respect to 0, we obtain 
 
 by which expression the tension c upon the lowest point of 
 the curve is determined, and thence the length y of the sus- 
 pending rod at any given distance x from the centre of the 
 span, by equation (628), and the section K of the chain at 
 that point by equation (627), which last equation gives by a 
 reduction similar to the above 
 
 (630). 
 
 405. The section of the chains being of uniform dimensions^ 
 as in the common suspension bridge, it is required to 
 determine the conditions of the equilibrium.^ 
 
 The weight of the suspending rods being neglected, and 
 the same notation being adopted as in the preceding arti- 
 cles, except that JA, is taken to represent the weight of one 
 foot in the length of the chains instead of a bar one square 
 inch in section, we have by equation (614), since K is here 
 constant, 
 
 (631). 
 
 * Church's Analyt. Geom. Art. 191. 
 
 f This problem appears first to have been investigated by Mr. Hodgkinson 
 in the fifth volume of the Manchester Transactions ; his investigation extends 
 to the case in which the influence of the weights of the suspending rods is 
 included. 
 
THE COMMON SUSPENSION BRIDGE. 
 
 517 
 
 Differentiating this equation in respect to a?, and observing 
 that ~=(l + )*=(l + 7)* (equation 615), and that 
 
 du _du cty __du u _ 
 
 == ~~-^- " 
 
 /cdu r udM 
 
 fs(c>+<0*+fV>' y = -/ f^+tO'+lV' 
 
 Tlie former of these equations may be rationalised by 
 assuming (tf+tf^G + zu, and the latter by assuming 
 u*y=z ; there will thus be obtained by reduction 
 
 The latter equation may be placed under the form 
 
 which expression being integrated and its value substituted 
 for z, we obtain 
 
 y=- 
 
 The method of rational fractions (Church's y . 
 
 Art. 135) being applied to the function under the integral 
 sign in the former equation, it becomes 
 
 
 
 The integral in the first term in this expression is repre- 
 sented by i log. s (j- J, and that of the second term by 
 
518 RUPTURE BY COMPRESSION. 
 
 f*, i ng . 
 
 -* 1 * 8 ' 
 
 according as ^ is greater or less than M- 2 , or according as the 
 weight of each foot in the length of the chains is greater or 
 less than the weight of each foot in- the length of the road- 
 way. 
 
 Substituting for z its value, we obtain, therefore, in the 
 two cases, 
 
 _ e ( . fo-c)-H^+o a )* _ ^ 
 
 ~Ji \ g>e KM-K+'')* W-fi 
 
 (M-A*0*H-fr-Ai)* {(u'+c^-c] ) 
 
 ' (M^i^-^-^i^iV-H 1 )*-*}" ) 
 
 (638). 
 
 g 
 
 If the given values, ^ and H, of a? and y at the points of 
 suspension, be substituted in equations (633) and (632), 
 equations will be obtained, whence the value of the constant 
 c and of u at the points of suspension may be determined by 
 approximation. A series of values of w, diminishing from 
 the value thus found to zero, being substituted in equations 
 (633) and (632), as many corresponding values of x and y 
 will then become known. The curve of the chains may thus 
 be laid down with any required degree of accuracy. 
 
 This common method of construction, which assigns a 
 uniform section to the chains, is evidently false in principle ; 
 the strength of a bridge, the section of whose chains* varied 
 according to the law established in Art. 401. (equation 619), 
 would be far greater, the same quantity of iron being 
 employed in its construction. 
 
 KTTPTURE BY COMPRESSION. 
 
 406. It results from the experiments of Mr. Eaton Hodg- 
 kinson,* on the compression of short columns of different 
 heights but of equal sections, first, that after a certain height 
 is passed the crushing pressure remains the same, as the 
 
 * Seventh Report of the British Association of Science. 
 
KUPTUKE BY COMPRESSION. 519 
 
 heights are increased, until another height is attained, when 
 they begin to break ; not as they have done before, by the 
 sliding of one portion upon a subjacent portion, but by 
 bending. Secondly, that the plane of rupture is always 
 inclined at the same constant angle to the base of the 
 column, when its height is between these limits. These two 
 facts explain one another ; for if K represent the transverse 
 section of the column in square inches, and a the constant 
 inclination of the plane of rupture to the base, then will 
 K sec. a represent the area of the plane of rupture. So that 
 if 7 represent the resistance opposed, by the coherence of 
 the material, to the sliding of one square inch upon the sur- 
 face of another,* then will 7K sec. a represent the resistance 
 which is overcome in the rupture of the column, so long as 
 its height lies between the supposed limits ; which resist- 
 ance being constant, the pressure applied upon the summit 
 of the column to overcome it must evidently be constant. 
 Let this pressure be represented by P, and let CD 
 be the plane of rupture. Now it is evident that 
 the inclination of the direction of P to the perpen- 
 dicular QK to the surface of the plane, or its 
 equal, the inclination a of CD to the base of the 
 column, must be greater than the limiting angle 
 of resistance of the surfaces ; if it were not, then 
 would no pressure applied in the direction of P 
 be sufficient to cause the one surface to slide upon the other, 
 even if a separation of the surfaces were produced along 
 that plane. 
 
 Let P be resolved into two other pressures, whose direc- 
 tions are perpendicular and parallel to the plane of rupture ; 
 the former will be represented by P cos. a, and the friction 
 resulting from it by P cos. a tan. 9 ; and the latter, repre- 
 sented by P sin. a, will, when rupture is about to take place, 
 be precisely equal to the coherence K/ sec. a of the plane of 
 rupture increased by its friction P cos. a tan. 9, or P sin. 
 a=K/ sec. a + P cos. a tan. 9, whence by reduction 
 
 p_ K? cos. 9 _ 2K/ cos. 9 
 
 sin. (a 9) cos. a sin. (2a 9) sin. 9 
 
 It is evident from this expression that if the coherence of 
 the material were the same in all directions, or if the unit of 
 
 * The force necessary to overcome a resistance, such as that here spoken 
 of, has been appropriately called by Mr. Hodgkinson the force necessary to 
 shear it across. 
 
THE PLANE OF RUPTURE. 
 
 coherence 7 opposed to the sliding of one portion of the 
 mass upon another were accurately the same in every direc- 
 tion in which the plane CD may be imagined to intersect 
 the mass, then would the plane of actual rupture be inclined 
 to the base at an angle represented by the formula 
 
 since the value of P would in this case be (equation 634) 
 a minimum when sin. (2a $) is a maximum, or when 
 
 If If 0) 
 
 2 a 9=-, or 0,=--}-- ; whence it follows that a plane in- 
 
 4: 2i 
 
 clined to the base at that angle is that plane along which the 
 rupture will first take place, as P is gradually increased be- 
 yond the limits of resistance. 
 
 The actual inclination of the plane of rupture was found 
 in the experiments of Mr. Hodgkinson to vary with the ma- 
 terial of the column. In cast iron, for instance, it varied 
 according to the quality of the iron from 48 to 58*, and 
 was different in different species. By this dependence of 
 the angle of rupture upon the nature of the material, it is 
 proved that the value of the modulus of sliding coherence 
 7 is not the same for every direction of the plane of rup- 
 ture, or that the value of 9 varies greatly in different quali- 
 ties of cast iron. 
 
 Solving equation (634) in respect to 7 we obtain 
 
 P 
 7=07- sin. (a 9) cos. a sec. 9 ..... (636) ; 
 
 from which expression the value of the modulus 7 may be 
 determined in respect to any material whose limiting angle 
 of resistance 9 is known, the force P producing rupture, 
 under the circumstances supposed, being observed, and also 
 the angle of rupture. f 
 
 THE SECTION OF RUPTURE IN A BEAM. 
 
 407. When a beam is deflected under a transverse strain, 
 
 * Seventh Report of British Association, p. 349. 
 
 f A detailed statement of the results obtained in the experiments of Mr. 
 Hodgkmson on this subject is contained in the Appendix to the " Illustrations 
 of Mechanics " by the author of this work. 
 
GENERAL CONDITIONS OF RUPTURE. 
 
 521 
 
 the material on that side of it on which it sustains the strain 
 is compressed, and the material on the opposite side 
 extended. That imaginary surface which separates the 
 compressed from the extended portion of the material is 
 called its neutral surface (Art. 354.), and its position has 
 been determined under all the ordinary circumstances of 
 flexure. That which constitutes the strength of a beam is 
 the resistance of its material to compression on the one side 
 of its neutral surface, and to extension on the other ; so that 
 if either of these yield the beam will be broken. 
 
 The section of rupture is that transverse section of the 
 beam about which, in its state bordering upon rupture, it is 
 the most extended, if it be about to yield by the extension 
 of its material, or the most compressed if about to yield by 
 the compression of its material. 
 
 In a prismatic beam, or a beam of uniform dimensions, it 
 is evidently that section which passes through the point of 
 greatest curvature of the neutral line, or the point in 
 respect to which the radius of curvature of the neutral line 
 is the least, or its reciprocal the greatest. 
 
 GENERAL CONDITIONS OF THE RUPTUKE OF A BEAM. 
 
 408. Let PQ be the section of rupture in a beam sustain- 
 ing any given pressures, whose 
 resultants are represented, if 
 they be more in number than 
 three, by the three pressures P 1? 
 P 2 , P 3 . Let the beam be upon 
 the point of breaking by the 
 yielding of its material to exten- 
 sion at the point of greatest ex- 
 tension P ; and let R represent, 
 in the state of the beam border- 
 ing upon rupture, the intersection of the neutral surface 
 with the section of rupture ; which intersection being in 
 the case of rectangular beams a straight line, and being in 
 fact the neutral axis, in that particular position which is 
 assumed by it when the beam is brought into its state bor- 
 dering upon rupture, may be called the axis of rupture ; 
 AK the area in square inches of any element of the section 
 of rupture, whose perpendicular distance from the axis of 
 rupture R is .represented by p ; S the resistance in pounds 
 
522 GENERAL CONDITIONS OF RUPTURE 
 
 opposed to the rupture of each square inch of the section at 
 P ; ^ and c a the distances PR and QR in inches. 
 
 The forces opposed per square inch to the extension and 
 compression of the material at different points of the sec- 
 tion of rupture are to one another as their several perpen- 
 dicular distances from the axis of rupture, if the elasticity 
 of the material be supposed to remain perfect throughout 
 the section of rupture, up to the period of rupture. 
 
 Now at the distance c l the force thus opposed to the 
 extension of the material is represented per square inch by 
 S ; at the distance p the elastic force opposed to the exten- 
 sion or compression of the material (according as that 
 distance is measured on the extended or compressed side), is 
 
 R 
 
 therefore represented per square inch by p, and the elastic 
 
 c i 
 
 force thus developed upon the element AK of the section of 
 
 S 
 
 rupture by pAK, so that the moment of this elastic force 
 G i 
 
 o 
 
 about R is represented by p a AK, and the sum of the mo- 
 
 c i 
 
 ments of all the elastic forces upon the section of rupture 
 
 Q 
 
 about the axis of rupture by 2p 2 AK ;* or representing the 
 
 moment of inertia of the section of rupture about the axis 
 of rupture by I, the sum of the moments of the elastic 
 forces upon the section of rupture about its axis of rupture 
 
 Q-r 
 
 is represented, at the instant of rupture, by ,f Now the 
 
 c i 
 
 elastic forces developed upon PQ are in equilibrium with 
 the pressures applied to either of the portions APQD or 
 BPQC, into which the beam is divided by that section ; the 
 sum of their moments about the point P is therefore equal 
 to the moment of R> about that point. Representing, 
 therefore, by p l the perpendicular let fall from the point B 
 upon the direction of P n we have 
 
 * It will be observed, as in Art. 358., that the elastic forces of extension 
 and those of compression tend to turn the surface of rupture in the same 
 direction about the axis of rupture. 
 
 f This expression is called by the French writers the moment of rupture ; 
 the beam is of greater or less strength under given circumstances according 
 as it has a greater or less value. 
 
BY TKANSVEKSE STEAIN. 523 
 
 409. If the deflexion "be small in the state bordering upon 
 rupture, and the directions of all the deflecting pressures be 
 perpendicular to the surface of the beam, the axis of rupture 
 passes through the centre of gravity of the section, and the 
 value of o l is known. Where these conditions do not obtain, 
 the value of c l might be determined by the principles laid 
 down in Arts. 355. and 381. This determination would, 
 however, leave the theory of the rupture of beams still in- 
 complete in one important particular. The elasticity of the 
 material has been supposed to remain perfect, at every point 
 of the section of rupture, up to the instant when rupture is 
 about to take place. Now it is to be observed, that by rea- 
 son of its greater extension about the point P than at any 
 other point of the section of rupture, the elastic limits are 
 there passed before rupture takes place, and before they are 
 attained at points nearer to the axis of rupture ; the forces 
 opposed to the extension of the material cannot therefore be 
 assumed to vary, at all points of PR, accurately as their dis- 
 tances from the point R, in that state of the equilibrium of 
 the beam which immediately precedes its rupture ; and the 
 sum of their moments cannot therefore be assumed to be ac- 
 
 QT 
 
 curately represented by the expression . This remark af- 
 fects, moreover, the determination of the values of h and R 
 (Arts. 355. and 381.), and therefore the value of c l 
 
 To determine the influence upon the conditions of rupture 
 by transverse strain of that unknown direction of the insistent 
 pressures, and that variation from the law of perfect elasti- 
 city which belongs to the state bordering upon rupture, we 
 must fall back upon experiment. From this it has resulted, 
 in respect to rectangular beams, that the error produced by 
 these different causes in equation (637) will be corrected if 
 a value be assigned to c t bearing, for each given material, a 
 constant ratio to the distance of the point P from the centre 
 of gravity of the section of rupture ; so that c representing 
 the depth of a rectangular beam, the error will be corrected, 
 in respect to a beam of any material, by assigning to c l the 
 value rajc, where m is a certain constant dependent upon 
 the nature of the material. It is evident that this cor- 
 rection is equivalent to assuming c,=ic, and assigning 
 to S the value ^S instead of that which it has hitherto 
 
524 
 
 GENERAL CONDITIONS OF KUPTTJKE 1 
 
 been supposed to represent, viz. the tenacity per square inch 
 of the material of the beam. 
 
 It is customary to make this assumption. The values of S 
 corresponding to it have been determined, by experiment, 
 in respect to the materials chiefly used in construction, and 
 will be found in a table at the end of this work. It is to 
 these tables that the values represented by S in all subse- 
 quent formulae are to be referred. 
 
 410. From the remarks contained in the preceding article, 
 it is not difficult to conceive the existence of some direct re- 
 lation between the conditions of rupture by transverse and by 
 longitudinal strain. Such a relation of the simplest kind ap- 
 pears recently to have been discovered by the experiments 
 of Mr. E. Hodgkinson*, extending to the conditions of rup- 
 ture by compression, and common to all the different varie- 
 ties of material included under each of the following great 
 divisions timber, cast iron, stone, glass. 
 
 The following tables contain the summary given by Mr. 
 Hodgkinson of his results : 
 
 Description of Material. 
 
 Assumed Crushing 
 Strength per Square 
 Inch. 
 
 Mean Tensile 
 Strength per Square 
 Inch. 
 
 Mean Transverse 
 Strength of a Bar 
 1 Inch Square and 
 1 Foot Long. 
 
 Timber .... 
 Cast-iron ... 
 Stone, including marble - 
 Glass (plate and crown) - 
 
 1000 
 1000 
 1000 
 1000 
 
 1900 
 158 
 100 
 123 
 
 85-1 
 19-8 
 9-8 
 
 10- 
 
 The following table shows the uniformity of this ratio in 
 respect to different varieties of the same material : 
 
 Description of Material. 
 
 Assumed Crushing 
 Strength per Square 
 Inch. 
 
 Mean Tensile 
 Strength per Square 
 Inch. 
 
 Mean Transverse 
 Strength of a Bar 
 1 Inch Square arid 
 1 Foot Long. 
 
 Black marble ... 
 Italian marble 
 Rochdale flagstone - 
 High Moorstone - 
 Yorkshire flag 
 Stone from Little Hulton, 
 
 near Bolton 
 
 
 
 1000 
 1000 
 1000 
 1000 
 1000 
 
 I 1000 
 
 143 
 84 
 104 
 100 
 
 70 
 
 10-1 
 
 10-6 
 9-9 
 
 9-5 
 S, 
 
 * This discovery was communicated to the British Association of Science at 
 their meeting in 1842 ; it opens to us a new field of theoretical research. 
 
THE STRONGEST FORM OF SECTION. 525 
 
 411. THE STRONGEST FORM OF SECTION AT ANY GIVEN POINT 
 IN THE LENGTH OF THE BEAM. 
 
 Since the extension and the compression of the material 
 are the greatest at those points which are most distant from 
 the neutral axes of the section, it is evident that the mate- 
 rial cannot be in the state bordering upon rupture at every 
 point of the section at the same instant (Art. 388.), unless all 
 the material of the compressed side be collected at the same 
 distance from the neutral axis, and likewise all the material 
 of the extended side, or unless the material of the extended 
 side and the material of the compressed side be respectively 
 collected into two geometrical lines parallel to the neutral 
 axis : a distribution manifestly impossible, since it would 
 produce an entire separation of the two sides of the beam. 
 
 The nearest* practicable approach to this form of section is 
 that represented in the accompanying figure, where the 
 material is shown collected in two thin but wide flanges, 
 united by a narrow rib. 
 
 13 That which constitutes the strength of the 
 beam being the resistance of its material to com- 
 pression on the one side of its neutral axis, and 
 its resistance to extension on the other side, it is 
 evidently (Art. 388.) a second condition of the 
 3 strongest form of any given section that when 
 the beam is about to break across that section by 
 extension on the one side, it may be about to break by com- 
 pression on the other. So long, therefore, as the distribution 
 of the material is not such as that the compressed and 
 extended sides would yield together, the strongest form of 
 section is not attained. Hence it is apparent that the 
 strongest form of the section collects the greater quantity 
 of the material on the compressed or the extended side of 
 the beam, according as the resistance of the material to 
 compression or to extension is the less. Where the material 
 of the beam is cast iron*, whose resistance to extension is 
 greatly less than its resistance to compression, it is evident 
 that the greater portion of the material must be collected on 
 the extended side. 
 
 Thus, then, it follows, from the preceding condition and 
 
 * It is only in cast iron beams that it is customary to seek an economy of 
 the material in the strength of the section of the beam ; the same principle of 
 economy is surely, however, applicable to beams of wood. 
 
526 
 
 THE STRONGEST FORM OF SECTION. 
 
 this, that the strongest form of section in a cast iron beam is 
 that by which the material is collected into two unequal 
 flanges joined by a rib, the greater flange being on the 
 extended side ; and the proportion of this inequality of the 
 flanges being just such as to make up for the inequality of 
 the resistances of the material to rupture by extension and 
 compression respectively. 
 
 Mr. Hodgkinson, to whom this suggestion is due, has 
 directed a series of experiments to the determination of that 
 proportion of the flanges by which the strongest form of 
 section is obtained.* 
 
 The details of these experiments are found in the following 
 table:- 
 
 Number of 
 Experiment. 
 
 Ratio of the Sections 
 of the Flanges. 
 
 Area of whole 
 Section in Square 
 Inches. 
 
 Strength per Square 
 Inch of Section in Ibs. 
 
 1 
 
 1 to 1- 
 
 2-82 
 
 2368 
 
 2 
 
 1 to 2- 
 
 2-87 
 
 2567 
 
 3 
 
 1 to 4- 
 
 3-02 
 
 2737 
 
 4 
 
 1 to 4-5 
 
 3-37 
 
 3183 
 
 5 
 
 1 to 5-5 
 
 5-0 
 
 3346 
 
 6 
 
 1 to 6-1 
 
 6-4 
 
 4075 
 
 In the first five experiments each beam broke by the tear- 
 ing asunder of the lower flange. The distribution by which 
 both were about to yield together that is, the strongest 
 distribution was not therefore up to that period reached. 
 At length, however, in the last experiment, the beam yielded 
 by the compression of the upper flange. In this experiment, 
 therefore, the upper flange was the weakest ; in the one be- 
 fore it, the lower flange was the weakest. For a form 
 between the two, therefore, the flanges were of equal strength 
 to resist extension- and compression respectively ; and this 
 was the strongest form of section (Art. 388.). 
 
 In this strongest form the lower flange had six times the 
 material of the upper. It is represented in the accompany- 
 ing figure. 
 
 A In the best form of cast iron beam or 
 
 girder used before these experiments, 
 there was never attained a strength of 
 more than 2885 Ibs. per square inch of 
 section. There was, therefore, by this 
 form, a gain of 1190 Ibs. per square inch 
 of the section, or of fths the strength of 
 the beam. 
 
 * Memoirs of Manchester Philosophical Society, vol. iv. p. 453. 
 tions of Mechanics, Art. 68. 
 
 Illustro- 
 
THE BEAM OF GREATEST STRENGTH. 527 
 
 412. THE SECTION OF RUPTURE. 
 
 The conditions of rupture being determined in respect to 
 any section of the beam by equation (637), it is evident that 
 the particular section across which rupture will actually take 
 place is that in respect to which equation (637} is first satis- 
 fied, as P, is continually increased ; or that section in respect 
 to which the formula 
 
 (638) 
 
 ftft 
 
 is the least. 
 
 If the beam be loaded along its whole length, arid x repre- 
 sent the distance of any section from the extremity at which 
 the load commences, and f* the load on each foot of the 
 length, then (Art. 371.) Pj?, is represented by 4*aj a . The 
 section of rupture in this case is therefore that section in 
 respect to which M- is first made to satisfy the equation. 
 
 QT 
 
 ; or in respect to which the formula 
 
 is the least. 
 
 If the section of the beam be uniform, is constant ; the 
 
 G \ 
 
 section of rupture is therefore evidently that which is most 
 distant from the free extremity of the beam. 
 
 4:13. THE BEAM OF GREATEST STRENGTH. 
 
 The beam of greatest strength being that (Art. 388.) which 
 presents an equal liability to rupture across every section, or 
 in respect to which every section is brought into the state 
 bordering upon rupture by the same deflecting pressure, is 
 evidently that by which a given value of Pis made to satisfy 
 equation (637) for all the possible values of I, p l9 and c l9 or 
 in respect to which the formula 
 
 ^T ( 64 ) 
 
 is constant. 
 
THE STRENGTH OF BEAMS. 
 
 If the beam be uniformly loaded throughout (Art. 371.), 
 this condition becomes 
 
 or constant, for all points in the length of the beam. 
 
 414. ONE EXTREMITY OF A BEAM is FIRMLY IMBEDDED IN 
 
 MASONRY, AND A PRESSURE IS APPLIED TO THE OTHER 
 EXTREMITY IN A DIRECTION PERPENDICULAR TO ITS LENGTH! 
 TO DETERMINE THE CONDITIONS OF THE RUPTURE. 
 
 If x represent the distance of any section of the beam 
 from the extremity A to which the load P 
 is applied, and a its whole length, and if the 
 section of the beam be everywhere the 
 same, then the formula ( 638 ) is least 
 at the point B, where x is greatest: at 
 this point, therefore, the rupture of the 
 beam will take place. Representing by 
 P the pressure necessary to break the 
 beam, and observing that in this case the 
 perpendicular upon the direction of P 
 from the section of rupture is represented 
 by #, we have (equation 637) 
 
 P=-|^ (642). 
 
 If the section of the beam be a rec- 
 tangle, whose breadth is & and its depth 0, 
 
 j 1 T - "" ~ 
 
 then 1= 
 
 (643). 
 
 If the beam be a solid cylinder, whose radius is 0, then 
 (Art. 364.) I=frc\ c.c. 
 
 G * . . (644). 
 
 a 
 
 If the beam be a hollow cylinder, whose radii are r l and 
 r,, I=^f(r*r*) ; which expression may be put under the 
 form fliw^-f ic*) (see Art. 86.), r representing the mean 
 
THE STRENGTH OF BEAMS. 529 
 
 radius of the hollow cylinder, and c its thickness. Also 
 
 .-.P=rS 
 
 (645). 
 
 415. The strongest form of beam under the conditions sup- 
 posed in the last article. 
 
 1st. Let the section of the beam be a 
 rectangle, and let y be the depth of 
 this rectangle at a point whose distance- 
 from its extremity A is represented by 
 #, and let its breadth 5 be the same 
 throughout. In this case I rV^V 
 c l =^y: therefore (equation 637) P=. 
 SI jf 
 =Sb . If therefore, P be taken 
 
 CjC X 
 
 to represent the pressure which the 
 beam is destined just to support, then 
 the form of its* section ABC is deter- 
 mined (Art. 413.) by the equation 
 
 6P 
 
 it is therefore a parabola, whose vertex 
 is at A.* 
 
 If the portion DO of the beam: do not rest against 
 masonry at every point, but only at its 
 extremity D, its form; should; evidently be 
 the same with that of ABC: 
 
 2d. Let the section, be a circle, and 
 let y represent its radius at distance x 
 from its extremity A, then I= 
 
 y' 
 
 c^=y\ therefore P=J*S~ so that the 
 
 oc 
 
 geometrical form of its longitudinal 
 section is determined by the equa- 
 tion 
 
 * The portion of the beam imbedded in the masonry should have the form 
 described in Art. 417. 
 
 34 
 
530 
 
 THE STRENGTH OF BEAMS. 
 
 (647), 
 
 P representing the greatest pressure to which it is destined 
 to be subjected- 
 
 416. THE CONDITIONS OF THE RUPTURE OF A BEAM SUPPORTED 
 AT ONE EXTREMITY, AND LOADED THROUGHOUT ITS WHOLE 
 
 LENGTH. 
 
 Kepresenting the weight resting upon each inch of its 
 
 length a by M., and observ- 
 ing that the moment of the 
 weight upon a length < of 
 the beam from A, about the 
 corresponding neutral axis, 
 is represented (Art. 371.) 
 by %we\ it is apparent (Art. 
 412.) that, if the beam be 
 of uniform dimensions, its 
 section of rupture is BD. 
 Its strength is determined 
 by substituting ^a? for P j^ 
 in equation (637), and solving in respect to ^ ; we thus obtain 
 
 2SI 
 
 (648); 
 
 by which equation is determined the uniform load to which 
 the beam may be subjected, on each inch of its length. 
 
 For a rectangular beam, whose width is 5 and its depth 
 c, this expression becomes 
 
 417. To determine the form of greatest strength (Art. 413.) 
 in the case of a beam having a rectangular section of uni- 
 form breadth ^a? 2 must be substituted for P J p 1 in equation 
 
 (637), and 
 reduction 
 
 for I, and \y for c l ; whence we obtain by 
 
 (650. 
 
THE STRENGTH OF BEAMS. 
 
 531 
 
 The form of greatest strength is therefore, in this case, the 
 straight line joining the points A and B ; the distance DB 
 being determined by substituting the distance AD for x in 
 the above equation. 
 
 That portion BED of the beam which is embedded in the 
 masonry should evidently be of the same form with DBA.* 
 
 418. If, in addition to the uniform load upon the beam, a 
 given weight "W" be suspended from A, Jf*aj*-|-"Waj must be 
 
 substituted for P j>, in equation (637) ; we shall thus obtain 
 lor the equation to the form of greatest strength 
 
 which is the equation to an hyperbola having its vertex 
 at A.f 
 
 * It is obvious that in all cases the strength of a beam at each point of its 
 length is dependent upon the dimensions of its cross section at that point, and 
 that its general form may in any way be changed without impairing its strength 
 provided those dimensions of the section be everywhere preserved. 
 
 f Church's Anal. Geom. Art. 124. 
 
532 
 
 THE STRENGTH OF BEAMS. 
 
 419. THE BEAM OF GREATEST STRENGTH IN REFERENCE TO THE 
 
 FORM OF ITS SECTION AND TO THE VARIATION OF THE 
 
 DIMENSIONS OF ITS SECTION, WHEN SUPPORTED AT ONE 
 
 EXTREMITY IN A HORIZONTAL POSITION, AND LOADED UNI- 
 FORMLY THROUGHOUT ITS LENGTH, 
 
 The general form of the section must evidently be that 
 
 described in Art. 411. Let 
 the same notation be taken 
 as in Art. 365., ^xcept that 
 the depth MQ of the plate 
 or^ rib joining the two 
 flanges is to be represented 
 by 2/, and its thickness by c. 
 
 therefore by equation (503), 
 
 so that d z =y, and A 3 =cy ; 
 
 Also representing by c l the distance of the centre of gravity 
 of the whole section from the upper surface of the beam, 
 we have c l (A J + A z + cy)=(jiy+d,)cy+(y + d,+%d 1 )A l +%dt 
 A a . Substituting for I and c l in equation (637), and for l p l its 
 value -J-fAa? 2 , x being taken to represent the distance AM, and 
 M. the load on each inch of tliat length, we have (Art. 
 413.) 
 
 3f* f 
 
 In = 
 
 ) 
 d, 
 
 (y + Zdjoy + 2(y + d, 4- K) A, + A & 
 
 ..... (652). 
 
 Let the area cy of the section of the rib now be neglected, 
 as exceedingly small when compared with the areas of the 
 sections of the flanges, an hypothesis which assigns to the 
 beam somewhat less than its actual strength ; let also the 
 area of the section of the upper flange be assumed equal to 
 n times that of the lower, or A a =^A 1 , 
 
 (653). 
 
 If the flanges be exceedingly thin, d l and d^ are exceed- 
 ingly small and may be neglected. The equation will then 
 
THE STRENGTH OF BEAMS. 533 
 
 become that to a parabola whose vertex is at A and its axis 
 vertical. This may therefore be assumed as a near approxi- 
 mation to the true form of the curve AQC. 
 
 Where the material is cast iron, it appears by Mr. Hodg- 
 kinson's experiments (Art. 411.) that n is to be taken=6. 
 
 420. A BEAM OF UNIFORM SECTION IS SUPPORTED AT ITS 
 EXTREMITIES AND LOADED AT ANY POINT BETWEEN THEM! 
 IT IS REQUIRED TO DETERMINE THE CONDITIONS OF RUPTURE. 
 
 The point of rupture in the case of a uniform section 
 
 is evidently (Art. 412.) the point 
 C, from which the load is sus- 
 pended; representing AB, AC, 
 EC, by a, a^ and # a ; and ob- 
 serving that the pressure P t 
 
 Wa 
 
 upon the point B of the beam = -, so that the moment 
 
 of Pj, in respect to the section of rupture C = *, we 
 
 SI 
 
 have, by equation (637), " ~ 1 ^ a = _ 
 
 a G I 
 
 (654). 
 
 If the beam be rectangular , I =-^50*, <?!=<?, 
 VW== 5 T- (655); 
 
 O tJjQ/2 
 
 where "W represents the breaking weight, S the modulus of 
 rupture, a the length, b the breadth, c the depth, and a l9 a % 
 the distances of the point C from the two extremities, all 
 these dimensions being in inches. 
 
 If the load be suspended in the middle, a 1 =a, t =^a y 
 
 " W =^ .....(656). 
 
 If the beam be a solid cylinder, whose radius =<?, then 1= 
 Jtf<? 4 , <?!=<?; therefore, equation (654), 
 
 W=^ -^ . . (657). 
 
534 THE STRENGTH OF BEAMS. 
 
 
 
 If the beam be a hollow cylinder, whose mean radius is r, 
 and its thickness c, I=tcr(r i +%c*), c 1 ^=r-\-^c] therefore, 
 equation (654), 
 
 ( 
 
 If the section of the beam be that represented in Art. 411., 
 being everywhere of the same dimensions, then, observing 
 that Ac l =^d 3 A a -}-d s A^ nearly, we have, (equations 503 and 
 654) 
 
 l12 
 -T~ (2A 1 +A,)a 1 a,4 
 
 where A 15 A 3 represent the areas of the sections of the upper 
 and lower flanges, and A 3 that of the connecting rib or plate, 
 and dv d d a their respective depths. 
 
 421. A BEAM IS SUPPORTED AT ITS EXTREMITIES, AND LOADED 
 AT ANT GIVEN POINT BETWEEN THEM J ITS SECTION IS OF A 
 GIVEN GEOMETRICAL FORM, BUT OF VARIABLE DIMENSIONS : 
 IT IS REQUIRED TO DETERMINE THE LAW OF THIS VARIA- 
 TION, SO THAT THE STRENGTH OF THE BEAM MAY BE A 
 MAXIMUM. 
 
 W representing the breaking load upon the beam, and 
 
 # 2 the distances of its point 
 of suspension C, from A and 
 B, the pressure P, upon A is 
 
 represented by 2 . If, there- 
 
 a 
 
 fore (Art. 388.), x represent 
 
 the horizontal distance of any section MQ from the point of 
 support A, and I its moment of inertia, and ^ the distance 
 from its centre of gravity to the point where rupture is about 
 to take place (in this case its lowest point) ; then by equa- 
 tion (637) 
 
 W "<~- SI . (660). 
 
 a 
 
 1st. Let the section be rectangular ; let its breadth b be 
 constant; and let its depth at the distance x from A be 
 
THE STRENGTH OF BEAMS. 535 
 
 represented by y ; therefore I=i-V^y'> c i%y- Substituting 
 in the above equation and reducing, 
 
 The curve AC is therefore a parabola, whose vertex is at 
 A, and its axis horizontal. In like manner the curve EC is 
 a parabola, whose equation is identical with the above, ex- 
 cept that a l is to be substituted in it for a y 
 
 2d. Let the section of the beam be a circle. Represent- 
 ing the radius of a section at distance x from A by y, we 
 have I^Jtf^ 4 , c 1 =y. ) therefore by equation (660) 
 
 (662 > 
 
 3d. Let the section of the beam be circular ; but let it be 
 hollow, the thickness of its material being every where the 
 same, and represented by o. If y= mean radius of cylinder 
 at distance x from A, then I=<7r<3?/(y 2 -f Jc 2 ), c 1 = 
 
 422. THE BEAM OF GREATEST ABSOLUTE STRENGTH WHEN 
 LOADED AT A GIVEN POINT AND SUPPORTED AT THE EXTRE- 
 
 MITIES. 
 
 Let the section of the beam be that of greatest strength 
 rt. 411.). Substituting in equation (66 
 as before in equation (652), and reducing, 
 
 (Art. 411.). Substituting in equation (660) the value of 
 
 &. . (664). 
 
 If the section cy of the rib be every where exceedingly 
 small as compared with the sections of the flanges, and if 
 
 j^ B= OH-9w + ^o+i*/ ( 
 
 There is a value of x in this equation for which y becomes 
 
536 THE STRENGTH OF BEAMS. 
 
 impossible. For values less than this, the condition of uni- 
 form strength cannot therefore obtain. It is only in respect 
 to those parts of the beam which lie between the values of 
 SB (measured from the two points of support) for which y 
 thus becomes impossible, that the condition of greatest 
 strength (Art. 388.) is possibly. If its proper value be 
 assigned to n (Art. 411.), this may be assumed as an approxi- 
 mation to the true form of beam of THE GREATEST ABSOLUTE 
 STRENGTH. When the material is cast iron, it appears by the 
 experiments of Mr. Hodgkinson (Art. 411.) that n=6. A 2 
 represents in all the above cases the section of the extended 
 flange ; in this case, therefore, it represents the section of 
 the lower flange. 
 
 The depth CD at the point of suspension may be deter- 
 mined by substituting a^ for x in equation (665) ; its value is 
 thus found to be represented by the formula 
 
 (666), 
 
 423. If instead of the depth of the beam being made to 
 vary so as to adapt itself to the condition (Art. 388.) of uni- 
 form strength, its breadth b be made thus to vary, the depth 
 c remaining the same ; then, assuming the breadth of the 
 upper flange at the distance x from the point of support A 
 to be represented by y, and the section of the lower flange 
 to be n times greater than that of the upper; observing, 
 moreover, that in equation (503) A t = yd^ A z =nA. 1 =nyd 1 ; 
 neglecting also A 3 as exceedingly small when compared with 
 A, and A a , and writing c for <# 3 , we have by reduction, 
 
 ftrr.l 
 
 Also <?, being the distance of the lower surface of the beam 
 from the common centre of gravity of the sections of the 
 two fl^iges, we have c l (n-\-\)=c. Eliminating, therefore, 
 the values of I and c, from equation (660), 
 
 = \ A(+l) (d^nd.^+ncd, y ..... (667), 
 
 the equation to a straight line. Each flange is therefore in 
 this case a quadrilateral figure, whose dimensions are deter- 
 mined from the greatest breadth ; this last being known, foi 
 
THE STRENGTH OF BEAMS. 
 
 537 
 
 the upper flange, by substituting ^ for x in the above equa- 
 tion, and solving in respect of y, and for the lower flange 
 from the equation nb^b^ in which &j, 9 represent the 
 greatest breadths of the two flanges, and a a d^ their depths 
 
 424. A BEAM IS LOADED UNIFORMLY THROUGHOUT ITS WHOLfl 
 LENGTH, AND SUPPORTED AT ITS EXTREMITIES I IT IS REQUIRED 
 TO DETERMINE, 1. TfiE CONDITIONS OF ITS RUPTURE WHEN ITS 
 CROSS SECTION IS UNIFORM THROUGHOUT ; 2. THE STRONGEST 
 
 FORM OF BEAM HAVING EVERY WHERE A RECTANGULAR CROSS 
 SECTION ; 3. THE BEAM OF GREATEST STRENGTH IN REFER- 
 ENCE BOTH TO THE FORM AND THE VARIATION OF ITS CROSS 
 SECTION. 
 
 1. If the section of the beam be uniform, its point of rup- 
 ture is determined by formula (639) 
 to be its middle point. Kepresenting, 
 therefore, in this case, the length of 
 the beam by 2, the weight on each 
 inch of its length by ^ and its breadth 
 by & ; and observing that in this case 
 p^ /xa 2 ^a?=%t*<a\ we have by 
 equation (637) 
 
 n: 
 
 281 
 
 (668), 
 
 where f* represents the load per inch of the length of the 
 beam necessary to produce rupture. In the case of a rectan- 
 gular beam, this equation becomes 
 
 P = %. ..(669). 
 
 2. To determine the form of the beam of greatest strength 
 having a rectangular section of 
 given breadth &, let y be taken to 
 represent its depth PQ at a point P, 
 and x its horizontal distance from 
 the point A. Then I = T \fy/\ 
 c 1 =^/'j also Pj? x (equation 637) 
 
 representing the moment of the resultant of the pressures 
 upon AP about the centre of gravity of ~PQ=^ax 
 therefore by equation (637) pax - 
 
538 
 
 THE STRENGTH OF BEAMS. 
 
 the equation to an ellipse, whose vertex is in A, and its 
 centre at C. 
 
 3. To determine the beam of absolute maximum strength, 
 let it be assumed, as in Art. 422, that the area of the section 
 of the rib is exceedingly small as compared with the areas 
 of the sections of the flanges ; and let the area of the section 
 of the lower or extended flange be n times that of the upper ; 
 
 A 4 AW I -A-i J ( 7l + l) (d?+nd*) + 12ny* ) 
 then, as in Art. 422, -=-JT ) - \_ , , 
 
 also Pj? 1 =M-oa? if^B 8 ; whence, by equation (637), 
 
 SA 
 
 ' + 2K 
 
 4. If it be proposed to make the rib or plate uniting the 
 two flanges everywhere of the same depth,* and so to vary 
 the breadths of the flanges as to give to the beam a uniform 
 strength at all points under these circumstances ; represent- 
 ing by y the breadth of the upper flange at a horizontal 
 distance x from the point of support, we shall obtain, as in 
 Art. 423, 
 
 Moreover, ~P l p 1 =^ax J^*==-Jfw?(2a a?); whence we obtain 
 by substitution in equation (637), and reduction, 
 
 }y ..... (673) ; 
 
 the equation to a parobola,f whose axis is in the horizontaJ 
 line bisecting the flange at right angles, its parameter repre- 
 
 * As in Mr. Hodgkinson's construction. 
 j- Church's Anal. Geom. Art. 171. 
 
THE STRENGTH OF BEAMS. 539 
 
 sented by the coefficient of y in the preceding equation, and 
 half the breadth of the flange in the middle determined by 
 the formula 
 
 (674) 
 
 The equation to the lower flange is determined by substi- 
 tuting for y, in equation (673), ^ ; whence it follows that 
 the breadth of the lower flange in the middle is equal to 
 that of the upper multiplied by the fraction -~ . 
 
 425. A RECTANGULAR BEAM OF UNIFORM SECTION, AND UNI- 
 FORMLY LOADED THROUGHOUT ITS LENGTH, IS SUPPORTED BY 
 TWO PROPS PLACED AT EQUAL DISTANCES FROM ITS EXTREM- 
 ITIES : TO DETERMINE THE CONDITIONS OF RUPTURE. 
 
 It is evident from formula (639) that the section of rup- 
 ture of the portion CA of the 
 beam is at A, and therefore that 
 the conditions of its rupture are 
 determined (Art. 416.) by the 
 equation 
 
 TO- -(675); 
 
 where ^, represents, as before, the 
 load upon each inch of the length of the beam, b its 
 breadth, c its depth, and a, the length of the portion AC. 
 
 Again, it is evident that the point of rupture of the por- 
 tion AB of the beam is at E. Now the value of P^, 
 (equation 637) is, in respect to the portion AE of the beam, 
 \^^a(a $j) i"^^ 2 5 ^ a representing the whole length of the 
 beam ^ the load upon each inch of the length of the beam 
 which would produce rupture at E, and therefore ^a the 
 resistance of each prop in the state bordering upon rupture } 
 
 also -=:|fo a . Whence, by equation (637), 
 
 0, 
 
54:0 THE STRENGTH OF BEAMS. 
 
 4 
 
 426. THE BEST POSITIONS OF THE PROPS. 
 
 If the load i^ be imagined to be continually increased, it 
 is evident that rupture will eventually take place at A or at 
 E according as the limit represented by equation (675), or 
 that represented by equation (676), is first attained, or 
 according as f* t or f* 2 is the less. 
 
 Let M-! be conceived to be tue less, and let the prop A be 
 moved nearer to the extremity C ; # t being thus diminished, 
 ju-j will be increased, and ^ diminished, Now if, after this 
 change in the position of the prop, f\ still remains less than 
 fx a , it is evident that the beam will bear a greater load than 
 it would before, and that when by continually increasing 
 the load it is brought into the state bordering upon rupture 
 at A it will not be in the state, bordering upon rupture at E. 
 The beam may therefore be strengthened yet further by 
 moving the prop A towards C; and thus continually, so 
 that the beam evidently becomes the strongest when the 
 prop is moved into such a position that i^ 1 may just equal 
 f* 2 . This position is readily determined from equations (675) 
 and (676) to be that in which 
 
 ~ (677). 
 
 427. A RECTANGULAR BEAM OF UNIFORM SECTION AND UNI- 
 FORMLY LOADED IS SUPPORTED AT ITS EXTREMITIES, AND BY 
 TWO PROPS SITUATED AT EQUAL DISTANCES FROM THEM : TO 
 DETERMINE THE CONDITIONS OF RUPTURE. 
 
 Adopting the same notation as in Art. 374., it appears by 
 
 equation (543) that the dis- 
 tance x, of the point of great- 
 est curvature of the neutral 
 line, and therefore of the sec- 
 tion of rupture in AB from 
 
 == A (Art, 407.) being that 
 where -T-, - t * is the greatest, is 
 
 ^ CUtHj 
 
 determined by the equation 
 
 * The curvature of the neutral line being everywhere exceedingly small, 
 ^ may be assumed =1. The expression for the radius of curvature in terms 
 
 of the rectangular co-ordinates resolves itself therefore, in this case, into the 
 second differential coefficient. 
 
THE STRENGTH OF BEAMS. 
 
 541 
 
 f/tf^P, it being observed that, at the section of rupture, the 
 neutral line is concave to the axis of #, and therefore the 
 second differential coefficient (equation 543) negative. The 
 value of P is that determined by equation (551) ; so that 
 
 n(8-8) 
 
 (678) ' 
 
 where a represents the distance AE, and na the distance AB. 
 Let P represent the intersection of the neutral line with 
 the plane of rupture, and i^ the load per inch of the whole 
 length of the beam which would produce a rupture at P. 
 Now the sum of the moments of the forces impressed on 
 AP (other than the elastic forces on the section of rupture) 
 is represented in the state bordering upon rupture, by 
 
 P^ i^a?, 8 ; or, since P 1 =i* 1 aj 1 , it is represented by^ P a 2 ; 
 
 J/x i 
 
 whence it follows by equation (637) that the conditions of 
 the rupture of the beam between A and B are determined 
 
 by the eq nation ?- P. 2 =jSfo 2 , or, 
 2f* t 
 
 .... (679). 
 
 Eliminating the value of 
 (679), we obtain 
 
 between equations (551) and 
 
 8n(2-3) 
 
 r 
 
 (680). 
 
 Substituting this value of f*, in equation (679), and 
 reducing 
 
 '- 3) } (681). 
 
 If the points B and C 
 coincide, or the beam be 
 supported by a single prop 
 in the middle, n=l; there- 
 fore, by equations (680) and 
 (681), 
 
 (682); 
 
54:2 THE STRENGTH OF BEAMS. 
 
 Similarly, it appears by equation (547) that the point of 
 greatest curvature between B and C is E ; if the rupture of 
 the beam take place first between these points, it will there- 
 fore take place in the middle. Let ^ represent the load, 
 per inch of the length, which would produce a rupture at E. 
 Now, the sum of the moments about E of the forces im- 
 pressed upon AE is P l( $-hP 2 (a no) foi> 9 a*=(P ' 
 rjia iiv& 9 = M-X (^a P,)^a if^X (since P 1 + 
 Therefore by equation (637) 
 
 + I > i^=iS5c a ..... (684). 
 
 Substituting for ~P l its value from equation (551), and 
 solving in respect to ^ 2 , 
 
 S6c' 271-3 , 
 
 ( 
 
 If the load be continually increased, the beam will break 
 between A and B, or between B and C, according as f*, 
 (equation 680) or f* a (equation 685) is the less. 
 
 428. THE BEST POSITIONS OF THE PROPS. 
 
 It may be shown, as in Art. 426., that the positions in 
 which the props must be placed so as to cause the beam to 
 bear the greatest possible load distributed uniformly over its 
 whole length, are those by which the values of f*, (equation 
 680) and ^ (equation 685) are made equal ; the former of 
 these quantities representing the load per inch of the length, 
 which being uniformly distributed over the whole beam 
 would just produce rupture between A and B, if it did not 
 before take place between B and C ; and the latter that 
 which would, under the same circumstances, produce rup- 
 ture between B and C if it had not before taken place 
 between A and B. 
 
 Let, then, na represent the distance at which the prop B 
 must be placed from A to produce this equality ; and let the 
 value of M-, given by equation (679) be substituted for (x a in 
 equation (684) ; we shall thus obtain by reduction 
 
 * 3(1-27*)- 9(1-27*) 
 Solving this quadratic in respect to P,a, 
 
THE STRENGTH OF BEAMS. 54:3 
 
 The negative sign must be taken in this expression, since 
 the positive would give P 1 =f* 1 a by equation (679), and cor- 
 responds therefore to the case n=Q. Assuming the negative 
 sign, and reducing, we have 3(2n l)P l a=Sbc*. Substitut- 
 ing in this expression for P x its value from equation (681), 
 and reducing, 
 
 1) (2n 3) 
 ~ 
 
 The three roots of this equation are 1-57087, '61078, and 
 26994. The first and last are inadmissible ; the one carry- 
 ing the point B beyond E, and the other assigning to Pj a 
 negative value.* the best position of the prop is therefore 
 that which is determined by the value 
 
 n= -61078 ..... (686). 
 
 429. THE CONDITIONS OF THE RUPTURE OF A RECTANGULAR 
 
 BEAM LOADED UNIFORMLY THROUGHOUT ITS LENGTH, AND 
 HAVING ITS EXTREMITIES PROLONGED AND FIRMLY IMBEDDED 
 IN MASONRY. 
 
 It has been shown (Art. 376.) that the conditions of the 
 deflexion of the beam are, in this case, the same as though 
 its extremities, having been prolonged to a point A (see Jig. 
 p. 540.), such that AB might equal '6202 AE, had been sup- 
 ported by a prop at B, and by the resistance of any fixed 
 surface at A. The load which would produce the rupture 
 of the beam is therefore, in this case, the same as that which 
 would produce the rupture of a beam supported by props 
 (Art. 427.) between the props, and is determined by that 
 value of f* a (equation 685) which is given by the value '6202 
 of n. It is, however, to be observed that the symbol a 
 
 
 * We may, nevertheless, suppose the extremity A, instead of being sup 
 ported from beneath, to be pinned down by a resistance or a pressure acting 
 from above. This case may occur in practice, and the best position of the 
 props corresponding to it is that whic^h is determined by the least root of the 
 equation, viz. '26994. 
 
544 
 
 THE STRENGTH OF COLUMNS. 
 
 represents in that equation 
 the distance AE (Jig. Art. 
 427.) ; and that if we take 
 it to represent the distance 
 BE in that or the accompa- 
 nying figure, we must sub- 
 
 stitute = -- for a in equa- 
 1n 
 
 tion (685), since 0=BE=AE AB=(1 w)AE ; so that 
 AE=:j -. This substitution being made, equation (685), 
 becomes 
 
 |Cj 
 
 h- 
 
 [ 
 
 
 i 
 
 
 
 i p_ 
 
 1 
 
 ^J 1 ' .- 
 
 ~l 
 
 i 
 
 , 
 
 
 C 1 
 
 
 K 
 
 U 
 
 1 
 
 1 1 1 
 
 
 
 
 
 1 1 1 
 
 
 
 i 
 
 
 1 1 1 
 
 
 
 
 
 
 1 
 
 i 
 
 \ 
 
 
 1 1 1 
 
 
 
 
 
 II i 
 
 
 i 
 
 \ 
 
 
 J 1 1 
 
 
 
 
 
 _ II i 
 
 \ 
 
 
 \ 
 
 
 _lJL__L__j 
 
 and substituting the value *6202 for n, we obtain by reduc- 
 tion 
 
 (687), 
 
 by which formula the load per inch of the length of the 
 beam necessary to produce rupture is determined. 
 
 If the beam had not been prolonged beyond the points of 
 support B and C and imbedded in the masonry, then the 
 load per inch of the length necessary to produce rupture 
 would have been represented by equation (669) : eliminat- 
 ing between that equation and equation (687), we obtain 
 |x a =3fx ; so that the load per inch of the length necessary to 
 produce rupture is 3 times as great, when the extremities 
 of the beam are prolonged and firmly imbedded in the ma- 
 sonry, as when they are free ; i. e. the strength of the beam 
 is 3 times as great in the one case as in the other. 
 
 430. THE STRENGH OF COLUMNS. 
 
 For all the knowledge of this subject on which any reli- 
 ance .can be placed by the engineer he is endebted to expe- 
 riment.* 
 
 * The hypothesis upon which it has been customary to found tlfe theoretical 
 discussion of it, is so obviously insufficient, and the results have been shown 
 by Mr. Hodgkinson to be so little in accordance with those of practice, that 
 the high sanction it has received from labours such as those of Euler, Lagrange, 
 Poisson, and Navier, can no longer establish for it a claim to be admitted 
 among the conclusions of science. (See* Appendix K.) 
 
THE STRENGTH OF COLUMNS. 
 
 545 
 
 The following are the principal results obtained in the 
 valuable series of experimental inquiries recently instituted 
 by Mr. Eaton Hodgkinson.* 
 
 FORMULAE REPRESENTING THE ABSOLUTE STRENGTH OF A CYL- 
 INDRICAL COLUMN TO SUSTAIN A. PRESSURE IN THE DIRECTION 
 OF ITS LENGTH. 
 
 D= external diameter or side of the square of the column 
 in inches. 
 
 D 1 =internal diameter of hollow cylinder in inches.. 
 
 L length in feet. 
 
 W= breaking weight in tons. 
 
 Nature of the Column. 
 
 Both Ends being 
 rounded, the Length of 
 the Column exceeding 
 fifteen times its 
 Diameter. 
 
 Both Ends being flat,, the 
 Length of, the Column 
 exceeding- thirty times i.ts 
 Diameter.. 
 
 Solid cylindrical column of ) 
 cast iron - - - - ) 
 Hollow cylindrical column of ) 
 
 W=U-9^ 
 D""-^"" 
 
 7)3.68, 
 
 ^=44-16^ 
 
 T)3-SB_D 3.66 
 
 W 44-^4. 
 
 cast iron - - - - ) 
 Solid cylindrical column of ) 
 
 L 1 ' 7 
 
 JJ3.78 
 
 w 42'fl 
 
 4 L" 
 
 JJS.66 
 
 "W ll^'*?^ 
 
 wrought iron - - ) 
 
 Solid square pillar of Dantzic ) 
 oak (dry) - - - - f 
 Solid square pillar of red deal ) 
 (dry) J 
 
 L a 
 
 W=10'95?j 
 
 W=n7'81^ 
 JL 
 
 In all cases the strength of a column, one of whose ends 
 was rounded and the other flat, was found to be an arith- 
 metic mean between the strengths of two other columns of 
 the same dimensions, one having both ends rounded and the 
 other having both ends flat. 
 
 The above results only apply to the case in which the 
 length of the column is so great that its fracture is produced 
 wholly by the bending of its material ; this limit is fixed by 
 Mr. 'Hodgkinson in respect to columns of cast iron at about 
 fifteen times the diameter when the extremities are rounded, 
 
 * From a paper by Mr. Hodgkinson, published in the second part of the 
 Transactions of the Royal Society for 1840, to which the royal medal of the 
 Society was awarded. The experiments were made at the expense of 
 Mr. Fairbairn of Manchester, by whose liberal encouragement the researches 
 of practical science have been in other respects so greatly advanced. 
 
 35 
 
546 TORSION. 
 
 and thirty times ths diameter when they arc flat. In 
 shorter columns fracture takes place partly by the crushing 
 and partly by the bending of the material. To these shorter 
 columns the following rule was found to apply with suf- 
 ficient accuracy : " If W, represent the weight in tons 
 which would break the column by bending alone (or if it 
 did not crush) as given by the preceding formula, and W 2 
 the weight in tons which would break the column by crush- 
 ing alone (or if it did not bend) as determined from the 
 preceding table, then the actual breaking weight "W of the 
 column is represented in tons by the formula 
 
 Columns enlarged in the middle. It was found that the 
 strengths of columns of cast iron, whose diameters were from 
 one and a half times to twice as great in the middle as at 
 the extremities, were stronger by one seventh than solid 
 columns, containing the same quantity of iron and of the 
 same length, when their extremities were rounded; and 
 stronger by one eighth or one ninth when their extremities 
 were flat and rendered immoveable by discs. 
 
 431. EELATIVE STRENGTH OF LONG COLUMNS OF CAST 
 
 WROUGHT IRON, STEEL, AND TIMBER OF THE SAME DIMENSIONS. 
 
 Calling the strength of the cast iron column 1000, the 
 strength of the wrought iron column wi ll, according to these 
 experiments, be 1745, that of the cast steel column 2518, of 
 the column of Dantzic oak 108*8, and of the column of red 
 deal 78-5. 
 
 Effect of drying on the strength of columns of timber. 
 It results from these experiments, that the strength of short 
 columns of wet timber to resist crushing is not one half that 
 of columns of the same dimensions of dry timber. 
 
 TORSION. 
 
 432. The elasticity of torsion. 
 Let ABCD represent a solid cylinder, ou<e -of whose trans- 
 
TORSION, 
 
 54:7 
 
 verse sections AEB is immoveably fixed, 
 and every other displaced in its own plane, 
 about its centre, by the action of a pres- 
 sure P applied, at a given distance a from 
 the axis, to the section CD of the cylinder 
 in the plane of that section and round its 
 centre ; the t cylinder is said, under these 
 circumstances, to be subjected to torsion, 
 and the forces opposed to the alteration of 
 its form, and to its rupture, constitute its 
 resistance to torsion. 
 
 Let aabfi be any section of the cylinder 
 whose distance from the section AEB is 
 represented by a?, and let a/3 represent that 
 diameter of the section aabfi which was 
 parallel to the diameter AB before the torsion commenced : 
 let ab be the projection of the diameter AB upon the sec- 
 tion aalfi, and let the angle aca be represented by 6. 
 
 Now the elastic forces called into action upon the section 
 aaibfi are in equilibrium with the pressure P. But these 
 elastic forces result from the displacement of the section 
 aab@ upon its immediately subjacent section. Moreover, 
 the actual displacement of any small element AK of the 
 section aabfi, upon the subjacent section, evidently depends 
 partly upon the angular displacement of the one section 
 upon the other, and partly upon the distance p of the 
 element in question from the axis of the cylinder. Now the 
 angle oca or 6 is evidently the sum of the angular displace- 
 ments of all the sections between aabfi and AEB upon their 
 subjacent sections ; and the angular displacement of each 
 upon its subjacent section is the same, the circumstances 
 affecting the displacement of each being obviously the same : 
 also the number of these sections varies as a?, and the sum 
 of their angular displacements is represented by 6 ; there- 
 fore the angular displacement of each section upon its sub- 
 
 & 
 jacent section varies as -, and the actual displacement of 
 
 & 
 the small element AK of the section a&bfi varies as -p. Now 
 
 the material being elastic, the pressure which must be 
 applied to this element in order to keep it in this state of 
 displacement varies as the amount of the displacement 
 
 (Art. 345.), or as -p. Let its actual amount, when referred 
 
548 TOESION. 
 
 8 
 to a unit of surface, be represented by G-p, where G is a 
 
 certain constant dependant for its amount on the elastic 
 qualities of the material, and called the modulus of torsion ; 
 then will the force of torsion required to keep the element 
 
 & 
 AK iii its state of displacement be represented by G-pAK, and 
 
 $ 
 its moment about the axis of the cylinder by G- p a AK. So 
 
 that the sum of the moments of all such forces of torsion in 
 respect to the whole section aubfi will be represented by 
 
 A A 
 
 G - 2p 2 AK, or by G -I, if I represent the moment of inertia 
 
 of the section about the axis of the cylinder. Now these 
 forces are in equilibrium with P ; therefore, by the principle 
 of the equality of moments, 
 
 P0=GI- (689). 
 
 <KJ 
 
 If r represent the radius of the cylinder, I=farr* (Art. 
 85.). Substituting this value, representing by L the whole 
 length of the cylinder, and by the angle through which 
 its extreme section CD is displaced or through which OP is 
 made to revolve, called the angle of torsion, and solving in 
 r,espect to , 
 
 Thus, then, it appears that when the dimensions of the 
 Y" cylinder are given, the angle of torsion varies 
 directly as the pressure P by which the torsion 
 is produced ; whence, also, it follows (Art. 97.) 
 that if the cylinder, after having been deflected 
 through any distance, be set free, it will oscil- 
 late isochronously about is position of repose, 
 the time T of each oscillation being represented 
 in seconds (equation 76) by the formula 
 
 since 
 
 /*T\ 
 by equation (690) P= I H-FT I (0) j m which expression 
 
TORSION. 549 
 
 (#) represents the length of the path described by the 
 point P from its position of repose, so that the moving force 
 upon the point P, when the pressure prducing torsion is 
 removed, varies as the path described by it from its position 
 of repose. 
 
 The above is manifestly the theory of Coulomb's Torsion 
 Balance.* "W represents in the formula the weight of the 
 mass supposed to be carried round by the point P, and the 
 inertia of the cylinder itself is neglected as exceedingly 
 small when compared with the inertia of this weight. 
 
 The torsion of rectangular prisms has been made the sub- 
 ject of the profound investigations of MM. Cauchyf, Lame, 
 et Clapeyron, and Poisson.g It results from these investi- 
 gations 1 that if 5 and c be taken to represent the sides of the 
 rectangular section of the prism, and the same notation be 
 adopted in other respects as before, then 
 
 
 M. Cauchy has shown the values of the constant G to 
 be related to those of the modulus of elasticity E by the 
 formula 
 
 G=fE (693). 
 
 In using the values of G deduced by this formula from 
 the table of moduli of elasticity, all the dimensions must be 
 taken in inches, and the weights in pounds. 
 
 433. ELASTICITY OF TORSION IN A SOLED HAVING A CIRCULAR 
 
 SECTION OF VARIABLE DIMENSIONS. 
 
 Let db represent an element of the solid contained by 
 
 * Illustrations of Mechanics, Art. 37. 
 
 j Exereices de Mathematiques, 4 e annee. 
 
 i Crelle's Journal. Memoires de 1' Academic, tome viii. 
 
 f Navier, Resume des Lecons, &c., Art. 159. 
 
550 
 
 TOBSION. 
 
 planes, perpendicular to the axis, whose dis- 
 tance from one another is represented by 
 the exceedingly small increment A&> of the 
 distance x of the section ab from the fixed 
 section AB, and let its radius be repre- 
 sented by y ; and suppose the whole of the 
 solid except this single element to become 
 rigid, a supposition by which the conditions 
 of the equilibrium of this particular element 
 will remain unchanged, the pressure P re- 
 maining the same, and being that which 
 produces the torsion of this single element. 
 Whence, representing by A& the angle of 
 torsion of this element, and considering it 
 
 a cylinder whose length is AOJ, we have by equation (689), 
 
 substituting for I its value 
 
 Passing to the limit, and integrating between the limits 
 and L, observing that at the former limit 0=0, and at the 
 latter 0=0. 
 
 2P0 r 
 
 =-^ - 
 
 . . (694.) 
 
 If the sides AC and BD of the solid be straight lines, its 
 form being that of a truncated cone, and if r v and r z repre- 
 sent its diameters AB and CD respectively ; then 
 
 Also, 
 
 dx 
 dy 
 
 i ryt 
 
 1 'a 
 
 .. 
 
TORSION. 551 
 
 4:34. THE RUPTURE OF A CYLINDER BY TORSION. 
 
 It is evident that rupture will first take place in respect 
 to those elements of the cylinder which are nearest to its 
 surface, the displacement of each section upon its subjacent 
 section being greatest about those points which are nearest 
 to its circumference. If, therefore, we represent by T the 
 pressure per square inch which will cause rupture by the 
 sliding of any section of the mass upon its contiguous sec- 
 tion,* then will T represent the resistance of torsion per 
 square inch of the section, at the distance r from the axis, at 
 the instant when rupture is upon the point of taking place, 
 the radius of the cylinder being represented by r. Whence 
 it follows that the displacement, and therefore the resistance 
 to torsion per square inch of the section, at any other dis- 
 tance p from the axis, will be represented at that distance by 
 
 , the resistance upon any element AK, by - p^K, -and the 
 r . r 
 
 sum of the moments about the axis, of the resistances of all 
 
 T T 
 
 such elements, by - 2p 2 AK, or by - I, r substituting for I 
 r r 
 
 its value (equation 64) by -JTV. But these resistances are 
 in equilibrium with the pressure P, which produces torsion, 
 acting at the distance a from the axis ; 
 
 /.Pa=JTW ____ (696). 
 
 It results from the researches of M. Cauchy, before referred 
 to, that in the case of a rectangular section whose sides are 
 represented by ~b and <?, the conditions of rupture are deter- 
 mined by the equation 
 
 (697). 
 
 The length of a prism subjected to torsion does not -affect 
 the actual amount of the pressure required to produce rup- 
 ture, but only the angle of torsion (equation 690) which 
 precedes rupture, and therefore the space through which 
 
 * Or the pressure per square inch necessary to shear it across (Art. 406.). 
 \ Navier, Resume d'un Cours, &c. Art. 167. 
 
552 TORSION 
 
 the pressure must be made to act, and the amount of WORK 
 which must be done to produce rupture. 
 
 According to M. Cauchy, the modulus of rupture by tor- 
 sion T is connected with that S of rupture by transverse 
 strain by the equation 
 
 T=|S (698). 
 
r vi. 
 
 IMPACT* 
 
 435. THE IMPACT OF TWO BODIES WHOSE CENTRES OF GRAVITS 
 
 MOVE IN THE SAME RIGHT LINE, AND WHOSE POINT OF CON- 
 TACT IS IN THAT LINE. 
 
 From the period when a body first receives the impact of 
 another, until that period of the impact when both move for 
 an instant with the same velocity, it is evident that the sur- 
 faces must have been in a state of continually increasing 
 compression : the instant when they acquire a common velo- 
 city is, therefore, that of their greatest compression. When 
 this common velocity is attained, their mutual pressures will 
 have ceased if they be inelastic bodies, and they will move 
 with a common motion ; if they be elastic, their surfaces 
 will, in the act of recovering their forms, be mutually 
 repelled, and the velocities will, after the impact, be dif- 
 ferent from one another. 
 
 436. A BODY WHOSE WEIGHT 18 W,, AND WHICH IS MOVING 
 IN A HORIZONTAL DIRECTION WITH A UNIFORM VELOCITY 
 REPRESENTED BY "V,, IS IMPINGED UPON BY A SECOND BODY 
 WHOSE WEIGHT IS W a , AND WHICH IS MOVING IN THE SAME 
 STRAIGHT LINE WITH THE VELOCITY Y a : IT IS REQUIRED TO 
 DETERMINE THEIR COMMON VELOCITY V AT THE INSTANT OF 
 GREATEST COMPRESSION. 
 
 Let/, represent the decrement per second of the velocity 
 of W, at any instant of the impact (Art. 94.), or rather the 
 decrement per second which its velocity would experience 
 if the retarding pressure were to remain constant ; then will 
 
 * Note (u\ Ed, App. 
 
554: IMPACT. 
 
 l fi represent (Art. 95.) the effective force upon W 1 ; and if 
 j be taken to represent, under the same circumstances, the 
 increment of velocity received by W,, then will ? / a repre- 
 
 sent the effective force upon "W,. Whence it follows, by the 
 principle of D'Alembert (Art. 103.), that if these effective 
 forces be conceived to be applied to the bodies in directions 
 opposite to those in which the corresponding retardation 
 and acceleration take place, they will be in equilibrium with 
 the other forces applied to the bodies. But, by supposition, 
 no other forces than these are applied to the bodies : these 
 are therefore in equilibrium with one another, 
 
 w W 
 
 W,=^/. ..... (699). 
 
 y & 
 
 Let now an exceedingly small increment of the time from 
 the commencement of the impact be represented by A, and 
 let A^ and A^ 2 represent the decrement and increment of 
 the velocities of the bodies respectively during that time, 
 
 .-.(Art. 
 /.(equation 699) W l . A-y 1 =W s . Av a ; 
 
 and this equality obtaining throughout that period of the 
 impact which precedes the period of greatest compression, it 
 follows that when the bodies are moving in the same direc- 
 tion 
 
 W 1 (V,-V)=W,(V-VO ..... (700); 
 
 since Y, - Y represents the whole velocity lost by W, during 
 that period, and Y Y, the whole velocity gained by "W,. 
 
 If the bodies be moving in opposite directions, and their 
 common motion at the instant of greatest compression be in 
 the direction of the motion of W,, then is the velocity lost 
 by W x represented as before by (Y x Y) ; but the sum of 
 the decrements and increments of velocity communicated to 
 "W 2 , in order that its velocity Y 2 may in the first place be 
 destroyed, and then the velocity Y communicated to it in an 
 opposite direction, is represented by (Y a -f Y). 
 
 Solving these equations in respect to Y, we obtain 
 
IMPACT. 555 
 
 V 
 
 the sign being taken according as the motions of tlu 
 bodies before impact are both in the same direction or in 
 opposite directions. 
 If the second body was at rest before impact, Y a =0, and 
 
 wy 
 
 v =^rtt 
 
 If the bodies be equal in weight, 
 
 The demonstration of this proposition is wholly indepen- 
 dent of any hypothesis as to the nature of the impinging 
 bodies or their elastic properties ; the proposition is there- 
 fore true of all bodies, whatever may be their degrees of 
 hardness or their elasticity, provided only that at the 
 instant of greatest compression every part of each body 
 partakes in the common velocities of the bodies, there being 
 no relative or vibratory motion of the parts of either body 
 among themselves. 
 
 437. To DETERMINE THE WORK EXPENDED UPON PRODUCING 
 THE STATE OF THE GREATEST COMPRESSION OF THE SUR- 
 FACES OF THE BODIES. 
 
 The same notation being taken as before, the whole work 
 accumulated in the bodies, before impact, is represented 
 
 "W W 2 
 
 by -J * Y x a 4- ? Y 2 a ; and the work accumulated in them 
 
 y y 
 
 at the period of greatest compression, when they move with 
 
 "W 
 
 the common velocity Y, is represented by -J- 
 
 "Now the difference between the amounts of work accumu- 
 lated in the bodies in these two states of their motion has 
 been expended in producing their compression ; if, there- 
 fore, the amount of work thus expended be represented bj 
 v, we have 
 
 WW 
 
 u V,' + V.'-f 
 
556 IMPACT. 
 
 or substituting for V its value from equation (701), and 
 reducing, 
 
 U=7T-\ 
 
 This expression represents the amount of work permanently 
 lost in the impact of two inelastic bodies, their common 
 velocity after impact being represented by equation (T01). 
 If W, be exceedingly great as compared with W x , 
 
 ,)' .... (704). 
 
 438. TWO ELASTIC BODIES IMPINGE UPON ONE ANOTHER : IT IS 
 REQUIRED TO DETERMINE THE VELOCITY AFTER IMPACT. 
 
 If the impinging bodies be perfectly elastic, it is evident 
 that after the period of their greatest compression is passed, 
 they will, in the act of expanding their surfaces, exert 
 mutual pressures upon one another, which are, in corres- 
 ponding positions of the surfaces, precisely the same with 
 those which they sustained whilst in the act of compression ; 
 whence it follows that the decrements of velocity expe- 
 rienced by that body whose motion is retarded by this 
 expansion of the surfaces, and the increments acquired by 
 that whose velocity is accelerated, will be equal to those 
 before received in passing through corresponding positions, 
 and therefore the whole decrements and increments thus 
 received during the whole expansion equal to those received 
 during the whole compression. 
 
 Now the velocity lost by W, during the compression is 
 represented by (Y, Y) ; that lost by it during the expan- 
 sion, or from the period of greatest compression to that 
 when the bodies separate from one another, is therefore 
 represented by the same quantity. But at the instant of 
 greatest compression both bodies had the velocity Y ; the 
 velocity v l of W, at the instant of separation is therefore 
 Y (Y! Y), or 2Y Y,. In like manner, the velocity 
 gained by W 2 during compression, and therefore during 
 expansion, being represented by (Y^ Y 2 ), and its velocity 
 at the instant of greatest compression by Y,'its velocity v t 
 at the instant of separation is represented by Y + (Y=Fv 2 ), 
 or by 2Y=pY a , the sign =F being taken according as the 
 
IMPACT. 557 
 
 motion of the bodies before impact was in the same or 
 opposite directions. 
 
 Substituting for Y its value in these expressions (equation 
 701), and reducing, we obtain 
 
 i _ ii , 
 w.+w, 
 
 w.+w, 
 
 If the bodies be perfectly elastic and ep,ual in weight, 
 v,=Y,, u a =Y 1 ; they therefore, in this case, interchange their 
 velocities by impact ; and if either was at rest before impact, 
 the other will be at rest after impact. 
 
 If W 3 be exceedingly great as compared with "W,, v^ = 
 Y 1 2Y 2 , a =V a . In this case v 1 is negative, or the 
 motion of the lesser body alters its direction after impact, 
 when their motions before impact were in opposite direc- 
 tions ; or when they were in the same direction, provided 
 that 2Y, be not greater than Y r 
 
 439. If the elasticities of the balls be imperfect, the force 
 with which they tend to separate at any given point of the 
 expansion is different from that at the corresponding point 
 of the compression ; the decrements and increments of the 
 velocities, produced during given corresponding periods of 
 the compression and expansion, are therefore different ; 
 whence it follows that the whole amounts of velocity, lost 
 by the one and gained by the other during the two periods, 
 are different : let them bear to one another the ratio of 1 to e. 
 ~Now the velocity lost during compression by "W, is under 
 all circumstances represented by (Y a Y); that lost during 
 expansion is therefore represented, in this case, by e (Yj Y); 
 therefore, v l =Y e(V l Y) (1 + 0)Y eV r In like man- 
 ner, the velocity gained by W 2 during compression is in all 
 cases represented by (Y=F Y a ) ; that gained during expansion 
 is therefore represented by ^(Y^Y,,); therefore, a =Y-f- 
 e( V qp Y,) = (1 + e} Y =F e Y a . Substituting for Y, and reducing, 
 
 . , 7(m . 
 
558 IMPACT. 
 
 440. IN THE IMPACT OF TWO ELASTIC BODIES, TO DETERMINE 
 THE ACCUMULATED WORK, OR ONE HALF THE VIS VIVA, LOST 
 BY THE ONE AND GAINED BY THE OTHER. 
 
 The vis viva lost by W\ during the impact is evidently 
 represented by L Y 1 2 -v* = -fVY v*) = - \ Y. a 
 
 g 9 g g \ 
 
 K l + ,)Y_,Y ir J^ K l- 
 
 Substituting in this expression its value for Y (equation 
 701) reducing and representing by u^ one half the vis viva 
 lost by W l in its impact, or the amount by which its accumu- 
 lated work is diminished by the impact (Art. 67.), 
 
 1= 
 
 ,V,j ....(709). 
 
 Similarly, if u^ be taken to represent one half the vis viva 
 gained by W 2 , or the amount by which its accum ulated work 
 is increased by the impact, then 
 
 (710). 
 
 441. Let u be taken to represent the whole amount of the 
 work accumulated in the two bodies before their impact, 
 which is lost during their impact. This amount of work is 
 evidently equal to the difference between that gained by the 
 one body and lost by the other; so that u=u 1 u. l . Substi- 
 tuting the values of u^ and u^ from the preceding equations, 
 and reducing, we obtain 
 
 _ 
 
 This expression is equal to one half the vis viva lost during 
 the impact of the bodies. If the bodies be perfectly elastic, 
 6=1, and i=0. In this case there is no real loss of vis viva 
 
IMPACT. 559 
 
 in the impact, all that which the one body yields, during the 
 impact, being taken up by the other.* 
 
 442. In the preceding propositions it has been supposed 
 that the motions of the impinging body, and the body im- 
 pinged upon, are opposed by no resistance whatever during 
 the period of the impact. There is no practical case in 
 which this condition obtains accurately. If, nevertheless, 
 the resistance opposed to the motion of each body be small, 
 as compared with the pressure exerted by each upon the 
 other, at any period of the impact, then it is evident that all 
 the circumstances of the impact as it proceeds, and the mo- 
 tion of each body at the instant when it ceases, will be very 
 nearly the same as though no resistance were opposed to the 
 motion of either, f 
 
 443. As an illustration of the principle established in the 
 last article, let it be required to determine the space through 
 
 * It has been customary, nevertheless, to speak of a loss of vis viva in the 
 impact of perfectly elastic bodies. This loss is in all such cases to be under- 
 stood only as a loss experienced by one of the bodies, and not as an absolute 
 loss. When the impinging bodies are perfectly elastic, it is evident that the 
 one flies away with all the vis viva which is lost in the impact by the other. 
 
 f Let PI and P 2 represent resistances opposed to the motions of two im- 
 
 TTT -m- 
 
 pinging bodies whose weights are Wi and W 2 ; also let -/ lf and -ft re- 
 
 y y 
 
 present the effective forces upon the two bodies at any period of the impact ; 
 then, by D'Alembert's principle, 
 
 or representing by t the time occupied in the impact, up to the period of 
 greatest compression, by V their common velocity at that period, and by Vi 
 and v 2 their velocities at any period of the impact, and substituting for fi and 
 / 2 their values (equation 72), 
 
 W l dv l W,dv 2 
 
 ~i~dt~ Fl ~ Y'd 
 
 Transposing and integrating between the limits and t, 
 
 t 
 
 Now if PI and P 2 be not exceedingly great, the integral in the second member 
 of the equation is exceedingly small as compared with its other terms, and may 
 be neglected ; the above equation will then become identical with equation 
 (700). 
 
560 IMPACT. 
 
 which a nail will be driven by the blow of a hammer ; and 
 let it be supposed that the resistance opposed to the driving 
 of the nail is partly a constant resistance overcome at its 
 point, and partly a resistance opposed by the friction of the 
 mass into which it is driven upon its sides, varying in amount 
 directly with the length of it #, at any time imbedded in the 
 wood. Let this resistance be represented bya-f/fo; then 
 will the work which must be expended in driving it to a 
 depth D be represented (Art. 51.) by 
 
 J(a+Px)dx, or by (a 
 
 Let W, represent the weight of the nail, and Y the velocity 
 with which a hammer whose weight is "W, must impinge 
 upon it to drive it to this depth, and let the surfaces of the 
 nail and hammer both be supposed inelastic ; then will the 
 work accumulated in the hammer before impact be repre- 
 
 W 
 
 sented by 'Y 2 , and the amount of this work lost during 
 
 y 
 the impact by the compression of the surfaces of contact will 
 
 1 / "W W \ 
 
 be represented (equation 711) by l^-i-^-JY 2 . The work 
 
 remaining, and effective to drive the nail, is therefore repre- 
 sented by the difference of these quantities ; and this work 
 being that actually expended in driving the nail, we have 
 
 1 Y a W 2 
 
 -^ ^.=2aP+^P a .... (712); 
 
 by the solution of which quadratic equation, D may be deter- 
 mined. 
 
 444. TWO SOLID PRISMS HAVE A COMMON AXIS; THE EXTREM- 
 ITY OF ONE OF THEM RESTS AGAINST A FIXED SURFACE, AND 
 ITS OPPOSITE EXTREMITY RECEIVES THE IMPACT, IN A HORI- 
 ZONTAL DIRECTION, OF THE OTHER PRISM : IT IS REQUIRED 
 TO DETERMINE THE COMPRESSION OF EACH PRISM, THE LIMITS 
 OF PERFECT ELASTICITY NOT BEING PASSED IN THE IMPACT. 
 
 Let "W represent the weight of the impinging prism, and 
 
IMPACT. 561 
 
 Y its velocity before impact ; L x and : L 3 the- lengths of the 
 prisms before compression ; E t and E 2 their moduli of elas- 
 ticity ; Kj and K 2 their sections ; ^ and Z 2 the greatest com- 
 pressions produced in them respectively by the impact ; 
 then will the amounts of work which must have been done 
 upon the prisms to produce these compressions be repre- 
 sented (equation (486) by the formulae 
 
 _ ) 
 
 and the whole work thus expended by 
 
 w 
 
 But this work has been done by the work -J Y*,, accumu- 
 lated (Art. 66) before impact in the impinging body, and 
 that work has been exhausted in doing it ; 
 
 Moreover, the mutual pressures upon the surfaces of con- 
 tact are at every period of the impact equal, and at the 
 instant of greatest compression they are represented respec- 
 
 K E I K E I 
 
 tively (equation 485) by t ' ' and I * 2 ; 
 
 Eliminating Z 2 between this equation and the preceding, and 
 reducing, 
 
 in which expressions Z, represents the greatest compression 
 
 36 
 
562 IMPACT. 
 
 of the prism whose section is K,, and P the driving pressure 
 at the instant of greatest compression. 
 
 445. The mutual pressures P of the surfaces of contact at 
 any period of the impact. 
 
 Let I represent the space described by that extremity of 
 the impinging prism, by which it does not impinge : it is 
 evident that this space is made up of the two corresponding 
 compressions of the surfaces of impact of the prisms ; so that 
 if these be represented by ^ and 1 99 then 1=^ + 1^ But 
 
 (equation 713) ^=^-^,4=^; therefore 1 = ^*1- 
 L, 
 
 446. A measure of the compressibility of the prisms. 
 
 If X be taken to represent the space through which that 
 extremity of the impinging prism by which it does not 
 impinge will have moved when the mutual pressure of the 
 surfaces of contact is 1 Ib. ; or, in other words, if X repre- 
 sent the aggregate space through which the prisms would 
 be compressed by a pressure of 1 Ib. ; then, by the preced- 
 ing equation, 
 
 ^ X may be taken as a measure of the aggregate compressi- 
 bility of the prisms, being the space through which their 
 opposite extremities would be made to approach one another 
 by a pressure of 1 Ib. applied in the direction of their 
 length. 
 
 If \ and \ represent the spaces through which the 
 prisms would severally be compressed by pressures of 1 Ib. 
 
 applied to each, then \= w -^-. X 2 ^- ; therefore X=X -f 
 
 \, or the aggregate compressibility of the two prisms is 
 equal to the sum of their separate compressibilities. 
 
IMPACT. 563 
 
 447. The work u expended upon the compression of the 
 prisms at any period of the impact. 
 
 The work expended upon the compression Z t is repre- 
 
 TT Tp 
 
 sented by -J- -r l ^ 9 ; or substituting its value for ^ (equation 
 -U 
 
 713), it is represented by ig-g-P 2 . And, similarly, the work 
 expended on the compression , is represented by 8 
 
 therefore u i(f^ +^4^ )P 8 ; or substituting for P its 
 M^iLj J\. 2 ii( a / 
 
 value from equation (716), 
 
 448. The velocity of the impinging body at any period of the 
 impact, the impact being supposed to take place vertically. 
 
 It is evident that at any period of the impact, when the 
 velocity of the impinging body is represented by -y, there 
 will have been expended, upon the compression of the two 
 bodies, an amount of work which is represented by the 
 work accumulated in the impinging body before impact, 
 increased by the work done upon it by gravity during the 
 impact, and diminished by that which still remains accu- 
 
 W W 
 
 mulated in it, or by Y 2 + Wl \ v\ 
 
 y y 
 
 Representing, therefore, by u the work expended upon 
 
 W 
 
 the compression of the bodies, we have -J Y a -f"W7 
 
 \ ^u. 
 
 V 
 
 Substituting, therefore, for u its value from equation 
 (718), 
 
IMPACT. 
 
 Or substituting for I its value in terms of P (equation 716), 
 
 THE PILE DEIVEB. 
 
 449. It is evident that the pile will not begin to be 
 driven until a period of the impact is at- 
 tained, when the pressure of the ram upon 
 its head, together with the weight of the 
 pile, exceeds the resistance opposed to its 
 motion by the coherence and the friction of 
 the mass into which it is driven. Let this 
 resistance be represented by P ; let Y repre- 
 sent the velocity of the ram at the instant of 
 impact, and v its velocity at the instant when 
 the pile begins to move, and W 15 W 2 the 
 weights of the ram and pile ; then, since the 
 pile will have been at rest during the whole 
 of the intervening period of the impact, since 
 moreover, the mutual pressures Q of the sur- 
 faces of contact are at the instant of motion 
 represented by P W 2J we have by equation 
 (720) 
 
 If the value of v determined by this equation be not a 
 possible quantity, no motion can be communicated to the 
 pile by the impact of the ram ; the following inequality is 
 therefore a condition necessary to the driving of the pile, 
 
 After the pile has moved through any given distance, one 
 portion of the work accumulated in the ram before its 
 impact will have been expended in overcoming, through 
 that distance, the resistance opposed to the motion of the 
 
IMPACT. 565 
 
 pile; another portion will have been expended upon the 
 compression of the surfaces of the ram and pile ; and the 
 remainder will be accumulated in the moving masses of the 
 ram and pile. The motion of the pile cannot cease until 
 after the period of the greatest compression of the ram and 
 pile is attained ; since the reaction of the surface of the pile 
 upon the ram, and therefore the driving pressure upon the 
 pile, increases continually with the compression. If the 
 surfaces be inelastic, having no tendency to recover the 
 forms they may have received at the instant of greatest 
 compression, they will move on afterwards with a common 
 velocity, and come to rest together ; so that the whole work 
 expended prejudicially during the impact will be that 
 expended upon the compression of the inelastic surfaces of 
 the ram and pile. If, however, both surfaces be elastic, 
 that of the ram will return from its position of greatest 
 compression, and the ram will thus acquire a velocity rela- 
 tively to the pile, in a direction opposite to the motion of 
 the pile. Until it has thus reached the position in respect to 
 the pile in which it first began to drive it, their mutual 
 reaction Q will exceed the resistance P, and the- pile will 
 continue to be driven. After the ram has, in its return, 
 passed this point, the pile will still continue to be driven 
 through a certain space, by the work which has been accu- 
 mulating in it during the period in which Q has been in 
 excess of P. When the motion of the pile ceases, the ram 
 on its return will thus have passed the point at which it 
 first began to drive the pile : if it has not also then passed 
 the point at which its weight is just balanced by the elas- 
 ticity of the surfaces, it will have been continually acquiring 
 velocity relatively to the pile from the period of greatest 
 compression; it will thus have a certain velocity, and a 
 certain amount of work will be accumulated in it when the 
 motion of the pile ceases : this amount of work, together 
 with that which must have been done to produce that com- 
 pression which the surfaces of contact retain at that instant, 
 will in no respect have contributed to the driving of the 
 pile, and will have been expended uselessly. If the ram in 
 its return has, at the instant when the motion of the pile 
 ceases, passed the point at which its weight would just be 
 balanced by the elasticity of the surfaces of contact, its 
 velocity relatively to the pile will be in the act of diminish- 
 ing ; or it may, for an instant, cease at the instant when the 
 pile ceases to move. In this last case, the pile and ram, for 
 an instant, coming to rest together, the whole work accumu- 
 
566 IMPACT. 
 
 lated in the impinging ram will have been usefully expended 
 in driving the pile, excepting only that by which the remain- 
 ing compression of the surfaces has been produced ; which 
 compression is less than that due to the weight of the rara. 
 This, therefore, may be considered the case in which a maxi- 
 mum useful effect is produced by the ram. The following 
 article contains an analytical discussion of these conditions 
 under their most general form. 
 
 450. A prism impinged upon ~by another is moveable in the 
 direction of its axis, and its motion is opposed by a con- 
 stant pressure P ; it is required to determine the con- 
 ditions of the motion during the period of impact, the 
 circumstances of the impact being in other respects the 
 same as in Article 448. 
 
 Let yj and f^ represent the additional velocities which 
 would be lost and acquired per second (see Art. 
 95) by the impinging prism and the prism 
 impinged upon, if the pressures, at any instant 
 operating upon them, were to remain from that 
 
 W W 
 
 instant constant ; then will - 1 /., VI represent 
 
 g j g t j 
 
 the effective forces upon the two bodies (Art. 103) 
 or the pressures which would, by the principle of 
 D'Alembert, be in equilibrium with the unbal- 
 anced pressures upon them, if applied in opposite 
 directions. 
 
 Now the unbalanced pressure upon the system 
 BP composed of the two prisms is represented by 
 
 (W 1+ W 2 -P), 
 
 "W 1 W 2 
 
 "~g~ y* ~ * 2 ~~ 
 
 also the unbalanced pressure upon the prism PQ=W 2 -f- 
 Q P, where Q represents the mutual pressure of the prisms 
 atQ; 
 
 Let A have been the position of the extremity B of the 
 impinging prism at the instant of impact ; and let x l repre- 
 eent the space through which the aggregate length BP of 
 the two prisms has been diminished since that period of the 
 
IMPACT. 567 
 
 impact, and a? 3 the space through which the point P has 
 moved ; then (equation 716) 
 
 L ' L 1 ^ 
 
 Also AB =2^+0?,,; therefore velocity of point B= 
 (Art. 96); therefore f*=-j-++-j-=-j-+- 
 
 \MV WV WV 
 
 Substituting these values of f t and Q in equations (723) and 
 (724), and eliminating/^ between the resulting equations, 
 
 Integrating this equation by the known rules,t we obtain 
 aj^A sin. /+B cos. 7^ + -r- ..... (727); 
 
 in which expression the value of 7 is determined by the 
 equation 
 
 ^(W+WJ =9 | L/Kl^+Sk)-' f ' ' ' ' 
 
 and A and B are certain constants to be determined by the 
 conditions of the question. Substituting in equation (724) 
 the value of Q from equation (725), and solving in respect 
 to/* 
 
 (729). 
 
 Substituting for x l its value from equation (727), and for f t 
 its value -r^, and reducing, 
 
 <#X Ag . B<7 
 
 ^ = w? sin - 7 * + w> cos - 
 
 Integrating between the limits and , and observing that 
 
 yy/v 
 
 when tf=0, -77=0 ; the time being supposed to commence 
 
 CM 
 
 with the motion of the prism PQ, 
 
 * Art. 96. Equations (72) and (74). 
 f Church's Int. Cal. Art. 183. 
 
568 IMPACT. 
 
 sn - 
 
 Integrating a second time between the same limits, 
 
 * = vw ^ ~ 8in * 7f} + ~ cos * 
 
 Now when the motion of the second prism ceases -^7=0 ; 
 
 d/t 
 
 whence, if the corresponding value of t be represented by T, 
 A(l-cos. yT)+B8in. 7 T + 1- 
 
 To determine the constants A and B, let it be observed 
 that the motion of the prism QP cannot commence until the 
 pressure Q of the impinging prism upon it, added to its own 
 weight W 2 , is equal to the resistance r opposed to its motion. 
 So that if c be taken to represent the value of x l (i. e. the 
 aggregate compression of the two prisms) at that instant, 
 
 then, substituting for Q its value from equation (725), - + 
 
 X 
 
 W,=P; 
 
 .... (732). 
 
 Now since the time t is supposed to commence at the 
 instant when this compression is attained, and the prism PQ 
 is upon the point of moving, substituting the above value of 
 c for x l in equation (727), and observing that when x=c, 
 
 t=Q, we obtain (P W a )X=:B+ 3 ^. ; whence by substitu- 
 
 7 "s 
 tion from equation (728), and reduction, 
 
 
 So long as the extremity P, of the prism impinged upon, 
 is at rest, the whole motion of the point B arises from the 
 
 compression of the two prisms, and is represented by -^. 
 
IMPACT. 569 
 
 The value of * ? when t=0, is represented therefore by v 
 
 dt 
 (equation 721). Differentiating, therefore, equation (727), 
 
 assuming =0, and substituting v for-^- 1 ; we obtain v=yA; 
 
 whence it appears that the value of A is determined by 
 dividing the square root of the second number of equation 
 (721) by r 
 
 Substituting for A and B their values in equations (731-3) 
 
 -lsin. r T+ 
 
 Reducing, and dividing by the common factor of the two 
 last terms, 
 
 Substituting for A and B their values in equation (730), and 
 representing by D the value of a? a , when =T, 
 
 .... (735). 
 
 The value of T determined by equation (734) being sub- 
 stituted in equation (735), an expression is obtained for the 
 whole space through which the second prism is driven by 
 the impact of the first.* 
 
 * The method of the above investigation is, from equation (726), nearly the 
 same with that given by Dr. Whewell, in the last edition of his Mechanics*; the 
 principle of the investigation appears to be due to Mr. Airey. If the value 
 of y, as determined by equation (734), were not exceedingly great, then, since 
 the value of T is in all practical cases exceedingly small, the value of yT would 
 in all cases be exceedingly small, and we might approximate to the value of 
 T in equation (735), by substituting for cos. yT and sin. yT, the two first terms 
 of the expansions of those functions, in terms of yT. 
 
EDITORIAL APPENDIX. 
 
 NOTE (a). 
 
 BESIDES its direction defined (Art. 1), we have also to take 
 into consideration, in estimating the effects of a force, its 
 point of application, or the point of the body where it acts, 
 either directly or through the medium of some other body, 
 as a rigid bar, or an inextensible cord in its line of direction ; 
 the point on its line of direction towards which the point of 
 application has a tendency to move ; and finally the inten- 
 sity, or magnitude of the force as expressed in terms of some 
 settled unit of measure. 
 
 NOTE (b\ 
 
 This result of experiment also admits of the following 
 proof: Let A be the point of appli- 
 
 * * *> > <P cation of a force P, and let this point 
 
 be invariably connected with another 
 
 point B, in the line of direction towards which A tends to 
 move from the action of P ; suppose now two other forces 
 P, and P 2 , each equal to P, to be applied ; the one at A, in 
 a direction opposite to P, and the other at B, in the same 
 direction as P ; the introduction of these two equal forces, 
 acting in opposite directions,will evidently in no wise change 
 the direction or intensity of P ; but as P a is equal and oppo- 
 site to P its eifect will be to balance the action of P at A, 
 whilst it leaves P 2 to exert an action at B precisely the same 
 as P was exerting at A before the introduction of J?, and P a . 
 
 NOTE (c). 
 Suppose two forces P 1 and P a applied to the same point A, 
 
 571 
 
572 EDITORIAL APPENDIX. 
 
 the direction of the one being AB, that of the 
 other AC ; no was these forces make an angle 
 with each other, it is evident, as the point of 
 application can move but in one direction, and 
 as it is solicited to move towards B and C at 
 the same time, that it must move in some 
 direction which is coincident with neither of 
 these; this direction, it is equally evident, 
 must be in the same plane as the directions AB and AC, for 
 there is no argument in favor of a direction assumed exterior 
 to the plane and on one side of it which would not equally 
 apply to a symmetrical direction assumed on the other side ; 
 it is also evident that this direction must be some one AF 
 within the angle formed by AB and AC, for the point, if 
 solicited by Pj alone, would take the direction AB, and as it 
 cannot take a direction to the left of BD, as there is no force 
 that solicits it on that side, and, for like reasons, cannot take 
 one to the right of CE, it must therefore take the one 
 assigned somewhere within the angle BAC. 
 
 ]Now suppose further that P x and P 3 are equal, it is evi- 
 dent that the direction assigned to their resultant, or that of 
 the motion of their point of application, must be the one 
 which bisects the angle BAC, for the argument for any 
 direction on the left of this line would be equally cogent for 
 the like position on the other side. 
 
 If Pj and P 2 are unequal then will the direction of their 
 resultant divide the angle BAC unequally, the 
 smaller portion being next to the greater force ; 
 for suppose Pj divided into two portions, one 
 of which P shall be equal to P 2 ; P and P 2 can 
 be replaced by their resultant Rj, the direction 
 of which AF bisects the angle BAC ; we shall 
 then have two forces K t and the remaining 
 portion of P 1? the resultant of which R must lie 
 somewhere within the angle BAF, and there- 
 fore nearer to P x than to P 2 ; but It is the resultant of the 
 two forces P l and P 2 . Therefore, &c. 
 
 Hence it is seen that two forces w r hose directions form an 
 angle between them and meet, 1st, have a resultant ; 2nd, that 
 the direction of this resultant lies in the plane of the two 
 forces ; 3d, that it passes through the point where the direc- 
 tions meet, and lies within the angle contained between 
 them ; 4th, that it bisects this angle when the forces are 
 equal ; 5th, that when the forces are unequal it divides this 
 angle unequally, the smaller portion being next to the greater 
 force. 
 
EDITOKIAL APPENDIX. 
 
 Now as the two forces P x and P 2 can be replaced by their 
 resultant R, and as the effect of this will be the 
 same if applied at any point F in its line of 
 direction as at the point of application of the 
 two forces, it is evident, if we transfer P, and P 7 
 also to F, preserving their new parallel to theii 
 original directions, that they, in turn, can be made 
 to replace R. It thus appears that the point of 
 application of two forces may be transferred to 
 any point of the line of direction of their result- 
 ant without changing the effects of these forces, provided 
 their new directions are kept parallel to their original ones. 
 It is upon the preceding propositions, in themselves self- 
 evident, that the mode of demonstration, known as Duchay- 
 la's, of the proposition, termed the parallelogram of forces, 
 or of pressures, is based. 
 
 NOTE (d). 
 
 When two parallel forces are applied to two points inva- 
 riably connected, their resultant can be found by applying 
 the propositions in (Arts. 1, 2, 3). 
 
 Let P, and P 2 be two parallel forces applied at the points 
 A and B invariably connected, as by a 
 rigid bar. Let two equal forces Q x and 
 Q 2 be so applied, the one at A the other 
 at B, as to act in opposite directions 
 along AB. These two will evidently 
 have no effect to change the action of 
 P, and P 2 . Now the two forces P, and 
 Q, applied at A will have a resultant R 1? 
 the intensity and direction of which can 
 be found by the preceding method. In like manner the 
 resultant R 2 of P 2 and Q 2 can be obtained. Now the forces 
 being replaced by their resultants, the equilibrium will still 
 subsist, and the effect will remain the same whether R x and 
 R 2 act at A and B, or at o their point of meeting. But as 
 R t and R 2 can each be replaced by their components at any 
 point of their direction, let these components be transferred 
 to the point o. In this position Q x and Q 2 will destroy each 
 other, whilst P, and P 3 will act in the same direction along 
 <?C and parallel to their original ones, with an intensity equal 
 to their sum Pj-f P 2 . 
 
 Now from the similar triangles A#C, rom / and B0C, son, 
 there obtains, 
 
574: EDITORIAL APPENDIX. 
 
 om : mr : : oC : CA, or P x : Q, : : oC : CA. 
 ns : on :: CB : 00, or Q, : P 2 :: CB : 00. 
 
 Multiplying the two last proportions, there obtains, 
 
 I\ : P 2 : : CB : CA, 
 and 
 
 P, : P a : P,+ P a :: CB : CA : CB+CA or AB. 
 
 From this we see that two parallel forces acting in the 
 same direction, 1st, have a resultant which is equal to their 
 sum ; 2nd, that the direction of this resultant is parallel to 
 that of the forces; 3d, that it divides the line joining the 
 points of application of the two forces into parts reciprocally 
 proportional to the forces ; 4th, that either force is to the 
 resultant as the portion of the line between the resultant and 
 the other force is to the total distance between the points of 
 application ; 5th, that the foregoing propositions hold true 
 for any position of the line AB with respect to the two 
 parallel forces and their resultant. 
 
 When the two forces act in opposite directions at the 
 
 points A and B, by following 
 
 \:;;:f* a like process, we obtain the 
 
 --"'"* two resultants E x and E 2 , 
 
 ...--"'" which being prolonged to 
 
 their point of meeting o we 
 can again replace them by 
 
 / / \ their components P 1? Q x , and 
 
 klJ P 2 , Q 2 ; of which P t and P 2 , 
 
 acting parallel to their ori- 
 ginal positions but in opposite directions, will have for their 
 resultant P, P 2 . 
 
 Now prolonging the direction of this resultant until it 
 meets AB prolonged at C, there obtains as in the preceding 
 case, from the similar triangles A0C, rom* and B0C, son, 
 
 om : mr : : 0C : CA, or P a : Q, : : 00 : CA, 
 ns : on :: CB : 00, or Q 3 : P, :: CB : 0C, 
 hence, 
 
 P, : P a : P.-P, :: CB : CA : CB-CA or AB. 
 
EDITORIAL APPENDIX. 
 
 575 
 
 Remark. Although it may be assumed, as self-evident, 
 that any resultant can be replaced 
 by two equivalent components, 
 without disturbing the equilibrium, 
 and that each of these in turn 
 may be replaced by two other 
 equivalent components, and so on 
 for any number of components ; still like compositions and 
 resolutions of forces are of such frequent occurrence in esti- 
 mating the pressures, or strains on the various points of any 
 mechanical contrivance, as a machine, a frame of timber, 
 &c., arising from a resultant pressure, that the student can- 
 not be too familiar with the processes of effecting such com- 
 positions and resolutions. 
 
 To show by a simple illustration this truth, let the result- 
 ant AR be replaced by its two equivalent components AP, 
 and AP 2 in any assumed positions ; and let each of these 
 components be resolved into two others, AQ 15 AE^ for APj ; 
 and AQ 2 , AR 2 for AP 2 , taken respectively perpendicular and 
 parallel to AR. Now it is evident, from the figure, that the 
 two components AQ 1? AQ 2 of this last resolution are equal 
 and opposite in direction, and therefore destroy each other ; 
 whilst the two AR 15 AR 2 act in the direction of AH, and 
 their sum is equal to AH. The same may in like manner be 
 shown for any number of sets of components by which AH 
 might be replaced. 
 
 NOTE (e). 
 
 If the point o from which perpendiculars are drawn to the 
 directions of two forces P : and P 2 , is 
 taken on the direction of their re- 
 sultant, then will m t P 1 =m t P 2 . 
 
 For from o draw the perpendicu- 
 lars om, on, to P a and "r,, and join 
 the points m and n of their inter- 
 section. The quadrilateral A.mon, 
 having the angles at m and n right 
 angles, can be inscribed in a circle, therefore the two angles 
 at m and A, subtended by the chord on, will be equal. In 
 the triangles mon and ABC, the angle o is the supplement 
 of the angle A of the quadrilateral, and B, being the adja- 
 cent angle of the parallelogram, is also the supplement of A ; 
 
576 EDITORIAL APPENDIX. 
 
 the two triangles, having two angles equal, are similar, 
 therefore, 
 
 AB : BC : : om : on, or P 3 : P : : : om : on ; 
 
 hence 
 
 =~Pi x on. Therefore, &c. 
 
 From this proposition the relations of two parallel forces 
 to their resultant can be readily deduced from the limiting 
 case of the angle mon of the triangle ; for from the two simi- 
 lar triangles there obtains as before 
 
 P 3 : P 1 : R or AC : : om : on : mn. 
 
 Now as this is true for any value of the angle 0, when it 
 becomes 180, the forces P,, P 2 having the same direction, 
 and their resultant E become parallel; the perpendiculars 
 om and on become portions of the line mn ; and, as mn=om 
 + on, it follows, from the above proportion, that R Pj + P,. 
 
 When Pj and P 2 have opposite directions, we can suppose 
 the force P a , for example, and its perpendicular turned about 
 the point o in the plane of the forces until the point m falls 
 on the prolongation of on on the opposite side from 0, in 
 which position P x and P 2 again become parallel, but act in 
 opposite directions. During this rotation of P 1? the resultant 
 still passes through o, and there still obtains 
 
 P a : P! : : R : om : on : mn ; 
 
 but, as mn now is equal to omon, it follows, from the 
 proportion, that R=P a P r Hence the same relations 
 between P,, P a and R as already established, NOTE (d). 
 
 ^ Otherwise, since in any number of forces in equilibrium 
 either of them is equal and directly opposed to the resultant 
 of all the rest, the whole may be replaced by these two 
 without disturbing the equilibrium. If now through the 
 point of application of these two we draw any two lines at 
 right angles to each other, and then resolve each of the two 
 forces into two components parallel to these two lines, it 
 will be at once seen, from the diagram, that the like com- 
 
EDITORIAL APPENDIX. 577 
 
 portents will be equal and opposite to each other, and this 
 would evidently be the same for the components of the ori- 
 ginal forces resolved in the same manner, otherwise there 
 would be a resultant for all the forces, which is contrary to 
 the supposition of an equilibrium. 
 
 .Remark. As this method of resolving a system of forces 
 into sets of components parallel to any assumed rectangular 
 axes, in order to determine their algebraical values, is of 
 frequent use, in simplifying the numerical calculations 
 necessary in the applications of the principles of statics, the 
 student should make himself perfectly familiar with the pro- 
 positions that precede and follow Art. 11. 
 
 (g). 
 
 Otherwise, join DE which will be parallel to AC^thus 
 forming with it and the lines AD and CE two equi-angular 
 triangles, from which there obtains 
 
 DE:DG:: AC: AG; 
 but DE=f AC, therefore DG=4AG=DA. 
 
 NOTE (A). 
 
 Otherwise, join GH which, as AG and CH intersect, will 
 be in the same plane with them and with the line AC. As 
 AH and CG are respectively -f of the lines drawn from A 
 and C to the middle of BD, it follows that GH is parallel to 
 AC and forms with it and the lines AG and CH, by their 
 intersection at K, the two equi-angular triangles CKH and 
 and AKC, from- which there obtains 
 
 GH:GK:: AC : AK, 
 but GH=|AC, therefore GK= AK=iAG. 
 
 As the methods employed in (Art. 45, &c.) to represent, 
 by geometrical diagrams, what are termed the laws of 
 motion, or the relations which exist at any two given 
 
 37 
 
m 
 
 -V 
 
 578 EDITORIAL APPENDIX. 
 
 instants between the velocity, the space, and the time of a 
 body's motion, although very simple in themselves, are 
 sometimes found to offer difficulties to the student, particu- 
 larly as to the representation of spaces by areas, a few addi- 
 tional marks on this point may not be here misplaced. 
 Taking, in the first place, the case of a body M moving in 
 D a rectilinear path from A towards 
 
 IB with a uniform motion. Ac- 
 cording to the definition, the 
 body will move over the suc- 
 
 5. cessive equal portions Ab, be, cd, 
 
 &c., of its path in equal succes- 
 sive portions of time, however small or great these portions 
 may be. Taking now any portion of time as a unit, as a second, 
 a minute, &c., and supposing Ab the portion of its path, or 
 the space through which the body has moved during this 
 unit, Ab will represent what is termed the velocity, or rate 
 of motion of the body ; and when the path itself is expressed 
 in terms of any linear unit, as a foot, a yard, a mile, &c., the 
 number of these units in Ab will measure the velocity ; for 
 example, if the unit of path, or space is a foot, and there 
 were four of these units in Ab, and the unit of time is a 
 second, then the velocity would be termed a velocity of four 
 feet per second, &c. Supposing the body to start from A, 
 with this velocity, it will successively move over distances, 
 each of four feet in length, along its path, in successive 
 seconds of time ; consequently any distance, or space, as Ad, 
 will be equal to Ab taken as many times as the number of 
 seconds elapsed from the time the body started from A until 
 it reached d\ or, in other words, the number of units in the 
 space Ad is expressed by the abstract number obtained by 
 multiplying the number of units in the velocity by the num- 
 ber of units in the time. This, like all other similar pro- 
 ducts, can be expressed algebraically, or geometrically ; but 
 by whatever symbol expressed, the signification is the same. 
 For example, on any two lines, as AB and AC, taken at 
 right angles, set off any number of equal parts as Ab, be, 
 cd, &c., as units of time, and on AC any number also of 
 equal, parts, which may be the same in length, or otherwise, 
 as those on AB, to represent the units in wnich the velocity 
 is expressed. Suppose the latter to be composed of the four 
 units Am, mn, &c. ; and that the number of units of time 
 considered is three ; on the lines Ad, AC construct the 
 rectangle AD; then is the area of the rectangle said to 
 express the space corresponding to the velocity and time 
 
EDITORIAL APPENDIX. 579 
 
 here assumed ; that is, the number of units in area of this 
 rectangle, expressed in terms of the unit of area on Ab and 
 Am for example, is equal to the number of units of space. 
 In like manner the area of the rectangle AE expresses the 
 space corresponding to the velocity and the time Ae, &c. 
 In uniformly varied motion, as the velocity increases in 
 
 the same proportion as the time 
 increases, or, in other words, as 
 the augmentations of the velocity 
 for equal intervals of time is the 
 same, these relations between the 
 B times, velocities and spaces, can, 
 in like manner, be expressed by 
 
 a geometrical diagram as follows : On any line, as AB, set 
 oft a number of equal parts as Ab, be, ed, &c., to represent 
 equal intervals of time ; at the points J, c, d, &c., having 
 drawn perpendiculars to AB, set off on them distances b?n, 
 en, do, &c., to represent the corresponding velocities ; in 
 which cn%bm do3bm ; or Ad : Ac : Ab :: do : en : bm, 
 &c. Now, as the same relations obtain between all the dis- 
 tances set off on AB and their corresponding perpendicu- 
 lars, it follows that the line AC, drawn through the points 
 in, n, o, &c., is a right line, and that the triangles Abm, Aon, 
 &c., are therefore similar. As the relations between the 
 times and velocities are true, however great, or however 
 small the equal portions of time may be assumed, let us sup- 
 pose these portions, as Ab, "be, cd, to be taken so small that 
 the velocity of the body during any one of them may be 
 considered uniform, and as a mean between what it actually 
 is at the commencement and end of this portion ; tkat is en 
 and do, for example, representing the actual velocities at the 
 beginning and end of the interval of time represented by cd, 
 then J (en + do) represents^ the mean, or uniform velocity 
 during this interval. This being premised, the number of 
 units of space over which the body will pass whilst moving 
 with a uniform velocity, expressed by % (cn + do), during the 
 interval cd, will be represented, according to the preceding 
 proposition, by cdx% (cn + do), but this also expresses the 
 area of the trapezoid cdno ; and as the same is true for all 
 the like trapezoids it will also be true for their sums, or for 
 the triangles, as Ado and Afq for example, the areas of which 
 are equal to the sum of the areas of the trapezoids of which 
 they are composed. Supposing the body to move from a 
 state of rest with a uniformly accelerated motion, and that 
 at the intervals of time, represented by Ad and Af, its 
 
580 EDITORIAL APPENDIX. 
 
 respective velocities are do andjfy, then will tlie number of 
 units of space which the body will have moved over in these 
 two intervals be respectively expressed by the number of 
 units of area in the triangles Ado and Afq. As the trian- 
 gles are similar their areas are as the squares of their like 
 sides ; it therefore follows that in uniformly varied motion, 
 the spaces are as the squares of the times, or as the squares 
 of the velocities. 
 
 As do represents the velocity acquired during the time 
 A<#, supposing the body to have moved from a state of rest, 
 and the number of units of area in the triangle Ado repre- 
 sents the corresponding number of units of space, it follows, 
 that if the body had moved, during the same interval, with 
 the velocity do which it actually acquired in it, the number 
 of units of space it would then have passed overVould have 
 been represented by the number of units of area in the rect- 
 angle A0, constructed on Ad and do. But, as the area of 
 the rectangle is double that of the triangle, the space that 
 would have been passed over in the supposed case would 
 have been double that passed over in the actual case. 
 
 If we take any portion, as A0, to represent the unit of 
 time, then the corresponding perpendicular ep will represent 
 the velocity, or the quantity f used in (Arts. 46. 47) fol- 
 lowing. 
 
 NOTE (j). 
 
 As the propositions under this head, and those under the 
 heads of Accumulation of Work in a Moving Body (Art. 64) 
 and Principle of Yis Yiva (Art. 129) constitute the basis of 
 what may be termed Industrial Mechanics, or the applica- 
 tions of the principles of abstract mechanics to the calcula- 
 tion of the effects of motive power transmitted by machines 
 and employed to produce some 'useful mechanical end, it is 
 very important that the student should have a clear and 
 definite apprehension of their signification in this point of 
 view. Work, as here defined, supposes two conditions as 
 essential to its production : a continued resistance, or obstacle 
 removed by the action of a force, and a motion of the point 
 of application of the force in a direction opposite to that in 
 which the resistance acts. Its measure is expressed by the 
 product arising from multiplying the number of units of the 
 resistance, or of its equivalent force directly opposed to it, 
 by the number of units of path which the point of applica- 
 tion of this force has described during the interval consi- 
 
EDITORIAL APPENDIX. 581 
 
 dered, in which the force acts to overcome the resistance. 
 It follows that the work will be when this .product is ; 
 that is, when either of the factors, the resistance, or its equi- 
 valent force, or the path described, is 0. 
 
 In estimating work, that which is external and which alone 
 generally we have the means of measuring, is alone consi- 
 dered. For example, if with a flexible bar a person attempts 
 to push before him any obstacle, the first effect observed will 
 be a certain deflection of the bar, during which the hand, at 
 one end of the bar, .will have moved forward a certain dis- 
 tance in the direction of the point of application at the other, 
 producing an amount of work which is expressed by the 
 product of the pressure, or force exerted by the hand, sup- 
 posing this pressure to remain constant during this period, 
 and the path described, in the line of direction of this pres- 
 sure, by the point where it is applied. During this period, 
 as the obstacle to be moved has remained at rest, no path 
 has been described by the point where the bar rests against 
 it, therefore, according to our definition, no work has been 
 done upon the resistance. The effect produced by the 
 pressure has been simply to bend the bar, and the work is 
 therefore due only to the resistance offered by the molecular 
 forces of the material composing the bar to the force that 
 tends to bend it. This portion of the work, although in this 
 case we have the means of measuring it, being what may be 
 termed internal, is not taken into the account in estimating 
 that due to the resistance to be overcome, which would have 
 been the same had a perfectly rigid bar been used instead of 
 the flexible one. 
 
 In like manner, when an animal carries a burthen on his 
 back from one point to another on a horizontal plane no 
 work is produced according to our definition ; for no resist- 
 ance has been overcome in the direction in which the bur- 
 then has been carried, and therefore the product that repre- 
 sents the work is 0. The work in this case, as in that of the 
 flexible bar, is internal ; and similar to that arising from a 
 burthen borne by an animal whilst standing still ; and there- 
 fore although both of them may be very useful operations 
 and have a marketable value, still they can neither be mea- 
 sured by the standard by which it is agreed to estimate 
 work. 
 
 Every mechanical operation performed by machinery pre- 
 sents a case of work. Take for example the simple opera- 
 tion of planing, in which the hand moves a plane, which is 
 but a rigid bar to which is fixed an iron tool like a chisel for 
 
582 EDITORIAL APPENDIX. 
 
 removing successive thin portions from the edge, or surface 
 of a board. In this case the resistance offered, and which is 
 sensibly in the same direction as the power applied, is that 
 arising from the cohesion of the fibres of the material, and 
 is measured by the pressure applied ; the path which the 
 point of application of the iron tool describes is the same as 
 that described by the hand ; and the work will be expressed 
 by the product of these two elements, each estimated in 
 terms of its own unit of measure. The case of the common 
 grindstone presents an example of a rather more complicated 
 character. Here the instrument to be ground is pressed 
 against the periphery of the stone with sufficient force to 
 cause a certain resistance to any power however applied to 
 put the stone in motion. The direction however in which 
 this resistance acts at the point of application is in the 
 direction of the tangent to the periphery at this point, and, 
 in one revolution of the stone, it will describe a path equal 
 in length to the circle described by the point of application. 
 The work therefore for each revolution will be the product 
 of the resistance, estimated in the direction of the tangent, 
 and the circumference described by the point of application. 
 It may be as well to remark, in this place, that although 
 the work done in overcoming the molecular resistances of 
 the materials by means of which the action of a force or 
 pressure is transmitted, as in the example above cited of a 
 flexible bar, is not taken into account in estimating the 
 external work, there are cases in which this work constitutes 
 the entire work done, and which again is reproduced in 
 external work ; as for example in the cases of the common 
 bow used for projecting arrows, and the springs by which 
 the machinery of some time-pieces is moved. In each of 
 these the resistance offered by the molecular forces of the 
 material is overcome by the action of some external force, 
 whose point of application is made to describe a given path ; 
 by this action a certain amount of work is expended in 
 bringing the spring to a certain degree of tension which, 
 when the force is withdrawn, will reproduce the same amount 
 of external work in an opposite direction to that in which the 
 force acted. 
 
 NOTE (&). 
 
 The work of a pressure of constant intensity acting in the 
 same direction as the path described by its point of applica- 
 tion may be represented by a geometrical diagram in the 
 
EDITORIAL APPENDIX. 
 
 583 
 
 same way as that used for representing the space described 
 by a body moving with a uniform velocity in any given 
 time ; by constructing a rectangle, one side of which repre- 
 sents the number of units of force, the other the number of 
 units of path ; the number of units of area of the rectangle 
 will express the number of units of work. 
 
 NOTE (I). 
 
 The method given (Art. 51) for estimating, by a geometri- 
 cal diagram, the work of a pressure which varies in inten- 
 sity at different points of the path described in its line of 
 direction by its point of application, finds its application and 
 has to be used whenever there is no geometrical law of con- 
 tinuity by which the pressure can be expressed in terms of 
 the path ; and, even when such a law obtains, it is some- 
 times found to be a more convenient method of obtaining an 
 approximate value of the amount of work than the more 
 rigorous one expressed by the formula 
 
 s 
 
 
 S, 
 
 in which U can be rigorously found whenever P, which being 
 a function of S can be expressed algebraically in terms 
 of it. 
 
 As an example of these two methods of estimating the 
 work of a variable pressure, acting^ in the 
 direction of the rectilinear path described by 
 its point of application, let the familiar case 
 of the action of steam on the piston of the 
 steam-engine be taken. 
 
 Let ABCD represent the steam-tight cy- 
 linder in which the piston is driven from the 
 position at #, at one end, to c at the other, in 
 the direction of the axis ac, of the cylinder, 
 by means of the pressure of the steam on the 
 end of the piston. Let us suppose that the 
 A ~~ ~~ B steam acts with a constant pressure, repre- 
 
 sented by Pj, whilst the piston is driven through the portion 
 ba of the path, and, having reached this point, the commu- 
 nication between the cylinder and the boiler being then cut 
 
584: EDITOEIAL APPENDIX. 
 
 off, that the steam already admitted acts, through the re- 
 mainder of the path described by the piston, by what is 
 termed its expansive force, in which the pressure continually 
 decreases as the piston approaches the point c. Let us sup- 
 pose that the law of variation of this pressure on the piston 
 at different points is such that the pressures at any two 
 points are inversely proportional to their distances from the 
 point a. P, then denoting the pressure when the piston is 
 at &, let P denote the pressure when it has reached another 
 point o at a distance S from #, and S 3 and S x denote the 
 lengths ac and ab, then according to the above law there 
 obtains 
 
 P> : P :: S : S 1? therefore P=P X ^. 
 
 Let the elementary portion of the path be denoted by <#S, 
 then by multiplying the variable force by the elementary 
 path there obtains 
 
 P^S=P a |^S, 
 
 b 
 
 which may be termed the elementary work, or in other 
 words, the work done whilst the variable pressure acts 
 through the elementary path, during which period the vari- 
 able pressure may be regarded as constant. 
 
 To obtain the total work whilst the variable pressure acts 
 from I to Cj or through the path S 2 S 1? there obtains 
 
 S, 8, 
 
 . 8 s,-io g . s,). 
 
 s, s, k 
 
 If instead of the exact work due to the expansive force of 
 the steam, and which is given by the foregoing formula, an 
 approximate value only was required, it could be obtained by 
 a geometrical diagram as follows. 
 Having set off to any scale a num- 
 ber of units representing the path 
 fo, calculate the pressures at the 
 points , <?, and at the middle point 
 0, for a first approximation. That 
 at l> will be simply P 1 ; that at c, 
 
 P,, and that at o, P, S > ' 
 
 2 S,) 
 
 TO. 
 
 X 
 
 T-- 
 
 or 
 
EDITOKIAL APPENDIX. 585 
 
 Having drawn perpendiculars to ~bc at &, <?, and <?, set off on 
 them the distances m, on and cp respectively equal to the 
 corresponding pressures, estimated in terms of the unit of 
 pressure, and according to any given scale. Join mn and 
 np ; the number of units of area, in the figure thus formed, 
 estimated in units of path and pressure, will be an approxi- 
 mate value of the required number of units of work. 
 
 The greater the number of parts into which T>c is divided and 
 the corresponding pressures calculated, the nearer will the 
 enclosed area approach to the true value of the work. 
 
 The mean pressure, or. that force which, acting with a 
 constant intensity along the same path as that described by 
 the point of application of the variable pressure, would give 
 the same work, is found either by dividing the result of the 
 integration by 838^ or by dividing the area in tlie last 
 method by ~bc. 
 
 NOTE (m). 
 
 As an example of the manner of obtaining the work done 
 ^. A.. by a constant pressure acting always 
 
 /''' ^\ in parallel directions whilst its inclina- 
 
 p ^ tion to the path described by its point 
 
 of application is continually varying, 
 let the well known mechanism of the 
 crank arm and connecting rod be taken. 
 Let O be the centre around which the 
 crank arm is made to revolve, by the 
 application of a constant pressure P x , 
 transmitted through a connecting rod 
 
 CD, all of whose positions during the motion are parallel to 
 the diameter AB. The path described by the point of ap- 
 plication C will be the circumference of which OC is the 
 radius, and the inclination of P a to this path will be the 
 variable angle DCN, between its direction and the tangent 
 to the circle at C, of which the variable angle AOC, that 
 measures the inclination of the crank arm to the diameter 
 AB, is the complement. Denote this last angle by a, and 
 the length of the crank arm OC by ~b. Now decomposing 
 P! into components in the direction of the tangent CN and 
 the radius OC, we obtain for the first Pj sin. a, and for the 
 second P 1 cos. a, of which P a sin. a is alone effective to pro- 
 duce work, since Pj cos. a acts constantly towards the fixed 
 point O without describing any path in the direction of its 
 
 f/ 
 
586 EDITORIAL APPENDIX. 
 
 action. But the elementary path described by the point 
 of application is evidently Ida, the infinitely small arc Cn of 
 the circle. The elementary work of the variable component 
 P, sin. a will therefore be expressed by 
 
 Pj sin. a x Ma. 
 
 The total work for any portion of the path, as AC, will 
 therefore be 
 .a 
 
 \ sin. a bda=T > J>(lco8. a)=PJ) ver. sin. a. 
 o 
 
 and for O=TT, it becomes 
 
 P a x25, orP,xAB; 
 
 a result which might have been foreseen, since AB is the 
 path described by the point of application of P, in its line 
 of direction, whilst the actual path is the semi-circumfe- 
 rence ACB. 
 
 As Cn=bda, if through n a perpendicular nm is drawn to 
 CD, the line of direction of P 1} the distance Cm is evidently 
 the projection of the elementary path actually described on 
 the line of direction of P,, and is therefore the corresponding 
 elementary path of P x in its line of direction ; but Cm=C? 
 sin. a=bda sin. a. Denoting AB by A, then Cm=dh ; and 
 there obtains 
 
 dh=bda sin. TT ; and P 1 dh=P 1 Ida sin. a ; 
 and 
 
 r 
 
 J 
 
 A result the same as is shown to obtain by the preceding 
 proposition. 
 
 To find the mean, or constant pressure which, acting in 
 the direction of the circular path, would produce the same 
 amount of work as the variable force does in acting through 
 the semi-circumference ; call Q this mean force, its path 
 being TT&, its work will be Q x tib ; and as this is to be equal 
 to the work of P t sin. a, there obtains 
 
 Q x n^=P l x 2ft, hence Q^P, - =0*6366 P, nearly, 
 for the value of the force. 
 
EDITORIAL APPENDIX. 587 
 
 It may be well to observe here that the mean pressures 
 have no farther relations to the actual pressures than as 
 numerical results which are frequently used instead of the 
 actual pressures to facilitate calculations; and also as a 
 means of comparing results, or work actually obtained from 
 a force of variable intensity, at different epochs of its action, 
 with what would have been yielded at the same epochs by 
 the equivalent mean force. 
 
 To show the manner of making the comparison in this 
 case, let us take the two expressions for the quantity of work 
 due the mean force, and also to the variable component, for 
 a portion of the path corresponding to any angle a. Since 
 
 2 
 Q^Pj-, its work corresponding to a will be 
 
 The corresponding work of the variable component Y l sin. a 
 will be 
 
 P^CL-cos. a). 
 
 The difference therefore between these two amounts of work 
 will be 
 
 . a). 
 
 Now this difference will be for the following values of o, 
 
 a=0, a= -, and a=7r. 
 
 2 
 
 The maximum value of this difference can be found by the 
 usual method of differentiation and placing the first differ- 
 ential coefficient equal to 0. Performing this operation, 
 there obtains 
 
 sin. a =-= 0-6366; 
 
 <r 
 
 the corresponding values of a being respectively 
 a 0-21964 ?r,and a = TT 0-21964 n. 
 
 Substituting these values of a and the corresponding values 
 of cos. a in the preceding expression for the difference there 
 obtains, for the first, 
 
588 EDITORIAL APPENDIX. 
 
 PJ, (^_l + cos. a) =?$ (2x0-21964-1+ 
 
 + 0-21039^6; 
 and for the second, 
 
 PJ> (^-1-f-cos. a)=PJ> (2x0-21964-1-1/1-1) = 
 - 0-21039 PA 
 
 From these two expressions it is seen that the greatest excess 
 of the work of the mean force over that of the other would 
 be + 0-21039 P 1 &=+ 0-1052 xP,2J, or about T V of the total 
 work of Pj corresponding to the path 25 ; whilst that of the 
 work of P, over the mean force, represented by 0'21039P 1 , 
 is the same in amount. 
 
 If now we suppose the direction of the constant force P a 
 to be changed, when its point of application reaches the point 
 B, so as to act parallel to the direction BA until the point of 
 application arrives at A, it is clear that the work r of P^ due 
 to the path described from B to A will also be expressed by 
 P t x 25, so that the work due to an entire revolution of the 
 point of application will be P t x 45. As the mean force will 
 evidently be the same for the entire revolution of the point 
 of application, it follows that the greatest positive, or nega- 
 tive excess, as stated above, will be 0-0526 xPj4J, or 2 1 of 
 the work for one entire revolution. 
 
 It is thus seen that although the work of the effective 
 variable component P, sin. a of r l is not, like that of the mean 
 force, uniform for equal paths, still it at no time falls short 
 of nor exceeds the work of the mean force by more than 
 about V of the entire work for each revolution. Were any 
 mechanism, as that for pumping water for example, so 
 arranged that either the constant force P 15 or a mean force 
 equal to 0*6366 P 15 acting as above described, were applied 
 to it, the quantity of water delivered by the one would at no 
 time exceed, in any one revolution, that delivered by the 
 other by more than V of the total quantity delivered by 
 either during the entire revolution. 
 
 NOTE (n). 
 
 If P 2 , for example, were the resultant of the other pres- 
 sures, its component P 3 cos. a a would be equal to the alge- 
 
EDITORIAL APPENDIX. 589 
 
 braic sum of the components P! cos. a l5 P 3 cos. 3 , &c., of the 
 other pressures P 15 P 3 , &c. ; the work therefore of P a , esti- 
 mated in the direction of the given path AB, and corres- 
 ponding to any portion of this path, will be equal to the 
 algebraic sum of the work of the other pressures P M P 3 , &c., 
 corresponding to the same portion of the given path. 
 
 NOTE (0). 
 
 Since at the point E, taken as the point of application, the 
 line of direction of the pressure becomes a tangent to the 
 arc described with the radius OE, it follows that the infi- 
 nitely small arc described with the radius OE may be taken 
 for the infinitely small path described by the point of appli- 
 cation in the direction of the tangent. Denoting by da the 
 infinitely small angle described by the radius OE, then 
 OE x da will express the infinitely small path, or arc ; and 
 P x OE<^a will represent the elementary work of the 
 pressure. ' 
 
 If the pressure remains constant in intensity and direction 
 during an entire revolution of the body about 0, then will 
 the work of P for this revolution be represented by 
 P x circum. OE. 
 
 (p). 
 
 The term living force is more generally used with us by 
 writers on mechanics instead of its Latin equivalent vis vwa, 
 to designate the numerical result arising from multiplying 
 the quantity denominated the mass of a body by the square 
 of the velocity with which the body is moving at any 
 instant. It will be readily seen that this product does not 
 represent a pressure, or force, but the numerical equivalent 
 of the product of a certain number of units of pressure and a 
 certain number of units of path. The one magnitude being 
 of as totally a distinct order from the other as an area is 
 different from a line, and therefore having no common unit 
 of measure. 
 
 Besides this expression, which serves no other really use- 
 ful purposes than as a name to designate a certain numerical 
 magnitude which is of constant occurrence in the subject of 
 mechanics, there is another also of frequent use, termed 
 
590 EDITORIAL APPENDIX. 
 
 quantity of motion, wnich is the product of the mass and 
 
 w 
 
 the velocity, or v. This is also termed the dynamical 
 
 measure of a force in contradistinction to pressure, as usually 
 estimated, which is termed the statical measure of a force. 
 
 In estimating the accumulated work in the pieces of a 
 machine which have either a continuous or a reciprocating 
 motion of rotation it is necessary to find expressions for the 
 moments of inertia of these pieces with respect to their axis 
 of rotation, and this may, in all cases, be done, within a cer- 
 tain degree of approximation to the true valu, by calculat- 
 ing separately the moment of inertia of each of the compo- 
 nent parts of each piece and taking their sum for its total 
 moment of inertia, on the principle that these moments may 
 be added to or subtracted from each other in a manner 
 similar to that in which volumes, or areas are found from 
 their component parts. 
 
 In making these approximate calculations, which in many 
 cases are intricate and tedious, it will be well to keep in view 
 the two or three leading points following, with the examples 
 given in illustration ol some of the more usual forms of 
 rotating pieces. 
 
 1st. The general form for the moment of inertia of a body 
 rotating around an axis parallel to the one passing through 
 its centre of gravity as given in equation 58, (Art. 79) is 
 
 Now if the distances of the extreme elements of the body 
 from the axis passing through its centre of gravity are small 
 compared with that of A, the distance between the two axes, 
 the second term I of the second member of this equation 
 may be neglected with respect to the first, and A 2 M be taken 
 as the approximate value of the required moment. This 
 consideration will find its application in many of the cases 
 referred to, as, for example, in that of finding the moment of 
 inertia of the portion of a solid, like the exterior flanch of the 
 beam of a steam-engine, the volume of which may be approx- 
 imately obtained by the method of Guldinus (Art. 39.). In 
 this case, A representing the area of the cross section of the 
 
EDITORIAL APPENDIX. 591 
 
 flanch, and s the path which its centre of gravity would 
 describe in moving parallel to itself in the direction of the 
 flanch around the beam, any elementary volume of the 
 flanch between two parallel planes of section will be ex- 
 pressed by Ads. Now the moment of inertia of this elemen- 
 tary volume from equation 58 is 
 
 in which the first term of the second member, which 
 expresses the sum of the elementary volumes Ads into the 
 squares of their respective distances r from the axis of rota- 
 tion, may be taken as the approximate value required ; inas- 
 much as I, the sum of their moments of inertia with respect 
 to the parallel axes through their centres of gravity, may be 
 neglected witn respect to the first term. The problem will 
 therefore reduce to finding the moment of inertia of the line 
 represented by s, which would be described by the centre of 
 gravity of A, with respect to the assumed axis of rotation, 
 and then multiplying the result by A. 
 
 2nd. As the line s is generally contained in a plane per- 
 pendicular to the axis of rotation, and is given in kind, as 
 well as in position with respect to this axis, being also gene- 
 rally symmetrically placed with respect to it, its required 
 moment of inertia may, in most cases, be most readily 
 obtained by finding the moment of inertia of s separately, 
 with respect to two rectangular axes contained in its plane, 
 and taken through the point in which the given axis of rotation 
 pierces this plane, and then adding these two moments. 
 
 The moment of inertia of a line taken in this way with 
 respect to a point in its plane has been called by some 
 writers the polar moment of inertia. 
 
 This method is also equally applicable to finding the mo- 
 ment of inertia of a plane thin disk revolving around an axis 
 perpendicular to its plane, and to solids which can be divided 
 into equal laminae by planes passed perpendicular to the axis 
 of rotation. 
 
 (a 1 ) The moment of inertia of the arc of a parabola with 
 respect to an axis perpendicular to the plane of the curve at 
 a given point on the axis of the curve. 
 
 Let BAG be the given arc ; A the vertex of the parabola 
 
592 
 
 EDITORIAL APPENDIX. 
 
 R the point on its axis at which the 
 axis is taken. Through R draw 
 the chord PQ. Represent the 
 D chord BC of the given arc by 5; 
 its corresponding abscissa AD by a\ 
 and AR by c. Let y represent the 
 ordinate pq, and x the correspond- 
 
 ing abscissa of any element dz of the arc. 
 
 From the preceding remarks, the moment of inertia of dz 
 
 with respect to the axis AD will be expressed by y*dz ; and 
 
 that of the entire arc BAG by 
 
 as from the equation of the parabola, y*= ^ x. 
 By integration f 
 
 in which Z is the length of the arc BAG. 
 
 In like manner the moment of inertia of dz with respect 
 to the chord PQ is 
 
 and for the entire arc BAG, 
 
 i& 
 
 *z 
 
 = 2 ^-f A = .- 
 
 
 
 which integrated as above, 
 
 
 6l?^ 9 / ^ Z U 
 
 * Church's Int. Cal. Art. 199. 
 
 f Ibid. Art. 150. 
 
EDITORIAL APPENDIX. 593 
 
 From the preceding remarks, the moment of inertia of Z, 
 with respect to the axis at the point E perpendicular to the 
 plane of Z, is 
 
 32 
 
 \ 
 
 / " 
 
 The value of Z in the above expressions is 
 
 Each of the preceding expressions may be simplified, and 
 an approximate value obtained, sufficiently near for practical 
 applications, when the ratios of I and c to a are given* For 
 example, when b /_ \a there obtains 
 
 the terms omitted, being small fractions with respect to. 
 unity, do not materially affect the result. 
 
 Having found the moment of inertia of a parabolic curve, . 
 that of a parabolic ring of uniform cross section, taken per- 
 pendicular to the direction of the curve at any point, and 
 having its centre of gravity at its point of intersection with 
 the curve, can be obtained by simply multiplying Ij+I, by 
 S the area of the given cross section. 
 
 (A 1 ) The moment of inertia of the segment of a parabola with 
 respect to an axis perpendicular to its plane at a given 
 point of the axis of the curve. 
 
 Let BAG be the given segment ; A the vertex ; AD the 
 
 * Church's Int. Cal. Art. 199. 
 38 
 
EDITORIAL APPENDIX. 
 
 axis of the curve ; and D the point on 
 the axis with respect to which the mo- 
 ments of inertia are estimated. Denote 
 the chord BC by 5; the abscissa AD 
 by a. 
 
 By (Art. 81) the moment of inertia of 
 an elementary area pq with respect to AD is T ^ (pq)*dx= 
 ' That of the segment therefore will be 
 
 ~ 
 
 In like manner the moment of inertia of an elementary 
 area asps, with respect to the axis BC, is -J (ps) 3 dy= -J(a a?) 8 
 dy. That of the segment therefore will be 
 
 16 *& 
 
 = f f 
 
 (a - 
 
 From this last expression we readily obtain the moment 
 of inertia of a disk having the segment for its base and its 
 thickness represented by c, with respect to an axis at D per- 
 pendicular to its base by simply multiplying I x + 1 2 by c ; 
 
 ft + 1.) c = A fo + T% 0'fo = I ^ 
 in which f Zc = Y, the volume of the disk. 
 
 or 
 
 l ) The moment of inertia of a parabolic disk, or prism,, 
 with respect to an axis parallel to the chords which termi- 
 nate the upper and lower bases and midway between the 
 chords. 
 
 Let pq be an elementary volume of the disk contained 
 between two planes parallel to the base 
 BC of the disk. Adopting the same 
 notation as in the preceding article, the 
 volume pq is expressed by 
 
 2t/ . c . dx. 
 
 The moment of inertia of this elemen- 
 tary volume with respect to an axis 
 through its centre of gravity and parallel 
 to BC is (Art, 83) 
 
EDITORIAL APPENDIX. 
 
 W . 2v . c . dx 
 
 595 
 
 and the moment of inertia of the same volume with respect 
 to the axis, parallel to the one through its centre of gravity, 
 taken on the base BC of the disk and midway between the 
 upper and lower chords is (Art. 79 Eq. 58) 
 
 . c . dx 
 
 . c . dx (axf ; 
 
 the moment of inertia of the entire disk with respect to the 
 same axis is 
 
 .-. I = T v C 2y . G . dx {c* + (dx)*\ + f Zy . c . dx (ax)\ 
 
 Substituting for x and dx in terms of y, omitting the term 
 containing (dx) 3 , and integrating as indicated, there obtains, 
 
 a 
 
 in which V= 
 
 (d r ) The moments of inertia of a right prism with a trape* 
 zoidal "base with respect to axes perpendicular and parallel 
 to the base at the middle point oj the face terminated by the 
 broader side of the trapezoid. 
 
 Let AG-HC be the trapezoid forming the base of the prism. 
 Represent the altitude EF of the trape- 
 zoid by a ; AG by I ; CH by b 1 ; and the 
 height GB of the prism by c. Let ^ be an 
 elementary volume of the prism between 
 two planes parallel to the face AB and 
 at a distance Ee=x from the face CD. 
 From C drawing Cc parallel to HG there 
 obtains 
 
 A /I 
 
 pr= ==- . Ac = - (b 
 
 EF 
 
 a 
 
 =pr 
 
 V. 
 
596 EDITORIAL APPENDIX. 
 
 The elementary volume pq is therefore 
 
 The moment of inertia of py with respect to an axis through 
 its centre of gravity and perpendicular to the base of the 
 prism is (Art. 83). 
 
 
 and that of the entire prism with respect to an axis at F, the 
 middle point of AGr, and parallel to the preceding axis, is 
 
 = A f | - Q-V)+V [ c. 
 
 / I # 
 
 omitting the term containing (dx}\ and integrating, as indi- 
 cated, there obtains 
 
 in which F= 
 
 By a like series of operations the moment of inertia of the 
 entire prism, with respect to an axis perpendicular to the 
 preceding one at its middle point between the upper and 
 lower bases of the prism, will be 
 
EDITORIAL APPENDIX. 
 
 597 
 
 (e) The moment of inertia of a right prismoid with rectan- 
 gular bases with respect to an axis XY through the centre 
 o gravity of the tower base and parallel to one of its 
 sides. 
 
 Let AB = J, BC = c be the sides of the rectangle of the 
 lower base ; ab = b\ be = c l the sides 
 of the upper base. Let pqrs be any 
 section of the prismoid parallel to 
 the lower base and at a distance x 
 from it* and let a be the altitude of 
 the prismoid, or the distance between 
 its upper and lower bases. 
 
 From the relations between the 
 dimensions of the prismoid there ob- 
 tains (Art. f ') 
 
 a 
 
 x. x(c c l ) + ac l 
 2r=-(c-c l )+c l =- ^ ; 
 
 and to express the elementary solid contained between two 
 planes parallel to the base of the prismoid and at the height 
 x above it, 
 
 x(b b l ) + ab 1 x(c c l ) + ac 1 
 
 x * - . dx. 
 
 The moment of inertia of this solid, with respect to an 
 axis xy through its centre of gravity and parallel to XY, is 
 (Art. 83) 
 
 x (b b 1 ) + afr x(c c l }+atf 
 
 -- -* 
 
 
 The moment of inertia of the prismoid (Art. 79 Eq. 58) is 
 
 \ (*c-^+ 
 
593 EDITORIAL APPENDIX. 
 
 o 
 
 fxg>- 
 J a 
 
 omitting the term containing (d)* and integrating as indi 
 cated, there obtains, 
 
 By integrating the expression for the elementary volume 
 between the same limits, there obtains to express the volume 
 of the prisinoid 
 
 which is the formula usually given in mensuration. 
 
 In each of the preceding examples, the quantities I, I 15 &c. 
 are expressed only in terms of certain linear dimensions ; to 
 obtain therefore the moments of inertia proper these results 
 
 must be multiplied by the quantity -, or the unit of mass 
 
 corresponding to the unit of volume, in which ^ represents 
 the weight of the unit of volume of the material and 
 g = 32 feet. 
 
 Each of the above values of I may be placed under more 
 simple forms for the greater readiness of numerical calcula- 
 tion by throwing out such terms as will visibly affect the 
 result in only a slight degree. But as such omissions depend 
 upon the numerical relations of the linear dimensions of the 
 parts no rule for making them can be laid down which will 
 be applicable to all cases. . 
 
 (/"') The moment of inertia of a trip hammer. 
 
 These hammers consist of a head of iron of which A repre- 
 sents a side and A' a 
 front elevation; of a 
 handle of wood B, 
 which is either of the 
 shape of a rectangular 
 parallelepiped, or of 
 
EDITORIAL APPENDIX. 599 
 
 
 
 two rectangular prismoids, having a common base at the 
 axis of rotation C where the trunnions, upon which the 
 hammer revolves, are connected firmly with the handle by 
 an iron collar. Another iron collar is placed at the end of 
 the handle, and is acted on by that piece of the mechanism 
 which causes the hammer to rotate. 
 
 To obtain the moment of inertia of the whole, that of each 
 part with respect to the axis is separately estimated and 
 the sum then taken. 
 
 The head A, A' may be regarded as a parallelopiped of 
 which the side A', reduced to its equivalent rectangle by 
 drawing two lines parallel to the vertical line that bisects 
 the figure, is the end, and the breadth of the side A is the 
 length. If then from the moment of inertia of this parallel- 
 opiped that of the void a, or eye of the -hammer, which is 
 also a parallelopiped, be taken, the difference will be the 
 moment of inertia of the solid portion of the head. The 
 moments of inertia of these parallelepipeds may be calcu- 
 lated, with respect to the axis C, by first estimating them 
 with respect to the axes through their respective centres of 
 gravities G and ^, parallel to C, by (Art. 83) and then witli 
 respect to C by (Art. 79. Eq. 58). Or if the moments of 
 inertia with respect to G and g are small with respect to the 
 product of their volumes and the squares of the distances 
 GO and ^C, then the difference of the latter products may 
 be taken as the approximate value. 
 
 The moment of inertia of the handle, if also a parallelo- 
 piped, will be found with respect to C by (Arts. 79, 83). If 
 it is composed of two rectangular prismoids, then the mo- 
 ment of the parts on each side of the axis must be found by 
 (') and their sum taken. 
 
 The moment of inertia of the trunnions and the iron hoop 
 to which they are attached may be found by (Arts. 85, 87) 
 and their sum taken. But as this quantity will be generally 
 small with respect to the others it may be omitted. 
 
 That of the hoop at the end of the handle may be taken 
 approximately as equal to the product of its volume and the 
 square of the distance between the axis through its centre 
 of gravity and that of rotation. 
 
 (g f ) The moment of inertia of a cast iron wheel. 
 These wheels usually consist of an exterior rim A A' of 
 
600 
 
 EDITORIAL APPENDIX. 
 
 uniform cross section con- 
 nected with the boss, or nave 
 C, C', which is a hollow 
 cylinder, by radial pieces, or 
 arms B, B', the cross section 
 of which is in the form of a 
 cross. Each arm having the 
 same breadths at top and 
 bottom in the direction of 
 the axis of the wheel as those of the rim and nave which it 
 connects; the thickness perpendicular to the axis being 
 uniform. The projection or ribs on the side of each arm, 
 and which give the cross form to the section, being of uni- 
 form breadth and thickness ; or else of uniform thickness 
 but tapering in -breadth from the nave to the rim. These 
 ribs join another of the same thickness that projects from the 
 inner surface of the rim. 
 
 Eepresent by E the mean radius of the rim, estimated from 
 the axis to the centre of gravity of its cross section ; J its 
 breadth, and d its mean thickness ; V its volume, and I its 
 moment of inertia with respect to the axis; ^ the weight of 
 its unit of volume, and y=3%% feet ; then by (Art. 86) 
 
 V=totW>d andl=^ FE 2 , 
 V 
 
 omitting J^ 2 as but a small fractional part of E 2 . 
 
 Eepresenting by b l the breadth of the arm at the axis, 
 supposing it prolonged to this line ; J 2 its breadth at the rim, 
 supposing it prolonged also to the mean circle of the rim, d 1 
 its thickness ; V. its volume ; I, its moment of inertia, then 
 
 2 
 
 Eepresenting by a^ the breadth at bottom, 2 the breadth at 
 top of the ribs, or projections on the sides of each arm, esti- 
 mated also at the axis and mean circle of the rim ; d^ their 
 thickness ; F 2 their volume ; I 2 their moment of inertia ; 
 then by (a') 
 
 F.=B* ?dp and !.= 
 
 The sum I +1,4-1, will be the moment of inertia of the 
 
EDITORIAL APPENDIX. 601 
 
 entire wheel approximately, since the moment of inertia of 
 the portions of the boss between the arms is omitted, this 
 being compensated for by supposing the arms prolonged to 
 the axis and to the mean circle of the rim. As the quanti- 
 ties Fj, F" 2 , Ij and I 2 are taken but for one arm, they must 
 be multiplied by the number of arms to have the entire 
 moment. 
 
 (h f ) The moment of inertia of a cast iron steam engine beam. 
 
 These beams usually consist of two equal arms symmetri- 
 cal with respect to a 
 line a a' through the 
 axis of rotation o. 
 Each arm, a V a' and 
 a 1) a', consists of a 
 parabolic disk of uniform thickness ; b and V being the ver- 
 tices of the exterior bounding curves, a a' their common 
 chord, and ob, ob' their axes. The disk is terminated on the 
 exterior by a flanch B of uniform breadth and thickness. A 
 rib C, either of uniform breadth and thickness, or else of 
 uniform thickness, and tapering in breadth from the centre o 
 to the ends &, 5', projects from each face of the plane disk 
 along the axis b b'. The beam is perforated at the centre, 
 near the two extremities and at intermediate points, to 
 receive the short shafts, or centres around which rotation 
 takes place. Around each of these perforations, projections, 
 or bosses D', D", &c., are' cast, to add strength and give a 
 more secure fastening for the shafts. 
 
 The beam being symmetrical with respect to a a', it will 
 be only necessary to calculate the moments of inertia of the 
 component parts of each arm with respect to the axis o and 
 take double their sum for the total moment of inertia of the 
 beam. These component parts are 1st, the parabolic flanch ; 
 2nd, the parabolic disk of uniform thickness enclosed by the 
 flanch ; 3d, the rib on each side of the disk, running along 
 the centralline bb f ; 4th, the projections, or bosses D' &c., 
 around the centres. 
 
 The moment of inertia of the flanch will be calculated by 
 (') as its thickness is small compared with the other linear 
 dimensions. That of the disk will be calculated by (&'). 
 That of the rib by (d f ). Those of the projections may be 
 obtained within a sufficient degree of approximation by 
 
602 EDITORIAL APPENDIX. 
 
 taking the product of their volumes and the squares of their 
 respective distances from the axis o. 
 
 The sum of these quantities being taken it must be multi- 
 plied by - as in the preceding cases ; v> being the weight of 
 the unit of volume of the material. 
 
 NOTE (s). 
 
 The increase of tension due to rigidity and which ig ex- 
 pressed by - - - - may be placed under the following 
 
 form, 
 
 c m . a+c m . I . P a c m (a+l . F a ) 
 
 K R 
 
 by writing c m . a for D, and c m . ~b for E, in which c repre- 
 sents the circumference of the rope, and m the power to 
 which c is raised. 
 
 The increase of tension of any other rope whose circumfer- 
 ence is c l bent over the same pulley and subjected to the 
 same tension P a is, in like manner, expressed by 
 
 E 
 
 Now representing by T and T l the two values above for the 
 respective increase of tension for c and c l there obtains, by 
 dividing the one by the other, 
 
 which expresses the rule given above for using the tables in 
 calculating the increase of rigidity due to a cord whose cir- 
 cumference is different from those in the tables. 
 
 NOTE (t). 
 
 As one of the chief ends of every machine designed for 
 industrial purposes is, under certain restrictions as to the 
 
EDITORIAL APPENDIX. 603 
 
 quality, to yield the greatest amount of its products for the 
 motive power consumed, it becomes a subject of prime 
 importance to see clearly in what way the work yielded by 
 the motive power to the receiver, at its applied point, is 
 diminished by the various prejudicial resistances, in its 
 transmission through the material elements of the machine 
 to the operator, or tool by which the products in question 
 are formed. 
 
 The most convenient method for doing this will be to 
 place (equation 112, Art. 145) which expresses the relation 
 between the work 2U 1 of the motive power at the applied 
 point and that 2U 3 the work of the operator at the working 
 
 point, with the portion 2U + - - 2w (v*v*) which repre- 
 sents the work consumed by the prejudicial resistances and 
 the inertia, under a form such that the work of each preju- 
 dicial resistance shall be separately exhibited, for the pur- 
 pose of deducing, from this new form of the equation, the 
 influence which each of these has in diminishing the work 
 yielded at the applied point and transmitted to the operator. 
 To effect this change of form in (equation 112) designate by 
 P! the motive power, and S x the path passed over by its 
 point of application in its line of direction between any two 
 intervals of time, during which Pj may be regarded as vari- 
 able both in intensity and direction ; P 2 and S a the resistance 
 and corresponding path at the working point ; R the various 
 prejudicial resistances which, like friction, the stiffness of 
 cordage, &c., act with a constant intensity, or are propor- 
 tional to P 1? and S their path ; w l the weight of the parts the 
 centre of gravity of which has changed its level during the 
 
 period considered, and H its path ; and w (v.?v*)= %m 
 
 2<7 
 
 (v^v^) the half of the difference between the living forces 
 or the accumulated work of the material elements in motion, 
 
 of which m - - is the mass, during the same period, in 
 
 which the velocity has changed from v 1 to v v 
 
 Now for an elementary period dt of time, during which 
 the forces P, &c., may be regarded as constant, and their 
 points of application to have described the elementary paths 
 dS l &c., in their lines of direction, (equation 112) will take 
 the form, 
 
 h, . . . (A), 
 
604: EDITORIAL APPENDIX. 
 
 in which the 1st member of the equation expresses the incre- 
 ment of the living force, or the elementary accumulated 
 work for the interval dt at any instant when the velocity of 
 the mass m, is v\ and the 2nd member the corresponding 
 algebraic sum of the elementary work of P n R, &c. This 
 equation being integrated between the limits ^ and 2 in 
 which v changes from v l to v^ there obtains, 
 
 2 lm - O = 
 
 2 fw.dh ..... (B). 
 
 This equation (B) is the same as (equation 112). The 
 symbol 2 designating the aggregate of the work of the 
 
 various forces of the same kind ; and that as / P^Sj &c. 
 
 the work of each force as P,, supposing it to be either con- 
 stant or variable. In either case whenever P x &c., can be 
 
 expressed in terms of B! the value of / Y 1 dS 1 can be found 
 
 by one of the methods in (Notes I and m) ; and supposing P 1 
 &c., to represent their mean values, and S x &c., the paths 
 described in their true directions during the interval con- 
 sidered, equation (B) may be written under the following form 
 for the convenience of discussion, 
 
 . . . .(0). 
 
 In this last equation 2 / w l dh = WH (Art. 60) represents 
 
 the work of the total weight of the parts whose centre of 
 gravity has changed its level during the interval considered, 
 and it takes the double sign , as the path H may be 
 described either in the same, or a contrary direction to that 
 in which W always acts. 
 
 Before proceeding to discuss the terms of (equation C), 
 it may be well to remark that the term RS does not take 
 into account the work expended by P, in overcoming the 
 molecular forces brought into play by the deflection, torsion, 
 extension, &c., of the parts of the machine ; for, owing to 
 the rigidity of these parts, this forms but a very small .frac- 
 tional part of the total work of the exterior forces whilst the 
 machine operates continuously for some time; as, during 
 
EDITORIAL APPENDIX. 605 
 
 this time, the tension of the parts, or the molecular resist 
 ances remain sensibly the same, and the molecular displace 
 ments are for the most part inappreciable, or else very small 
 compared with the paths described by the points of applica- 
 tion of the other forces. 
 
 This remark, however, does not apply to the expenditure 
 of work by the motive power where the operation of the 
 machine requires that some of the parts in motion shall be 
 brought into contact with others which are either at rest, or 
 moving with a slower velocity so as to produce a shock. 
 In this case there may be a very appreciable amount of 
 living force, or accumulated work destroyed by the shook, 
 owing to the constitution of the material of which the parts 
 are composed where the shock takes place ; and, if the shocks 
 are frequent during the interval considered, and in which 
 the other forces continue to act, the accumulated work 
 destroyed during this interval may form a large portion of 
 the work expended, or to be supplied by the motive power. 
 In calculating this amount of accumulated work destroyed, 
 we admit what is in fact frue in such machines, that the 
 interval in which the shock takes place is infinitely small 
 compared with the interval in which the other forces act 
 continuously, and therefore, in estimating the accumulated 
 work destroyed in each shock, that we can leave out of 
 account the work of the other forces during this infinitely 
 small interval. In this way, considering also that the parts 
 where the shock takes place are usually formed of materials 
 which undergo an almost inappreciable change of form from 
 the shock, and that therefore the mechanical combinations 
 of the machine are sensibly the same after the shock as 
 before it, we readily see that, to obtain the total expenditure 
 of work by the motive power, for any finite interval, we must 
 calculate that which is consumed by all the other resistances 
 during this interval, and add to this that destroyed by the 
 shocks during the same interval, the latter being calculated 
 irrespective of the work of the other forces during the short 
 duration in which each shock occurs. 
 
 We thus see that, except in some cases where the great 
 velocity of the parts in motion may give rise to an appreci- 
 able expenditure of work caused by the resistance of the 
 medium in which these parts may be moving, as the air, &c., 
 the forces which act upon any machine in motion are the 
 motive power ; the resistances, such as friction, stiffness of 
 cordage, &c., which act either with a constant intensity 
 during the motion, or are proportional to the motive power , 
 
606 EDITORIAL APPENDIX. 
 
 the weight of the parts whose centres of gravity do not remain 
 on the same level during this interval ; the useful^ resistance 
 arising from the mechanical functions the machine is designed 
 for ; and the forces of inertia which either give rise to accu- 
 mulated work, or the reverse, as the velocity increases, or 
 decreases during the interval considered. 
 
 Eesuming equation (C) we obtain, by transposition, 
 
 P&=P& I^WH-Jwv.' + iw^ 1 . 
 
 That is the useful work, or that yielded at the working point 
 and which it is generally the object of the machine to make 
 as great as possible consistently with the quality of the 
 required products, will be the greater as the terms in the 
 second member of the equation affected with the negative 
 sign are the smaller. 
 
 Taking the term ES, it is apparent that all that can be 
 done is to endeavor in the case of each machine to give 
 such forms, dimensions and velocities to those parts where 
 these resistances are developed as will make it the least 
 possible. 
 
 With respect to WH it will entirely disappear from the 
 equation when H=o ; in which case the centre of gravity of 
 the entire system will remain at the same level ; or else 
 only that portion of this term will disappear which belongs 
 to those parts of the machine whose centres of gravity either 
 remain at rest, as in the case of wheels exactly centered, end- 
 less bands and chains, &c. ; or in the case of those pieces 
 which receive a motion simply in a horizontal direction. 
 This term will also disappear in whole, or in part, in those 
 cases where the centre of gravity ascends and descends 
 exactly the same vertical distance in the interval correspond- 
 ing to the work Pfi l ; for during the ascent, as the direction 
 of the path H is opposite to that of the weight "W", the work 
 consumed will be WH, whereas, in the descent, it will 
 restore the same amount or -f-WH, and the sum of the two 
 will therefore be 0. This takes places in the parts of many 
 machines, for example in crank arms, and in wheels which 
 are not, accurately centered; in both of which cases the 
 centre of gravity ascends and descends the same distance 
 vertically in the interval corresponding to each revolution 
 of these parts whilst in motion ; also in those parts of a ma- 
 chine, like the saw and its frame in the saw mill, which rise 
 and fall alternately the same distance. 
 
 In all of these cases then the useful work P 2 S a will not be 
 
EDITORIAL APPENDIX. 607 
 
 affected by the work due to the weight of the parts in 
 question. 
 
 It may be well to observe that the preceding remarks refer 
 only to the direct influence of the weight of the parts on the 
 amount of useful work ; but whilst directly it may produce 
 no effect however great its amount, the weight, indirectly, 
 may cause a considerable diminution of this work, by 
 increasing the passive resistances and thus the term US. 
 The same holds with regard to the accumulated work, repre- 
 sented by the term %mv*, from which a considerable dimi- 
 nution may be made in P 2 S 2 if this accumulated work cannot 
 be converted into useful work, and thus be made to form a 
 portion of P 2 S 2 , when the action of the motive power is either 
 withdrawn, or ceases, by variations in its intensity, to yield 
 an amount of work which shall suffice for the work consumed 
 by the resistances. 
 
 These last remarks naturally lead us to the consideration 
 of the two terms fawv*, and -~ jmi> 2 2 , or half the living forces. 
 or accumulated work at the commencement and end of the 
 interval considered. As the machine necessarily starts from 
 a state of rest under the action of the motive power P l5 it 
 follows that fynv*, the accumulated work due to this action 
 tends to increase P 2 S 2 , whilst that fynv* is so much accu- 
 mulated in the moving parts by which P 2 S 2 is lessened. 
 This diminution of P 2 S 2 is but inconsiderable in comparison 
 with the total useful work when the interval in question, and 
 during which the machine operates without intermission, is 
 great ; also in cases where the velocity attained by the parts 
 in motion is inconsiderable, as for example in machines em- 
 ployed for raising heavy weights, in which &nv* will in 
 most cases be but a small fraction of the useful work which 
 is the product of the weight raised and the vertical height it 
 passes through. In this last example we also see the incon- 
 veniences which would result from allowing bodies raised by 
 machinery to acquire any considerable amount of velocity ; 
 or to quit the machine with any acquired velocity, as, in 
 this case, the accumulated work generally would be entirely 
 lost so far as the required useful effect is concerned. 
 
 Except in the case where the accumulated work fynv* 
 can be usefully employed in continuing the motion of the 
 machine and gradually bringing it to a state of rest when the 
 motive power P, has either ceased to act, or has so far 
 decreased in intensity as to be incapable of overcoming the 
 resistances, whatever tends to any augmentation of living 
 force should be avoided, for the teim which represents this 
 
608 EDITORIAL APPENDIX. 
 
 being composed of two factors the one representing the mass 
 of the parts in motion and the other the square of its velo- 
 city, it is evident that the prejudicial resistances such as 
 friction on the one hand and the resistance of the air on the 
 other will increase as either of these factors is increased, and 
 thus a very appreciable amount of this accumulated work 
 may be consumed in useless work caused by the very in- 
 crease in question. If, moreover, the machine from the nature 
 of its operations is one that requires to be brought suddenly 
 to a state of rest, any considerable amount of accumulated 
 work might so increase the effects' of shocks at the points of 
 articulation as to endanger the safety of the parts. 
 
 The foregoing remarks apply only to those parts of a ma- 
 chine where the direction of motion remains the same whilst 
 the machine is in operation. Where any of the parts have 
 a reciprocating motion, in which case whilst the part is 
 moving in one direction the velocity increases from up to 
 a certain limit and then decreases until it again becomes 
 at the moment when the change in the direction of motion 
 takes place, and so on for each period of change, it will be 
 readily seen that where the velocity varies by insensible 
 degrees, the accumulated work of these parts for each period 
 of change will be and will therefore have no influence on 
 the amount P 2 S 2 of useful work. 
 
 The avoidance of abrupt changes of velocity in any of the 
 parts of a machine is of great importance. The mechanism 
 therefore should, as a general rule, be so contrived that there 
 shall be the least play possible at the articulations of the 
 various parts, and that the articulations shall receive such 
 forms as to procure a continuous motion. In cases also 
 where any of the parts have a reciprocating motion such 
 mechanical contrivances should be used as will cause the 
 variations of velocity in these parts, within the range of 
 their paths, to take place in a very gradual manner ; such 
 for examples as what obtains in the cranks and eccentrics 
 which are mostly employed to convert the continuous circu- 
 lar motion of one part into reciprocating motion in another, 
 or the reverse. 
 
 There are some industrial operations however which are 
 performed by shocks, as in stamping machines, trip ham- 
 mers, &c., and in these cases the useful work is due to the 
 work developed by the motive power in raising the pestle 
 of the stamping machine, or the head of the trip hammer 
 through a certain vertical distance from which it again falls 
 upon the matter to be acted on, having acquired in its 
 
EDITORIAL APPENDIX. 609 
 
 descent an amount of living force, or accumulated work due 
 to the height through which it has been raised. In such 
 cases it is to be noted that, independently of the work due 
 to the motive power consumed by the resistances whilst the 
 hammer or pestle is kept in motion by the other parts of the 
 mechanism, and which is so much uselessly consumed so for 
 as the useful work is concerned, there will be a portion of 
 the accumulated work in the pestle, or hammer also uselessly 
 consumed, arising from the want of perfect rigidity and 
 elasticity in the material of which these two pieces are 
 usually composed. Besides this, both the pestle and matter 
 acted on may and generally do have relative velocities after 
 the shock between them, which as they are foreign to the 
 purpose of the operation, will also represent an amount o* 
 accumulated work lost to the useful work. From this, w$- 
 may infer that, as a general rule, other industrial modes, of 
 operating a change of form in matter will be preferable; to- 
 those by shocks, whenever they can be employed ; and that 
 such modes are moreover advantageous, as they avoid those 
 jars to the entire mechanism which accompany abrupt 
 changes in the velocity of any of the parts, and which, by 
 loosening the articulations more and more, increase the evil, 
 and ultimately render the machine unfit for service. 
 
 Having examined the influence of all the various hurtful 
 resistances brought into action in the motion of machines 
 upon the work PjS, expended by the motive power, and 
 pointed out generally how the consumption of the work may 
 be lessened, and the useful work to the same extent increased, 
 we readily infer that like observations are applicable to the 
 term P 2 S 2 the work of the resistance at the working point. 
 As the prime object in all industrial operations performed by 
 machinery is to produce the greatest result of a certain kind 
 for the amount of work expended by the motive power, it 
 will be necessary to this end that the velocity, the form, &c., 
 of the operator, or tool by which the result sought is to be 
 obtained, should be such as will not cause any useless expen- 
 diture of work. On this point experiment has shown that 
 for certain operators there is a certain velocity of motion 
 by which the result produced will be the most advantageous 
 both as to the quality and quantity. 
 
 With respect to the work of the motive power itself repre- 
 sented by the product PjS, it admits of a maximum value ; 
 for when the receiver to which Pj is applied is at rest, 'P l will 
 act with its greatest intensity, but the velocity then being 
 the product P l S l will also be ; but as the velocity increases 
 
 39 
 
610 EDITORIAL APPENDIX. 
 
 after the receiver begins to move the intensity of the action 
 of P x upon it decreases, until finally the velocity of the 
 applied point may receive such a value Y that P 1 will become 
 0, and the product P^ in this case will then also he 0. As 
 the work PjS, thus becomes in these two states of the velo- 
 city, it is evident that there is a certain value of the velo- 
 city which will make FjS, a .maximum. To attain this 
 maximum the mode of action of the motive power selected 
 on each form of receiver to which it is applicable will require 
 to be studied, and such an arrangement of its mechanism 
 adopted as will prevent any decompositions of the motive 
 power that would tend in any manner to increase the hurt- 
 ful resistances and thus diminish the useful work. 
 
 It will be very easy to show that the laws of motion of all 
 machines, that is the relations between the times, spaces and 
 velocities of the motion of any one of the moving parts are 
 implicitly contained in the general equation of living forces 
 as applied to machines w T hich has just been discussed. 
 Resuming (equation B) with this view, and representing by 
 dm any elementary mass in motion whose velocity is v 9 at 
 any instant w r hen it has described the path, or space s, if we 
 take any other elementary mass dm in a given position and 
 denote by u its velocity at tke same instant, we shall have 
 i' a =M a (9,9), and v l =u l ($$ t ) ; in which 95 is a purely geome- 
 trical function, since, from the connection of the parts of a 
 machine, in which, any motion given to one part is trans- 
 mitted in an invariable manner to the other, the space passed 
 over by any one point can always be expressed in terms of 
 that passed over by any other assumed at pleasure. 
 
 From the relations v t =u t (95), and v^ dtds, we obtain 
 
 v* and u^ d^ ^ 
 
 (tt 
 
 Substituting these values of v? and i> 2 dv t in (equations B and 
 A), and letting m still represent the sum of the elementary 
 masses as dm, there obtain the two equations 
 
 tf^ /^P^S, 2 fudS - 
 f P 2 d S 2 zfwdh. (B') 
 
EDITORIAL APPENDIX. 611 
 
 ds = sP^-sRdS - 
 . (A'), 
 
 the first showing the relations between any two states of the 
 velocities u^ and u t for any definite interval, and the second 
 for the infinitely small interval dt. ]STow as the relations 
 between the quantities dS^ dS &c., or the elementary 
 paths described by the points of application of P 15 P 2 , &c., 
 and the elementary space ds, from the connection of the 
 parts of the machine, can be expressed in functions of s and 
 of the constants that determine the relative magnitudes and 
 positions of those parts ; and as, moreover, P 15 P 2 , &c., are 
 either constant, or vary according to certain laws by which 
 they are given in functions of the paths S 1? S 2 , &c., we see 
 that all the relations in question are implicitly contained in 
 the two preceding equations. 
 
 Let us examine the kinds of motion of which a machine is 
 susceptible and the conditions attendant upon them. We 
 observe, in the first place, supposing the machine to start 
 from a state of rest, that the elementary work ^ l d^> l of the 
 motive power must be greater than that of the resistances 
 combined, or P^Sj R^S &c. >0, so long as the velocity 
 is on the increase. The living force is thus increased at 
 each instant by a quantity d (mv*)=2mvdv, or by an amount 
 which is equal to twice the elementary work of the motive 
 power and resistances combined ; and this increase will go 
 on so long as the elementary work of the motive power is 
 greater than that of the resistances. But, from the very 
 nature of the question, this increase cannot go on indefinitely, 
 for the point of application of the motive power would in the 
 end acquire a velocity so great that P 1 would exert no effort 
 on the receiver, whereas the resistances still act as at the 
 commencement, and some of them even increase in intensity 
 with the velocity. The living force therefore will, at some 
 period of the motion, attain a limit beyond which it will not 
 increase, a fact which the operation of all known machines 
 confirms, and, having thus reached this state, it must either 
 continue the same during the remainder of the time that the 
 machine continues in motion, or else it must commence to 
 decrease until the velocity attains some inferior limit from 
 which it will again commence to increase, and so on for each 
 successive period of motion, during which the action of the 
 forces remains the same. 
 
612 EDITORIAL APPENDIX. 
 
 Supposing the machine to continue its motion with the velo- 
 city it has attained at this maximum state of the living force, 
 we shall then have 
 
 and 
 
 inasmuch as the motion being now uniform the difference 
 between the living forces corresponding to any finite inter- 
 val of time is 0. Considering the manner in which the parts 
 of machines are combined to transmit motion from point to 
 point, we infer that this condition with respect to the 
 increase of living force, and which constitutes uniform 
 motion, can only obtain when the velocities of all the differ- 
 ent parts bear a constant ratio to each -other. Representing 
 by t/, v", /", &c., these velocities which are respectively 
 
 equal to , _, -, &c., we see that the ratios of ds, 
 dt dt dt 
 
 ds", ds'", &c., will also be constant when those of -y', v", &c., 
 are so ; that is, this constancy of the ratio of the effective velo- 
 cities and of the quantities ds', ds", &c., must subsist together 
 for all positions of the parts of machines to which they refer ; 
 but as the latter, which are the virtual velocities, or ele- 
 mentary paths described, depend entirely on the geometrical 
 laws that govern the motion of the parts, a little consideration 
 of the various mechanical combinations by which motion is 
 transmitted will show that, in order that their ratios shall 
 respectively remain constant, no pieces having a reciprocat- 
 ing motion can enter into the composition of the machine, 
 as the velocities of such pieces evidently cannot be anade to 
 bear a constant ratio to the others. This condition it will be 
 seen refers exclusively to the mechanism of the machine, or 
 the geometrical conditions by which the parts are connected, 
 and has nothing to do with the action of the forces them- 
 selves. 
 
 But when the condition of uniform motion is satisfied 
 there obtains also 
 
 that is, according to the principle of virtual velocities, an 
 equilibrium obtains between the forces which act on the 
 machine irrespective of the inertia of the parts. As a gene- 
 ral rule this condition requires that not only must the forces 
 
EDITORIAL APPENDIX. 613 
 
 P 15 11, &c., be constant both in intensity and direction and 
 act continuously, but that the term Wc$tl must be sepa- 
 rately equal to 0, or the centre of gravity of each part must 
 preserve the same level during the motion ; for were this 
 not so any piece whose weight is w would evidently impress 
 an elementary work represented by wdh which would be 
 variable in the different positions of the mechanism ; unless 
 w, having itself a uniform velocity, formed, as might be the 
 case, a part of the motive power P^ or of the useful resist- 
 ance P 2 . 
 
 It thus appears that to obtain uniform motion not only 
 must the mechanism used for transmitting the motion con- 
 tain no reciprocating pieces, and therefore consist solely 
 of rotating parts, as wheels, &c., and parts moving continu- 
 ously in the same direction, as endless bands, and chains, &c. ; 
 but that the centres of gravity of these pieces shall remain 
 at the same level during the motion, which will require that 
 the wheels and other rotating pieces shall be accurately cen- 
 tered so as to turn truly about their axes. 
 
 The difficulty of obtaining a strictly uniform motion in 
 machines is thus apparent, for it involves conditions in them- 
 selves practically unattainable, that is, applied forces acting 
 continuously and with a constant intensity and direction, and 
 that the ratio of the virtual velocities of the different parts 
 should be constant and independent of the positions of the 
 mechanism, a condition which requires that the terms ($s) 
 and I,d7n(s)' 2 in the preceding equations shall also be con. 
 stant for all of these positions. But even were these condi- 
 tions satisfied, it can be shown that rigorously speaking a 
 machine starting from a state of rest will attain a uniform 
 velocity only in a time infinitely great. This will appear 
 from geometrical considerations of a very simple character, 
 or from the form taken by equation. By the first method, 
 _ let OT, OY be two co-ordinate 
 
 axes, along the one set off the 
 abscissas Qt'^ O", & c . ? to re- 
 present the times elapsed from 
 the commencement of the mo- 
 tion, and the ordinates V, t"v', 
 -f &c., the corresponding veloci- 
 ties, the curve Qv'v", &c., will 
 give the relation between the times and the velocities. Now, 
 from the circumstances of the motion, the increments of the 
 velocities will continually decrease, and the curve, from the 
 law of continuity, will approach more nearly to a right line 
 
614: EDITOEIAL APPENDIX. 
 
 as the time increases ; having for its assymptote a right line 
 parallel to OT, drawn at a distance Ov from it, which is the 
 limit the velocity attains when the motion becomes uniform. 
 We moreover see from the form the curve may assume that 
 this limit will be approached more or less rapidly. 
 
 From (equation B'), representing by c the quantity 
 we obtain 
 
 #* dv * - vT> ^L V-P ^ P ^ 
 
 -~~ l ' * r ** 
 
 Now, from the preceding discussion, the forces being sup- 
 posed to act continuously, and with a constant intensity and 
 
 7Q 70 
 
 direction, and the quantities - 1 , -^- 2 being constant, the 
 
 ds ' ds 
 
 function expressed by the second member -of this equation 
 has its greatest value when ^ 2 0, or when the machine is 
 about to move, and that after motion begins it decreases 
 more or less rapidly as the velocity increases, until it be- 
 comes for a certain finite value of the velocity. Hence it 
 follows that the function must be of the following, or some 
 equivalent form, 
 
 in which ~k is essentially positive and a function of t> 2 and 
 certain constants, and V is the limit of the velocity in ques- 
 tion. We shall therefore obtain from (equation B'), by sub- 
 stituting this function for the second member, 
 
 v 
 
 dv z r cdv^ 
 
 c ~dt ^v^Vty 5 an d t J kTVv Y ' 
 
 The second member of this last equation, when integrated 
 between the limits v 9 =Q, arid 9 = y. , must contain, according 
 to the known rules applicable to it, at least one term of the 
 form of a log. (Vv t ) if the exponent n is odd ; or 
 a(Y / y 2 )~ w+1 , if n is even; either of which functions will 
 become infinite for V v^= 0, or when v 2 attains its limit. 
 
 From the conditions requisite to attain uniformity of mo- 
 tion in a machine, the advantages attendant upon it, so far 
 as it aifects the mechanism are self-apparent ; not only will 
 there be none of that jarring which attends abrupt transi- 
 tions in the velocity, but, from the manner in which the 
 
EDITOEIAL APPENDIX. 615 
 
 forces act, the strains on all the parts- will be equable, and 
 the respective form and strength of each can thus be regu- 
 lated in accordance with the strain to be brought upon it, 
 thus reducing the bulk and weight of each to what is strictly 
 requisite for the safety of the machine. But advantages not 
 less important than these result from the use of mecnanism 
 susceptible of uniform motion, owing to the fact that for 
 each receiver and operator there is a velocity for the applied 
 and working points with which the functions of the machine 
 are best performed as respects the products ; and these 
 respective velocities can be readily secured in uniform 
 motion by a suitable arrangement of the mechanism inter- 
 mediate between these two pieces. 
 
 The advantages resulting from uniform motion in machines 
 has led to the abandonment of mechanism that necessarily 
 causes irregularity of motion, in many processes where the 
 character of the operation admits of its being done ; and 
 where, from the manner in which the motive power acts on 
 the receiver and is transmitted to the operator, parts with a 
 reciprocating motion have to be introduced, every possible 
 care is taken to so regulate the action of these parts and to 
 confine the working velocity within the narrowest limits that 
 the character of the operation may seem to demand. Many 
 ingenious contrivances have been resorted to for this pur- 
 pose, but as they belong to the descriptive part of mechanism 
 rather than to the object of this discussion, and, to be under- 
 stood, would require diagrams and explanations beyond the 
 limits of this work, they can only be here alluded to. There 
 is one however of general application, the fly wheel, the 
 general theory and application of which to one of the sim- 
 plest cases of irregularity are given in (Arts. 75, 76, 265, &c.) 
 The functions of this piece are to confine the change of velo- 
 city, arising from irregularities caused either by the mechan- 
 ism, or the mode of action of the motive power within certain 
 limits ; absorbing, by the resistance offered by its inertia, 
 or accumulating work whilst the motion is accelerated, and 
 the work of the motive power is therefore greater than that 
 of the other resistances, and then yielding it w^hen the reverse 
 obtains ; thus performing in machinery like functions to 
 those of regulating reservoirs in the distribution of water. 
 It should however not be lost sight of that whatever resources 
 the fly wheel may offer in this respect they are accompanied 
 with drawbacks, inasmuch as the weight of the wheel, its 
 bulk and the great velocity with which it is frequently 
 required to revolve, add considerably to the prejudicial 
 
616 EDITOKIAL APPENDIX. 
 
 resistances, as friction and the resistance of the air, and thug 
 cause a useless consumption of a portion of the work of the 
 motive power. Whenever therefore, by a proper adjustment 
 of the motive power and the resistances, and a suitable 
 arrangement of the mechanism, a sufficient degree of regu- 
 larity can be attained for the character of the operation, the 
 use of a fly wheel would be injudicious. In cases also where, 
 from the functions of the machine, its velocity is at times 
 rapidly diminished, or sudden stoppages are requisite, the 
 fly wheel might endanger the safety of the machine, or be 
 liable itself to rupture, it should either be left out, or else 
 the mass of the material should be concentrated as near as 
 practicable around the axis of rotation ; thus supplying the 
 requisite energy of the fly wheel by an augmentation of its 
 mass. In all other cases the matter should be thrown as far 
 from the axis as safety will permit, as the same end will be 
 attained with less augmentation of the prejudicial resist- 
 ances. 
 
 From this general discussion some idea may be gathered 
 of the relations between the work of the power and that of 
 the resistances in machines, and of the means by which the 
 latter may be so reduced as to secure the greatest amount of 
 the former being converted into useful work. It must not 
 however be concealed that the problem, as a practical one, 
 presents considerable difficulty, and requires, for its satis- 
 factory solution, a knowledge of the various operators and 
 receivers of power, as to their forms and the best modes of 
 their action. This knowledge it is hardly necessary to 
 observe must, for the most part, be the result of experiment; 
 theory serving to point out the best roads for the experi- 
 menter to follow. Both of these have shown that the work 
 of the motive power consumed by the resistances, caused by 
 the parts through which motion is communicated from the 
 receiver to the operator, is but a small fractional part of the 
 total work uselessly consumed, whenever the mechanism has 
 been arranged with proper attention to the functions required 
 of it ; but that the principal loss takes place at the receiver 
 and operator, and this is owing to the difficulty of so arrang- 
 ing the receiver that the motive power shall expend upon it 
 all its work without loss from any cause ; and in like manner 
 of causing the operator to act in the most advantageous way 
 upon the resistance opposed to it. Some of the general con- 
 ditions to which these two pieces must be subjected, as to 
 uniformity and continuity of action of the motive power and 
 the resistances, and the avoidance of jarring and shocks have 
 
EDITORIAL APPENDIX. 
 
 617 
 
 been pointed out. as well as the fact that to each corresponds 
 a certain velocity by which the greatest amount of useful 
 effect will be attained. 
 
 This discussion will make apparent that, comparatively 
 speaking, but a small amount of the work due to the motive 
 power is expended on the useful resistance, or the matter to 
 be operated on. In some of the best contrived receivers, as 
 the water wheel, for example, where the motive power can 
 be made to act with the greatest regularity, and the receiver 
 be brought to as near an approach to uniformity of motion 
 as attainable, the quantity of work it is capable of yielding 
 seldom exceeds eight tenths of that due to what the water 
 expends upon it, under the most careful arrangement of the 
 wheel and the velocity of its motion. 
 
 As an example under this head (Art. 149) equation (115), 
 and an illustration of the circumstances attending the attain- 
 ment of uniformity of motion Note (t) in machines ; suppose 
 the axle A carrying two arms B, B, to the 
 extremities of which two thin rectangular 
 disks C, C, are attached, their planes pass- 
 ing through the axis of rotation, to be put 
 in motion by the descent of a weight P, at- 
 tached to a cord wound round the axle. 
 In this case the resistances to the moving 
 force during the acceleration will be 
 that of the air acting against the disks and 
 the two arms, the inertia of the parts in 
 motion, and the friction on the gudgeons 
 of the axle. 
 
 Represent by A the sum of the areas of 
 the two disks, a the distance of their centres from the axis, 
 dm an elementary mass of the machine at the distance r 
 from the axis, w the angular velocity of the system, a 1 the 
 radius of the axle measured to the axis of the cord, p the 
 radius of the gudgeon, 9 the limiting angle of resistance, 
 l v the total length of the cord, I the length of the part 
 unwound, w the weight of the unit in length of the cord, W 
 the total weight of the machine excepting P r 
 
 From experiment we have for the resistance of the air to 
 the motion of the two discs cA-y 2 cAw 2 <2 2 , in which v=d)O> 
 
618 EDITOEIAL APPENDIX. 
 
 expresses the velocity of the centre of the disk and c a con- 
 stant determined by experiment. The resistance offered by 
 the inertia of dm during the acceleration of the motion is 
 
 represented (Art. 95) equations (Y2) (73) by dmr ^-, in 
 
 Cut 
 
 which ^ is the acceleration of the angular velocity in the 
 
 dt 
 
 element of time dt, the resistances offered by the inertia of 
 the weight P x and that of the pendant portion of the cord 
 
 represented by wl are, in like manner, expressed -i" 1 "^ a l *** , 
 
 g dt 
 
 the total pressure upon the gudgeons will evidently be ex- 
 
 pressed by Pj + W- l - 0,1 =-, since, during the accele- 
 
 9 dt 
 
 ration of the motion, the resistance of the inertia of the 
 weights Pj and wl act in an opposite direction to these 
 weights. 
 
 In the state bordering upon motion at each instant there 
 obtains 
 
 (-p . w 
 r x 4- W 
 
 dt g dt 
 
 du \ 
 
 a, --- I p sm. 9. 
 g dt J 
 
 Representing by n^ the coefficient of w a , by m 2 that of 1 
 
 dt 
 
 and by <f the algebraic sum of the other terms, there obtains 
 
 dt 
 
 <i.i) 
 
 From this last equation we se that w approaches rapidly the 
 
 limit i which it only attains when fcoo . As this limit cor- 
 n 
 
 responds to that in which the motion would become uni- 
 
EDITORIAL APPENDIX. 619 
 
 form, it might have been deduced directly from .the first of 
 
 these equations; for when w becomes constant =0, 
 
 cLt 
 
 (v). 
 
 Manner of estimating the amount of work consumed Tyy the 
 trip hammer. 
 
 The trip hammer is used in forging heavy iron work, 
 motion being given to it for this purpose by teeth, termed 
 
 cams, hrmly fixed in an axle A, termed the cam shaft, 
 around which they are arranged at equal intervals apart. 
 The tail of the hammer is furnished with an iron band, the 
 upper surface of which receives a suitable form to work 
 truly with the surface of the cam whilst the two remain in 
 contact during the ascent of the head of the hammer, on the 
 same principle as the teeth are fashioned in other cases. 
 The interval between the cams is so calculated that each 
 cam shall take the band at rest at the point t on the hori- 
 zontal line C 2 C, joining the centres of rotation of the cam 
 shaft and hammer. 
 
 To estimate the work consumed in the play of this ma- 
 chine, it must be observed that it consists of three distinct 
 parts ; the first is that consumed by the impact or shock ; 
 the second that due to the period after the shock, in which 
 the cam and tail of the hammer remain in contact ; the third 
 that consumed by the cam shaft in the interval between the 
 separation of the cam and hammer and the moment when 
 the succeeding cam takes the hammer. 
 
 Denote by R 2 the radius of the primitive circle C 2 of the 
 cams ; by 2 the angular velocity of the cam shaft at any 
 period of the shock; by p a the radius of the gudgeon on 
 
620 EDITOKIAL APPENDIX. 
 
 which the shaft revolves ; by <p a the limiting angle of resist- 
 ance for the surfaces of the gudgeon and its bed ; by m an 
 elementary mass of the shaft ; by r the distance of m from 
 C a ; by R^C/, w l5 p l5 p x m l9 and T I the corresponding quan- 
 tities for the hammer. 
 
 Now if we represent by P the mutual pressure between 
 the surfaces of the cam and band at any period of the impact, 
 there must be an equilibrium at each instant between P and 
 the forces of inertia and the passive resistances developed 
 in the play of the machine. Considering the equilibrium 
 around the axis of rotation C x of the hammer in the first 
 place, we have for the velocity of any element ra t , at any 
 instant, 7-^, and for the increment of velocity impressed 
 upon it by the cam rjlu l ; the force of inertia therefore deve- 
 loped by this increment is expressed by 
 
 and its moment with respect to the axis Q l is 
 
 and the sum of the moments of all the forces of inertia is 
 (Arts. 95, 106) 
 
 To obtain the friction on the trunnions of the hammer due 
 
 to P and the resultant of the forces of inertia m.r.^. we 
 
 at 
 
 have for the resultant of the latter (Art. 108) equation (82) 
 
 in which M represents the mass of the hammer, its handle, 
 &c., and G the distance of its centre of gravity from Cj the 
 axis of rotation. Now, decomposing this resultant into two 
 components perpendicular and parallel to the line C a C I repre- 
 senting by a the angle between this line and the one C,G 
 through the centre of gravity of the hammer, &c., we have 
 for the perpendicular component 
 
EDITORIAL APPENDIX. 621 
 
 cos. a 
 at 
 
 and for the parallel one 
 
 sn. #. 
 
 The total pressure on the trunnions, from P and the forces 
 of inertia, will therefore be 
 
 As however, in most cases of practice, the angle a is either 0, 
 or very small, the value of the quantity under the radical 
 may be taken without sensible error 
 
 The equation of equilibrium about the axis G l is therefore 
 
 ^' 
 
 (P + -'-MG) Pl sin. ?, 
 
 . 
 ~ dt ' B I -p 1 sin.<p l 
 
 Now with respect to the cam shaft we have, to express the 
 sum of the moments of the forces of inertia with respect to 
 the axis C a , 
 
 As the pressure on the trunnions of this shaft is due to the 
 force P alone, the moment of the friction on them will be 
 expressed by P p sin. <p 2 . 
 
 The equation of equilibrium of all the forces with respect 
 to C 2 will therefore be 
 
 . sin. <(,,.... (B). 
 Eliminating P between equations (A) and (B) there obtains 
 
622 EDITORIAL APPENDIX. 
 
 .^ . : . .(0) 
 
 Hie coefficient of can be written as follows, 
 dt 
 
 placing K for the coefficient of MjRjRj. Making these sub- 
 stitutions in equation (C) there obtains 
 
 in which ft represents the greatest, and w 2 the least angular 
 
 velocity of the cam shaft ; and w 1 =0, 6) 1= ^-^j the angu- 
 
 -"i 
 
 lar velocities of the hammer ; since before the impact it is 
 at rest, and finally attains the same velocity as the cam has, 
 in which, from the circumstance of the mechanism, w 1 R 1 = 
 JL 
 
 From the preceding equation there obtains 
 
 "* = ^ 
 
 Now, as a general rule, the quantities p2 sin ' ^ pl sm> (pl and 
 
 R 2 Rj 
 
 P -" S1 M-b are ver ^ sma ^ W ^ tn Aspect to unity, and may 
 
EDITORIAL APPENDIX. Gl'3 
 
 therefore be disregarded, and'the quantity K will differ but 
 very little from unity also. From this it will be seen that 
 w 2 will differ the less from 12 as M ? is greater than M^ But, 
 as the mass of the cam shaft ordinarily very much exceeds 
 that of the hammer, w r e can assume, without liability to any 
 great error, that the mean angular velocity of the cam shaft. 
 deduced from observing the number of revolutions made by 
 it in a given time, is sensibly the arithmetical mean of 12 
 
 and w 2 . Designating this mean by 12, we have Q t = + G) \ 
 From this relation and equation (D) there obtains 
 
 _2a.(M.+gM.). dfc) 20.M. 
 2M,+KM, "'-2M.+ KM,- 
 
 From these two relations the living force destroyed by the 
 impact can be deduced as follows. Before the impact the 
 living force of the cam shaft was 12 2 M 2 R Q ; after the impact, 
 as the point of contact of the cam and band moved with the 
 same velocity, the living force of the whole machine is 
 
 The living force destroyed therefore is expressed by 
 
 a'M.B.'-u.'I^OM.+M,); 
 or, substituting for w 2 from equation (D), by 
 
 finally, substituting for 12 and w a their values in 12,, there 
 obtains 
 
 It is now readily seen, from the form of this last expression 
 for the loss of living force by the impact, that, since K may 
 
624: EDITORIAL APPENDIX. 
 
 be assumed as sensibly equal to unity, the numerical value 
 of this expression will depend upon the ratio L. Taking 
 
 M^M, the value of the expression becomes fC^MjB,*; and 
 for M a =oo it becomes i2j 2 M,R 2 2 . Therefore between these 
 limits the difference is | only of the living force lost under 
 the supposition of M 2 =o> . 
 
 In the ordinary arrangement of this machine it rarely 
 occurs that M, is not less than T VM 2 . Assuming this as the 
 limit, and substituting in the preceding expression IDIM^ for 
 M 2 , there obtains for the required loss of living force 
 O'OTT^VMjIV. It is therefore seen that, in all usual cases, 
 M 2 may be assumed as infinite without causing any notice- 
 able error in the result. 
 
 To estimate the accumulated work expended by the cam 
 shaft for each shock, fi, a> 2 and fi, being the same as in the 
 preceding expression, this work is expressed by 
 
 As the cam shaft expends this amount of accumulated work 
 at each impact, a quantity of work equal to the half of this 
 must be yielded by the motive power at each impact, or 
 
 If therefore there are N cams on the shaft, and it makes n 
 revolutions in one minute, then the work consumed by the 
 number of shocks in one second will be expressed by 
 
 Nn 2n i 2 MJVI 1 R 2 2 K 
 60 '' 
 
 This then is the work consumed by the impact in one second 
 fo v the first period of the play of the machine ; and it has 
 been calculated according to what was laid down in Note (t) 
 on the subject of shocks, by disregarding the work of the 
 other forces as inappreciable during the short interval of the 
 impact. 
 
 To estimate now the work expended during the second 
 period, or whilst the cam and band are in contact after the 
 , let C 1 G 1 be any position of the line C,G, during this 
 
EDITORIAL APPENDIX. 625 
 
 period, making an angle G^G^a with its position when 
 the hammer is at rest. Represent by Pj the normal pressure 
 at the surface of contact of the cam and band which will 
 balance all the resistances developed in the motion of the 
 hammer, leaving out of consideration that of inertia, as the 
 change of velocity between the end of the impact and when 
 the cam disengages from the band is so small that the living 
 force due to this interval may be neglected in comparison 
 with the work of the other forces ; by W x the weight of the 
 hammer, its handle, &c. 
 
 When the line GC t is in the position G^, the line C t t will 
 oe in that C^ making the angle 0^ a with its original 
 position. The force Pj acting at ^ in this position and per- 
 pendicular to the line t l C l since the surface of the band 
 produced passes through the axis C n the surface of the cam 
 being an epicycloid has for its vertical and horizontal com- 
 ponents P, cos. a and P x sin. a. The pressure on the trun- 
 nions of the hammer, which is the resultant of P l and W^ 
 therefore will be expressed by 
 
 , + P x cos. a) 2 + P t * sin. 2 a ; 
 
 and since the first term of the radical is in all cases greater 
 than the second, the value of the radical itself Haav, be 
 expressed by (KOTE B) 
 
 ! cos. a + 4 sn. a. 
 
 The equation of equilibrium between ~P l and the other forces^ 
 will therefore be 
 
 PjB^WjGcos. (+)+ {/(W.+P^os.^H-^sin.^psin.^. 
 
 The moment of the friction at the point 15 due.-to P x with 
 respect to the point C 15 in this case from the form .of the cam 
 and band, being 0. 
 
 As the pressure P, varies with the angle a, we can -, only 
 obtain its mean value by first finding its quantity of work 
 for the angle OL=OL I described whilst the cam and band are an 
 contact. Multiplying the last equation by da, and then 
 integrating between a=0 and a ^ there obtains 
 
 cos. a + m ?1 sn. 
 40 
 
62G EDITORIAL APPENDIX. 
 
 representing by P OT the mean value of P, or tne constant 
 force applied vertically at tf, which multiplied by !, the 
 path described by the point of application, will give the 
 amount of work of the variable pressure R for the same 
 path ; and introducing this mean value into the term of the 
 preceding equation that represents the moment of the fric- 
 tion on the trunnions, as this will not produce any sensible 
 error in the results. 
 
 Now observing that the quantity G jsin. (# + a) sin. aj is 
 the vertical height through which the centre of gravity of 
 the hammer, &c. is raised during the period in question, and 
 that PmR^ is the work of the mean force ; calling this ver- 
 tical height A, and substituting the work of the mean for 
 that of the variable force in the last equation ; there obtains 
 
 m sn. ai - m cos. a, 
 Pl sin. <p. 
 
 ^ \y sin. c^ + fi (1 cos.a,) \ ^ sin. <p, 
 
 ^ ^ / 
 ' 
 
 If we now multiply the second member of equation (E) by 
 Rjttj we shall obtain the approximate value of the work of 
 the variable force P l during the period in question ; or the 
 value of P m H 1 a 1 as determined from equation (E). 
 
 To find now the work that the motive power must supply 
 to the cam shaft for this expenditure P^R^-due to the 
 motion given to the hammer during the period in question, 
 and also that/ arising from the resistances developed by the 
 motion of the cam shaft itself during this period, represent 
 by P 3 a force which, acting at a distance R 3 from the axis C 2 
 of the cam shaft, will balance all the resistances around C 2 ; 
 by W 2 the weight of the cam shaft and its fixtures : by 6 any 
 angle described by the cam shaft during the period con- 
 sidered ; and 9 the limiting angle of resistance at the point 
 of contact of the cam and band. 
 
 The pressure on the tnmnions of the cam shaft is evidently 
 expressed by 
 
 W 2 -fP 3 -P; 
 
 and the equation that expresses the work of P 8 for the ele- 
 mentary angle dO is 
 
EDITORIAL APPENDIX. 627 
 
 tan. 
 
 Now representing by P M the mean value of P s , and substi- 
 tuting it for P 8 in the last term of the second member of thia 
 equation, which may be done without causing any sensible 
 error in the result; observing, from the conditions of the 
 mechanism that R^^E^ ; and integrating this equation 
 
 between the limits = and 0=^ _^b.; there obtains, to 
 
 K a 
 
 express the total work of P 3 for the angle 15 
 
 * Omitting the work consumed by the friction of the axles in equation (251) 
 (Art. 220), that which is expended on the teeth in contact whilst the arc r^ is 
 described is represented by the term of the equation 
 
 Now if we suppose a 2 =r a , or that P 2 acts at the point of contact and normal 
 to the surface, this term, modified to suit the supposition, becomes 
 
 tan. 0= 
 
 2 
 Dividing this last expression by r 2 i/>, there obtains, 
 
 P 2 ( ri + r *\ r ^L tan. 4> 
 
 as the value of a mean or constant force which applied tangentially to the 
 circumference having the radius r 2 will expend, whilst the point of application 
 describes the arc r 2 i/>, the same quantity of work as that consumed by the fric- 
 tion of the teeth in contact whilst this arc is described. In this expression 
 the value of P 2 is less than the true value. 
 
 The foregoing is the theorem of M. Poncelet referred to on page xii. 
 Author's Preface. The direct manner of deducing it is found on page 192 
 Navier. Resume des Lefons^ &c. Troisieme Partie. Paris, 1838. 
 
628 EDITORIAL APPENDIX. 
 
 (F). 
 R 3 -p 2 sm.<p a . 
 
 The work therefore that the motive power must supply to 
 the cam shaft during this period is found by multiplying the 
 
 second member of equation (F) by R 3 1 :=:R 3 -j|- i or the path 
 
 passed over by the point of application of the mean force P n 
 during this period. 
 
 Representing in like manner, by -^- the number of times 
 
 the hammer is raised per second, the quantity of work that 
 the motive power must supply for this expenditure will be 
 expressed by 
 
 .(2). 
 
 S 
 60 60 8 R, 
 
 During the last period, or whilst the hammer is down, the 
 motive power will only have to supply the expenditure of 
 work caused by the friction on the trunnions of the cam 
 shaft, arising from the weight of this shaft and its fixtures 
 and the power ; any accumulation of work in this shaft 
 during this period being neglected as small in amount. 
 Representing byj?=N the number of cams on the shaft, their 
 distance apart on the primitive circumference whose radius 
 
 is R 2 is evidently - -> and, as the arc described on this 
 circumference whilst the cam shaft and hammer are engaged 
 
 is R 2 0j, that described whilst the hammer is down is -- a 
 
 P 
 RjCtj. Calling P p the power which acting at the distance R, 
 
 will balance the friction arising from the weight W 2 of the 
 cam shaft and fixtures and P 2 , the value of P p will be found 
 according to the conditions stated as follows, 
 
 The work of P p is 
 
 W 2 p 2 sin. 9, 
 
 trl 2 Ri a i) 
 
 R,\ P I 
 
EDITORIAL APPENDIX. 629 
 
 as the path passed over by its point of application is evi- 
 dentlythearc 
 
 The work which the motive power must supply therefore 
 per second during this last period is expressed by 
 
 By taking the sum of the quantities expressed by the 
 formulas (1), (2), and (3) there obtains 
 
 K 3 
 
 to express the total work that the motive power must yield 
 to the cam shaft per second to supply the work consumed 
 by all the resistances. 
 
 Tiiat consumed by the useful resistances, which consist of 
 half the living force transmitted to the hammer and the 
 work consumed in raising the centre of gravity of the ham- 
 mer, &c., through the vertical height h is represented by 
 
 o^MJV w . _2n i 2 M,M 1 R ' w , 
 -(2M 8 + KM 1 ) + 
 
 From the preceding expressions, it is easy to deduce the 
 work which must be expended in producing a given depth 
 of indentation by the hammer upon the metal when brought 
 to a given state of heat. For this purpose, we observe that 
 to half the living force acquired by the hammer there cor- 
 responds a certain amount of work, estimated in terms of 
 the weight of the hammer and a certain height h l to which 
 its centre of gravity has been raised, and expressed by 
 
 0) a M, R, 2 TTT 7 
 
 2 
 
 the total work therefore expended by the hammer in 
 indenting the metal is expressed by W^-fW^ ; since, from 
 the state of the metal the molecules which are displaced by 
 the impact acquire velocities which are not appreciable from 
 their smallness ; the resistances therefore offered by the 
 metal to indentation may be regarded as independent of the 
 
630 EDITOEIAL APPENDIX. 
 
 velocity and, from the laws of the penetration of solids into 
 different media, proportional simply to the area of the inden- 
 tation. Representing then by a and ~b the sides of the area 
 of the indentation, supposed rectangular, at the surface of 
 the metal impinged on, d the depth of the indentation, and 
 C the constant ratio of the resistance and the area of the 
 indentation, the following relation obtains between the work 
 expended by the hammer in its fall and that offered by the 
 resistance of the metal 
 
 an equation from which C may be determined by experi- 
 ment 'in any particular case. 
 
 It will be readily seen that the preceding expressions will 
 be rendered applicable to the cases where the cam catches 
 the hammer on the same side of its axis of rotation as its 
 
 centre of gravity, by writing ^L MG for + ^-MG, and 
 
 at at 
 
 moreover in this case when P 5_ L MG=0, there will be no 
 
 dt 
 
 shock on the trunnions (Arts. 108, 109), and there then 
 obtains, to find the point where the cam should catch the 
 hammer corresponding to this case, 
 
 E= MG 
 
 * Morin, Suite des Nouvelles Experiences sur le Frottenient, p. 67. Paris, 1836. 
 
APPENDIX. 
 
 NOTE A. 
 
 THEOBEM. The definite integral J fxdx is the limit of the sums of tht 
 
 a 
 
 values severally assumed by the product fx . Ax, as x is made to vary by 
 successive equal increments of Ax, from a to b, and as each such equal 
 increment is continually and infinitely diminished, and their number there- 
 fore continually and infinitely increased. 
 
 To prove this, let the general integral be represented by Fa; ; let us sup- 
 pose that fx does not become infinite for any value of x between a and b, 
 and let any two such values be x and x + Ax ; therefore, by Taylor's the- 
 orem, F (x + Ace) = Fx + Axfx + (Ax) '+XM, where the exponent 1 + a, is 
 given to the third term of the expansion instead of the exponent 2, that the 
 
 case may be included in which the second differential coefficient of Fa;, -^ , 
 
 dx 
 
 is infinite, and in which the exponent of Aa; in that term is therefore a 
 fraction less than 2. 
 
 Let the difference between a and b be divided into n equal parts ; and 
 let each be represented by Aa;, so that 
 
 & a 
 
 = Aa;. 
 n 
 
 Giving to a;, then, the successive values a, a+ Aa;, a + 2 A a; . . a+(n 1) 
 A a;, and adding, 
 
 .'. F& Ya=Ax% l *f{a + (n l)Ax} 
 Now none of the values of M are infinite, since for none of these values is 
 fx infinite. If, therefore, M be the greatest of these values, then is SM, less 
 than riM. : and therefore 
 
 F& _ Ya Aa;2 tf{a + (n 1) Aa;} < (oa) M (Aa;)X. 
 
 The difference of the definite integral F5 Fa, and the sum JZi n (Ax)f{a-{' 
 (n 1) Aa;} is always, therefore, less than (b a) M (A#)A. Now M is finite, 
 and (& a) is given, and as n is increased Aa; is diminished continually ; 
 and therefore (Aa;)x is diminished continually, a, being positive. 
 
 Thus by increasing n indefinitely, the difference of the definite integral 
 
32 APPENDIX. 
 
 and the sum may be diminished indefinitely, and therefore, in the limit, the 
 definite integral is equal to the sum (i. e.) 
 
 FJF0 = limit 2 t B (Atf) ./{a + (n 1) Az} ; 
 
 or, interpreting this formula, F5 F<z is the sum of the values of AJK ./a?, 
 when x is made to pass hy infinitesimal increments, each represented by 
 Afc from a to &. 
 
 NOTE B. 
 PONOELET'S FIEST THEOEEM. 
 
 * The values of a and & in the radical -v/^ 2 + ^ a being linear and rational, 
 let it be required to determine the values of two indeterminate quantities 
 a and ]5, such that the errors which result from assuming v/a a + &*= a# + j3&, 
 
 f-\ 
 
 through a given range of the values of the ratio ( & ), may be the least pos- 
 
 sible in reference to the true value of the radical ; or that *** + &> a*_ + V 
 
 or - . =1- i^ may be the least possible in respect to all that range of 
 values which this formula may be made to assume between two given 
 extreme values of the ratio T. Let these extreme values of the ratio ^ 
 
 be represented by cot. ifo and cot. ^ 2 , and any other value by cot. 4. Sub- 
 
 d __ 
 
 stituting cot. 4- for T in the preceding formula, and observing thaty a 2 -f& a 
 
 = -v/& 2 cot.. 2 4/ + & 2 = & cosec. 4, also that a+|3& = a& cot. 4/+|3&=(a cos. 
 sin. 4)& cosec. 4, the corresponding error is represented by 
 
 a cos. 4-+ 18 sin. 41 ..... (1); 
 
 which expression is evidently a maximum for that value <// 3 of i// which is 
 determined by the equation 
 
 a 
 cot.T//,=jg ..... (2); 
 
 so that its maximum value is 
 
 -V/^HS 2 1 ..... (3). 
 
 Moreover, the function admits of no other maximum value, nor of any 
 minimum value. The values of a and being arbitrary, let them be 
 
 assumed to be such that - or cot. i// 3 may be less than cot. ^ and greater 
 
 * The method of this investigation is not the same as that adopted by M. 
 Poncelet ; the principle is the same. 
 
PONCELET'S THEOREM. G33 
 
 than cot. 4/ a . Now, so long as all the values of the error (formula 1) 
 remain positive, between the proposed limits, they are all manifestly di- 
 minished by diminishing a and j3 ; but when by this diminution the error 
 is at length rendered negative in respect to one or both of the extreme 
 values 4,!, or 4/ 2 of 4, and to others adjacent to them, then do these nega- 
 tive errors continually increase, as a and |3 are yet farther diminished, 
 whilst the positive maximum error (formula 3) continually diminishes. 
 Now the most favorable condition, in respect to the whole range of the 
 errors between the proposed limits of variation, will manifestly be attained 
 when, by thus diminishing the positive and thereby increasing the negative 
 errors, the greatest positive error is rendered equal to each of the two 
 negative errors; a condition which will be found to be consistent with 
 that before made in respect to the arbitrary values of a arid j3, and which 
 supposes that the values of the error (formula 1) corresponding to the 
 values 4, and 4 2 are each equal, when taken negatively, to the maximum 
 error represented by formula 3, or that the constants a and ]3 are taken 
 so as to satisfy the two following equations. 
 
 1 (a cos. *,+]3 sin. ,)= V'aM-jS 8 1. 
 1 (a cos. *,+ sin. *,)=! (a cos. V^p sin. *). 
 The last equation gives us by reduction 
 
 and a = j8 cot. (*, + *). 
 
 Substituting these values in the first equation, and reducing, 
 2 sin. K*i+^) sin, j+y, 
 
 
 1+008. *(*, *,) COS. 2 i(, 
 2 COS - 
 
 1 + cos. i(% ,) cos. H(*i ,) " 
 
 These values of a and ]3 give for the maximum error (formula 3) the ex- 
 pression 
 
 tan. '*(*! *0 ..... (6). 
 
 Thus, then, it appears that the value of the radical vV + 5 3 is represented, 
 in respect to all those values of j- which are included between the limits 
 cot. , and cot. by the formula 
 
 with a degree of approximation which is determined by the valu of 
 
 tan.'K*. ^)- 
 
 If in the proposed radical the value of a admits of being increased in- 
 finitely in respect to 5, or the value of 5 infinitely diminished in respect to 
 a, then cot. , = infinity ; therefore , = 0. In this case the formula of 
 approximation becomes 
 
634 
 
 APPENDIX. 
 
 a (I 
 and the maximum error 
 
 (9). 
 
 If the values of a and & are wholly unlimited, so that a may be infinitely 
 small or infinitely great as compared with &, then cot. ^ = infinity, 
 cot. , = ; 
 
 therefore ^i=0, =o- Substituting these values, the formula of approx- 
 
 imation becomes 
 
 82840+ -82845 ..... (10); 
 and the maximum error 
 
 1716, or |th nearly. 
 
 'If & is essentially less than a, but may be of any value less than it, so 
 that T is always greater than unity, but may be infinite, then cot. ^ = in- 
 
 finity, cot. ^2=1 5 therefore ^r=0, ^2=0' Substituting these values in the 
 formula of approximation, and reducing, it becomes 
 
 960460 + -39783& ..... (11); 
 and the maximum error 
 
 03945, or ^\th nearly. 
 
 It is in its application to this case that the formula has been employed in 
 the preceding pages of this work. 
 
 The following table, calculated by M. Gosselin, contains the values of 
 the coefficients a, and ]3 for a series of values of the inferior limit cot. the 
 superior limit being in every case infinity. 
 
 EelationofatoJ. 
 
 1? 
 
 Value of o. 
 
 Value of/?. 
 
 Maximum Error. 
 
 Approximate Value 
 
 a and 6 any > 
 whatever J 
 
 
 
 0-82840 
 
 0-82840 
 
 0-17160 or | 
 
 0-8284 (a + 6) 
 
 a> b 
 
 1 
 
 0-96046 
 
 0-39783 
 
 0-03954 or J y 
 
 96046a + -397836 
 
 o> 26 
 a > 36 
 
 2 
 3 
 
 0-98592 
 0-99350 
 
 0-23270 
 0-16123 
 
 0-01408 or ~!_ 
 0-00650 or T j T 
 
 98592a+ -232706 
 99350a+ -161236 
 
 a> 46 
 
 4 
 
 0-99625 
 
 0-12260 
 
 0-00376 or ^ 
 
 996250 + -122606 
 
 a> 56 
 
 5 
 
 0-99757 
 
 0-09878 
 
 0-00243 or r } ? 
 
 997570 + -098786 
 
 a> 66 
 
 6 
 
 0-99826 
 
 0-08261 
 
 0-00174 or y] T 
 
 99826a + -082616 
 
 i> 76 
 
 7 
 
 0-99875 
 
 0-07098 
 
 0-00125 or ^ j^ 
 
 99875a + -070986 
 
 a> 86 
 
 8 
 
 0-99905 
 
 0-06220 
 
 0-00095 or T ^ 
 
 99905a+ -062206 
 
 a> 96 
 
 9 
 
 0-99930 
 
 0-05535 
 
 0-00070 or -j-^ 
 
 99930a+ -055356 
 
 a> 106 
 
 10 
 
 0-99935 
 
 0-04984 
 
 0-00065 or ^^ 
 
 99935a+ -049846 
 
POWCELET'S SECOND THEOKEM. 635 
 
 PONCELET'S SECOND THEOEEM. 
 
 To approximate to the value of Va? & 2 , let aa 35 be the formula of 
 approximation, then will the relative error be represented by 
 
 Now, let it be observed that a a being essentially greater than &*->!; 
 
 let j, therefore, be represented by cosec. 4/, then will the relative error ba 
 
 (a cosec. 4/ 3) 
 represented by 1 - ~, or by 
 
 V cosec. "4-1 
 
 1 asec.4/+3tan. 4/ ..... (12), 
 
 which function attains its maximum when sin. 4 = -. Substituting this 
 
 a 
 
 value in the preceding formula, and observing that a sec. 4/ + 6 tan. 4/ = 
 
 (-9 
 
 sec.4* (a 3 sin4)= . _= \^L.^ we obtain for the maximum 
 1- 
 
 error the expression 
 
 1V3=& ..... (13), 
 Assuming 4/, and 4/ 2 to represent the values of 4, corresponding to the 
 
 greatest and least values of -, and observing that in this case, as in the 
 
 preceding, the values of a and 3, which satisfy the conditions of the 
 question, are those which render the values of the error corresponding to 
 these limits equal, when taken with contrary signs, to the maximum error, 
 we have 
 
 1 +a sec. 4, 3 tan. 4, = 1 tV 3 a ---- (14). 
 1 a sec. 4, + 3 tan. 4,,=! asec.4/,+3tan.4/, .... (15). 
 The latter equation gives, by reduction, 
 
 cos. i(4/. 4*) /-^ 
 
 _ 2 
 
 ~ ' 
 
 And a sec. , + /3 tan. //,= cot. i (^i + <//,) . . . . (17). 
 Substituting these values in equation (14), and solving in respect to j(? 
 
636 APPENDIX. 
 
 cos. i (//, + i/> 2 ) + Vcos. />, cos. </>j 
 2 cos. i (?//, ^ 2 ) 
 
 a := i 
 
 cos. i (^, + 1^ 2 ) + r cos. i//i cos. ?//, 
 The maximum error is represented by the formula 
 
 2 4/cos. T//J cos. t// 2 
 
 These formulaa will be adapted to logarithmic calculation, if we assume 
 
 d (^t + ^)="^n and c . 08 ' ; ; = cosec. 2 ; we shall thus obtain from 
 
 sin. % (j^i + t// 2 ) 
 
 equations (16) and (17) a = j3 cosec. ^ 2 , Vo? j3 2 = j3 cot. ^ 2 , and a sec. ^ 
 j3 tan. i//i = 3 cot. ^i ; therefore, by equation (14), 
 
 2 2 sin. 
 
 ' cot. ^r l + cot. ^2 sin. (^ + a/ 
 
 2 cosec. ^ 2 2 sin. "SP", 
 
 cot. ^, + cot. , = sin. ( V K 1 + ^ 
 
 sn. , 
 Max,mum errror = 
 
 sn + _ _ _ _ 
 
 The form under which this theorem has been given by M. Poncelet is 
 different from the above. Assuming, as in the previous case, the limiting 
 
 values of - to be represented by cot. ^, and cot. i/^ and proceeding by a 
 & 
 
 geometrical method of investigation, he has shown that if we assume 
 tan. h = cos. tan. i// z = cos. 2 , ! + w 2 = 2y,, ! u> 2 = 25, and cos. y 8 = 
 
 ; then 
 
 cos. 5 
 
 2 COS. y, 2 COS. 2 y, sin. (y, y 2 ) 
 
 a -. ^ &=-. -, - r-= - , maximum error = - -- ~. 
 
 Sin. (y, + y 2 y Sin. (y, + y 2 ) COS. 8 Sin. (y , + y 2 ) 
 
 If the least possible value of a be l T l o&? and its greatest possible value 
 be infinite as compared with 5, M. Poncelet has shown the formula of 
 approximation to become 
 
 Va? & 8 = 1-13190 0-72636& (23), 
 
 with a possible error of 0-1319 or | nearly. 
 
 If the least possible value of a be 25, and its greatest possible valua 
 infinite compared with 5 ; then 
 
 4/^Zy = 1-0186230 0-2T2944& (24), 
 
 with a possible error of -0186 or ^d nearly. 
 
ON THE ROLLING OF SHIPS. 637 
 
 NOTE 0. 
 ON THE ROLLING OF SHIPS. 
 
 (First published ty the Author in the Transactions of the Royal Society 
 
 for 1850, Part //.) 
 
 Let a body be conceived to float, acted upon by no other forces than its 
 weight "W, and the upward pressure of the water (equal to its weight) ; 
 which forces may be conceived to be applied respectively to the centre of 
 gravity of the body and to the centre of gravity of the displaced fluid ; 
 and let it be supposed to be subjected to the action of a third force whose 
 direction is parallel to the surface of the fluid. Let AHi represent the ver- 
 tical displacement of the centre of gravity of the body thereby produced*, 
 and AH, that of the centre of gravity of its immersed part. Let more- 
 over the volume of the immersed part be conceived to remain unaltered f 
 whilst the body is in the act of displacement. If each centre of gravity 
 be assumed to ascend, the work of the weight of the body will be repre- 
 sented by W.AH,, and that of the upward pressure of the fluid by + 
 W.AH 2 , the negative sign being taken in the former case because the force 
 acts in a direction opposite to that in which the point of application is 
 moved, and the positive sign in the latter, because it acts in the same direc- 
 tion, so that the aggregate work 2u 2 (see equation 1, p. 122.) of the forces 
 which constituted the equilibrium of the body in the state from which it 
 has been disturbed is represented by 
 
 Moreover, the system put in motion includes, with the floating body, the 
 particles of the fluid displaced by it as it changes its position, so that if 
 the weight of any element of the floating body be represented by to,, and 
 of the fluid by w z , and if their velocities be 0, and 2 , the whole vis viva is 
 represented by 
 
 * "When a floating body is so made to incline from any one position into any 
 other as that the volume of fluid displaced by it may in the one position be 
 equal to that in the other, its centre of gravity is also vertically displaced ; 
 for if this be not the case, the perpendicular distance of the centre of gravity 
 of the body from its plane of flotation must remain unchanged, and the form 
 of that portion of its surface, which is subject to immersion, must be determined 
 geometrically by this condition ; but by the supposition the form of the body 
 is undetermined. It is remarkable what currency has been given to the error, 
 that whilst a vessel is rolling or pitching, its centre of gravity remains at rest 
 I should not otherwise have thought this note necessary. 
 
 f This supposition is only approximately true. 
 
 \ If the centre of gravity of the body or of the displaced fluid descends ( 
 property which will be found to characterise a large class of vessels), AH, in 
 the one case, and ^ILj in the other, will of course take the negative sign. 
 
t)38 APPENDIX. 
 
 ~2w,f>? + -5Wi > 
 and we have by equation 1 (p. 122), 
 
 W(AH AH^=S I ! + Sw 1 ; .... (25). 
 
 In the extreme position into which the body is made to roll and IE 
 which Xw,flJ=0, 
 
 or if the inertia of the displaced fluid be neglected, 
 
 U(5)=W.(AH 1 AH 2 ) ..... (27). 
 
 Whence it follows that the work necessary to incline a floating body 
 through any given angle is equal to that necessary to raise it bodily through 
 a height equal to the difference of the vertical displacements of its centre 
 of gravity and of that of its immersed part ; so that other things being 
 the same, that ship is the most stable the product of whose weight by this 
 difference is the greatest. 
 
 In the case in which the centre of gravity of the displaced fluid descends, 
 the sum of the displacements is to be taken instead of the difference. 
 
 This conclusion is nevertheless in error in the following respects : 
 
 1st. It supposes that throughout the motion the weight of the displaced 
 fluid remains equal to that of the floating body, which equality cannot 
 accurately have been preserved by reason of the inertia of the body and 
 of the displaced fluid.* 
 
 From this cause there cannot but result small vertical oscillations of the 
 body about those positions which, whilst it is in the act of inclining, cor- 
 respond to this equality, which oscillations are independent of its principal 
 oscillation. 
 
 2ndly. It involves the hypothesis of absolute rigidity in the floating 
 body, so that the motion of every part and its vis viva may cease at once 
 when the principal oscillation terminates. The frame of a ship and its 
 masts are, however, elastic, and by reason of this elasticity there cannot 
 
 * The motion of the centre of gravity of the body being the same as though 
 all the disturbing forces were applied directly to it, it follows, that no elevation 
 of this point is caused in the beginning of the motion, by the application of a 
 horizontal disturbing force, or by a horizontal displacement of the weight of 
 the body, which, if it be a ship, may be effected by moving .its ballast. The 
 motion of rotation thereby produced takes place therefore, in the first instance, 
 about the centre of gravity, but it cannot so take place without destroying the 
 equality of the weight of the displaced fluid to that of the body. From this 
 inequality there results a vertical motion of the centre of gravity, and anothei 
 axis of rotation. 
 
ON THE ROLLING OF SHIPS. 639 
 
 but result oscillations, which are independent of, and may not synchro- 
 nise with, the principal oscillation of the ship as she rolls, so that the VIA 
 viva of every part cannot be assumed to cease and determine at one and 
 the same instant, as it has been supposed to do. 
 
 3rdly. No account has been taken of the work expended in communi- 
 cating motion to the displaced fluid, measured by half its vis viva and 
 
 represented by the term ^w^vl in equation 26. 
 
 From a careful consideration of these causes of error, the author was 
 led to conclude that they would not affect that practical application of the 
 formula which he had principally in view in investigating it, especially as 
 in certain respects they tended to neutralise one another. The question 
 appeared, however, of sufficient importance to be subjected to the test of 
 experiment, and on his application, the Lords Commissioners of the Admi- 
 ralty were pleased to direct that such experiments should be made in Her 
 Majesty's Dockyard at Portsmouth, and Mr. FINCHAM, the eminent Master 
 Shipwright of that dockyard, and Mr. KAWSON, were kind enough to 
 undertake them. 
 
 These experiments extended beyond the object originally contemplated 
 by him ; and they claim to rank as authentic and important contributions 
 to the science of naval construction, whether regard be had to the prac- 
 tical importance of the question under discussion, the care and labor 
 bestowed upon them, or the many expedients by which these gentlemen 
 succeeded in giving to them an accuracy hitherto unknown in experiments 
 of this kind. 
 
 That it might be determined experimentally whether the work which 
 must be done upon a floating body to incline it through a given angle be 
 that represented by equation 27, it was necessary to do upon such a body 
 an amount of work which could be measured ; and it was further neces- 
 sary to ascertain what were the elevations of the centres of gravity of the 
 body and of its immersed part thus produced, and then to see whether 
 the amount of work done upon the body equalled the difference of these 
 elevations multiplied by its weight. 
 
 To effect this, the author proposed that a vessel should be constructed 
 of a simple geometrical form, such that the place of the centre of gravity 
 of its immersed part might readily be determined in every position into 
 which it might be inclined, that of its plane of flotation being supposed to 
 be known ; and that a mast should be fixed to it, and a long yard to this 
 mast, and that when the body floated in a vertical position a weight 
 suspended from one extremity of the yard should suddenly be allowed to 
 act upon it causing it to roll over; that the position into which it thus 
 rolled should be ascertained, together with the corresponding elevations 
 of its centre of gravity and the centre of gravity of its immersed part, 
 and the vertical descent of the weight suspended from the extremity of 
 its arm. The product of this vertical descent by the weight suspended 
 
640 APPENDIX. 
 
 from the arm ought then, by the formula, to be found nearly equal to the 
 difference of the elevations of the two centres of gravity multiplied by 
 the weight of the body ; and this was the test to which it was proposed 
 that the formula should be subjected, with a view to its adoption by prac- 
 tical men as a principle of naval construction. 
 
 To give to the deflecting weight that instantaneous action on the ex- 
 tremity of the arm which was necessary to the accuracy of the experiment, 
 a string was in the first place to be affixed to it and attached to a point 
 vertically above, in the ceiling. When the deflecting weight was first 
 applied this string would sustain its pressure, but this might be thrown 
 at once upon the extremity of the arm by cutting it. A transverse sec- 
 tion of the vessel, with its mast and arm, was to be plotted on a large 
 scale on a board, and the extreme position into which the vessel rolled 
 being by some means observed, the water-line corresponding to this 
 position was to be drawn. The position of the yard, in respect to the 
 surface of the water in that position, would then be known, and the vertical 
 descent of the deflecting weight could be measured, and also the vertical 
 ascent of the centre of gravity of the immersed part or displacement. 
 
 To determine the position of the centre of gravity of the vessel, it was 
 to be allowed to rest in an inclined position under the action of the deflect- 
 ing weight ; and the water-line corresponding to this position being drawn 
 on the board, the corresponding position of the deflecting weight and of 
 the centre of gravity of the immersion were thence to be determined. 
 The determination of the position of the vertical passing through the 
 centre of gravity of the body would thus become an elementary question 
 of statics ; and the intersection of this line, with that about which the 
 section was symmetrical, would mark the position of the centre of gravity. 
 This determination might be verified by a second similar experiment with 
 a different deflecting weight. 
 
 These suggestions received a great development at the hands of Mr. 
 RAWSON, and he adopted many new and ingenious expedients in carrying 
 them out. Among these, that by which the position of the water-line 
 was determined in the extreme position into which the vessel rolls, is 
 specially worthy of observation. A strip of wood was fastened at right 
 angles to that extremity of the yard to which the deflecting weight was 
 attached, of sufficient length to dip into the water when the vessel rolled ; 
 on this slip of wood, and also on the side of the vessel nearest to it, a 
 strip of glazed paper was fixed. The highest points at which these strips 
 of paper were wetted in the rolling of the vessel, were obviously points 
 in the water line in its extreme position, and being plotted upon the board, 
 a line drawn through them determined that position with a degree of 
 accuracy which left nothing to be desired. 
 
 Two forms of vessels were used ; one of them had a triangular and the 
 other a semicircular section. The following table contains the general 
 results of the experiments. 
 
ON THE ROLLING OF SHIPS. 
 
 641 
 
 
 
 
 
 
 'f 'f! 
 
 II 
 
 ft| * 
 
 **4 
 
 IJ1J 
 
 
 
 
 
 L 
 
 * w 
 
 
 si w 
 
 *=jj 
 
 *l ! u 
 
 Form of thi- 
 model ex- 
 perimented 
 
 No. of 
 experi- 
 
 Weieht of 
 motlr! and 
 loading. 
 
 Disturb- 
 &. 
 
 i! 
 
 : L 
 ii 
 ii< 
 
 3^ 
 
 1 
 
 IP L 
 
 \\*& 
 
 -'li* 
 
 n in whir 
 v rested wh 
 
 I- action of 
 ght. 
 
 iSjfl 
 
 g -.SB'S 
 
 ji;ii 
 
 
 
 
 
 ii 
 
 Q 
 
 l|ll 
 
 III 
 
 1 
 
 % z 
 
 nif ii 
 
 wf 
 
 sSi 
 
 Pa! 
 
 fill 
 
 " 112 
 
 * l*s 
 |lf*l 
 
 Sill* 
 fllj 
 
 
 
 Ibs. 
 
 Ibs. 
 
 
 
 , 
 
 , 
 
 , 
 
 
 
 1 
 
 38-8626 
 
 5485 
 
 5161 
 
 5361 
 
 23 80 
 
 
 12 80 
 
 8961 
 
 Triangular 
 
 2. 
 
 36-8690 
 
 8450 
 
 4887 
 
 4951 
 
 15 30 
 
 
 8 
 
 98114 
 
 
 3 
 
 87-8568 
 
 5877 
 
 1-1724 
 
 1-4503 
 
 24 
 
 
 13 
 
 88512 
 
 
 4. 
 
 88-2911 
 
 5789 
 
 1-2678 
 
 1-8460 
 
 25 
 
 
 
 13 80 
 
 9380 
 
 Circular 
 model. 
 
 1. 
 2. 
 8 - 
 
 19T-18 
 197-18 
 255-43 
 
 2-8225 
 1-9570 
 1-9570 
 
 7-8761 
 8-2486 
 1-7727 
 
 7-894 
 8-122 
 1-7667 
 
 26 
 17 
 10 
 
 24 20 
 16 22 
 10 
 
 18 
 9 
 4 30 
 
 
 In the experiments with the smaller triangular model the differences^ 
 between the results and those given by the formula are much greater tba&i 
 in the experiments with the heavier cylindrical vessel. 
 
 In explanation of this difference, it will be observed, first, that t.he <aon-- 
 ditions of the experiment with the cylindrical model more nearly approach 
 to those which are assumed in the formula than those with, the, other ; the 
 disturbance of the water in the change of the position of the former being 
 less, and therefore the work expended upon the inertia of the water, of 
 which the formula takes no account, less in the one case than . the other ; 
 and, secondly, that the weight of the model being greater, this inertia 
 bears a less proportion to the amount of work required' for inclining it 
 than in the other case. 
 
 The effect of this inertia adding itself to the buoyancy of the fluid, 
 cannot but be to lift the vessel out of the water ; and to cause the. -displace- 
 ment to be less at the termination of each rolling oscillation than at its 
 commencement.* This variation in volume of the displacement .was appa- 
 rent in all the experiments. Its amount was measured and is recorded 
 in the last column of the Table ; its tendency is to produce in the body 
 vertical oscillations, which are so far independent of its rolling motion 
 that they will not probably synchronise with it. The body, displacing, 
 when rolling, less fluid than it would at rest, the effect of the weight* 
 used in the experiments to incline it is thereby increased, and thus is 
 explained the fact (apparent in the eighth and ninth columns of the Table) ; 
 that the inclination by experiment is somewhat greater than the formula 
 would make it. 
 
 The dynamical stability of a vessel whose athwart sections (where they 
 
 * This result connects itself with the well-known fact of the rise of a vessel 
 out of the water when propelled rapidly, which is so great in the case of fast 
 track-boats, as considerably to reduce the resistance upon them. 
 
 41 
 
APPENDIX. 
 
 are subject to immersion and emersion) are circular, 
 a common axis. 
 
 Fig.l. 
 
 in 
 
 Fig. 2. 
 
 Let EDF, fig. 1. or 2., be an athwart section of such a vessel, the 
 parts of whose periphery ES and FR, subject to immersion and emersion, 
 are parts of the same circular arc ETF, whose centre is 0. Let G, repre- 
 sent the projection of the centre of gravity of the vessel on this section, 
 and G 2 that of the centre of gravity of the space whose section is SDRT, 
 supposing it filled with water. The space lies wholly within the vessel in 
 fig. 1. and without it in fig. 2. Let 
 
 A, = CG,, Ji t = CG,. 
 
 W, = weight of vessel. 
 
 W 2 = weight of water occupying, or which would occupy, the space 
 whose section is STRD. 
 
 6 = the inclination from the vertical. 
 
 Since in the act of the inclination of the vessel the whole volume of 
 the displaced fluid remains constant, and also that volume of which STRD 
 is the section,* it follows that the volume of that portion of which the 
 circular area PSRQ is the section remains also constant, and that the 
 water-line PQ, which is the chord of that area, remains at the same dis- 
 tance from 0, so that the point neither ascends nor descends. Now the 
 forces which constituted the equilibrium of the vessel in its vertical posi- 
 tion were its weight and that of the fluid it displaced. Since the point 
 is not vertically displaced, the work of the former force, as the body 
 inclines through the angle 0, is represented by Wj A, vers. 8. The work 
 of the latter is equal to that of the upward pressure of the water which 
 would occupy the space of which the circular area PTQ is the section 
 increased, in the case represented in fig. 1., by that of the water which 
 would occupy STRD; and diminished by it in the case represented in 
 fig. 2. 
 
 But since the space, of which the circular area PTQ is the section, 
 remains similar and equal to itself, its centre of gravity remains always 
 at the same distance from the centre 0, and therefore neither ascends 
 nor descends. "Whence it follows that the work of the water which 
 would occupy this space is zero ; so that the work of the whole displaced 
 fluid is equal to that- of the part of it which occupies the space STRD, 
 
 * It will be observed that the space STRD is supposed always to be under 
 water. 
 
ON THE ROLLING OF SHIPS. -. 6-t3 
 
 taken in the case represented in fig. 1 . with the positive, and in that re- 
 presented in fig. 2. with the negative sign. It is represented, therefore, 
 generally by the formula W, A 2 vers. 0. On the whole, therefore, the 
 work 2u 2 of those forces, which in the vertical position of the body con- 
 stituted its equilibrium, is represented by the formula 
 SUz = W, A, vers. 6 W 2 A 2 vers. 8. 
 
 Representing, therefore, the dynamical stability 2^ by U (0), we have by 
 equation (2. p. 122.) 
 
 U (9) = (W, A, T W 2 & 2 ) vers. 9, 
 
 in which expression the sign ~-f is to be taken according as the circular 
 area ATB lies wholly within the area ADB, as in fig. 1 , or partially with- 
 out it, as in fig. 2. Other things being the same, the latter is therefore a 
 more stable form than the other. 
 
 13. The work of the upward pressure of the water upon the vessel 
 represented in fig. 2. being a negative quantity, W 2 A Z vers. 0, it follows 
 that the point of application of the pressure must be moved in a direction 
 opposite to that in which the pressure acts ; but the pressure acts upwards, 
 therefore its point of application, i. e. the centre of gravity of the displaced 
 fluid, descends. This property may be considered to distinguish mechani- 
 cally the class of vessels whose type is fig. 1., from that class whose type is 
 fig. 2. ; as the property of including wholly or only partly, within the area 
 of any of their athwart sections, the corresponding circular area ETF, dis- 
 tinguishes them geometrically. 
 
 The dynamical stability of a vessel of any given form subjected to a roll- 
 ing or pitching motion. 
 
 Conceive the vessel, after having completed an oscillation in any given 
 direction being then about to return towards its vertical position to 
 be for an instant at rest, and let RS represent the 
 intersection of its plane of flotation then, and PQ 
 of its flotation when in its vertical position, with 
 a section CAD of the vessel perpendicular to the 
 mutual intersection of these planes. The sec- 
 tion CAD will then be a vertical section of the 
 vessel. 
 
 Let G be the projection upon it of the vessel's 
 centre of gravity when in its vertical position. 
 
 H, that of the centre of gravity of the fluid displaced by the vessel in the 
 vertical position. 
 
 g, that of the fluid displaced by the portion of the vessel of which QOS 
 is a section. * 
 
 h, that of the fluid which would be displaced by the portion, of which 
 FOR is a section, if it were immersed. 
 
 GM, HN, gm, Jin, KL, perpendiculars upon the plane RS. 
 
 W = weight of vessel or of displaced fluid. 
 
 to = weight of water displaced by either of the equal portions of the 
 vessel of which FOR and QOS are sections. 
 
APPENDIX. 
 
 H! = depth of centre of gravity of vessel in vertical position. 
 
 H, = depth of centre of gravity of displaced water in vertical 
 
 position. 
 
 Hj = elevation of centre of gravity of vessel. 
 Hj = elevation of centre of gravity of displaced water. 
 P = area of plane PQ. 
 = inclination of planes PQ and RS. 
 *l = inclination of line in which planes PQ and RS intersect, 
 
 to that line about which the plane PQ is symmetrical. 
 h = perpendicular distance of line O from centre of gravity of 
 
 plane PQ. 
 = inclination to horizon of line about which the plane PQ is 
 
 symmetrical. 
 
 x = distance of section CAD, measured along the line whose 
 projection is O, from the point where that line intersects 
 the midship section. 
 
 y, = RS. 
 z = Tin + mg. 
 jt = KL. 
 
 I = moment of inertia of plane PQ about axis 0. 
 
 A and B = moments of inertia of PQ about its principal axes. 
 p, = weight of a cubic unit of water. 
 
 Suppose the water actually displaced by the vessel to be, on the contrary, 
 contained by it; and conceive that which occupies the space QOS to pass 
 into the space POR, the whole becoming solid. Let AH 3 represent the 
 corresponding elevation of the centre of gravity of the whole contained 
 fluid. Then will AH 2 + AH 3 represent the total elevation of the centre of 
 gravity of this fluid as it passes from the position it occupied when the 
 vessel was vertical into the position PAQ. But this elevation is obviously 
 the same as though the fluid had assumed the solid state in the vertical 
 position of the body, and the latter had revolved with it, in that state, into 
 its present position. It is therefore represented by KH NH ; 
 
 /. AH 2 + AH 3 = KH NH and AH 3 = KH NH AH 2 . 
 
 Since, moreover, by the elevation of the fluid in QOS, whose weight is o, 
 into the space OPR, and of its centre of gravity through (gm + Tin\ the 
 centre of gravity of mass of fluid of which it forms a part, and whose weight 
 is W, is raised through the space AH 3 ; it follows, by a well-known property 
 of the centre of gravity of a system,* *that 
 
 * The line joining the centres of gravity of the vessel and its immersed part, 
 in its vertical position, is parallel to the plane CAD, for it is perpendicular to 
 the plane PQ, to vhose intersection with the plane RS the plane CAD is per- 
 pendicular ; . . GK = H, and HK = H* 
 
ON THE EOLLING OF SHIPS. 645 
 
 W. A H 3 = w (gm + hri) 
 .-. W(KH NH ~ A H,) = w (gm + hri). 
 Bnt 
 
 NH = KHcos.0 KL = H 2 cos. 021 ; 
 
 .-. KH N"H = H 2 vers. 9 + *, 
 and 
 
 .-. W (H 2 vers. + ?i A H 2 ) = wz ; 
 
 .-. W. A H, = W (H 2 vers. +x) wz . . . . (28). 
 Also A H! = KG MG = H, (H, cos. ji) = H, vers. + x ; 
 
 /. W (A H, A H 2 ) = W (H, H 2 ) vers. +wz ; 
 
 /. (equation 2T.) U (0, ?) = W (H, H a ) vers. +;... (29). 
 If aj3 be a vertical prismatic element of the space QOS, whose base 
 is dx dy cos. 0, and height y sin. then will w .mg be represented, in 
 
 respect to that element, by p,y sin. 0. c c?y cos. 0-%y sin. 0, or by ^ ju sin. a 
 
 cos. 6 y*dx dy ; and wz will be represented, in respect to the whole space 
 of which PrsQ is the section, by 
 
 2 p sin. 8 e cos. Qjjy^ dy, 
 
 or by 2^ sm - a cos - e - 1- 
 
 If therefore we represent by $ the value of wz, in respect to the spaces 
 of which the mixtilinear areas PRr and QSs are the sections, we have 
 
 wz = ^fil sin.* e cos. e + <j,. 
 
 But the axis O, about which the moment of inertia of the plane PQ is 
 I, is inclined to the principal axes of that plane at the angles y and ^ tj, 
 
 about which principal axes the moments of inertia are A and B, 
 /. I = A cos. 2 n + B sin. 2 tj + PA 2 , 
 
 W (H t H 2 ) vers. + ' 2 > (A cos. 2 ^ + B sin. 2 ; + PA 2 ) sin 2 9 cos. + 9 ... (30). 
 It has been shown by M. DUPIN* that when 6 is small the line in 
 
 * Sur la Stabilite des Corps Flottarits, p. 32. In calculations having refer- 
 ence to the stability of ships, it is not allowable to consider extremely small, 
 except in so far as they have reference to the form of the ship immediately 
 about the load-water line. The rolling of the ship often extends to 20 or 30, 
 and is therefore largely influenced by the form of the vessel beyond these 
 limits. Generally, therefore, equation 30. is to be taken as that applicable to 
 the rolling of ships, those which follow being approximations only applicable 
 to small oscillations, and not sufficiently near (excepting equation 37) for 
 practical purposes 
 
APPENDIX. 
 
 which the planes PQ or RS intersect passes through the centre of gravity 
 
 of each ; in this case 
 
 /. I = A cos. 2 q + B sin. 2 q ; 
 
 therefore by equation (30), 
 
 If e be so small that the spaces PrR and QsS are evanescent in compari- 
 son with POr and QO, then, assuming <j> = and cos. = 1, 
 
 U (0, rj) = W (H, H 2 ) vers. e + - p (A cos. 2 n + B sin. 2 n } sin. 2 0, ... (3 1), 
 which may be put under the form 
 
 U (0, if ) = j W (H, H 2 ) + JK ( A cos. 2 n + B sin. 2 *?) j vers. 0. 
 Again, since 
 
 sin. = sin. flsin.j?, .... (32), 
 and (A cos. 2 ^ + B sin. 2 17) sin. 2 = {A + (B A)sin. 8 ;}sm. 2 0, 
 
 .-. (A cos. 2 n + B sin.'jf) sin. 2 = A sin. 2 e -f (B - A) sin. a C ; 
 /.by equation 31, 
 
 H 2 )vers.0-f >{Asin. 2 + (B A)sin. 2 ^}, ---- (33), 
 
 ty which formula the dynamical stability of the ship is represented, loth 
 as it regards a pitching and a rolling motion. 
 
 If in equation 31. y = -, the line in which the plane PQ (parallel to the 
 2 
 
 deck of the ship) intersects its plane of flotation is at right angles to the 
 length of the ship, and we have, since in this case = (see equation 32.), 
 
 U(0 = W(H, HO vers+ I^Bsin^ ..... (34), 
 
 which expression represents the dynamical stability, in regard to a pitch- 
 ing motion alone, as the equation 
 
 U(0) = W(H, H 2 ) vers 0+ ~p Asin 2 ..... (35), 
 
 represents it in regard to a rolling motion alone. 
 
 16. If a given quantity of work represented by TJ() be supposed to 
 be done upon the vessel, the angle through which it is thus made to 
 
 roll may be determined by solving equation 35. with respect to sin.-. 
 We thus obtain 
 
 2 2^ A.U(0). . . (36). 
 
ON THE ROLLING OF SHIPS. 617 
 
 17. If PR and QS be conceived to be straight lines, so that POR and 
 QOS are triangles, then w. z, taken in respect to an element included 
 between the section CAD, and another parallel to it and distant by the 
 small space dx, is represented by 
 
 or, since mg + nh=- y l sin.0, 
 
 3 
 
 by _ 
 
 12 
 
 .-.wz = p sin. 2 I y] 
 
 12 J 
 
 and, equation 29 
 
 U(0,)= W(Hj H 2 ) vers.o + n&v.?e fy]y$x, . . . (37), 
 24 v 
 
 which formula may be considered an approximate measure of the stability 
 of the vessel under all circumstances. 
 
 If, as in the case of the experiments of Messrs. FINCHAM and RAWSON, 
 the vessel b| prismatic and the direction of the disturbance perpendicular 
 to its axis, 
 
 y = constant = a, and z = a sin.fi ; 
 3 
 
 ..wz = a/w sin. 0, and 
 3 
 
 !ow> sinA 
 3 
 
 A rigid surface on which the vessel may ~be supposed to rest whilst in the 
 act of rolling. 
 
 If we imagine the position of the centre of gravity of a vessel afloat 
 to be continually changed by altering the positions of some of its con- 
 tained weights without altering the weight of the whole, so as to cause 
 the vessel to incline into an infinite number of different positions dis- 
 placing, in each, the same volume of water, then will the different planes 
 of flotation, corresponding to these different positions, envelope a curve 1 
 surface, called the surface of the planes of flotation (surface des flotaisons), 
 whose properties have been discussed at length by M. DUPIN in his ex- 
 cellent memoir, Sur la Stabilite des Corps Flottants, which forms part of 
 his Applications de Geometric.* So far as the properties of this surface 
 concern the conditions of the vessel's equilibrium, they have been ex- 
 hausted in that memoir, but the following property, which has reference 
 
 * BACHBLIEB, Paris, 1822. 
 
648 
 
 rather to the conditions of its dynamical stability than its equilibrium, is 
 not stated by M. DUPIN : 
 
 If we conceive the surface of the planes of flotation to become a rigid 
 surface, and also the surface of the fluid to become a rigid plane without 
 friction, so that the former surface may rest upon the latter and roll and 
 slide upon it, the other parts of the vessel being imagined to be so far im- 
 material as not to interfere with this motion, but not so as to take away 
 their weight or to interfere with the application of the upward pressure of 
 the fluid to them, then will the motion of the vessel, when resting by this 
 curved surface upon this rigid but perfectly smooth horizontal plane, be 
 the same as it was when, acted upon by the same force, it rolled and pitched 
 in the fluid. 
 
 In this general case of the motion of a body resting by a curved sur- 
 face upon a horizontal plane, that motion may be, and generally will be, 
 of a complicated character, including a sliding motion upon the plane, 
 and simultaneous motions round two axes passing through the point of 
 contact of the surface with the planes and corresponding with the rolling 
 and pitching motion of a ship. It being however possible to determine 
 these motions by the known laws of dynamics, when the form of the 
 surface of the planes of flotation is known, the complete solution of the 
 question is involved in the determination of the latter surface. 
 
 The following property*, proved by M. DUPIN in the memoir before 
 referred to (p. 32), effects this determination : 
 
 " The intersection of any two planes of flotation, infinitely near to each 
 other, passes through the centre of gravity of the area intercepted upon 
 either of these planes by the external surface of the vessel." 
 
 If, therefore, any plane of flotation be taken, and the centre of gravity 
 of the area here spoken of be determined with reference to that plane of 
 flotation, then that point will be one in the curved surface in question, 
 called the surface of the planes of flotation, and by this means any number 
 of such points may be found and the surface determined. 
 
 The axis about which a vessel rolls may be determined, the direction in 
 which it is rolling being given. 
 
 If, after the vessel has been inclined through any angle, it be left to 
 itself, the only forces acting upon it (the inertia of the fluid being neglected) 
 will be its weight and the upward pressure of the fluid it displaces ; the 
 motion of its centre of gravity will therefore, by a well-known principle 
 of mechanics, be wholly in the same vertical line. 
 
 Let HE represent this vertical line, PQ the surface of the fluid, and 
 aMb the surface of the planes of flotation. As the centre of gravity G 
 traverses the vertical HK, this surface will partly roll and partly slide 
 by its point of contact M on the plane PQ. 
 
 If we suppose, therefore, PRQ to be a section of the vessel through 
 
 * This property appears to have been first given by EULBR. 
 
ON THE ROLLING OF SHIPS. 
 
 619 
 
 the point M, and perpendicular to the axis about which it is rolling, and 
 if we draw a vertical line MO through the point M, and through G a 
 horizontal line GO parallel to the plane PRQ, then 
 the position of the axis will be determined by a line 
 perpendicular to these, whose projection on the plane 
 PRQ is O. 
 
 For since the motion of the point G is in the verti- 
 cal line HK, the axis about which the body is revolv- 
 ing passes through GO, which is perpendicular to 
 HK ; and since the point M of the vessel traverses 
 the line PQ, the axis passes also through MO, which 
 is perpendicular to PQ ; and GO is drawn parallel to, and MO in the 
 plane PRQ, which, by supposition, is perpendicular to the axis, therefore 
 the axis is perpendicular to GO and MO. 
 
 If HK be in the plane PRQ, winch is the case whenever the motion is 
 exclusively one of rolling or one of pitching, the point is determined by 
 the intersection of GO and MO. 
 
 The time of the rolling through a small angle of a vessel whose athwart 
 sections are (in respect to the parts subject to immersion and emersion) 
 circular, and have their centres in the same longitudinal axis. 
 
 Let EDF (fig. 1. or fig. 2J> represent the midship section of such a 
 
 vessel, in which section let the centre of gravity GI be supposed to be situ- 
 ated, and let HK be the vertical line traversed by G, as the vessel rolls. 
 Imagine it to have been inclined from its vertical position through a given 
 angle QI and the forces which so inclined it then to have ceased to act 
 upon it, so as to have allowed it to roll freely back again towards its posi- 
 tion of equilibrium until it had attained the inclination OOD to the verti- 
 cal, which suppose to be represented by 0. 
 
 Referring to equation 1. page 123. let it be observed that in this case 
 2w 2 =0, so that the motion is determined by the condition 
 
 Zu^ Zwv* (38). 
 
 But the forces which have displaced it from the position in which it 
 was, for an instant, at rest are its weight and the upward pressure of the 
 
650 APPENDIX. 
 
 water; and the work of these, U(0,) U(0), done between the inclinations 
 6 and 0, when the vessel was in the act of receding from the vertical, was 
 shown to be represented by (W A =F Wz^j) (vers. vers. 0,) ; therefore 
 the work, between the same inclinations, when the motion is in the 
 opposite direction, is represented by the same expression with the sign 
 changed ; 
 
 (vers. 0, vers. 0), 
 
 and since the axis about which the vessel is revolving is perpendicular to 
 the plane EDF, and passes through the point O, if W,& 2 represents its 
 moment of inertia about an axis perpendicular to the plane EDF, and 
 passing through its centre of gravity G 1} 
 
 Substituting in equation 38. and writing for OG, its value A, sin. <?, we 
 have 
 
 vers. 0, vers. fl)= - 1 (*+Amn.0) ( ) ; 
 
 . . . (39). 
 
 or assuming to be so small that the fourth and all higher powers ol 
 sin. may be neglected, and observing that, this being the case, 
 
 y 
 
 * sec. 2 
 
 . 
 
ON THE ROLLING OF SHIPS. 651 
 
 ~ lc \J 1 + ~V~ sin -V-' 
 
 W r tfnr^ I ' / d sin. i 
 
 . 8 -^ sin. a - 
 
 and 
 
 4 
 
 / 
 
 
 The sign + being taken according as the centre of gravity of the displaced 
 fluid ascends or descends. 
 
 The time of a vessel's rolling or pitching through a small angle, its form 
 and dimensions ~being any whatever. 
 
 Let EDF (figs. 1. or 2.) represent the midship section of such a vessel, 
 supposed to be rolling about an axis whose projection is ; and let 
 represent the centre of the circle of curvature of the surface of its planes 
 of flotation at the point M where that surface is touched by the plane PQ, 
 being above the load water-line AB in fig. 1, and beneath it in fig. 2. Let 
 the radius of curvature CM be represented by p ; then adopting the same 
 notation as in the last article, and observing that the axis O about which 
 the vessel is turning is perpendicular to EDF, we shall find its moment of 
 inertia to be represented by 
 
 where H, represents the depth of the centre of gravity in the vertical posi- 
 tion of the vessel. 
 
652 APPENDIX. 
 
 Also, by equation 35. 
 ^ i=U0i )__U(0)=W I (H 1 H 2 )(cos.0 cos.eO + - 
 
 ..by equation 38. 
 
 IP + (H, p) 2 sin. 2 
 
 (H,-H 2 ) (cos. 0-cos. e t ) +: cos. 8 cos. 1 *,) 
 
 p) a 8iD.'Q 
 
 j H! H 2 + ^% (COS. + COS. 0,) U COS. COS. 0, j. 
 
 Assuming and 9 l to be so small that cos. + cos. e l = 2, and observing 
 that 
 
 cos> e cos. 0j = vers. 0! vers. 0, 
 
 Supposing, moreover, p to remain constant between the limits 0, and 
 and integrating as in equation 39. 
 
 iD ,, . . . (41). 
 
 Since the value of sin.* ~ Q l is exceedingly small, the oscillations are 
 
 A . 
 
 nearly tautochronous, and the period of each is nearly represented by the 
 formula 
 
 .... (42.) 
 
EQUILIBRIUM OF PRESSURES. 653 
 
 The following method is given by M. DTJPIN for determining the value 
 ofp*: 
 
 " If the periphery of the plane of flotation be imagined to be loaded at 
 every point with a weight represented by the tangent of the inclination of 
 the sides of the vessel at that point to the vertical, then will the moments 
 of inertia of that curve, so loaded, about its two principal axes, when 
 divided by the area of the plane of flotation, represent the radii of greatest 
 and least curvature of the envelope of the planes of flotation." 
 
 If p be taken to represent the radius of greatest curvature, the formula 
 41. will represent the time of the vessel's rolling; if the radius of least 
 curvature (B being also substituted for A), it will represent the time of 
 pitching. 
 
 NOTE D. 
 
 On the conditions of the equilibrium of any number of pressures in the 
 same plane, applied to a body movedble about a cylindrical axis in the state 
 "bordering upon motion. (From a memoir on the Theory of Mechanics, 
 printed in the second part of the Transactions of the Royal Society for 1841.) 
 
 LET PI, P 2 , P 3 , &c. represent these pressures, and R their resultant. Also 
 let a l: a 2 , 3 , represent the perpendiculars let fall upon them severally from 
 the centre of the axis, those perpendiculars being token with the positive 
 signs whose corresponding pressures tend to turn the system in the same 
 direction as the pressure P,, and those negatively which tend to turn it in 
 the opposite direction. Also let A, represent the perpendicular distance of 
 the direction of the resultant R from the centre of the axis, then, since R 
 is equal and opposite to the resistance of the axis, and that this resistance 
 and the pressures P,, P 2 , P 3 , &c. are pressures in equilibrium, we have by 
 the principle of the equality of moments, 
 
 PII + P 2 2 + P^ 3 + &c. = *,R. 
 
 Representing, therefore, the inclinations of the directions of the pressures 
 PI, P 2 , P 3 , &c. to one another by < r2 , , . i 23 , f, &c., &c., and substituting 
 for the value of R.J 
 
 * Applications de Geometrie, p. 47. 
 
 f The inclination i,. a of the directions of any two pressures in the above ex- 
 pression is taken on the supposition that both the pressures act from, or both 
 towards the point in which they intersect, and not one towards, and the other 
 from, that point ; so that in the case represented in the figure in the note at p. 
 171., the inclination ,., of the pressures P, and P 2 , represented by the arrows, 
 is not the angle P, IP 2 , but the angle P,IQ, since IQ and IP, are directions of 
 these pressures, both tending from this point of intersection, whilst the direc- 
 tions of P 2 T and IP, are one of them towards that point, and the other from it 
 
 \ POISSON, Mecanique, Art. 33. 
 
APPENDIX. 
 
 .p.= 
 
 + 2 P,P 2 cos. t 1<2 + 2 PjPs cos. i,., + 
 + 2 P 2 P 3 cos. i 2 . 3 + 2 P 2 P 4 cos. t2M + 
 
 + &C. &C. 
 
 f P, 2 + 2 P! (P 2 COS. ,., + P 3 COS. 
 
 + &C. &C. 
 
 If the value of P, involved in this equation be expanded by Lagrange's 
 theorem *, in a series ascending by powers ofa,, and terms involving powers 
 above the first be omitted, we shall obtain the following value of that 
 quantity : 
 
 P _ 
 
 1 * 
 
 a, 
 
 or reducing, 
 P,= 
 
 (P 2 COS.t,. 2 
 
 + 2 P a P 3 cos. i 2 . 3 + 2 P 2 P 4 cos. <2. 4 
 + 2P 3 P 4 cos. 3 . 4 + ---- 
 
 . t r2 + a,, 2 ) 
 
 P 2 2 (, 2 
 
 + &c. &c. 
 
 + 2 P, P 3 {0 2 3 ,(, cos.t 2 . 3 + 2 cos. t,. 8 + 3 cos. i, 
 
 4-2P t P 4 {aa 4 
 
 + &C. &C. 
 
 Now a 7 2,a 2 cos. ,. 2 + 2 represents the square of the line joining the 
 feet of the perpendiculars <z, and 2 let fall from the centre of the axia 
 upon Pj and P 2 ; similarly a\ 2a,a 3 cos. 1,., + ag represents the square of 
 the line joining the feet of the perpendiculars let fall upon P! and P 3 , and 
 
 * This expansion may be effected by squaring both sides of the equation, 
 solving the quadratic in respect to P,, neglecting powers of > above the first 
 and reducing ; this method, however, is exceedingly laborious. 
 
ROLLING MOTION OF A CYLINDER. 655 
 
 BO of the rest. Let these lines be represented by L L2 , L t 3 , L,. 4 , &c., and let 
 the different values of the function 
 
 a l #, cos. 
 be represented by M 2 . 3 , M 2 . 4 , M 3 . 4 , &c. 
 
 NOTE E. 
 
 ON THE KOLLING MOTION OF A CYLINDER. 
 
 (From a memoir printed in the Transactions of the Royal Society for 
 1851, part II.) 
 
 THE oscillatory motion of a heterogeneous cylinder rolling on a horizontal 
 plane has been investigated by EULER.* He has determined the pressure 
 of the cylinder on the plane at any period of the oscillation, and the time 
 of completing an oscillation when the arcs of oscillation are small. 
 
 The forms under which the cylinder enters into the composition of 
 machinery are so various, and its uses so important, that I have thought it 
 desirable to extend this inquiry, and in the following paper I have sought 
 to include in the discussion the case of the continuous rolling of the cylin- 
 der, and to determine 
 
 1st. The time occupied by a heterogeneous cylinder in rolling continu- 
 ously through any given space. 
 
 2ndly. The time occupied in its oscillation through any given arc. 
 
 Srdly. Its pressure, when thus rolling continuously, on the horizontal 
 plane on which it rolls. 
 
 Under the second and third heads this discussion has a practical appli- 
 cation to the theory of the pendulum ; determining the time occupied in 
 the oscillations of a pendulum through any given arc, whether it rests 
 on a cylindrical axis or on knife-edges, and the circumstances under 
 which it will jump or slip on its bearings ; and under the first and third, 
 to the stability and the lateral oscillations of locomotive engines in rapid 
 motion, whose driving-wheels are, by reason of their cranked axles, untruly 
 balanced. 
 
 * Nova Acta Acad. Petropol. 1788. " De motu oscillatorio circa axcm cylin- 
 dricum piano horizontal! incumbentem." 
 
APPENDIX, 
 
 Let AMB represent the section of a heterogeneous cylinder through 
 its centre of gravity G and perpendicular to its axis ; and let M be its 
 point of contact, at any time, with the hori- 
 zontal plane BD on which it is rolling. 
 Assume 
 
 a = AC, h = CG, = ACM. 
 W = weight of cylinder. W&* = momen- 
 tum of inertia of the cylinder ahout 
 an axis passing through G and 
 parallel to the axis of the cylinder. 
 
 w =- given value of the angular velocity ( ) when e has the given 
 
 . \ dt / 
 
 value 0j. 
 
 y, = given value of 6 when the angular velocity has the given value . 
 I = given value of GM corresponding to the value 0, of 9. 
 
 Then W (#* + GM 2 ) = W(P + a? + h* 2ah cos. 0) = moment of inertia 
 ahout M. Since moreover the cylinder may be considered to be in the act 
 of revolving about the point M by which it is in contact with the plane, 
 one-half of its vis viva is represented by the formula 
 
 and one-half of the vis viva acquired by it in rolling through the angle 
 0, 0,by 
 
 _ 2aA cos. + #)* 
 
 But the vertical descent of the centre of gravity while the cylinder is 
 passing from the one position into the other, is represented by 
 
 h (cos. cos. 0j). 
 Therefore, by the principle of vis viva,* 
 
 I -j (V + a 9 2ah cos. e + F/ -) (F + 1 7 ) w 2 i = W (cos. e - cos. 0,), 
 
 whence we obtain 
 
 (cos. 6 cos. 0Q + (ff + P) co 8 
 
 & V _ 
 
 dt/ ~ 
 
 cos. e ( cos. 0, ^-1 w 2 
 
 fff\ V * 2gh 
 
 \aj I (ft a h\ 
 
 * POK60N, Dynamique, 2 me partie, 565. ; PONCELET, Mecanique Induatrielle, 
 or Art. (129.) of this Work. 
 
ROLLING MOTION OF A CYLINDER. 657 
 
 ^ 
 
 .3 
 
 
 
 where t represents the time of the body's passing from the inclination 0j to 
 zero. 
 
 Now let it be observed that in this function a>3 so long as a is less than 
 g, since 
 
 # + ?>_(# + p) M ? or ff + a* 2ah cos. 0, + #> 
 and . . #* + a 2 + A 8 > 2A cos. (F + Z> 8 , 
 
 a h 
 and 
 
 1+a 1 a a COS. 
 
 1_0 cos.0 p 
 
 Then when = 0, # 2 sec. 8 = 7 - = ?*, . ' . sec. = 1 and = 0. 
 
 1 p 
 
 When 9 = 0! let = 0,, 
 
 af Tfc4 
 
 _ a cos. 0i 
 
 also 
 . __ l-q_ 
 
 = 
 
 
 a cos. 
 And since - 
 
 t 2 cos. 
 
 (a j8) ~cos.0 + ? 
 
 (a + 3)(cos. 8 ^ + g 2 ) + (a J3)(C08.' 9f) 
 -.2008.0 = (COS. 2 ^^ 2 ) 
 
658 APPENDIX. 
 
 2 . (cos. 2 y + g*) 2 -(<* cos. 2 
 
 (cos.* + * a cos. 2 
 
 Now 
 
 ^0 _ <Z0 <? cos. <? cos. ^ _ sin. ^ <? cos. 5 
 
 ^ ~~^ cos. * d cos. 9 ' ^> ~ sin. " ^ cos. $ ' ' " " ' *' 
 
 Also by equation (6.), 
 
 2(aC03. 2 +j3^ 2 ) COS. ? >_2(a - 13)^008. 
 
 ^ cos. $~~ (g* + cos. 2 
 
 . . by equations (7.) and (8.), 
 <?9 __ 2(g 3)g 2 
 
 <^~ (1 
 
 2(g J3)g COS. 
 
 /a COS. ') y^_2fg j3)^ 2 j _ 1 _ I 
 
 ' ' V cos. j3 ) Tt>~ (1 j3*)*" ' ( (^4-cos. 2 ?.) (^ 2 +^> 2 cos. 8 0)j j 
 
 2(a p)g 2 j _ 1 _ | 
 (1 (3 2 )J ' j^ + l-sin.^X^+p 2 ^ 2 sin. 2 ^)i j" 
 
 2(>-p)g' j _ 1 _ ) 
 
 ~(1 3'O i (^ + ? 2 K 1 + ^)( (1 wsin. 2 ^)(l c 2 sin. 2 ^)i f 
 
 and 
 
 1 +a 
 
ROLLING MOTION OF A CYLINDER. 950 
 
 0i 
 
 y' /a cos. 0\*'< 
 ( ~0~^T# ) " 
 
 2(a /,_ 
 
 n sin. 2 0)(1 c 2 sin. 2 
 o 
 
 2(a % 2 
 
 where n( nc^) is that elliptic function of the third order whose par* 
 meter is n and modulus c. 
 
 1 /T-^" 
 
 
 # a 
 
 . . by equations 11. and 4. 
 
 where (9.) (2.) (3.) 
 
 1 cos. 0, 
 
 
 
 * I cannot find that this function has before been integrated, except in the 
 case in which is exceedingly small. 
 
APPENDIX, 
 and (10.) (2.) (3.) 
 
 .j 
 
 (a/?) 
 
 The value of n( TMJ^) being determinable by known methods 
 DBE, Fonctions Elliptiques, vol. i. chap, xxxiii.), the time of rolling is given 
 by equation 13. 
 
 In the case in which the rolling motion is not continuous but oscillatory, 
 
 we have w = ; and therefore (equation 5.) $, = - ; n ( nc<p^) becomes 
 
 therefore in this case a complete function. 
 
 To express the value of this complete elliptic function of the third order 
 in terms of functions of the first and second orders, let 
 
 Then* 
 
 Eepresenting therefore the time of a semi-oscillation by 
 
 *!= 
 
 where (15.) t = JvZn vers - i (18). 
 
 Since the values of elliptic functions of the first and second orders, 
 having given amplitudes and moduli, are given by the tables of LEGENDEE,- 
 it follows that the value of t is given by this formula for all possible values 
 of <j and 4,. 
 
 If the angle of oscillation 0, be very small e is very small, so that its 
 square may be neglected in comparison with unity. In this case 
 
 * LBOBNDRE, Calcus des Fonctions Elliptiques, vol. i. chap, xxni Art. 116. 
 
ROLLING MOTION OF A CYLINDER. 661 
 
 = Ee* = 4, and Fc ? = Ec^ = 
 2 22 
 
 /. Fe - 04, Ec - Fo^ = 0. 
 2 2 
 
 For small oscillations therefore 
 
 2 
 
 If the pendulum oscillate on knife-edges a = 0, I h, and we obtain the 
 well-known theorem of LEGENDEK (Fonctions Elliptiques, vol. i. chap. 
 viii.) 
 
 where (18.) <?, = | vers." 0, = sin. 3 i 
 
 . . c = sin. - BI . . . . (21). 
 
 In the case of the small oscillations of a pendulum resting on knife-edge, 
 equation 20. becomes 
 
 which is the well-known formula applicable to that case. 
 If the pendulum be one which for small arcs beats seconds (21.), 
 
 .-.(20.) 2* = , (23). 
 
 by which equation the time of the oscillation through any arc, of a pen- 
 dnluni which oscillates through a small arc in one second, may be deter- 
 mined. I have caused the following table to be calculated from it. 
 
662 
 
 APPENDIX. 
 
 Table of the time occupied in oscillating through every two degrees of a 
 complete circle, by a pendulum which oscillates through a small arc in 
 one second. 
 
 Arc of oscillation 
 on each side of 
 the vertical 
 in degrees. 
 
 Time of one 
 complete 
 oscillation in 
 seconds. 
 
 Arc of oscillation 
 on each side of 
 the vertical 
 in degrees. 
 
 Time of one 
 complete 
 . oscillation in 
 seconds. 
 
 Arc of oscillation 
 on each side of 
 
 in degrees. 
 
 Time of one 
 complete 
 oscillation in 
 seconds. 
 
 2 
 
 1-1899 
 
 32 
 
 1-3905 
 
 62 
 
 1-8032 
 
 4 
 
 1-2000 
 
 34 
 
 1-4089 
 
 64 
 
 1-8478 
 
 6 
 
 1-2123 
 
 36 
 
 1-4283 
 
 66 
 
 1-8963 
 
 8 
 
 1-2210 
 
 38 
 
 1 -4486 
 
 68 
 
 1-9491 
 
 10 
 
 1-2322 
 
 40 
 
 1-4698 
 
 70 
 
 2-0075 
 
 12 
 
 1 2439 
 
 42 
 
 1-4922 
 
 72 
 
 2-0724 
 
 14 
 
 1-2560 
 
 44 
 
 1-5157 
 
 74 
 
 2-1453 
 
 16 
 
 1-2686 
 
 46 
 
 1-5405 
 
 76 
 
 2-2285 
 
 18 
 
 1-2817 
 
 48 
 
 1-5667 
 
 78 
 
 2-3248 
 
 20 
 
 1-2953 
 
 50 
 
 1-5944 
 
 80 
 
 2-4393 
 
 22 
 
 1-3099 
 
 52 
 
 1-6238 
 
 82 
 
 2-5801 
 
 24 
 
 1-3249 
 
 54 
 
 1-6551 
 
 84 
 
 2-7621 
 
 26 
 
 1-3400 
 
 56 
 
 1-6884 
 
 86 
 
 3-0193 
 
 28 
 
 1-3560 
 
 58 
 
 1-7240 
 
 88 
 
 3-4600 
 
 30 
 
 1-3729 
 
 60 
 
 1-7622 
 
 90' 
 
 Infinite. 
 
 The pressure of the cylinder on its point of contact with the plane on which 
 
 it rolls. 
 
 Let A' be the point where the point *A of the cylinder was in contact 
 with the plane. 
 
 Let A'N = x, NQ = y. 
 
 X = horizontal pressures on M in direction 
 A'M. 
 
 Y = vertical pressure on M in direction MC. 
 
 M M A' D Since the centre of gravity G moves as it would 
 
 do if, the whole mass being collected there, all the impressed forces were 
 applied to it, we have, by the principle of D'ALEMBEKT, 
 
 dt* 
 
 v 
 
 .... (28). 
 
 But since CA = a, OG = h, MOA = 0, 
 . * . x = ae h sin. 0, 
 y = a h cos. ; 
 dx dd 
 
 Art. 96, equations (73.), (74.). 
 
ROLLING MOTION OF A CYLINDER. 
 
 dy _ . dd 
 
 d'e 
 
 > . . . . (29). 
 
 . . by equation (29.), 
 
 = MA cos. ISta sin. ; 
 
 by equation (28.), 
 
 X= | 
 
 
 
 | cos.eU 
 
 (30). 
 
 But by equation (1.), substituting and B l for and 
 
 QQB.gQ + (y+P)o a 
 2a& cos. + A 2 ' * ; * 
 
 2afc(cos. cos. 0,) + (7c 2 + V} 
 
 g_ 
 
 2ah cos. 9 + h 2 
 
 * 
 
 k 2 + a? 2ah cos. + h 2 
 
 ' ~ 
 
 ..(32). 
 
 Observing that a 2 + ft 1 2ah cos. 0, = I 2 . 
 Differentiating this equation and dividing Ibtfl \ 
 
 
 Ism. 
 
 _ 
 
 (F + a 7 + 7?-2aA cos. 0) 2 
 
 C33X 
 
664 APPENDIX. 
 
 Substituting these values of M and N in equation (30.), and reducing, 
 
 _. WAsm.0( (F + Z'X&' + A 2 ahcos.e)(g + au> T )> 
 ~t~~ * * " 
 
 Whia 
 
 '- 2oA ooa. 
 
 The rotation of a "body about a cylindrical axis of small diameter. 
 
 Assuming a = in equations (31.), (33.), and 0!=0, we have 
 
 2gft(cos.0 1) 2 gr^ sin, e 
 
 2 = > 
 
 Therefore, by equation (30.), 
 WAjgr^ 3cos.e) 
 
 X =Y \ 
 
 The last equation may be placed under the form 
 
 If --I , w 2 1 J be numerically less than unity, whether it be positive 
 
 or negative, there will be some value of between and n for which this 
 expression will be equalled, with an opposite sign, by cos. 0, and for which 
 the first term under the bracket in the value of Y will vanish. This cor- 
 responds to a minimum value of Y represented by the formula 
 
 But if -/ w 2 1 J be numerically greater than nnity, then the 
 
 minimum of Y will be attained when = *, and when 
 
ROLLING Ml/i'ION OF A CYLINDEE. G65 
 
 The Jump of an Axis. 
 
 If Y be negative in any position of the body, the axis will obviously 
 jump from its bearings, unless it be retained by some mechanical expe- 
 dient not taken account of in this calculation. But if Y be negative in 
 any position, it must be negative in that in which its value is a minimum. 
 If a jump take place at all, therefore, it will take place when Y is a mini- 
 mum; and whether it will take place or not, is determined by finding 
 whether the minimum value of Y is negative. If therefore the expression 
 (42.) or (43.) be negative, the axis will jump in the corresponding case. 
 An axis of infinitely small diameter, such as we have here supposed, 
 becomes a fixed axis; and the pressure upon a fixed axis, supposed to 
 turn in cylindrical bearings without friction, is the same, whatever may 
 be its diameter; equations (40.) and (41.) determine therefore that pres- 
 sure, and equation (42.) or (43.) determines the vertical strain upon the 
 collar when the tendency of the axis to jump from its bearings is the 
 greatest. 
 
 The Jump of a Rolling Cylinder. 
 
 Whether a jump will or will not take place, has been shown to be deter- 
 mined by finding whether the minimum value of Y be negative or not. 
 
 Substituting a for-/ + - + f | and reducing, equation (35.) becomes 
 Z\ah a hi 
 
 y-W/1 *> WF + C+a* 8 cos. 2 e-2acos.0 + l 
 
 Wl- 
 
 .-.-^=0, Is ^ ^ ^ 
 
 Srdly, when = 0. 
 The first condition evidently yields a positive value of -r^-, since it 
 
666 APPENDIX. 
 
 causes the first term of the preceding equation to vanish ; and the second 
 term is essentially positive, a being always greater than unity. 
 
 If, therefore, the first condition be possible, or if there be any value of 
 6 which satisfies it, that value corresponds to a position of minimum pres- 
 sure. Solving, in respect to cos. 0, we obtain 
 
 The first condition will therefore yield a position of minimum pres- 
 sure, if 
 
 o i / ~rrz ^ . i or if 
 
 or if 
 
 or if 
 
 J*\ ( iv ^ (i* _L 7-^ ^ il ^ 
 
 t^a 1J ^AJ + I ) (a, I) a 
 
 and 
 
 or 
 
 whence, substituting for a and reducing, we obtain finally, the conditions' 
 \ {& 2 + (a + A)T Sg\ (g\ {#+( &)'}' f Q \ 
 
 k^m^^-)^^ 
 
 Of these inequalities the second always obtains, because 
 
 whatever be the values of &, a and A. And the first is always possible, 
 since 
 
 {F + (a + )<}* >(&* + ?*) }# + (<, _&)}. 
 
 ^ If the/rsi obtain, there are two corresponding positions of OA on either 
 ide of the vertical, determined by equation (46.), in which the pressure Y 
 of the cylinder upon the plane is a minimum. 
 
ROLLING MOTION OF A CYLINDER. 667 
 
 Substituting the other two values (rt and 0) of 6 which cause -j- to 
 
 vanish in the value of ^-^ we obtain the values 
 
 w 
 
 < h 
 
 a 2ga\* + 1) 2 la 2gd\a I) 2 
 
 or 
 
 _ a . oa 
 
 2 I) 2 $'* 
 
 which expressions are both negative if the inequalities (47.) obtain. The 
 same conditions which yield minimum values of Y in two corresponding 
 oblique positions of CA, yield, therefore, maximum values in the two ver- 
 tical positions ; so that if the inequalities (48.) obtain, there are two posi- 
 tions of maximum and two of minimum pressure. 
 
 Substituting the values of cos. (equation 46.) in equation (44.), and 
 reducing, we obtain for the minimum value of Y in the case in which the 
 inequalities (48.) obtain, 
 
 +3 
 
 If this expression be negative the cylinder will jump. 
 In the case in which = 0, which is that of a, pendulum having a cylin- 
 drical axis of finite diameter, it becomes 
 
 Y 5 
 
 If the first of the inequalities (48.) do not obtain, no position of mini- 
 mum pressure corresponds to equation (46.) ; and the inequalities (47.) do 
 
 not obtain, so that the values (49.) of -=-5-, given respectively by the sub- 
 stitution of rt and for 0, are no longer both negative, but the second only. 
 In this case the value rt of is that, therefore, which corresponds to a posi- 
 tion of minimum pressure, which minimum pressure is determined by 
 substituting rt for in equation (35.), and is represented by 
 
 * When the pendulum oscillates on knife-edges a=0, and this expression 
 assumes the form of a vanishing fraction, whose value may be determined by 
 the known rules. See the next article. 
 
668 APPENDIX. 
 
 . . . (51). 
 
 g ' w + (a+h)* 
 
 The cylinder will jump if this expression be negative, that is, if 
 
 i 
 
 or, substituting and reducing, if 
 
 If the angular velocity be assumed to be that acquired in the highest 
 position of the centre of gravity, 0i=rt, and cos. - 81 = 0. In this case, 
 therefore (equation 61.) 
 
 and there will be a jump if w ! > | . . . (53). 
 
 The Pendulum oscillating on Knife-edges. 
 
 In this case a is evanescent, and w=0. Equations (31.) and (33.) 
 become, therefore, 
 
 s. cos. 0,) __ 
 
 - 
 
 * + # 
 
 Substituting these values of M and K" in equation (30.), 
 
 _ ~Wh* r \ 
 
 # + # ] 2 ( cos ' 9 cos ' ') sm ' cos - 9 sin ' 9 1 
 
 Y= W + F+A 7 | ( cos - 9 ~ cos ' e ^ cos ' ~ sin ' 20 } ? 
 
 .". X = -^T^- 2 (2 cos. 0, 3 cos. 0) sin. ... (54). 
 
ROLLING MOTION OF A CYLINDER. 669 
 
 P\ 
 . 2 2008.0008.0,+-! * ' ' (55) ' 
 
 Y is a minimum when cos. 9 = -cos. 0,, in which case 
 
 3 
 
 There will therefore be a jump of the pendulum upon its b'earings at 
 each oscillation if the amplitude 0, of the oscillation be such, that 
 
 1 Z* Q 1* 
 
 icos. 2 0, > _, or cos. 2 0,>riL. 
 8 A 2 A 8 
 
 of the falsely -balanced Carriage-wheel. 
 
 The theory of the falsely-balanced carriage- wheel differs from that of 
 the rolling cylinder, 1st, in that the inertia of the carriage applied at its 
 axle influences the acceleration produced by the weight of the wheel, as 
 its centre of gravity descends or ascends in rolling; and, 2ndly, in that 
 the wheel is retained in contact with the plane by the weight of the car- 
 riage. The first cause may be neglected, because the displacement of the 
 centre of gravity is always in the carriage-wheel very small, and because 
 the angular velocity is, compared with it, very great. 
 
 If W, represent that portion of the weight of the carriage which must 
 be overcome in order that the wheel may jump (which weight is supposed 
 to be borne by the plane), and if Y! be taken to represent the pressure 
 upon the plane, then (equation 52.) 
 
 Y 1 = W,+Y=W 1 + w(l ) (57). 
 
 V ff ' 
 
 In order that there may be a jump, this expression must be negative, 
 or 
 
670 APPENDIX. 
 
 The Driving-Wheel of a Locomotive Engine. 
 
 The attention of engineers was some years since directed to the effects 
 which might result from the false balancing of a wheel by accidents on 
 railways, which appeared to be occasioned by a tendency to jump in the 
 driving-wheels of the engines. The cranked axle in all cases destroys the 
 balance of the driving-wheel unless a counterpoise be applied ; at that time 
 there was no counterpoise, and the axle was so cranked as to displace the 
 centre of gravity more than it does now. Mr. GEOKGE HEATON, of Bir- 
 mingham, appears to have been principally instrumental in causing the 
 danger of this false-balancing of the driving-wheels to be understood. By 
 means of an ingenious apparatus*, which enabled him to roll a falsely- 
 balanced wheel round the circumference of a table with any given velocity, 
 and to make any required displacement of the centre of gravity, he showed 
 the tendency to jump, produced even by a very small displacement, to be 
 so great, as to leave no doubt on the minds of practical men as to the 
 danger of such displacement in the case of locomotive engines, and a coun- 
 terpoise is now, I believe, always applied. To determine what is the 
 degree of accuracy required in such a counterpoise, I have calculated from 
 the preceding formulae that displacement of the centre of gravity of a 
 driving-wheel of a locomotive-engine, which is necessary to cause it to 
 jump at the high velocities not unfrequently attained at some parts of the 
 journey of an express train ; from such information as I have been able to 
 obtain as to the dimensions of such wheels, and their weights, and those 
 of the engines f. The weight of a pair of driving-wheels, six feet in 
 diameter, with a cranked axle, varies, I am told, from 2 to 3 tons ; and 
 that of an engine on the London and Birmingham Kailway, when filled with 
 water, from 20 to 25 tons. If n represent the number of miles per hour 
 at which the engine is travelling, it may be shown by a simple calculation, 
 
 that the angular velocity, in feet, of a six-feet wheel is represented by -=1 
 or by -n very nearly. In this case we have, therefore, since "W represents 
 
 the weight of a single wheel and its portion of the axle, and "VV, represents 
 the weight, exclusive of the driving-wheels, which must be raised that 
 
 * This apparatus was exhibited by the late Professor Co WPEK to illustrate his 
 Lectures on Machinery at King's College. It has also been placed by General 
 MORIN among the apparatus of the Conservatoire des Arts et Metiers at Paris. 
 
 \ I have not included in this calculation the inertia of the crank rods, of the 
 slide gearing, or of the piston and piston rods. The effect of these is to increase 
 the tendency to jiimp produced by the displacement of the centre of gravity 
 of the wheel ; and the like effect is due to the thrust upon the piston rod, 
 The discussion of these subjects does not belong to my present paper. 
 
ROLLING MOTION OF A CYLINDER. 
 
 G71 
 
 either side of the engine may jump*, that is, half the weight of the engine 
 exclusive of the driving-wheels, W = ! to H tons, W, = 8f to Hi tons, 
 
 w = -TO, ^ = 32-19084 whence I have made the following calculations from 
 formula (59.). 
 
 Height of 
 the engine in 
 tons, includ- 
 ing the driv- 
 ing wheels. 
 
 Weight of a 
 pair of wheels 
 with cranted 
 axle, in tons. 
 
 Formula (59.) 
 reduced. 
 
 J2 ,re(,4') 
 
 Displacement of the centre of gravity 
 of a six-feet driving-wheel which 
 will cause a jump of the wheel on 
 the rail. 
 
 Eate of travelling in miles per hour. 
 
 .* 
 
 50. 
 
 60. 
 
 70. 
 
 20 
 
 2-5 
 3 
 
 1030-08 
 
 4128 
 3434 
 
 2867 
 2384 
 
 2106 
 , '1751 
 
 n* 
 
 858-4 
 
 .' 
 
 25 
 
 2-5 
 
 3 
 
 1287.6 
 
 5150 
 4292 
 
 3576 
 2908 
 
 2628 
 2189 
 
 n* 
 
 1073 
 ijiiS 
 
 It appears, by formula (59.), that the displacement of the centre of 
 gravity necessary to produce a jump at any given speed, is not dependent 
 on the actual weight of the engine or the wheels, but on the ratio of their 
 weights ; and, from the above table, that when the weight of the engine 
 and wheels is 6$- times that of the driving-wheels, a displacement of 2f 
 inches in the centre of gravity is enough to create a jump when the train 
 is travelling at sixty miles an hour, or of two inches when it is travelling 
 at seventy miles ; this displacement varying inversely as the square of the 
 velocity is less, other things being the same, as the square of the diameter 
 of the wheel is less ; for the radius of the wheel being represented by 0, 
 
 the angular velocity is represented by = z-^ > and substituting this value, 
 formula (59.) becomes 
 
 * It will be observed, that the cranks being placed on the axle at right angles 
 to one another, when the centre of gravity on the one side is in a favourable po- 
 
APPENDIX. 
 
 If the weight "W of the wheel be supposed to vary as the square of its 
 diameter and be represented by pd*, this formula will become 
 
 h>( M 
 
 still showing the displacement of the centre of gravity necessary to pro- 
 duce a jump to diminish with the diameter of the wheel. These conclu- 
 sions are opposed to the use of light engines and small driving-wheels; 
 and they show the necessity of a careful attention to the true balancing of 
 the wheels of the carriages as well as the driving-wheels of the engine. 
 It does not follow that every jump of the wheel would be high enough to 
 lift the edge of the flange off the rail ; the determination of the height of 
 the jump involves an independent investigation. Every jump nevertheless 
 creates an oscillation of the springs, which oscillation will not of necessity 
 be completed when the jump returns ; but as the jumps are made alter- 
 nately on opposite sides of the engine, it is probable that they may, and 
 that after a time they will, so synchronise with the times of the oscillations, 
 as that the amplitude of each oscillation shall be increased by every jump, 
 and a rocking motion be communicated to the engine attended with 
 danger. 
 
 Whilst every jump does not necessarily cause the wheel to run off the 
 rail, it nevertheless causes it to slip upon it, for before the wheel jumps 
 it is clear that it must have ceased to have any hold upon the rail or any 
 friction. 
 
 The Slip of the Wheel. 
 
 If /be taken to represent the coefficient of friction between the surface 
 of the wheel and that of the rail, the actual friction in any position of the 
 wheel will be represented by Y, /. But the friction which it is necessary 
 the rail should supply, in order that the rolling of the whel maybe main- 
 tained, is X. It is a condition therefore necessary to the wheel not slip- 
 ping that 
 
 If, therefore, taking the maximum value of in any revolution, we find 
 
 that /exceeds it, it is certain that the wheel cannot have slipped in that 
 revolution; whilst if, on the other hand, /falls short of it, it must have 
 
 sition for jumping, it is in an unfavourable position on the other side, so that 
 It can only jump on one side at once, and the efforts on the two sides alternate- 
 
EOLLING MOTION OF A CYLINDER. 673 
 
 slipped.* The positions between which the slipping will take place con- 
 tinually, are determined by solving, in respect to cos. 0,, the equation 
 
 /=?.... (61). 
 
 The application of these principles to the slip of the carriage-wheel is 
 rendered less difficult by the fact, that the value of h is always in that case 
 
 so small, as compared with the values of Tc and a, that - may be neglected 
 
 a 
 
 in formulae (34.) and (35.), as compared with unity. Those equations 
 then become 
 
 1 X. 2+ 7 l (62). 
 
 and 
 
 H I+ ^-[, 
 
 = --cos. a+ 
 
 (A 
 
 whence we obtain 
 
 X 
 
 Assume 
 
 /i + !L'W 
 
 ( + wj^+cos. 
 
 e d?u__{ j3 (jS + C03. 0) +2 (1 +|8 cos. &)} sin. 
 ~ 
 
 " d9 (3 + cos. 0) 2 ^ 2 ~ 03 + cos. e ) 3 
 
 Now if j3> 1, there will be some value of e for which ^ + cos. = 0, and 
 
 therefore 1+3 cos. = 0; and since for this value of 0, _ = 0, and ^ 
 
 ad ojr 
 
 * Of course, the slipping, in the case of the driving-wheels of a locomotive, 
 is diminished by the fact, that whilst one wheel IB not biting upon the rail 
 the other is. 
 
 43 
 
APPENDIX. 
 
 = t M - 3 it follows that it corresponds to a maximum value of u, and 
 
 therefore of 5. 
 Y, 
 
 But if (3 < 1, then there is some value of cos. for which |3 + cos. = 0, 
 and therefore for which u infinity, which value corresponds therefore in 
 
 this case to the maximum of 5 
 Y, 
 
 Thus then it appears that according as 
 
 the maximum value of = is attained when cos. 0= j3 or = ^ ; that 
 is, when 
 
 Y 
 
 In the one case the maximum value of y will be infinity, .... (67). 
 
 and in the other case it will be represented by the formula 
 
 (68). 
 
 < 
 
 In the first case, i. e. when /3< 1, the wheel will slip every time that it 
 revolves, whatever may be the value of f. In the second case, or when 
 3 > 1, it will slip if /do not exceed the number represented by formula 
 (68.). The conditions (65.) are obviously the same with those (59.) which 
 determine whether there be a jump or not, which agrees with an obser- 
 vation in the preceding article, to the effect, that as the wheel must cease 
 to bite upon the rail before it can jump, it must always slip before it 
 can jump. "When the conditions of slipping obtain, one of the wheels 
 always biting when the other is slipping, and the slips of the two wheels 
 alternating, it is evident that the engine will be impelled forwards, at 
 certain periods of each revolution, by one wheel only, and at others, by 
 the other wheel only; and that this is true irrespective of the action of 
 the two pistons on the crank, and would be true if the steam were thrown 
 off. Such alternate propulsions on the two sides of the train cannot but 
 
DESCENT UPON INCLINED PLANE. 675 
 
 communicate alternate oscillations to the buffer-springs, the intervals 
 between which will not be the same as those between the propulsions; 
 but they may so synchronise with a series of propulsions as that the 
 amplitude of each oscillation may be increased by them until the train 
 attains that fish-tail motion with which railway travellers are familiar. 
 It is obvious that the results shown here to follow from a displacement of 
 the centres of gravity of the driving-wheels, cannot fail also to be pro- 
 duced by the alternate action of the connecting rods at the most favorable 
 driving points of the crank and at the dead points,* and that the operation 
 of these two causes may tend to neutralize or may exaggerate one another. 
 It is not the object of this paper to discuss the question under this point 
 of view. 
 
 NOTE F. 
 
 ON THE DESCENT UPON AN INCLINED PLANE OF A BODY SUBJECT TO VARIA- 
 TIONS OF TEMPERATURE, AND ON THE MOTION OF GLACIERS. 
 
 IF we conceive two bodies of the same form and dimensions (cubes, for 
 instance), and of the same material, to be placed upon a uniform horizon- 
 tal plane and connected by a substance which alternately extends and 
 contracts itself, as does a metallic rod when subjected to variations of 
 temperature, it is evident that by the extension of the intervening rod 
 each will be made to recede from the other by the same distance, and, 
 by its contraction, to approach it by the same distance. But if they be 
 placed on an inclined plane (one being lower than the other) then when 
 by the increased temperature of the rod its tendency to extend becomes 
 sufficient to push the lower of the two bodies downwards, it will not have 
 become sufficient to push the higher upwards. The effect of its exten- 
 sion will therefore be to cause the lower of the two bodies to descend 
 whilst the higher remains at rest. The converse of this will result from 
 contraction ; for when the contractile force becomes sufficient to pull the 
 upper body down the plane it will not have become sufficient to pull 
 the lower up it. Thus, in the contraction of the substance which inter- 
 venes between the two bodies, the lower will remain at rest whilst the 
 upper descends. As often, then, as the expansion and contraction is 
 repeated the two bodies will descend the plane until, step by step, they 
 reach the bottom. 
 
 * A slip of the wheel may thus be, and probably is, produced at each revo- 
 lution. 
 
C7G APPENDIX. 
 
 Suppose the uniform bar AB placed on an inclined plane, and subject 
 to extension from increase of temperature, a por- 
 tion XB will descend, and the rest X A will ascend ; 
 the point X where they separate being determined 
 by the condition that the force requisite to push 
 XA up the plane is equal to that required to push 
 XB down it. 
 
 Let AX = a?, AB = L, weight of each linear unit = /*, t = inclination 
 of plane, $ = limiting angle of resistance. 
 
 .-.(jix = weight of AX. 
 
 Now, the force acting parallel to an inclined plane which is necessary 
 to push a weight W up it, is represented by W li^JLyj and that ne- 
 
 COS. 
 
 eessary to push it down the plane by W . (Art. 241.) 
 
 COS. 
 
 ^. 
 sin. (? Q 
 
 COS. <f> COS. <p 
 
 .-.x {sin. (0 + t) + sin. (0 t)} = L sin. (p 
 . . 2 x sin. cos. t = L sin. (^ t) 
 
 _. L 
 
 sm. <j> cos. 
 
 , _ f 
 
 = iL < 
 
 ( 
 
 tan. t 
 tan. 
 
 When contraction takes place, the converse of 
 the above will be true. The separating point X 
 will be such, that the force requisite to pull XB up 
 __^________ the plane is equal to that required to pull AX 
 
 down it. BX is obviously in this case equal to AX 
 in the other. 
 
 Let a. be the elongation per linear unit under any variation of tempera- 
 ture; then the distance which the point B (fig. l.)will be made to descend 
 by this elongation = x.BX 
 
DESCENT UPON INCLINED PLANE. 677 
 
 If we conceive the bar now to return to its former temperature, con- 
 tracting by the same amount (X) per linear unit; then the point B 
 (fig. 2.) will by this contraction be made to ascend through the space 
 ~ 
 
 Total descent I of B by elongation and contraction is therefore determined 
 by the equation 
 
 To determine the pressure upon a nail driven 
 through the rod at any point P fastening it to the 
 plane. 
 
 It is evident, that in the act of extension the part BP of the rod will 
 descend the plane and the part AP ascend; and conversely in the act of 
 contraction ; and that in the former case the nail B will sustain a pressure 
 upwards equal to that necessary to cause BP to descend, and a pressure 
 downwards equal to that necessary to cause PA to ascend ; so that, as- 
 suming the pressure to be downwards, and adopting the same notation as 
 before, except that AP is represented by p, AB by a, and the pressure 
 upon the nail (assumed to be downwards) by P, we have in the case of 
 extension 
 
 _ sin. (0 + e,) . sin. 
 and in the case of contraction, 
 
 p=, ( *-p) sin - 
 
 Reducing, these formulae become respectively, 
 
 p_ J 2p sin. cos. i a sin. (0 i) > (8). 
 
 COS. <f> ( j 
 
 P = < a sin. (<p + 1) 2p sin. <t> cos. >.- ' (4). 
 
 COS. ( J 
 
 EXAMPLE OF THE DESCENT OF THE LEAD ON THE ROOF OF BEISTOL 
 
 CATHEDEAL. 
 
 My attention was first drawn to the influence of variations in tempera- 
 ture to cause the descent of a lamina of metal resting on an inclined plane 
 
678 APPENDIX. 
 
 by observing, in the autumn of 1853, that a portion of the lead which 
 covers the south side of the choir of Bristol Cathedral, which had been 
 renewed in the year 1851, but had not been properly fastened to the ridgo 
 beam, had descended bodily eighteen inches into the gutter; so that if 
 plates of lead had not been inserted at the top, a strip of the roof of that 
 length would have been left exposed to the weather. The sheet of lead 
 which had so descended measured, from the ridge to the gutter, 19ft. 4in., 
 and along the ridge 60ft. The descent had been continually going on 
 from the time the lead had been laid down. An attempt made to stop it 
 by driving nails through it into the rafters had failed. The force by 
 which the lead had been made to descend, whatever it was, had been 
 found sufficient to draw the nails.* As the pitch of the roof was only 
 164- it was sufficiently evident that the weight of the lead alone could not 
 have caused it to descend. Sheet lead, whose surface is in the state of 
 that used in roofing, will stand firmly upon a surface of planed deal when 
 inclined at an angle of 30f, if no other force than its weight tends to 
 cause it to descend. The considerations which I have stated in the pre- 
 ceding articles, led me to the conclusion that the daily variations in the 
 temperature of the lead, exposed as it was to the action of the sun by its 
 southern aspect, could not but cause it to descend considerably, and the 
 only question which remained on my mind was, whether this descent 
 could be so great as was observed. To determine this I took the follow- 
 ing data : 
 Mean daily variation of temperature at Bristol in the 
 
 month of August ; assumed to be the same as at Leith 
 
 (Kcemtz Meteorology, by Walker, p. 18.) - - - 8 21' Cent. 
 Linear expansion of lead through 100 Cent. - - - '0028436. 
 Length of sheets of lead forming the roof from the ridge 
 
 to the gutter - - 232 inches. 
 
 Inclination of roof 16 32'. 
 
 Limiting angle of resistance between sheet lead and deal - 80 
 
 Whence the mean daily descent of the lead, in inches, in the month of 
 August, is determined by equation (2.) to be 
 
 * The evil was remedied by placing a beam across the rafters, near the ridge, 
 and doubling the sheets round it, and fixing their ends with spike-nails. 
 
 j- This may easily be verified. I give it as the result of a rough experiment 
 of my own. I am not acquainted with any experiments on the friction of lead 
 made with sufficient care to be received as authority in this matter. The 
 friction of copper on oak has, however, been determined by General MORIN 
 (see a table in the preceding part of this work) to be 0'62, and its limiting angle 
 of resistance 31 48' ; so that if the roof of Bristol Cathedral had been inclined 
 at 31 instead of 16", and had been covered with sheets of copper resting on 
 oak boards, instead of sheets of lead resting on deal, the sheeting would not 
 have slipped by its weight only. 
 
DESCENT UPON INCLINED PLANE. 679 
 
 Z=-027848 inches. 
 
 This average daily descent gives for the whole month of August a descent 
 of -863288. If the average daily variation of temperature of the month 
 of August had continued throughout the year, the lead would have 
 descended 10-19148 inches every year. And in the two years from 
 1851 to 1853 it would have descended 20-38296 inches. But the daily 
 variations of atmospheric temperature are less in the other months of the 
 year than in the month of August. For this reason, therefore, the cal- 
 culation is in excess. For the following reasons it is in defect: 1st., 
 The daily variations in the temperature of the lead cannot but have been 
 greater than those of the surrounding atmosphere. It must have been 
 heated above the surrounding atmosphere by radiation from the sun in 
 the day-time, or cooled below it by radiation into space at night. 2ndly., 
 One variation of temperature only has been assumed to take place every 
 twenty-four hours, viz. that from the extreme heat of the day to the 
 extreme cold of the night; whereas such variations are notoriously of 
 constant occurrence during the twenty-four hours. Each cannot but have 
 caused a corresponding descent of the lead, and their aggregate result 
 cannot but have been greater than though the temperature had passed 
 uniformly (without oscillations backwards and forwards) from one extreme 
 to the other. 
 
 These considerations show, I think, that the causes I have assigned are 
 sufficient to account for the fact observed. They suggest, moreover, the 
 possibility that results of importance in meteorology may be obtained 
 from observing with accuracy the descent of a metallic rod thus placed 
 upon an inclined plane. That descent would be a measure of the aggre- 
 gate of the changes of temperature to which the metal was subjected 
 during the time of observation. As every such change of temperature is 
 associated with a corresponding development of mechanical action under 
 the form of work,* it would be a measure of the aggregate of such changes 
 and of the work so developed during that period. And relations might be 
 found between measurements so taken in different equal periods of time 
 successive years for instance tending to the development of new 
 meteorological laws. 
 
 * Mr. JOULE has shown (Phil. Trans., 1850, Part I.) that the quantity of heat 
 capable of raising a pound of water by 1 Fah. requires for its evolution 772 
 units of work. 
 
680 APPENDIX. 
 
 THE DESCENT OF GLAOIEES. 
 
 The following are the results of recent experiments * on the expansion 
 of ice : 
 
 Linear Expansion of Ice for am, Interval of 100 of the Centigrade 
 Thermometer. 
 
 0-00524 Schumacher. 
 
 0-00513 Pohrt. 
 
 0-00518 Moritz. 
 
 Ice, therefore, has nearly twice the expansibility of lead ; so that a 
 sheet of ice would, under similar circumstances, have descended a plane 
 similarly inclined, twice the distance that the sheet of lead referred to in 
 the preceding article descended. Glaciers are, on an increased scale, 
 sheets of ice placed upon the slopes of mountains, and subjected to 
 atmospheric variations of temperature throughout their masses by varia- 
 tions in the quantity and the temperature of the water, which, flowing 
 from the surface, everywhere percolates them. That they must from this 
 cause descend into the valleys, is therefore certain. That portion of the 
 Mer de Glace of Chamouni which extends from Montanvert to very near 
 the origin of the Glacier de Lechaud has been accurately observed by 
 Professor James Forbes.t Its length is 22,600 feet, and its inclination 
 varies from 4 19' 22" to 5 5' 53". The Glacier du Geant, from the 
 Tacul to the Col du Geant, Professor Forbes estimates (but not from his 
 own observations, or with the same certainty) to be 24,700 feet in length, 
 and to have a mean inclination of 8 46' 40". 
 
 According to the observations of De Saussure, the mean daily range 
 of Reaumur's thermometer in the month of July, at the Col du Geant, is 
 4 -257}, and at Chamouni 10 -09 2. The resistance opposed by the 
 rugged channel of a glacier to its descent cannot but be different at dif- 
 ferent points, and in respect to different glaciers. The following passage 
 from Professor Forbes's work contains the most authentic information I 
 am able to find on this subject. Speaking of the Glacier of la Brenva 
 he says : u The ice removed, a layer of fine mud covered the rock, not 
 composed, however, alone of the clayey limestone mud, but of sharp sand 
 derived from the granitic moraines of the glacier, and brought down with 
 it from the opposite side of the valley. Upon examining the face of the 
 ice removed from contact with the rock, we found it set all over with 
 sharp angular fragments, from the size of grains of sand to that of a 
 cherry, or larger, of the same species of rook, and which were so firmly 
 
 Vide Archir. , Wissenschaftl. Kunde v. Russland, Bd. vii s. 883. 
 Travels through the Alps of Savoy. Edinburgh, 1853. 
 Quoted by Professor FORBES, p. 231. 
 
DESCENT OF GLACIERS. 681 
 
 fixed in the ice as to demonstrate the impossibility of such a surface being 
 forcibly urged forwards without sawing any comparatively soft body 
 which might be below it. Accordingly, it was not difficult to discover in 
 the limestone the very grooves and scratches which were in the act of 
 being made at the time by the pressure of the ice and its contained frag- 
 ments of stone." (Alps of the Savoy, pp. 203 4.) It is not difficult 
 from this description to account for the fact that small glaciers are some- 
 times seen to lie on a slope of 30 (p. 35.). The most probable supposition 
 would indeed fix the limiting angle of resistance between the rock and 
 the under surface of the ice set all over, as it is described to. be, with 
 particles of sand and small fragments of stone, at about 30 ; that being 
 nearly the slope at which smooth surfaces of calcareous stone will rest on 
 one another. If we take then 30 to be the limiting angle of resistance 
 between the under surface of the Mer de Glace and the rock on which it 
 rests, and if we assume the same mean daily variation of temperature 
 (4-257 Keaumur, or 5'321 Centigrade) to obtain throughout the length 
 of the Glacier du Geant, which De Saussure observed in July, at the 
 Col du Geant ; if, further, we take the linear expansion of ice at 100 
 Centigrade to be that ('00524) which was determined by the experiments 
 of Schumacher, and, lastly, if we assume the Glacier de Geant to descend 
 as it would if its descent were unopposed by its confluence with the 
 Glacier de Lechaut; we shall obtain, by substitution in equation (2.) 
 for the mean daily descent of the Glacier du Geant at the Tacul, the 
 formula 
 
 1 = 24700 X 5-321 x '^24 tan- 8 46' 
 
 100 tan. 30 
 1= 1-8395 feet. 
 
 The actual descent of the glacier in the centre was T5 feet. If the 
 Glacier de Lechaut descended, at a mean slope of 5, singly in a sheet of 
 uniform breadth to Montanvert without receiving the tributary glacier of 
 the Talefre, or uniting with the Glacier du Geant, its diurnal descent would 
 be given by the same formula, and would be found to be -95487 feet. 
 Eeasoning similarly with reference to the Glacier du Geant ; supposing it 
 to have continued its course singly from the Col du Geant to Montanvert 
 without confluence with the Glacier de Lechaut, its length being 40,420 
 feet, and its mean inclination 6 53', its mean diurnal motion I at Montan- 
 vert would, by formula (2.) have been 2'3564* feet. The actual mean 
 daily motion of the united glaciers, between the 1st and the 28th July, was 
 at Montanvert,f 
 
 * On the 1st of July the centre of the actual motion of the Mer de Glace at 
 Montanvert was 2*25 feet. 
 
 f Forbes' " Alps of Savoy," p. 140. 
 
682 APPENDIX. 
 
 Near the side of the glacier - - 1-441 feet. 
 Between the side and the centre - 1 '750 " 
 Near the centre 2-141 " 
 
 The motion of the Glacier de Lechaut was therefore accelerated by their 
 confluence, and that of the Glacier du Geant retarded. The former is 
 dragged down by the latter. 
 
 I have had the less hesitation in offering this solution of the mechanical 
 problem of the motion of glaciers, as those hitherto proposed are con- 
 fessedly imperfect. That of De Saussure, which attributes the descent of 
 the glacier simply to its weight, is contradicted by the fact that isolated 
 fragments of the glacier stand firmly on the slope on which the whole 
 nevertheless descends. It being obvious that if the parts would remain at 
 rest separately on the bed of the glacier, they would also remain at rest 
 when united. 
 
 That of Professor J. Forbes, which supposes a viscous or semirfiuid 
 structure of the glacier, is not consistent with the fact that no viscosity is 
 to be traced in its parts when separated. They appear as solid fragments, 
 and they cannot acquire in their union properties in this respect which 
 individually they have not. 
 
 Lastly, the theory of Oharpentier, which attributes the descent of the 
 glacier to the daily congelation of the water which percolates it, and the 
 expansion of its mass consequent thereon, whilst it assigns a cause which, 
 so far as it operates, cannot, as I have shown, but cause the glacier to 
 descend, appears to assign one inadequate to the result ; for the congelation 
 of the water which percolates the glacier does not, according to the obser- 
 vations of Professor Forbes,* take place at all in summer more than a few 
 inches from the surface. Nevertheless, it is in the summer that the daily 
 motion of the glacier is the greatest. 
 
 The following remarkable experiment of Mr. Hopkins of Cambridge,! 
 which is considered by him to be confirmatory of the sliding theory of 
 De Saussure as opposed to De Oharpentier's dilatation theory, receives 
 a ready explanation on the principles which I have laid down in this 
 note. It is indeed a necessary result of them. Mr. Hopkins placed a 
 mass of rough ice, confined by a square frame or bottomless box, upon 
 a roughly chiselled flag-stone, which he then inclined at a small angle ; 
 and found that a slow but uniform motion was produced, when even it 
 was placed at an inconsiderable slope. This motion, which Mr. Hopkins 
 attributed to the dissolution of the ice in contact with the stone, would, 
 I apprehend, have taken place if the mass had been of lead instead of ice ; 
 
 * "Travels in the Alps," J>. 413. 
 
 f I have quoted the following account of it from Professor Forbes's book, 
 p. 419. 
 
DIMENSIONS OF A BUTTRESS. (>S3 
 
 and it would have been but about half as fast, because the linear expan- 
 sion of lead is only about half that of ice. 
 
 NOTE G. 
 THE BEST DIMENSIONS OF A BTJTTBESS. 
 
 IF mi (Art. 299.) represent the modulus of stability of the portion AG of 
 the wall, it may be shown, as before, that 
 
 P{ (&, A 2 ) sin. a (I a 3 m,)cos. a} = (a, w,) (A, A 2 )a,^t ; 
 
 . . P{(^i A 2 )sin. a (I a 2 )cos. a} 
 == K&i ^2)^1 V w i{P C08 - + (^i A*X/*} 
 
 If m l =m^ the stability of the portion AG of the structure is the same 
 with that of the whole AC ; an arrangement by which the greatest 
 strength is obtained with a given quantity of material (see Art. 388.). 
 This supposition being made, and m eliminated between the above equa- 
 tion and equation (392.), that relation between the dimensions of the 
 buttress and those of the wall which is consistent with the greatest 
 economy of the material used will be determined. The following is that 
 relation : 
 
 , + ~afh z } P (A, sin. a I cos. a) 
 
 P cos. a + nt(a A + ~A) 
 
 AsK* P{(Ai A) sin, a (I a,) cos, a} 
 P cos. a + ^ (hi h 3 ) 
 
 It is necessary to the greatest economy of the material of the Gothic 
 buttress (Art. 301.) that the stability of the portions Qa and Q5, upon 
 their respective bases ao and fo, should be same with that of the whole 
 buttress on its base EC. If, in the preceding equation, h l h 3 be 
 substituted for A,, and 7i 2 h 3 for A 2 , the resulting equation, together 
 with that deduced as explained in the conclusion of Art. 301., will deter- 
 mine this condition, and will establish those relations between the dimen- 
 sions of the several portions of the buttress which are consistent with the 
 greatest economy of the material, or which yield the greatest strength to 
 the structure from the use of a given quantity of material. 
 
684: 
 
 APPENDIX. 
 
 NOTE H. 
 DIMENSIONS OP THE TEETH OF WHEELS. 
 
 THE following rules are extracted from the work of M. Morin, entitled 
 Aide Memoire de Mecanique Pratique : If we represent by a the width 
 in parts of a foot of the tooth measured parallel to the axis of the wheel, 
 and by I its breadth or thickness measure* parallel to the plane oi 
 rotation upon the pitch circle ; then, the teeth being constantly greased, 
 the relation of a and 5 should be expressed, when the velocity of the pitch 
 circle does not exceed 5 feet per second, by a = 45 ; when it exceeds 
 5 feet per second, by a = 55 : if the wheels are constantly exposed to wet, 
 by a=6&. 
 
 These relations being established, the width or thickness of the tooth 
 will be determined by the formulae contained in the columns of the follow- 
 ing table: 
 
 Material 
 
 French measures, cents 
 and kils. 
 
 English measures, feet 
 and pounds. 
 
 Cast iron ... 
 Brass ... 
 Hard wood 
 
 b=-I05V~P' 
 613H/F 
 6 = U5|/P 
 
 b = '00231 9 V~? 
 6= -002894 f/F 
 6= -003203 4/F 
 
 Assuming that when the teeth are carefully executed the space between 
 the teeth should be T ^th greater than their thickness, and y'^th greater 
 when the least labor is bestowed on them, the values of the pitch T will 
 in these two cases be represented by 6(2 + T s) and 6(2 + T V), or by 2-0676 
 and 2 '15. Substituting in these expressions the values of 6 given by the 
 formulas of the preceding table, then determining from the resulting 
 values of c (see equation 233.) the corresponding values of the coefficient 
 (see equation 234.), the following table is obtained : 
 
 Material. 
 
 Value of o (equation 238.). 
 
 Value of C (equation 234.). 
 
 For teeth of 
 the best work- 
 manship. 
 
 For teeth of 
 interior work- 
 manship. 
 
 For teeth of 
 the hest work- 
 manship. 
 
 For teeth of 
 inferior work- 
 manship. 
 
 Cast iron - 
 Brass 
 Hard wood 
 
 004795 
 005982 
 006621 
 
 004870 
 006077 
 006726 
 
 0-912 
 1-057 
 1-131 
 
 0-922 
 1 -068 
 1-143 
 
TRACTION OF CARRIAGES. 685 
 
 The following are the pitches commonly in use among mechanics : 
 in. in. in. in. in. in. in. 
 1, li, li, H, 2, 2i, 3. 
 
 Prof. Willis considers the following to be sufficient below inch pitch : 
 
 in. in. in. in. in. 
 
 i, t, *, *, *- 
 
 Having, therefore, determined the proper pitch to be given to the tooth 
 from formula 234., the nearest pitch is to be taken from the above series 
 to that thus determined. 
 
 NOTE I. 
 EXPERIMENTS OF M. MOKIN ON THE TRACTION OF OAEBIAGES. 
 
 THE following are among the general results deduced by M. Morin from 
 his experiments : 
 
 1. The traction is directly proportional to the load, and inversely pro- 
 portional to the diameter of the wheel. 
 
 2. Upon a paved or a hard Macadamised road, the resistance is independ- 
 ent of the width of the tire when it exceeds from 3 to 4 inches. 
 
 3. At a walking pace the traction is the same, under the same circum- 
 stances, for carriages with springs and without them. 
 
 4. Upon hard Macadamised and upon paved roads the traction increases 
 with the velocity ; the increments of traction being directly proportional 
 to the increments of velocity above the velocity 3'28 feet per second, or 
 about 2i miles per hour. The equal increment of traction thus due to 
 each equal increment of velocity is less as the road is more smooth, and 
 the carriage less rigid or better hung. 
 
 4. Upon soft roads of earth, or sand or turf, or roads fresh and thickly 
 graveled, the traction is independent of the velocity. 
 
 5. Upon a well-made and compact pavement of hewn stones the traction 
 at a walking pace is not more than three-fourths of that upon the beat 
 Macadamised road under similar circumstances; at a trotting pace it 
 is equal to it. 
 
 6. The destruction of the road is in all cases greater as the diameters 
 of the wheels are less, and it is greater in carriages without than with 
 springs. 
 
APPENDIX. 
 
 NOTE K. 
 ON THE STKENGTH OF COLUMNS. 
 
 MB. HODGKINSON has obligingly communicated the following observations 
 on Art. 430. : 
 
 1. The reader must be made to understand that the rounding of the 
 ends of the pillars is to make them moveable there, as if they turned by 
 means of a universal joint ; and the flat-ended pillars are conceived to be 
 supported in every part of the ends by means of flat surfaces, or otherwise 
 rendering the ends perfectly immoveable. 
 
 2. The coefficient (13) for hollow columns with rounded ends is deduced 
 from the whole of the experiments first made, including some which were 
 very defective on account of the difficulty experienced in the earlier 
 attempts to cast good hollow columns so small as were wanted. The 
 first castings were made lying on their side ; and this, notwithstanding 
 every effort, prevented the core being in the middle ; some of the columns 
 were reduced, too, in thickness, half way between the middle and the 
 ends, and near to the ends, and this slightly reduced the strength. These 
 causes of weakness existed much more among the pillars with rounded 
 ends than those with flat ones ; they are alluded to in the paper (Art. 47.). 
 Had it not been for them, the coefficient (13) would, I conceive, have 
 been equal to that for solid pillars (or 14- 9). 
 
 3. The fact of long pillars with flat ends being about three times as 
 strong as those of the same dimensions with rounded ends is, I conceive, 
 well made out, in cast iron, wrought iron, and timber; you have, how- 
 ever, omitted it, being perhaps led to do it through the low value of the 
 coefficient (13) above mentioned. 
 
 The same may be mentioned with respect to the near approach in 
 strength of long pillars with flat ends, and those of half the length with 
 rounded ends. It may be said that the law of the 1*7 power of the length 
 would nearly indicate the latter ; but this last, and the other powers 3 '76 
 and 3'55, are only approximations, and not exactly constant, though 
 nearly so, and I do not know whether the other equal quantities are not, 
 with some slight modifications, physical facts. 
 
 4. The strength of pillars of similar form and of the same materials 
 varies as the 1*865 power, or near as the square of their like linear 
 dimensions, or as the area of their cross section. 
 
COMPLETE ELLIPTIC FUNCTIONS. 
 
 687 
 
 TABLE I. 
 
 The Numerical Values of COMPLETE Elliptic Functions of the FIRST and 
 SECOND Orders for Values of the Modulus k corresponding to each Degree 
 of the Angle sin. '&. 
 
 Sin. 1 *. 
 
 Fi. 
 
 Ei. 
 
 Sin. ik. 
 
 F,. 
 
 E,. 
 
 
 1 
 2 
 3 
 
 4 
 5 
 
 1-57079 
 1-57091 
 1-57127 
 1-57187 
 1-57271 
 1-57379 
 
 1-57079 
 1-57067 
 1-57031 
 1-56972 
 1-56888 
 1-56780 
 
 46 
 
 47 
 48 
 49 
 50 
 
 1-86914 
 
 1-88480 
 1-90108 
 1-91799 
 1-93558 
 
 1-34180 
 1-33286 
 1 -32384 
 1-31472 
 1-30553 
 
 51 
 52 
 53 
 54 
 55 
 
 1-95386 
 1-97288 
 1-99266 
 2-01326 
 2-03471 
 
 1-29627 
 1-28695 
 1-27757 
 1-26814 
 1-25867 
 
 6 
 7 
 8 
 9 
 10 
 
 1-57511 
 
 1-57667 
 1-57848 
 1-58054 
 1*58284 
 
 1-56649 
 1-56494 
 1-56316 
 1-56114 
 1-55888 
 
 56 
 67 
 58 
 59 
 60 
 
 2-05706 
 2-08035 
 2-10465 
 2-13002 
 2-15651 
 
 1-24918 
 1-23966 
 1-23012 
 1-22058 
 1-21105 
 
 11 
 12 
 13 
 14 
 15 
 
 1-58539 
 1-58819 
 1-59125 
 1-59456 
 
 1-59814 
 
 1-55639 
 1-55368 
 1-550>J8 
 1-54755 
 1-54415 
 
 61 
 62 
 63 
 64 
 65 
 
 2-18421 
 2-21319 
 2-24354 
 2-27537 
 2-30878 
 
 1-20153 
 1-19204 
 1-18258 
 1-17317 
 1-16382 
 
 16 
 17 
 18 
 19 
 20 
 
 1-60197 
 1-60608 
 1-61045 
 1-61510 
 1-62002 
 
 1-54052 
 1-53666 
 1-53259 
 1 -52830 
 1-52379 
 
 66 
 67 
 68 
 69 
 70 
 
 2-34390 
 2-38087 
 2-41984 
 2-46099 
 2-50455 
 
 1-15454 
 1-14534 
 1-13624 
 1-12724 
 1-11837 
 
 21 
 22 
 23 
 24 
 25 
 
 1-62523 
 1-63072 
 1-63651 
 1 -64260 
 1-64899 
 
 1-51907 
 1-51414 
 1-50900 
 1-50366 
 1-49811 
 
 71 
 
 72 
 73 
 74 
 75 
 
 2-55073 
 2-59981 
 2-65213 
 2-70806 
 2-76806 
 
 1-10964 
 1-10106 
 1 -09265 
 1 -08442 
 1-07640 
 
 26 
 
 27 
 
 28 
 29 
 30 
 
 1 -65569 
 1-66271 
 1-67005 
 1-67773 
 1-68575 
 
 1-49236 
 1-48642 
 1-48029 
 1-47396 
 1-46746 
 
 76 
 
 77 
 78 
 79 
 80 
 
 2*83267 
 2-90256 
 2-97856 
 3-06172 
 3-15338 
 
 1-06860 
 1-06105 
 1-05377 
 1-04678 
 1-04011 
 
 31 
 32 
 33 
 34 
 35 
 
 1-69411 
 1-70283 
 1-71192 
 1-72139 
 1-73124 
 
 1-46077 
 1-45390 
 1-44686 
 1-43966 
 1-43229 
 
 81 
 
 82 
 83 
 
 84 
 85 
 
 3-25530 
 3-36986 
 3-50042 
 3-65186 
 3-83174 
 
 1-03378 
 1-02784 
 1-02281 
 1-01723 
 1-01266 
 
 3''> 
 37 
 
 3S 
 39 
 40 
 
 1-74149 
 1-75216 
 1-76325 
 1-77478 
 I -7807 8 
 
 1-42476 
 1-41707 
 1-40923 
 1-40125 
 1-39314 
 
 88 
 
 87 
 88 
 89 
 
 4-05275 
 4-33865 
 4-74271 
 5-43490 
 
 1-00864 
 1-00525 
 1-00258 
 1-00075 
 
 41 
 42 
 43 
 44 
 
 45 
 
 1-79922 
 l-R121o 
 1-82500 
 1-83956 
 1-85407 
 
 1-38488 
 1-37650 
 1-36799 
 1-35937 
 1 -35064 
 
4PPENDIX. 
 
 THE TABLES OF M. GAEIDEL. 
 
 TABLE II. 
 
 Showing the Angle of Rupture of an Arch whose Loading is of the same 
 Material with its Voussoirs, and whose Extrados is inclined at a given 
 Angle to the Horizon. (See Art. 344.)' 
 
 a = ratio of lengths of voussoirs to radius of intrados. 
 
 c = ratio of depth of load over crown to radius of intrados, so thut 
 
 e = /3(l+a). (Art. 838.) 
 t = inclination of extrados to horizon. 
 
 a 
 
 c=0 
 
 o=0-l 
 
 c=0-2 
 
 c=0-8 
 
 c=0-4 
 
 c=0-5 
 
 c=l-0 
 
 0-05 
 
 68-0 
 
 69-19 
 
 54-04 
 
 51-15 
 
 49-35 
 
 48-20 
 
 45-74 
 
 o-io 
 
 65-4 
 
 60-48 
 
 57-70 
 
 56-01 
 
 54-93 
 
 54-17 
 
 52-34 
 
 0-15 
 
 64-0 
 
 61-3 
 
 59-7 
 
 58-69 
 
 58-0 
 
 57-49 
 
 56-21 
 
 0-20 
 
 631 
 
 61-7 
 
 60-88 
 
 60-30 
 
 59-90 
 
 59-60 
 
 58-80 
 
 0-25 
 
 62-24 
 
 61-76 
 
 61-44 
 
 61-22 
 
 61-05 
 
 60-94 
 
 60-59 
 
 0-30 
 
 61-3 
 
 61-42 
 
 61-54 
 
 61-60 
 
 61-66 
 
 61-67 
 
 61-81 
 
 0-35 
 
 60-17 
 
 60-80 
 
 61-21 
 
 61-54 
 
 61-78 
 
 61-98 
 
 62-56 
 
 0-40 
 
 58-8 
 
 69-8 
 
 60-52 
 
 61-05 
 
 61-48 
 
 61-67 
 
 62-9 
 
 0-45 
 
 67-32 
 
 58-53 
 
 59-45 
 
 60-19 
 
 60-80 
 
 61-28 
 
 62-85 
 
 0-50 
 
 65-63 
 
 56-97 
 
 58-09 
 
 58-98 
 
 59-72 
 
 60-34 
 
 62-40 
 
 = T 30'. 
 
 a 
 
 c=0 
 
 c=0-l 
 
 c=0-2 
 
 c=0-3 
 
 c=0-4 
 
 c=0-5| 
 
 c=l-0 
 
 005 
 
 68-3 
 
 57-3 
 
 51-69 
 
 48-61 
 
 47-84 
 
 46-11 
 
 44-85 
 
 0-10 
 
 64-3 
 
 58-68 
 
 55-95 
 
 54-52 
 
 53-64 
 
 53-03 
 
 51-68 
 
 0-15 
 
 62-43 
 
 59-67 
 
 58-33 
 
 57-55 
 
 57-00 
 
 56-61 
 
 55-66 
 
 0-20 
 
 61-48 
 
 60-42 
 
 59-72 
 
 59-35 
 
 59-07 
 
 58-87 
 
 58-29 
 
 0-25 
 
 60-75 
 
 60-55 
 
 60-44 
 
 60-38 
 
 60-33 
 
 60-21 
 
 60-17 
 
 0-30 
 
 60-09 
 
 60-49 
 
 60-77 
 
 60-95 
 
 61-08 
 
 61-18 
 
 61-48 
 
 0-35 
 
 59-27 
 
 60-12 
 
 60-62 
 
 61-02 
 
 61-33 
 
 61-59 
 
 62-31 
 
 0-40 
 
 58-25 
 
 59-33 
 
 60-11 
 
 60-72 
 
 61-18 
 
 61-57 
 
 62-7 
 
 0-45 
 
 67-11 
 
 58-35 
 
 59-29 
 
 60-05 
 
 60-67 
 
 61-16 
 
 62-78 
 
 0-50 55-82 
 
 57-13 
 
 58-21 
 
 59-08 
 
 59-81 
 
 60-41 
 
 62-45 
 
ANGLE OF RUPTURE OF AN AKCH, 
 
 689 
 
 = 15'. 
 
 a 
 
 0=0 
 
 o=0-l 
 
 c=0-3 
 
 c=0-8 
 
 c=0-4 
 
 c=0-5 
 
 o-l-O 
 
 0-05 
 
 64-8 
 
 50-5 
 
 46-95 
 
 45-69 
 
 45-03 
 
 44-67 
 
 43-9 
 
 O'lO 
 
 59-3 
 
 55-07 
 
 53-34 
 
 52-47 
 
 51-99 
 
 51-69 
 
 50-93 
 
 0-15 
 
 59-08 
 
 57-32 
 
 56-65 
 
 56-05 
 
 55-75 
 
 55-55 
 
 55-05 
 
 0-20 
 
 59-06 
 
 68-60 
 
 58-35 
 
 58-20 
 
 58-10 
 
 68-02 
 
 57-84 
 
 0-25 
 
 59-05 
 
 59-28 
 
 59-42 
 
 o9-53 
 
 59-60 
 
 59-65 
 
 59-79 
 
 0-30 
 
 58-90 
 
 59-57 
 
 59-98 
 
 60-26 
 
 60-48 
 
 60-66 
 
 61-15 
 
 0-35 
 
 58-53 
 
 59-41 
 
 60-09 
 
 60-57 
 
 60-93 
 
 61-17 
 
 62-0 
 
 0-40 
 
 57-99 
 
 59-08 
 
 59-87 
 
 60-48 
 
 60-95 
 
 61-36 
 
 62-6 
 
 0-45 
 
 67-26 
 
 58-43 
 
 59-34 
 
 60-06 
 
 60-67 
 
 61-15 
 
 62-7 
 
 0-50 
 
 56-38 
 
 57-61 
 
 58-58 
 
 59-36 
 
 60-06 
 
 60-64 
 
 62-5 
 
 = 22 30'. 
 
 a 
 
 c=0 
 
 c=0-l 
 
 0=0-2 
 
 c=0-8 
 
 c=0-4 
 
 o=0-5 
 
 0=1-0 
 
 0-05 
 
 36-1 
 
 41-2 
 
 42-0 
 
 42-3 
 
 42-6 
 
 42-7 
 
 42-9 
 
 o-io 
 
 50-5 
 
 50-3 
 
 50-19 
 
 5017 
 
 50-14 
 
 50-13 
 
 50-11 
 
 0-15 
 
 54-25 
 
 54-31 
 
 54-35 
 
 54-35 
 
 54-36 
 
 54-36 
 
 54-38 
 
 0-20 
 
 56-17 
 
 56-60 
 
 56-82 
 
 56-95 
 
 57-04 
 
 57-11 
 
 57-28 
 
 0-25 
 
 57-27 
 
 57-93 
 
 58-33 
 
 58-61 
 
 58-79 
 
 58-95 
 
 59-33 
 
 0-30 
 
 57-85 
 
 58-68 
 
 59-23 
 
 59-60 
 
 59-93 
 
 60-16 
 
 60-83 
 
 Q-35 
 
 68-07 
 
 59-01 
 
 59-70 
 
 60-21 
 
 60-61 
 
 60-91 
 
 61-85 
 
 0-40 
 
 58-02 
 
 59-02 
 
 69-79 
 
 60-38 
 
 60-87 
 
 61-25 
 
 62-2 
 
 0-45 
 
 57-74 
 
 58-78 
 
 69-60 
 
 60-26 
 
 60-82 
 
 61-27 
 
 62-7 
 
 0-50 
 
 57-30 
 
 58-31 
 
 59-16 
 
 5988 
 
 60-47 
 
 61-00 
 
 62-9 
 
 = 30. 
 
 
 
 c=0 
 
 o=0-l 
 
 0=0-3 
 
 c=0-8 
 
 c=0-4 
 
 0=0-5 
 
 0=1-0 
 
 0-05 
 
 31-3 
 
 36-2 
 
 38-4 
 
 39-57 
 
 40-28 
 
 40-77 
 
 41-9 
 
 o-io 
 
 43-3 
 
 46-06 
 
 47-25 
 
 47-90 
 
 48-30 
 
 48-59 
 
 49-24 
 
 015 
 
 50-07 
 
 51-46 
 
 52-18 
 
 52-63 
 
 62-94 
 
 53-14 
 
 53-68 
 
 0-20 
 
 53-66 
 
 54-69 
 
 55-27 
 
 55-67 
 
 55-96 
 
 56-16 
 
 56-72 
 
 0-25 
 
 55-80 
 
 56-72 
 
 57-30 
 
 57-72 
 
 58-01 
 
 58-23 
 
 58-89 
 
 0-30 
 
 57-13 
 
 58-01 
 
 58-62 
 
 59-06 
 
 59-40 
 
 59-69 
 
 60-48 
 
 0-35 
 
 57-93 
 
 58-80 
 
 59-43 
 
 59-94 
 
 60-33 
 
 60-66 
 
 61-64 
 
 0-40 
 
 58-33 
 
 59-20 
 
 59-89 
 
 60-42 
 
 60-87 
 
 61-23 
 
 62-39 
 
 0-45 
 
 58-47 
 
 59-33 
 
 60-03 
 
 6061 
 
 61-08 
 
 61-48 
 
 62-87 
 
 0-50 
 
 58-38 
 
 59-22 
 
 59-93 
 
 60-53 
 
 61-03 
 
 61-47 
 
 63-0 
 
 44 
 
690 
 
 APPENDIX. 
 
 = 37 30'. 
 
 a 
 
 0=0 
 
 0=01 
 
 e=0-2 
 
 o=0-3 
 
 c=0-4 
 
 0=0-5 
 
 0=1-0 
 
 0-05 
 
 3M 
 
 34-3 
 
 36-28 
 
 37-59 
 
 38-48 
 
 39-16 
 
 40-82 
 
 o-io 
 
 40-98 
 
 43-59 
 
 45-09 
 
 46-01 
 
 46-67 
 
 47-14 
 
 48-35 
 
 0-15 
 
 47-71 
 
 49-40 
 
 50-43 
 
 51-12 
 
 51-61 
 
 51-96 
 
 52-93 
 
 0-20 
 
 52-01 
 
 52-23 
 
 54-01 
 
 54-54 
 
 54-94 
 
 55-24 
 
 56-10 
 
 0-25 
 
 54-87 
 
 55-80 
 
 56-45 
 
 56-94 
 
 57-29 
 
 57-59 
 
 58-41 
 
 0-30 
 
 56-77 
 
 57-o8 
 
 58-16 
 
 58-62 
 
 58-98 
 
 59-26 
 
 60-16 
 
 0-35 
 
 58-04 
 
 58-78 
 
 59-34 
 
 59-81 
 
 60-17 
 
 60-47 
 
 61-45 
 
 0-40 
 
 58-89 
 
 59'58 
 
 60-13 
 
 60-60 
 
 60-97 
 
 61-30 
 
 63-2 
 
 0-45 
 
 59-38 
 
 60-06 
 
 60-62 
 
 61-07 
 
 61-47 
 
 61-83 
 
 63-0 
 
 0-50 
 
 56-69 
 
 -60-29 
 
 60-84 
 
 61-26 
 
 61-72 
 
 62-07 
 
 63-3 
 
 a 
 
 c=0 
 
 c=01 
 
 0=0-2 
 
 0=0-8 
 
 o=0-4 
 
 c=0-5 
 
 c=l-0 
 
 0-05 
 
 31-3 
 
 33-68 
 
 35-46 
 
 36-36 
 
 37-22 
 
 38-0 
 
 39-9 
 
 o-io 
 
 40-6 
 
 42-4 
 
 43-7 
 
 44-64 
 
 45-35 
 
 45-92 
 
 47-45 
 
 0-15 
 
 46-77 
 
 48-20 
 
 49-18 
 
 49-93 
 
 60-47 
 
 50-92 
 
 52-15 
 
 0-20 
 
 51-23 
 
 52-27 
 
 53-05 
 
 63-64 
 
 54-07 
 
 54-42 
 
 55-47 
 
 0-25 
 
 54-42 
 
 55-22 
 
 55-84 
 
 56-31 
 
 56-70 
 
 57-01 
 
 57-97 
 
 0-30 
 
 56-72 
 
 57-38 
 
 57-90 
 
 58-30 
 
 58-65 
 
 58-94 
 
 59-85 
 
 0-35 
 
 58-35 
 
 58-94 
 
 59-40 
 
 59-79 
 
 60-11 
 
 60-38 
 
 61>30 
 
 0-40 
 
 69-56 
 
 60-09 
 
 60-52 
 
 60-89 
 
 61-19 
 
 61-46 
 
 62-4 
 
 0-45 
 
 60-40 
 
 60-89 
 
 61-29 
 
 61-67 
 
 61-97 
 
 62-24 
 
 63-2 
 
 0-50 
 
 60-99 
 
 61-43 
 
 61-8 
 
 62-2 
 
 62-5 
 
 62-8 
 
 63-8 
 
HORIZONTAL THRUST OF AN ARCH. 
 
 691 
 
 THE TABLES OF M. GARIDEL. 
 TABLE III. 
 
 Showing the Horizontal Thrust of an Arch, the Radiw of whose Intrados 
 is Unity, and the weight of each Cubic Foot of its Material and that of 
 its Loading, Unity. (See Art. 344.) 
 
 N.B. To find the horizontal thrust of any other arch, multiply that given 
 in the table by the square of the radius of the intrados and by the weight 
 of a cubic foot of the material. 
 
 = 0. 
 
 a 
 
 c=0 
 P 
 
 72- 
 
 c=0-l 
 P 
 
 r 2 
 
 c=0-2 
 P 
 
 75- 
 
 c=0-3 
 P 
 7e* 
 
 c=0-4 
 P 
 
 7T 
 
 c=0-5 
 P 
 r2 
 
 0=1-0 
 
 _P_ 
 r* 
 
 0-05 
 
 0-08174 
 
 0-14797 
 
 0-21762 
 
 0-28877 
 
 0-36060 
 
 0-43277 
 
 0-79541 
 
 o-io 
 
 0-10279 
 
 0-16370 
 
 0-22588 
 
 0-28862 
 
 0-35164 
 
 0-41481 
 
 0-73161 
 
 0-15 
 
 0-11894 
 
 0-17480 
 
 0-23111 
 
 0-28764 
 
 0-34429 
 
 0-40100 
 
 0-68504 
 
 0-20 
 
 0-13073 
 
 0-18191 
 
 0-23322 
 
 0-28460 
 
 0-33603 
 
 0-38747 
 
 0-64488 
 
 0-25 
 
 0-13871 
 
 0-18553 
 
 0-23237 
 
 0-27922 
 
 0-32607 
 
 037293 
 
 0-60727 
 
 0-30 
 
 0-14333 
 
 0-18604 
 
 0-^*2874 
 
 0-27145 
 
 0-31416 
 
 0-35687 
 
 0-57041 
 
 0-35 
 
 0-14504 
 
 0-18379 
 
 0-22258 
 
 0-26140 
 
 0-30023 
 
 0-33907 
 
 0-53335 
 
 0-40 
 
 0-14422 
 
 0-17913 
 
 0-21415 
 
 0-24924 
 
 0-28437 
 
 0-31953 
 
 0-49560 
 
 0-45 
 
 0-14124 
 
 0-17240 
 
 0-20374 
 
 0-23520 
 
 0-26674 
 
 0-29835 
 
 0-45693 
 
 0'50 
 
 0-13649 
 
 0-16396 
 
 0-19168 
 
 0-21957 
 
 0-24760 
 
 0-27573 
 
 0-41728 
 
 < = 7 30'. 
 
 i 
 
 c=0 
 P 
 
 r2 
 
 c=0-l 
 P 
 "JST 
 
 c=0-2 
 P 
 JS 
 
 o=0-3 
 P 
 r2 
 
 c=0-4 
 P 
 
 Ta" 
 
 c=0-5 
 P 
 
 7T 
 
 c 1-0 
 P 
 
 r 2 
 
 0-05 
 
 0-06180 
 
 0-12867 
 
 0-19937 
 
 0-27125 
 
 0-34356 
 
 0-41606 
 
 0-77944 
 
 o-io 
 
 0-08514 
 
 0-14666 
 
 0-20930 
 
 0-27237 
 
 0-33561 
 
 0-39895 
 
 0-71618 
 
 0-15 
 
 0-10380 
 
 0-16001 
 
 0-21657 
 
 0-27326 
 
 0-33003 
 
 0-38683 
 
 0-67110 
 
 0-20 
 
 0-11813 
 
 0-16948 
 
 0-22089 
 
 0-27237 
 
 0-32384 
 
 0-37533 
 
 0-63286 
 
 0-25 
 
 0-12870 
 
 0-17557 
 
 0-22244 
 
 0-26932 
 
 0-31619 
 
 0-36306 
 
 0-59743 
 
 030 
 
 0-13598 
 
 0-17866 
 
 0-22134 
 
 0-26403 
 
 0-80673 
 
 0-34943 
 
 0-56'295 
 
 0-35 
 
 0-14040 
 
 017909 
 
 0-21783 
 
 0-25661 
 
 0-29542 
 
 0-33424 
 
 0-52846 
 
 0-40 
 
 0-14234 
 
 0-17718 
 
 0-21215 
 
 0-24720 
 
 0-28230 
 
 0-31744 
 
 0-49344 
 
 0-45 
 
 0-14211 
 
 0-17323 
 
 20454 
 
 0-23598 
 
 0-26751 
 
 029910 
 
 0-45763 
 
 0-50 
 
 0-14003 
 
 0-16753 
 
 0-19528 
 
 0-22319 
 
 0-25124 
 
 0-27938 
 
 0-42096 
 
 
 
 
 
 
 
 
 1 
 
692 
 
 APPENDIX. 
 
 a 
 
 c=0 
 P 
 
 I5r 
 
 c=0-l 
 P 
 
 "ft" 
 
 ' c=0-2 
 P 
 
 7*" 
 
 c=0-8 
 P 
 
 r% 
 
 c=0-4 
 P 
 
 r2 
 
 c=0-5 
 P 
 
 7T 
 
 c=l-0 
 P 
 
 r2 
 
 0-05 
 
 0-05310 
 
 0-12265 
 
 0-19488 
 
 0-26748 
 
 0-34018 
 
 0-41293 
 
 0-77681 
 
 o-io 
 
 0-07903 
 
 0-14170 
 
 0-20493 
 
 0-26832 
 
 0-33176 
 
 0-39524 
 
 0-71277 
 
 0-15 
 
 '09990 
 
 0-15658 
 
 0-21336 
 
 0-27022 
 
 0-32708 
 
 0-38395 
 
 0-66840 
 
 0-20 
 
 0-11631 
 
 0-16781 
 
 0-21931 
 
 0-27083 
 
 0-32234 
 
 0-37386 
 
 0-63145 
 
 0-25 
 
 0-12894 
 
 0-17582 
 
 0-22268 
 
 0-26955 
 
 0-31643 
 
 0-36330 
 
 0-59767 
 
 0-30 
 
 0-13835 
 
 0-18096 
 
 0-23361 
 
 0-26627 
 
 0-30895 
 
 0-35163 
 
 0-56510 
 
 0-35 
 
 0-14494 
 
 0-18355 
 
 0-22224 
 
 0-26098 
 
 0-29976 
 
 0-33855 
 
 0-53271 
 
 0-40 
 
 0-14905 
 
 0-18384 
 
 0-21878 
 
 0-25380 
 
 0-28888 
 
 0-32399 
 
 0-49995 
 
 G'45 
 
 0-15097 
 
 0-18212 
 
 0-21344 
 
 0-24488 
 
 0*27641 
 
 0-30800 
 
 046652 
 
 0-50 
 
 0-15099 
 
 0-17860 
 
 0-20642 
 
 0-23439 
 
 0-26247 
 
 0-29065 
 
 0-43232 
 
 = 22 30'. 
 
 a 
 
 c=0 
 P 
 
 7-2 
 
 c=0-l 
 P 
 
 Te' 
 
 c=0-2 
 P 
 
 r& 
 
 c=0-3 
 P 
 
 r 
 
 c=0-4 
 P 
 
 ~r* 
 
 c=0-5 
 P 
 
 r- 
 
 c=l-0 
 P 
 
 7T 
 
 0-05 
 
 0-06102 
 
 0-13346 
 
 0-20621 
 
 0-27899 
 
 0-35178 
 
 0-42458 
 
 0-78857 
 
 o-io 
 
 0-08700 
 
 0-15053 
 
 0-21407 
 
 0-27760 
 
 0-34113 
 
 0'4"466 
 
 0-72233 
 
 0-15 
 
 0-10877 
 
 0-16567 
 
 0-22257 
 
 0-27947 
 
 0-33638 
 
 0-39328 
 
 0-67778 
 
 0-20 
 
 0-12635 
 
 0-17785 
 
 0-22936 
 
 0-28087 
 
 0-33239 
 
 0-38391 
 
 0-64150 
 
 0-25 
 
 0-14037 
 
 0-18716 
 
 0-23399 
 
 0-28082 
 
 0-32767 
 
 0-37453 
 
 0-60886 
 
 0-30 
 
 0-15129 
 
 0-19381 
 
 0-23640 
 
 0-27902 
 
 0-32166 
 
 0-36432 
 
 0-57773 
 
 0-35 
 
 0-15948 
 
 0-19804 
 
 0-23669 
 
 0-27540 
 
 0-31415 
 
 0-35292 
 
 0-54700 
 
 0-40 
 
 0-16525 
 
 0-20005 
 
 0-23497 
 
 0-26999 
 
 0-30506 
 
 0-34017 
 
 0-51608 
 
 0-45 
 
 0-16883 
 
 0-20005 
 
 0-23141 
 
 0-26289 
 
 0-29444 
 
 0-32604 
 
 0-48460 
 
 0-50 
 
 0-17047 
 
 0-19824 
 
 0-22617 
 
 0-25423 
 
 0-28238 
 
 0-31060 
 
 0-45241 
 
 a 
 
 o=0 
 P 
 
 ~r% 
 
 c=0-l 
 P 
 
 r2 
 
 c=0-2 
 P 
 
 r 
 
 c=0-8 
 P 
 
 75" 
 
 c=0-4 
 P 
 
 rB 
 
 c=0-5 
 P 
 
 r 
 
 o 1-0 
 p 
 
 72~ 
 
 0-05 
 
 0-09355 
 
 0-16408 
 
 0-23605 
 
 0-30845 
 
 0-38101 
 
 0-45365 
 
 0-81731 
 
 o-io 
 
 0-11297 
 
 0-17592 
 
 0-23922 
 
 0-30263 
 
 0-36609 
 
 0-42957 
 
 0-74711 
 
 0-15 
 
 0-13295 
 
 0-18962 
 
 0-24640 
 
 0-30323 
 
 0-36009 
 
 0-41696 
 
 0-70138 ' 
 
 0-20 
 
 0-15038 
 
 0-20172 
 
 0-25314 
 
 0-30459 
 
 0-35606 
 
 0-40755 
 
 0-66506 
 
 0-25 
 
 0-16493 
 
 0-21160 
 
 0-25834 
 
 0-30513 
 
 0-35193 
 
 0-39876 
 
 0-63299 
 
 0-30 
 
 0-17673 
 
 0-21917 
 
 0-26170 
 
 0-S0427 
 
 0-34688 
 
 0-38951 
 
 0-60282 
 
 0-35 
 
 0-18599 
 
 0-22452 
 
 0-26314 
 
 0-30182 
 
 0-34055 
 
 0-37930 
 
 0-57332 
 
 0-40 
 
 0-19293 
 
 0-22777 
 
 0-26271 
 
 0-29773 
 
 0-33280 
 
 Q -367 91 
 
 0-54380 
 
 0-45 
 
 0-19774 
 
 0-22906 
 
 0-26050 
 
 0-29202 
 
 0-32361 
 
 0-35524 
 
 0-51385 
 
 0-50 
 
 0-20060 
 
 0-22854 
 
 0-25661 
 
 0-28476 
 
 0-31299 
 
 0-34128 
 
 0-48327 
 
HORIZONTAL THRUST OF AN ARCH. 
 
 ,=370 30'. 
 
 693 
 
 
 c=0 
 
 c=0-l 
 
 0=0-2 
 
 c=0-3 
 
 o=0-4 
 
 c=0-5 
 
 c=l-0 
 
 a 
 
 F 
 
 P 
 
 P 
 
 P 
 
 P 
 
 P 
 
 P 
 
 
 re 
 
 re 
 
 re 
 
 re 
 
 T* 
 
 r2 
 
 re 
 
 0-05 
 
 0-14749 
 
 0-21733 
 
 0-28854 
 
 0-36038 
 
 t)-48255 
 
 0-50490 
 
 0-86784 
 
 o-io 
 
 0-15949 
 
 0-22174 
 
 0-28457 
 
 0-34768 
 
 0-41093 
 
 0-47426 
 
 0-79141 
 
 0-15 
 
 0-17605 
 
 0-23233 
 
 0-28886 
 
 0-34553 
 
 0-40226 
 
 0-45904 
 
 9-74322 
 
 0-20 
 
 0-19209 
 
 0-24321 
 
 0-29448 
 
 0-34583 
 
 0-39722 
 
 0-44865 
 
 0-70598 
 
 0-25 
 
 0-20627 
 
 0-25282 
 
 0-29948 
 
 0-34619 
 
 0-39294 
 
 0-43972 
 
 0-67382 
 
 0-30 
 
 0-21827 
 
 0-26066 
 
 0-30314 
 
 0-34568 
 
 0-38825 
 
 0-43085 
 
 0-64406 
 
 0-35 
 
 0-22805 
 
 0-26659 
 
 0-30521 
 
 0-34388 
 
 0-38259 
 
 042133 
 
 0-61529 
 
 0-40 
 
 0-23570 
 
 0-27060 
 
 0-30558 
 
 0-34062 
 
 0-37571 
 
 0-41088 
 
 0-58673 
 
 0-45 
 
 0-24130 
 
 0-27275 
 
 0-30427 
 
 0-33586 
 
 0-36749 
 
 0-39916 
 
 0-55787 
 
 0-50 
 
 0-24499 
 
 0-27312 
 
 0-30132 
 
 0-32958 
 
 0-35789 
 
 0-38625 
 
 0-52845 
 
 45. 
 
 
 c=0 
 
 c=0-l 
 
 c=0.2 
 
 c=0-8 
 
 c=0-4 
 
 c=0-5 
 
 c=l-0 
 
 a 
 
 P 
 
 P 
 
 P 
 
 P 
 
 P 
 
 P 
 
 P 
 
 
 re 
 
 re 
 
 ? 
 
 r2 
 
 re 
 
 r2 
 
 r2 
 
 0-05 
 
 0-23105 
 
 0-30081 
 
 0-37162 
 
 0-44305 
 
 0-51485 
 
 0-58688 
 
 0-94881 
 
 o-io 
 
 0-23318 
 
 0-29507 
 
 0-35754 
 
 0-42034 
 
 0-48333 
 
 0-54646 
 
 0-86300 
 
 0-15 
 
 0-24478 
 
 0-30079 
 
 0-35708 
 
 0-41355 
 
 0-47013 
 
 0-62678 
 
 0-81059 
 
 0-20 
 
 0-25819 
 
 0-30915 
 
 0-36028 
 
 0-41151 
 
 0-46281 
 
 0-51416 
 
 0-77124 
 
 0-25 
 
 0-27104 
 
 0-31752 
 
 0-36410 
 
 0-41074 
 
 0-45744 
 
 0-50417 
 
 0-73809 
 
 0-30 
 
 0-28248 
 
 0-32486 
 
 0-36731 
 
 0-40981 
 
 045235 
 
 0-49493 
 
 0-70803 
 
 0-35 
 
 0-29216 
 
 0-33073 
 
 0-36935 
 
 0-40803 
 
 0-44674 
 
 0-48547 
 
 0-67939 
 
 0-40 
 
 0-29997 
 
 0-33494 
 
 0-36998 
 
 0-40506 
 
 0-44016 
 
 0-47530 
 
 0-65123 
 
 0-46 
 
 0-30589 
 
 0-33745 
 
 0-36907 
 
 0-40072 
 
 0-43240 
 
 0-46412 
 
 0-62294 
 
 0-50 
 
 0-30996 
 
 0-33824 
 
 0-36657 
 
 0-39494 
 
 0-42334 
 
 0-45177 
 
 0-59419 
 
 1 
 
 
 
 
 
 
 
 ! 
 
691 
 
 APPENDIX. 
 
 TABLE IY. 
 
 Mechanical Properties of the Materials of Construction. 
 
 2fote.The capitals affixed to the numbers In 
 
 B. Barlow, Report to the Commissioners of 
 
 the Navy, &c. 
 Be. Be van. 
 
 Br. Belidor, Arch, ffydr. 
 Bru. Brunei. 
 
 C. Couch. 
 
 D. W. Daniell and Wheatstone, Report on the 
 
 Stone for the Houses of Parliament. 
 
 F. Fairbairn. 
 
 H. Hodgkinson, Report to the British As- 
 sociation of Science. &c. 
 
 K. Kirwan. 
 
 this table refer to the following authorities: 
 La. Lame. 
 
 M. Muschenbroek, Inirod. ad Phil. Nat. i. 
 Mi. Mitis. 
 Mt. Mushct. 
 Pa. Colonel Pasley. 
 R. Eondelet, VArt de Eatir, iv. 
 Re. Rennie, Phil. Trans. &e. 
 T. Thomson, Chemistry. 
 Te. Telford. 
 Tr. Tredgold, Essay on the Strength oj 
 
 Cast Iron. 
 W. Watson. 
 
 Names of Material!. 
 
 Specific 
 gravity. 
 
 Weight of 1 
 cubic foot 
 in Ibi. 
 
 Tenacity 
 per 
 
 -TnT 1 
 
 Crushing 
 force p-r 
 square in,:h 
 in DM. 
 
 Modulus of 
 elasticity . 
 
 Modulu.1 of 
 rupture S. 
 
 Acacia (Eng. growth) 
 
 710 B. 
 
 4487 
 
 16000 Be. 
 
 
 1152000 B 
 
 11202 B. 
 
 Air (atmospheric) . . 
 
 001228 
 
 0-0768 
 
 
 
 
 
 Alabaster (yel. Malta) 
 
 2-699 
 
 168-68 
 
 
 
 
 
 Do. (stained brown) 
 
 2-744 
 
 171-50 
 
 
 
 
 
 Do. (oriental white) 
 
 2-780 
 
 17062 
 
 
 
 
 
 j Alder .... 
 
 800 M. 
 
 50-00 
 
 14186 M. 
 
 6895 H. 
 
 
 
 Antimony (cast) 
 
 4-500 M. 
 
 281-25 
 
 1066 M. 
 
 
 
 
 Apple-tree 
 
 798 M. 
 
 49-56 
 
 19500 Be. 
 
 
 
 
 Ash . . . j 
 
 690 
 to -845 
 
 48-12 
 58-81 
 
 j- 17207 B. 
 
 j 8688 H. 
 \ 9868 H. 
 
 j- 1644800 B 
 
 12156 B. 
 
 Bay-tree . 
 
 882 M. 
 
 51-87 
 
 12396 
 
 7158 H. 
 
 
 
 Bean (Tonquin) 
 
 1-080 
 
 67-50 
 
 
 
 2601600 B. 
 
 20886 B. 
 
 Beech . J 
 
 854 
 
 5887 
 
 15784 B.' 
 
 7733 H. 
 
 ' 1 Q^QAAA "D 
 
 OQQA T? 
 
 
 to -690 
 
 4312 
 
 17S50 B. 
 
 9863 dry H. 
 
 / loOoOUU 15. 
 
 yooo x>. 
 
 Birch (common) 
 
 792 B. 
 
 4950 
 
 15000 | 
 
 4538 H. 
 6402 dryH. 
 
 j- 1562400 B. 
 
 10920 B. 
 
 Do. (American) 
 
 648 B. 
 
 40-50 
 
 . 
 
 11663 H. 
 
 1257600 B. 
 
 9624 B. 
 
 Bismuth (cast) . 
 
 9-810 M. 
 
 61887 
 
 ' 3250 M. 
 
 
 
 
 Bone of an ox 
 
 1-656 M. 
 
 103-50 
 
 5265 
 
 
 
 
 Box (dry) . 
 
 960 B. 
 
 6000 
 
 19891 B. 
 
 1C299 H. 
 
 
 
 Brass (cast) 
 
 8-899 
 
 52500 
 
 17968 Re. 
 
 10804 Re. 
 
 8930000 
 
 
 T>o. (wire-drawn) . 
 
 8-544 
 
 584-00 
 
 
 
 
 
 Brick (red) 
 
 2-168 Re. 
 
 135-50 
 
 280 
 
 807 Re. 
 
 
 
 Do. (pale red) 
 
 2-085 Re. 
 
 130 81 
 
 800 
 
 562 Re. 
 
 
 
 Brick-work 
 
 1'800 
 
 112'50 
 
 
 
 
 
 Bullet-tree (Berbice). 
 
 1 C29 B. 
 
 6431 
 
 . 
 
 . 
 
 2610600 B. 
 
 15686 B. 
 
 Cane .... 
 
 0-400 
 
 2500 
 
 6800 Be. 
 
 
 
 
 Cedar (Canadian,fresh) 
 
 0-909 0. 
 
 5681 
 
 11400 Be' 
 
 5674 H. 
 
 
 
 Do. (seasoned) 
 
 0-758 
 
 47-06 
 
 
 4912 H. 
 
 
 
 Chalk . . 
 
 2-784 
 to 1-869 
 
 17400 
 116-81 
 
 \- ' 
 
 884 Re. 
 
 
 
 Chestnut . 
 
 0-657 R 
 
 41 -06 
 
 -lOOflA T> 
 
 
 
 
 Clay (common) . 
 
 1-919 Br. 
 
 119-98 
 
 looUU XV. 
 
 
 
 
 Coal (Welsh furnace). 
 
 1-887 Mt. 
 
 88-56 
 
 
 
 
 
 Do. (coke) . 
 
 1-OOOMt 
 
 62-50 
 
 
 
 
 
 Do. (Alfreton) . 
 
 1-285 Mt. 
 
 77-18 
 
 
 
 
 
 Do. (Butterly) . 
 
 1 264 Mt. 
 
 79-00 
 
 
 
 
 
 Do. (coke) 
 
 1-100 Mt. 
 
 6875 
 
 
 
 
 
 
 Do. (Welsh stone) . 
 
 1-368 Mt. 
 
 85-50 
 
 
 
 
 1 
 
 Do. (coke). 
 
 1-890 Mt. 
 
 8687 
 
 
 
 
 
 Do. (Welsh slaty) . 
 Do. (Derby, cannel) . 
 
 1 409 Mt. 
 1-278 Mt. 
 
 88-06 
 79-87 
 
 
 
 
 
 Do. (Kilkenny) 
 
 1-6C2 Mt. 
 
 100-12 
 
 
 
 
 
 Do. (coke). 
 
 1-657 Mt. 
 
 108-56 
 
 
 
 
 
 Do. (slaty). 
 
 1-448 Mt. 
 
 90-18 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
PROPERTIES OF MATERIALS OF CONSTRUCTION. 
 
 695 
 
 
 
 
 Tenacity 
 
 Crushing 
 
 
 
 
 Specific 
 
 We ght of 
 
 per 
 
 force per 
 
 Modulu* of 
 
 Modulus of 
 
 Names of Material*. 
 
 gravity. 
 
 1 cubic foo 
 in Ibs. 
 
 square inch 
 in Ibs. 
 
 square inch 
 in Ibs. 
 
 elasticity K. 
 
 rupture S. 
 
 Coal (Bonlavooneen) . 
 Do. (coke) . 
 
 1-436 Mt 
 1-596 Mt. 
 
 89-75 
 99-75 
 
 
 
 
 
 Do. (Corgee) 
 
 1-403 Mt 
 
 87-68 
 
 
 
 
 
 Do. (coke) . 
 
 1-656 Mt. 
 
 108-50 
 
 
 
 
 
 Do. (Staffordshire) . 
 Do. (Swansea) . 
 
 1-240 
 1-357 K. 
 
 78-12 
 84-81 
 
 
 
 
 
 Do. (Wigan) 
 
 1-268 K. 
 
 79-25 
 
 
 
 
 
 Do. (Glasgow) . 
 
 1-290 
 
 80-62 
 
 
 
 
 
 Do. (Newcastle) 
 
 1-257 K. 
 
 78-56 
 
 
 
 
 
 Do. (common cannel) 
 
 1-282 K. 
 
 77-00 
 
 
 
 
 
 Do. (slaty cannel) 
 
 1-426 K. 
 
 89-12 
 
 
 
 
 
 Copper (cast) 
 
 8-607 
 
 587-93 
 
 19072 
 
 
 
 
 Do. (sheet) 
 
 8-785 
 
 549-06 
 
 
 
 
 
 Do. (wiredrawn) 
 
 8-878 
 
 560-00 
 
 61228 
 
 
 
 
 Do. (in bolts) . 
 
 
 
 48000 
 
 
 
 
 Crab-tree . 
 
 0-765 
 
 '47-80 
 
 
 6499 H. 
 
 
 
 Deal (Christiana mid- 
 
 
 
 
 
 
 
 dle) . 
 
 0-698 B. 
 
 48-62 
 
 12400 
 
 
 1672000 B 
 
 9864 B. 
 
 Do. (Memel middle) . 
 
 0-590 B. 
 
 86-87 
 
 < 
 
 
 1535200 B 
 
 10886 B. 
 
 Do. (Norway spruce) 
 Do. (English) . 
 
 0-340 
 0-470 
 
 21-25 
 29-37 
 
 ' 17600 
 7000 
 
 
 
 
 Earth (rammed) 
 
 1-584 Pa. 
 
 99-00 
 
 
 
 
 
 Elder .... 
 
 0-695 M. 
 
 48-43 
 
 10230 * 
 
 8467 H. 
 
 
 
 Elm (seasoned) . 
 
 0-588 C. 
 
 86-75 
 
 18489 M. 
 
 10831. H. 
 
 699840,B, 
 
 6078 B< 
 
 Fir (New England) . 
 
 0-553 B. 
 
 84-56 
 
 . 
 
 
 2191200- B. 
 
 6612 B. 
 
 Do. (Riga) . . . 
 
 0-753 B. 
 
 47-06 1 
 
 ' 11549 B. 
 to 12857 B. 
 
 5748 H. 
 o 688ft H.. 
 
 1328800 B. 
 869600 B. 
 
 6648 B. 
 7572 B. 
 
 Do. (Mar Forest) 
 
 0-693 B. 
 
 48-31 
 
 
 
 
 
 Flint .... 
 
 2-630 T. 
 
 164-87 
 
 
 
 
 
 Glass (plate) 
 
 2453 
 
 158-81 
 
 9420 
 
 
 
 
 Gravel 
 
 1'920 
 
 120-00 
 
 
 
 
 
 Granite (Aberdeen) . 
 
 2'526 
 
 164-00 
 
 
 
 
 
 Do. (Cornish) . 
 
 2-662 
 
 16680 
 
 
 
 
 
 Do. (red Egyptian) . 
 
 2-654 
 
 165-80 
 
 
 
 
 
 Hawthorn . 
 
 0-91 Be. 
 
 88-12 
 
 10500 Be. 
 
 
 
 
 Hazel .... 
 
 0-86 Be. 
 
 53-75 
 
 18000 Be. 
 
 
 
 
 Holly .... 
 
 0-76 Be. 
 
 47-5 
 
 16000 Be. 
 
 
 
 
 Horn of an ox . 
 
 1-689 M. 
 
 105-56 
 
 8949 
 
 
 
 
 Hornbeam (dry) 
 
 0-760 R. 
 
 47-50 
 
 20240 Be. 
 
 7289 H. 
 
 
 
 Iron (wrought Eng.) . 
 
 7-700 
 
 481-20 
 
 25itons, La. 
 
 
 
 
 Do. (in bars) 
 
 j -7600 
 1 to7-800 
 
 475-50 
 487-00 
 
 25itons, La. 
 
 
 
 
 Do. (hammered) 
 
 
 
 30 tons, Bru. 
 
 
 
 
 Do (Russian) in bars 
 
 m 
 
 
 27 tons, La. 
 
 
 
 
 Do (Swedish) in bars 
 
 . 
 
 
 12 tons, R. 
 
 
 
 
 Do. (English) in wfre 
 
 l 
 
 
 j 36 to 43 
 
 
 
 
 l-10th inch diameter 
 
 1 * 
 
 
 
 ; tons, Te. 
 
 
 
 
 Do. (Russian) in wire 
 l-20th to 1-SOth inch 
 diameter . 
 
 | 
 1 
 
 . 
 
 I 60 to 91 
 tons, La. 
 
 
 
 
 Do. rolled in sheets 
 
 
 
 
 
 
 
 and cut lengthwise . 
 Do. cut crosswise 
 
 
 
 
 
 14 tons, Mi. 
 18 tons, Mi. 
 
 
 
 
 Do. in chains, oval 
 
 
 
 
 
 
 
 links, 6 inches clear, 
 
 
 
 
 
 
 
 iron ^ inch diam. . 
 
 4 
 
 
 2Htons,Br. 
 
 
 
 
 Do. (Brunton's) with 
 
 
 
 
 
 
 
 stay across link 
 Iron, cast (old Park) 
 
 
 
 
 
 25 tons, B. 
 
 
 18014400 T. 
 
 48240 T 
 
 Do. (Adelphi) 
 
 
 
 
 
 18858600 T. 
 
 5860 T. 
 
 Do (Alfreton) 
 
 
 
 
 
 17686400 T. 
 
 4046 T. 
 
 Do. (scrap) 
 
 
 
 
 
 18082000 T. 
 
 5828 T! 
 
 Do. (Carron, No. 2 
 
 
 
 
 
 
 
 cold blast) 
 Do. (hot blast) . 
 
 7-066 H. 
 7-046 H. 
 
 441-62 
 440-87 
 
 16683 H. 
 18505 II. 
 
 06875 H. 
 08540 H. 
 
 17270500 H. 
 16085000 H. 
 
 J8556 H .*! 
 7508 H.*! 
 
 The numbers marked thus * are calculated from the experiments of Messrs. Hodgkinson & Fairbairn, 
 
600 
 
 APPENDIX. 
 
 1 
 
 N*met of Material.. 
 
 Specific 
 gravity. 
 
 Weight of 1 
 cubic foot 
 in Ibs. 
 
 Tenacity 
 per 
 square inch 
 in Ib.. 
 
 Crushing 
 force per 
 )uare inch 
 in Ib, 
 
 Modulus of 
 elasticity E. 
 
 Modunii of 
 rupture 8. 
 
 Iron, cast (do. No. 8, 
 
 
 
 
 
 
 
 Carron cold blast . 
 
 7-094 F. 
 
 443-37 
 
 14200 H. 
 
 115442 H. 
 
 16246966 F. 
 
 35980 F.* 
 
 Do. (hot blast) . 
 
 7-056 F. 
 
 441-00 
 
 17755 H. 
 
 183440 H. 
 
 17873100 F. 
 
 42120 F.* 
 
 Do. Devon. No. 8, 
 
 
 
 
 
 
 
 cold blast) 
 
 7-295 H. 
 
 455-93 
 
 
 
 22907700 H. 
 
 86288 H.* 
 
 Do. (hot blast) . 
 
 7-229 H. 
 
 451-81 
 
 29107 H. 
 
 145485 H. 
 
 22473650 H. 
 
 43497 H." 
 
 Do (Buflery, No. 1, 
 
 
 
 
 
 
 
 cold blast) 
 
 7-079 H. 
 
 442-48 
 
 17466 H. 
 
 93366 H. 
 
 15381200 H 
 
 37503 H.* 
 
 Do. (hot blast) . 
 
 6 998 H. 
 
 487-37 
 
 18434 H. 
 
 86397 H. 
 
 18730500 H. 
 
 85316 H.* 
 
 Do. (Coed Talon, No. 
 
 
 
 
 
 
 
 2, cold blast) . 
 
 6-955 F. 
 
 484-06 
 
 18855 H. 
 
 81770 H. 
 
 14813500 F. 
 
 33104 F * 
 
 Do. (hot blast) . 
 
 6-968 F. 
 
 435-50 
 
 16676 H. 
 
 82739 H. 
 
 14822500 F. 
 
 83145 F.* 
 
 Do. (Coed Talon, No. 
 
 
 
 
 
 
 
 8, cold blast) . 
 
 7-I94F. 
 
 449-62 
 
 
 . 
 
 17102000 F. 
 
 43541 F* 
 
 Do. (hot blast) . 
 
 6-970 F. 
 
 435-62 
 
 > 
 
 t 
 
 14707900 F. 
 
 40159 F.* 
 
 Do. (Elsicar, No. 1, 
 
 
 
 
 
 
 
 cold blast) . . 
 
 7-080 F. 
 
 489-87 
 
 
 
 13981000 F. 
 
 84862 F.* 
 
 Do (Milton, No. 1, 
 
 
 
 
 
 
 
 hot blast). 
 
 . 6-976 F. 
 
 486-00 
 
 t 
 
 t 
 
 11974500 F. 
 
 28552 F.* 
 
 Do. (Muirkirk, No. 1, 
 
 
 
 
 
 
 
 cold blast) 
 
 7-118 F. 
 
 444-56 
 
 t 
 
 t 
 
 14003550 F. 
 
 85923 F.* 
 
 Do. (hot blast) . 
 
 6-953 F. 
 
 434-56 
 
 
 m 
 
 13294400 F. 
 
 33850 F.* 
 
 Ivory .... 
 
 1-826 P. 
 
 114-12 
 
 16-626 
 
 
 
 
 Laburnum . 
 
 0-92 Be. 
 
 57-50 
 
 10500 Be. 
 
 
 
 
 Lance-wood 
 
 1-022 
 
 63-87 
 
 24696 
 
 
 
 
 Larch .... 
 
 0-522 B. 
 
 32-62 
 
 10220 E. 
 
 8201 H. 
 
 (green) 
 
 897600 B. 
 
 4992 B. 
 
 Do. (second specimen) 
 
 0-560 B. 
 
 85-00 
 
 8900 B. 
 
 5568 H. 
 (dry) 
 
 1052800 B. 
 
 6894 B. 
 
 Lead (cast English) . 
 
 11-446 M. 
 
 717-45 
 
 1824 Ee. 
 
 V>"V ) 
 
 720000 Tr. 
 
 
 Do. (milled sheet) . 
 
 11-407 T. 
 
 712-93 
 
 3328 Tr. 
 
 
 
 
 Do. (wire) . 
 
 11-317 T. 
 
 705-12 
 
 2581 M. 
 
 
 
 
 Lignum vitae [ous) 
 
 1-220 
 
 76-25 
 
 11800 M. 
 
 
 
 
 Limestone (arenace- 
 
 2-742 
 
 171-37 
 
 
 
 
 
 Do. (foliated) . 
 
 2-887 
 
 177-81 
 
 
 
 
 
 Do. (white fluor) . 
 
 8-156 
 
 197-25 
 
 
 
 
 
 Do. (green) . 
 
 3-182 
 
 198-87 
 
 
 
 
 
 Lime-tree . 
 
 0-760 
 
 47-50 
 
 23500 Be. 
 
 
 
 
 Lime (quick) 
 
 0-488 Br. 
 
 52-68 
 
 
 
 
 
 Mahogany (Spanish) . 
 Maple (Norway) 
 
 0-800 
 0-793 
 
 50-00 
 49-56 
 
 16500 
 10584 
 
 8198 H. 
 
 
 
 Marble (white Italian) 
 Do. (black Galway) . 
 Mercury (at 82) 
 
 2-638 H. 
 2-695 H. 
 13-619 
 
 16487 
 168-25 
 851-18 
 
 
 . 
 
 2520000 T. 
 
 1062 
 2664 
 
 Do. (at 60) . 
 
 13-580 
 
 848-75 
 
 
 
 
 
 Marl . . -j 
 
 1-600 
 
 100-00 
 
 
 
 
 
 
 to 2-877 T. 
 
 118-81 
 
 
 
 
 
 Mortar . . ' 
 
 1 751 Br. 
 
 107-18 
 
 50 
 
 
 
 
 Oak (English) . 
 
 0-934 B. 
 
 58-87 
 
 17-800 M.-! 
 
 4684 H. 
 9509 H. 
 
 [ 1451200 B. 
 
 0082 B. 
 
 Do. (Canadian). 
 
 0-872 B 
 
 54-50 
 
 I 
 
 1 0-253 M.j 
 
 (dry) 
 4231 H. 
 9509 H. 
 
 Ul48800B. 
 
 0596 B. 
 
 Do. (Dantzic) . 
 Do. (Adriatic) . 
 Do. (African middle). 
 
 0-756 B. 
 0-993 B. 
 0-972 B. 
 
 47-24 
 62-06 
 6075 
 
 12-780 
 
 (dry) 
 
 1191200 B. 
 974400 B. 
 2283200 B. 
 
 8742 B. 
 8298 B. 
 18566 B. 
 
 Pear-tree . 
 Pine (pitch. 
 
 0-661 M. 
 0-660 B. 
 
 41-81 
 4125 
 
 '7818M 
 
 7518 H. 
 
 1225600 B. 
 
 9792 B. 
 
 Do. (red) . . ] 
 Do. (Amer. yellow) . 
 Plane-tree . 
 
 0-657 B. 
 0-461 C. 
 0-64 Be. 
 
 4106 
 
 28-81 
 40-00 
 
 11700 Be. 
 
 5375 H. ' 
 5445 H. 
 
 1840000 B. 
 1600000 Tr. 
 
 8946 B. 
 
 Plum-tree . 
 
 0-785 M. 
 
 49-06 
 
 11-851- 
 
 9367 H. 
 8657 H. 
 
 
 
 
 
 
 
 (wet) 
 
 
 
 Poplar. . . . 
 
 0-388 M. 
 
 23-98 
 
 7200 Be.- 
 
 8107 H. 
 5124 H. 
 
 
 
 Pozzolano . 
 
 2-677 K. 
 
 169-87 
 
 
 (dry) 
 
 
 
 Sand (river) 
 
 1*886 
 
 117-R7 
 
 
 
 
 
 Serpentine (green) 
 
 2-574 K. 
 
 11 i Ol 
 
 168-87 
 
 
 
 
 
PROPERTIES OF MATERIALS OF CONSTRUCTION. 
 
 097 
 
 1 
 
 Names of mat. r al>. 
 
 Specific 
 grav.ty. 
 
 Weight of 
 1 cubic foot 
 
 in iha. 
 
 Tenacity 
 
 ^JTU 
 
 Crushing 
 /ua're'hich 
 
 Modulu. of 
 ehuticity E. 
 
 Modnlu. of 
 rupture 8. 
 
 Shingle - - - 1 424 Pa. 
 
 89-00 
 
 
 
 
 
 Silver (standard) - 10'812 T. 
 
 644-50 
 
 40902 M. 
 
 
 
 
 Slate (Welsh) - - 2'888 
 do. (Westmoreland) - 2-791 W. 
 
 180-50 
 
 12800 
 
 - 
 
 15800000 Tr 
 
 1 OQAAAAA m_ 
 
 11766 Re. 
 
 do. (Valontia) - - 2-880 Re'. 
 
 180*00 
 
 
 
 iiyuvuUu AJ 
 
 KOOA "PA 
 
 
 
 9600 
 
 
 1r7fWWWA T 
 
 0/^0 IwO. 
 
 Steel (soft) - - 7-780 
 
 486-25 
 
 120000 
 
 
 UOOOO Tr 
 
 
 do. (razor-tempered) - 7-840 
 Stone (Ancaster) - 2'182 D.W 
 
 490-00 
 186-37 
 
 150000 
 
 
 29000000 T. 
 
 
 do. Barnack - - 2*090 D.W 
 
 18062 
 
 
 
 
 
 do. Binnie - - 2-194 D.W 
 
 187*12 
 
 
 
 
 
 do. Bolsover - - 2-316 D.W 
 
 144-75 
 
 
 
 
 
 do. Box - - 1-889 D.W 
 
 114-98 
 
 
 
 
 
 do. Bramham Moor - 2-008 D.W 
 
 125-50 
 
 
 
 
 
 do. Brodsworth - 2-093 D.W 
 
 130*81 
 
 
 
 
 
 do. Cadeby - - 1-951 D.W 
 do. Oslthncss - - *2*764 R<? 
 
 121-98 
 
 
 
 
 
 do.' Craigleith- - 2-266 D.W 
 
 141-62 
 
 
 
 
 5142 Ee. 
 
 do. Chilmark (A) - 2-366 D.W 
 
 147-87 
 
 
 
 
 
 do. Chilmark (B) - 2-383 D.W 
 
 148-98 
 
 
 
 
 
 do. Chilmark (C) - 2-481 D.W 
 
 155-06 
 
 
 
 
 
 do. Darby Dale (Stan-! 
 
 
 
 
 
 
 cliffe) - - 2-626 D.W 
 
 164-25 
 
 
 
 
 
 do. Giffneuk - - 2 280 D.W 
 
 189-37 
 
 
 
 
 
 do. Gunbarrel (Stanley) 2 260 D.W 
 
 141-25 
 
 
 
 
 
 do. Ham Hill- - ,2260 D.W 
 
 14125 
 
 
 
 
 
 do. Haydon - - ' 2*040 D W 
 
 127-50 
 
 
 
 
 
 do. Heddon - - J2'229 D.W 
 
 139-81 
 
 
 
 
 
 do. Hildenly - 
 
 2-098 D.W 
 
 131-12 
 
 
 
 
 
 do. Hookstone 
 
 2-253 D.W 
 
 14081 
 
 
 
 
 
 do. Huddlestone 
 
 2 147 D.W 
 
 134*18 
 
 
 
 
 
 do. Little Hulton 
 
 2'857 H. 
 
 -J4.IT.0-J 
 
 
 
 
 774 TT 
 
 do. Keuton - 
 
 2-247 D!W 
 
 J.4( oi 
 , 140-43 
 
 
 
 
 < I3k -H. 
 
 do. Ketton 
 
 2-045 D.W 
 
 127-81 
 
 
 
 
 
 do. Ketton rag 
 
 2-490 D.W. 
 
 155-62 
 
 
 
 
 
 do. Mansfield, or Lind- 
 
 
 
 
 
 
 
 ley's red 
 
 2-838 D.W. 
 
 14612 
 
 
 
 
 
 do. white 
 
 2-277 D.W. 
 
 14281 
 
 
 
 
 
 do. Morley Moor 
 
 2-053 D.W. 
 
 128-81 
 
 
 
 
 
 do. Park Nook 
 
 2-138 D.W. 
 
 183-62 
 
 
 
 
 
 do. Park Spring 
 
 2-321 D.W. 
 
 145-06 
 
 
 
 
 
 do Portland (Waycroft 
 
 
 
 
 
 
 
 Quarry 
 do. Redgate - 
 
 2-145 D.W. 
 2 239 D.W. 
 
 184-06 
 189-93 
 
 
 
 
 
 do. Roach Abbey 
 
 2-134 D.W. 
 
 O.K77 1 TT 
 
 183-87 
 
 
 
 
 2868 H. 
 
 do. Stanley - 
 
 a 0( 1 II. 
 
 2-227 D.W. 
 
 161 "06 
 189-18 
 
 
 
 
 
 do. Taynton - 
 
 2-108 D.W. 
 
 181-48 
 
 
 
 
 
 do. Totterwhoe 
 
 2-891 D.W. 
 
 118-18 
 
 
 
 
 9 
 
 do. Jackdaw crag 
 
 2-070 D.W. 
 
 O.QOA TT 
 
 129-37 
 
 
 
 
 1116 H 
 
 do. Yorkshire nag 
 do. Green Moor 
 
 z oM n.. 
 2-584 Re. 
 
 140 "00 
 158-37 
 
 . 
 
 . 
 
 . 
 
 2010 Re. 
 
 Sycamore 
 
 0-69 Be. 
 
 43-12 
 
 18000 Be. 
 
 
 
 
 'eak (dry) 
 
 0*657 C. 
 
 41-06 
 
 15000 B. 
 
 12101 H. 
 
 2414400 B. 
 
 4772 B. 
 
 Mle (common) - 
 Tin (cast) 
 
 1-815 Br. 
 7*291 Tr. 
 
 113-48 
 455-68 
 
 6822 M. 
 
 - 
 
 4608000 Tr. 
 
 
 Water (sea) 
 
 1027T. 
 
 64-18 
 
 
 
 
 
 do. (rain) 
 
 1-000 
 
 62-50 
 
 
 
 
 
 Walnut - 
 
 0-671 M. 
 
 41-93 
 
 8130 M. 
 
 6645 H. 
 
 
 
 
 
 
 7667 
 
 . 
 
 820000 TT. 
 
 
 Willow (dry) - 
 
 0-890 
 
 24-87 
 
 14000 Be. 
 
 
 
 
 Yew (Spanish) - 
 
 0-807 M. 
 
 50-48 
 
 8000 Be. 
 
 
 
 
 Zinc 
 
 7-028 W. 
 
 48925 
 
 - 
 
 - 
 
 8680000 Tr. 
 
 
 Note. The experiments of Mr. Hodgkinson, from which the moduli of rupture of stones contained 
 in the last column of the above table have been calculated, together with those detailed in Art. 410., 
 i.j. in part of a more extended inquiry, which, when completed, will be laid before the Royal Society. 
 Tin- crushing forces given in the above tables are in every case determined by the compression of 
 j>; i.-ms of such a height as to allow the rupture to take place by the sliding of the upper portion freely 
 olf. along its plane of separation from the lower (see Art. 406). The experiments of Mr. Hodgkinson 
 have shown that all results obtained without reference to this circumstance are erroneous. (Se 
 ^castrations of Mechanics, p. 402.) 
 
APPENDIX. 
 
 TABLE V. 
 
 Useful Numbers. 
 
 it . 
 
 Log.* 
 
 Log. rt 
 
 1 
 
 ft 
 
 x* . . 
 
 1_ 
 
 *'_' 
 
 4/* 
 
 4^ 
 
 4/2 
 
 =3-1415927 
 =0-4971499 
 :1-1447299 
 
 =0-3183099 
 =9-8696044 
 =0-1013212 
 =1-7724538 
 =0-5641896 
 :1-4142136 
 
 1 
 
 V5 
 
 71 
 
 Vi 
 V! 
 
 =0-7071068 
 =4-4428829 
 =2-2214415 
 =0-4501582 
 =1-2533141 
 =0-7978846 
 
 . =2-7182818 
 
 Log. c . . . =0-4342945 
 
 Modulus of common logarithms =-434294482 
 
 Log. of ditto =9-6377843 
 
 g =32-19084 
 
 4/0 =5-67363 
 
 Log. g =1-5077222 
 
 Inches in a French m^tre =39*37079 
 
 Log. of ditto =1-5951741 
 
 Feet in ditto =3-2808992 
 
 Log. of ditto =0-5159929 
 
 Square feet in the square metre =10*764297 
 
 Acres in the Are =0-024711 
 
 Lbs. in a kilogramme =2-20548 
 
 Log. of ditto =0-3435031 
 
 Imperial gallons in a litre =0-2200967 
 
 Lbs. per square inch in 1 kilogramme per square millimetre =1422 
 
 Owts. ditto, ditto =12'7 
 
 Volume of a sphere whose diameter is 1 ... =0-5235988 
 
 Arc of 1 to rad. 1 =0-017453293 
 
 Arc of 1' to rad. 1 =0-000290888 
 
 Arc of 1" to rad. 1 =0-000004848 
 
 Degree in an arc whose length is 1 =57-295780 
 
 Grains in 1 oz. avoirdupois =437i 
 
USEFUL NUMBERS. 699 
 
 Grains in 1 Ib. ditto =7000 
 
 Grains in a cubic inch of distilled water, Bar. 30 in., Th. 62 =252-458 
 
 Cubic inches in an ounce of water =1*73298 
 
 Cubic inches in the imperial gallon =277'276 
 
 Feet in a geographical mile =6075*6 
 
 Log. of ditto =3-7835892 
 
 Feet in a statute mile =5280 
 
 Log. of ditto =3-7226339 
 
 Length of seconds' pendulum in inches ..... =39-19084 
 
 Cubic inches in 1 cwt. of cast iron =430-25 
 
 Bar iron . . . . . =397'60 . 
 
 Cast brass .... =368-88 
 
 Cast copper .... =352'41 
 
 Cast lead =272-80 
 
 Cubic feet in 1 ton of paving stone ..... =14-835 
 
 Granite =13'505 
 
 Marble =13-070 
 
 Chalk =12-874 
 
 Limestone ..... =11-273 
 Elm =64-460 
 
 Honduras mahogany . . . =64-000 
 Mar Forest fir .... =51-650 
 
 Beech =51-494 
 
 Kigafir =47'762 
 
 Ash and Dantzic oak . . . =47'158 
 
 Spanish mahogany . . . =42*066 
 English oak =36-205 
 
 To find the weight in Ibs. of 1 foot of common rope, multi- 
 ply the square of its circumference in inches by . *044 to -04C 
 Ditto for a cable -027 
 
 Note. The numerical values of the function of * in this table were calcu- 
 lated by Mr. Goodwin. These, together with the numbers of cubic inches and 
 feet per cwt. or ton of different materials, are taken from the late Dr. Gregory's 
 excellent treatise, entitled Mechanics for Practical Men. The other numbers 
 of the table are principally taken from Mr. Babbage's Tables of Logarithm* 
 and the Aide Memoire of M. Morin. 
 
 THE END. 
 
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