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AN 
 
 ELEMENTARY TREATISE 
 
 OF 
 
 MECHANICAL PHILOSOPHY, 
 
 &c. &c. 
 
AN 
 
 ELEMENTARY TREATISE 
 
 MECHANICAL PHILOSOPHY, 
 
 WRITTEN FOR THE USE OF THE 
 
 UNDERGRADUATE STUDENTS 
 
 THE UNIVERSITY OF DUBLIN 
 
 BY 
 
 BARTHOLOMEW LLOYD, D.D., M.R.LA., 
 
 PROVOST OF TRINITY COLLEGE, DUBLIN. 
 
 VOL. I. ^^^ ^-.^^ ^^ 
 
 SECOND EDITION. \\ -Ld^Ortiry^ 
 
 S^ Calif orBi^L 
 DUBLIN : 
 
 PRINTED AT THE UNIVERSITY PRESS, 
 
 FOB MILLIKEN AND SON, 104, GKAFTON-STEEET, 
 
 BOOKSELLERS TO THE UNIVERSITY. 
 
 M.DCCC.XXXV. 
 

 .?" 
 
 \> 
 
 ffo 
 
 R. Graisberry, Printei: to the University. 
 
TO 
 
 THE UNDERGKADUATE STUDENTS 
 
 OF 
 
 THE UNIVERSITY, 
 
 THE FOLLOWING TREATISE, 
 
 INTENDED FOR THEIR USE, 
 
 IS DEDICATED 
 
 BY THEIR VERY SINCERE FRIEND, 
 
 THE AUTHOR. 
 
PREFACE. 
 
 It has ever been expected of an author, that in 
 presenting himself to the notice of the public, he 
 should state his claims to the attention he seems to 
 solicit, and afford the means of judging how far it is 
 likely to be rewarded. Anticipating these reason- 
 able expectations, he endeavours to satisfy them by 
 a declaration of the motives which may have led to 
 the undertaking, or influenced his judgment respect- 
 ing any peculiarity of plan or execution, by which 
 his work may be distinguished. By aff'ording this 
 satisfaction to others, he also consults for his own 
 interest or reputation ; being well aware, that it is 
 only by a due regard to the particular ends and ob- 
 jects of the writer, it can be ascertained whether his 
 efforts are called for, his methods well chosen ; or, 
 even after the perusal, whether the work itself is 
 fairly executed. This first duty to the public and 
 
 b 2 
 
VIU PREFACE. 
 
 liimself, the writer of the following Treatise shall 
 endeavour to discharge as well as he is able. 
 
 Having been called to the chair of Natural 
 Philosophy in the University of Dublin, he naturally 
 felt it to be peculiarly his duty, in addition to his 
 course of public lectures, to furnish a manual for the 
 instruction of the students in his own department ; 
 the work long in their hands, consisting of lectures 
 formerly delivered by Professor Helsham, being ex- 
 tremely imperfect, even as a system of Statics, to 
 which branch it is almost exclusively devoted. In 
 making this remark, it is not intended to cast any 
 imputation on the memory of that writer. His 
 volume conveys, in the most clear and familiar style, 
 nearly all that was known on the subject when it 
 first appeared. But whatever may be its peculiar 
 merits, he who is aware of the vast improvements 
 which have been made in the establishment and de- 
 velopment of the principles of Mechanical Philoso- 
 phy, needs only to be informed, that the work here 
 spoken of appeared at the beginning of the eighteenth, 
 and was probably written before the close of the 
 seventeenth, century; and it is presumed that he 
 will deem it unnecessary to inquire whether it con- 
 tains, or not, a course of elementary instruction fitted 
 for the present age. 
 
PREFACE. IX 
 
 Some attempt to supply the deficiency bad in- 
 deed been made by the late Bishop Hamilton, author 
 of a work on Conic Sections, fit to be placed beside 
 the most finished productions of the ancient Geome- 
 ters. But Natural Philosophy is chiefly of recent 
 growth ; and Elementary Treatises are the latest 
 which arrive at perfection. During the progress of 
 science, the task of its cultivators is that of ex- 
 ploring some region of vast extent : and it is not 
 until their discoveries shall have swelled so as to 
 approximate, that the whole can be combined into 
 one continuous system. It is not, therefore, detract- 
 ing from the merited reputation of that writer to as- 
 sert, that the four supplementary lectures, first pub- 
 lished about fifty years ago, are now only fitted to 
 satisfy the curious as to the state of the science at 
 that period. 
 
 Such was the acknowledged character of the 
 manuals from which the students were to be in- 
 structed in this department of science : and the 
 tutors, generally too much occupied in the discharge 
 of their laborious duties to engage in a work for 
 which some leisure was essentially requisite, con- 
 tented themselves with supplying the deficiency of 
 written treatises by the oral instruction of their 
 
Xll PREFACE. 
 
 the author in this view, will also agree with him in 
 supposing, that as the present attempt was not un- 
 called for, the only charge which can lie against him 
 must relate to the quality of the work which he pro- 
 poses to substitute for those already in use. For 
 this indeed he has some apologies to offer, and a 
 large share of indulgence to solicit. Such he hopes 
 will not be withheld, when the purposes for which 
 the work was more particularly intended are con- 
 sidered. Some explanation of these purposes may 
 be requisite for the stranger to the plan of education 
 adopted in the Dublin University. He is to be in- 
 formed, that whilst it invites by honors to the highest 
 attainments, and provides the most able assistance 
 for those who may be so allured, it forbears from 
 compelling attention to the subjects of its instruc- 
 tions beyond very moderate limits. The sound dis- 
 cretion manifested in this treatment of the younger 
 members, is chiefly conspicuous in what relates to 
 their mathematical studies. In a course of academic 
 instruction, by which the youth of the country are 
 to be qualified for the various professional duties of 
 active life, and, therefore, necessarily embracing a 
 considerable variety of subjects, it was not to be 
 proposed that all should become profound mathe- 
 
PREFACE. XIU 
 
 maticians. Neither could it be deemed expedient to 
 interfere with the tastes of individuals, by which they 
 may be directed into some other of the many walks 
 of literature, perhaps equally useful, and certainly 
 to many minds more inviting. In a society so diver- 
 sified by tastes and objects, the number of those who 
 do not enter on the higher branches of Mathematics 
 must, at all times, be considerable. The qualifica- 
 tions of such, as well as of those of higher attain- 
 ments among the students of the University, were 
 necessarily to be attended to in a work proposed for 
 their general use. Accordingly, in the following 
 Treatise, the mathematical reasoning is sparingly 
 resorted to, and as much as possible confined to the 
 mere elements. Writing under these restrictions, 
 and, in a certain degree, obliged to forego the aids 
 peculiarly adapted to the science here treated of, 
 the writer is apprehensive that he may have af- 
 forded to the mathematical reader some ground of 
 complaint ; yet it is to be hoped that he will not be 
 dissatisfied with those sections which are more par- 
 ticularly intended for his perusal. 
 
 Such were the views of the author, and such the 
 modifications of character which he has endeavoured 
 to give to his performance. Should it be favourably 
 
XIV PREFACE. 
 
 received by the University, as a work suited to the 
 purposes for which it was intended, he shall feel 
 encouraged to complete what is now done by the 
 addition of a second volume : and for the judgment 
 whereby it is to be decided, whether he is to pro- 
 ceed, or here to close his labours, he waits with 
 some anxiety. 
 
 After explaining his views to the reader, the next 
 duty of an author is to offer some acknowledgments 
 for the aids he may have derived from his predeces- 
 sors. Yet much in this way cannot be required 
 from the writer of an Elementary Treatise, where 
 there is little room for any pretension to originality. 
 Most works of merit, wherein the writer could ex- 
 pect to find any thing connected with his subject, he 
 has consulted j and by many of them he has profited. 
 Among those to whom he is most largely indebted is 
 Poinsot, whom he has followed by the adoption of 
 his theory of couples, and his use of that doctrine for 
 the establishment of the conditions of equilibrium. 
 To this name must be added that of Poisson, an au- 
 thor not less remarkable for the depth of his views 
 than for the elegance with which he unfolds them. 
 
 January, 1835. 
 
ADVERTISEMENT 
 FOR THE SECOND EDITION. 
 
 The writer of the following pages had indulged the 
 hope of completing this work on Mechanical Philosophy 
 by the addition of another volume, before that a second 
 edition of the first could be demanded. That hope has 
 been frustrated by the engrossing duties of the situation 
 to which he has been in the mean time called. The 
 gentlemen for whose use it was chiefly intended, being 
 aware of the nature and extent of his other engage- 
 ments, will readily believe that he has done as much in 
 this way as present circumstances would permit, by re- 
 vising the work so far as it has been already accom- 
 plished, and which he now presents to them with seve- 
 ral corrections, and such further improvements as were 
 deemed compatible with the plan of the work. 
 
CONTENTS, 
 
 Page. 
 Introduction, ...... xxi 
 
 STATICS. 
 
 SECT. I. 
 
 Of forces applied to a material point, ... 1 
 
 SECT. II. 
 
 Of the composition of parallel forces, ... 24 
 
 SECT. Ill, 
 
 Of equal and parallel forces acting on an invariable sys- 
 tem towards opposite sides of a line transverse to 
 their directions, ..... 34 
 
 SECT, IV. 
 
 Conditions of equilihrium among forces applied to diffe- 
 rent points of an invariable system, and in any di- 
 rections, ...... 46 
 
 SECT. V. 
 
 Of the conditions of equilibrivim when the body is in part 
 restrained by fixed obstacles ; and of the pressures on 
 the points of contact, . , , . 61 
 
XVm CONTENTS. 
 
 SECT VI. 
 
 Page. 
 Of the centre of gravity, .... 83 
 
 SECT. VII. 
 
 Of the Mechanic Powers. 
 
 The lever. — The balance for equal weights. — The steel 
 yard. — The bent lever balance. — The weighing ma- 
 chine. — The Danish balance. — The axle in the 
 wheel. — The pully. — The inclined plane. — The wedge. 
 — The screw, . . . . .124 
 
 SECT. VIII. 
 
 The funicular polygon and catenary, . . . 183 
 
 SECT. IX. 
 
 Of the equilibrium of roofs, arches, and domes, . . 197 
 
 SECT. X. 
 
 Of strength and strains direct and transverse ; and of the 
 
 principles of frame work, . . . . 215 
 
 SECT. XI. 
 
 Of the elastic curve, ..... 238 
 
 SECT. XII. 
 
 Of the fundamental principles of statics, . , 244 
 
 DYNAMICS. 
 
 SECT. I. 
 
 The measures of motion and force, . . . 265 
 
CONTENTS. XIX 
 
 SECT. II. 
 
 Page. 
 Of forces producing rectilinear motion, . . . 280 
 
 SECT. III. 
 
 Offerees producing curvilinear motion, . . 302 
 
 SECT. IV. 
 
 Of central forces, . . . . . 318 
 
 SECT. V. 
 
 Of constrained motion in general, . . . 347 
 
 SECT. VI. 
 
 Of motion constrained by a given curve, . . 355 
 
 SECT. VII. 
 
 Of the simple pendulum, .... 364 
 
 SECT. VIII. 
 
 Ofmotion constrained by a surface, . . . 384 
 
INTRODUCTION. 
 
 The systems of matter which constitute this globe 
 are ever liable to changes, with respect to their charac- 
 teristic properties, their figures and motions ; and it is to 
 the observation of the influences, by which these changes 
 are effected, that man is indebted for the foresight with 
 which he anticipates the results of natural operations, 
 and for the skill with which he avails himself of the va- 
 rious capabilities of matter. The processes which are 
 constantly before his view, or those which are casually 
 offered to his notice, would be the foundation of his first 
 theories ; and they would also furnish the light, for the 
 direction of his earliest efforts, towards the improvement 
 of his condition; for he would have only to subject the 
 bodies at his disposal, to the same influences, in order to 
 have the same results. 
 
 But the aim of the philosopher is not the servile 
 imitation of nature ; neither is the knowledge to which 
 he aspires, limited to that of her most obvious proceed- 
 ings. The actual operations of nature, are indeed the 
 
 c 
 
XXll INTRODUCTION. 
 
 subjects of his attentive observation ; but he finds it re- 
 quisite, by experimental investigation, greatly to enlarge 
 the catalogue of these operations, when he attempts to 
 discover the laws by which they are conducted, or the 
 powers by which they are effected. Such is the scope 
 of his endeavours ; and having attained this point, he is 
 enabled to anticipate the consequences of conditions, 
 which nature had never presented ; or of those, to which 
 he is, himself, at liberty to subject the systems of matter 
 placed at his disposal. 
 
 But the reward of his labours is not withheld, until 
 the end of his researches is fully attained by him. 
 Every step by which he advances leads him to more en- 
 larged views of the power and wisdom of the Great 
 Author and Contriver of all things, and of his provident 
 concern for the well being of his Creation ; and for 
 every step, he is rewarded by an enlargement of his own 
 power, over that world, which his Creator hath put un- 
 der his dominion. 
 
 To trace his progress in this course, would be to 
 write a history of the arts and sciences. The purpose 
 of this introductory chapter, is to mark the subject 
 treated of in those which follow : for the field of philo- 
 sophical inquiry being of such vast extent, it is obviously 
 requisite that it should be divided into separate com- 
 partments ; and this division is the same with that of the 
 phenomena to be considered. 
 
 Of these phenomena, the changes which occur In 
 bodies as to rest and motion, constitute one important 
 class ; and the consideration of these changes, together 
 
INTRODUCTION. XXlil 
 
 with that of their external causes, belongs to that de- 
 partment of Science, denominated Mechanical Phi- 
 losophy. 
 
 We have spoken of these changes as being the ef- 
 fects of the actions of certain powers external to the 
 bodies themselves ; and without, for the present, decid- 
 ing the question, whether there is in matter any innate 
 tendency to rest or motion, to which such changes 
 of condition might in certain cases be ascribed, it is 
 obviously certain, that they are often the effects of ex- 
 ternal influences. We know that motion is frequently 
 produced by an act of the will ; and then as we are 
 conscious of an effort which we designate by the name 
 of force, so when the like effect is produced by any 
 other external influence, it is natural to extend to that 
 influence, the name with which we are already so fa- 
 miliar ; and this we do, without intimating that there is 
 any analogy whatever, between the powers exerted in 
 the two cases. By the word force in mechanical philo- 
 sophy, we denote, merely, the external cause of the 
 change of condition as to rest or motion, without af- 
 fecting to convey any notion of the more intimate 
 nature of the energy so exerted : and happily, this 
 acquaintance with the intimate nature of the force is not 
 requisite for our present purpose ; for in the investiga- 
 tion of the effects of certain powers, it is plainly unneces- 
 sary that we should know how they come to reside in tlie 
 bodies by which they seem to be exerted ; or even whe- 
 ther they actually do reside in those bodies. Thus when a 
 bar of iron is disturbed on the approach of a magnetized 
 
 c2 
 
XXIV INTRODUCTION. 
 
 needle, it is possible that the efficient cause of the dis- 
 turbance may not reside in the needle ; the presence of 
 this latter, for aught we know, may be nothing more 
 than a condition, according to which, something exter- 
 nal to the needle is brought into action ; and when we 
 speak of the energy of the needle, it is not to be su}> 
 posed that we intimate any opinion on a subject, with 
 which we have no concern in this branch of philosophy. 
 Nay in our investigations, we leave out much of what is 
 discoverable, or even already known, respecting these 
 forces. Certain bodies, under particular excitement, 
 are known to exert such influences on other bodies ; yet 
 the peculiar properties of such bodies, or the methods 
 by which these powers are called into action, are here 
 overlooked, as not pertaining to the subject of abstract 
 mechanics. 
 
 It must not, however, be supposed, that our specu- 
 lations, because of their abstractedness, are incapable of 
 any useful application. We should derive but little be- 
 nefit from the force of wind or water, did we not know 
 how to vary, at pleasure, the magnitude and direction of 
 the impression to be received from it : how to direct a 
 force against an obstacle capable of sustaining it, and 
 how to economize the effiart by which this is to be ef- 
 fected : how to modify the motion, which is the effect 
 of the force, to convert reciprocating into rotatory mo- 
 tion, and vice versa ; and of the elements of motion, 
 which are the mass and its velocity, how to augment 
 either, at the expense of the other. 
 
 But the value of mechanical philosophy is to be esti- 
 
INTRODUCTION. XXV 
 
 mated, not merely by the light it affords, as to the dis- 
 posal of the forces at our command : for by establishing 
 the connexion between motions and forces, in all their 
 modes of application, it enables us to ascend, from the 
 motions observed in the celestial bodies, to the forces 
 by which they are animated ; and showing how to dis- 
 cover their intensities, and laws of action, to follow them 
 in all their consequences, past, present, and future, of 
 which, many w^ould have otherwise eluded our keenest 
 observation. It is thus that Physical Astronomy has 
 become a science, scarcely yielding, in evidence, to ab- 
 stract mathematics. The planetary system is, therein, 
 regarded as a vast machine, which exhibits to our ob- 
 servation, the effects of the forces, by which these bo- 
 dies are mutually influenced : and having traced up the 
 apparent to the real motions, and these to the forces by 
 which the whole is enlivened, we become possessed of 
 the principle, from which we are enabled to derive, not 
 only the motions actually observed, but all that shall 
 occur, for ages to come, among the various bodies of 
 this vast assemblage. 
 
 To manifest the importance of the science of Theo- 
 retic Mechanics, it is sufficient to point to the fruits it 
 has borne in Physical Astronomy, and in Practical Me- 
 chanics, of which, the former enlarges, so wonderfully, 
 our conceptions of the power and wisdom of the Crea- 
 tor ; whilst the latter extends, in a manner no less won- 
 derful, the power of man himself over the materials of 
 this globe, which he was given to inhabit and com- 
 manded to cultivate. And it is manifest, that if ever 
 
XXVI INTRODUCTION. 
 
 he shall attahi to the same knowledge of the forces by 
 which the other operations of nature are conducted, and 
 to the same skill in applying them to his uses, it is to 
 experimental investigation, guided and illustrated by 
 Theoretic Mechanics, that he must be indebted for this 
 further advancement. 
 
 To return from this digression ; the problems to be 
 solved in this branch of science, are such as the follow- 
 ing. " To determine the conditions of equilibrium 
 among forces simultaneously applied. To ascertain the 
 force which results from their actions when unbalanced ; 
 i. e. the energy and direction, with which the body is 
 effectively impelled ; or conversely, to resolve a given 
 force, into other forces from whose combined actions it 
 would result. To deduce the motions which would fol- 
 low from the action of certain forces applied under cer- 
 tain conditions; or on the contrary, to trace back cer- 
 tain motions to the forces by which they are produced." 
 With respect to the forces concerned in such questions 
 as these, there are but three particulars which claim our 
 attention, viz. : 1. The intensity of the force; 2. Its 
 point of application ; 3. Its direction : and it is now 
 to be shewn how these things are estimated and desig- 
 nated. 
 
 1 . With respect to intensity, it is obvious that there 
 are two ways of comparing forces, viz. ; by means of the 
 motions which are the effects of their unobstructed ac- 
 tions ; and immediately, by opposing the forces to each 
 other. Each of these methods shall be applied in its 
 proper place ; but it is evident that we should begin 
 
INTRODUCTION. XXVU 
 
 with that which is the more simple, and such is the me- 
 thod of immediate comparison. The other method 
 of estimating the energy of forces proceeds on certain 
 physical principles, which therefore are to be pre- 
 viously established by experience. Thus ; before that 
 motion or change of motion, can be regarded as the 
 measure of force, it must be proved that there is in 
 matter, no innate tendency to rest or motion ; and 
 therefore, that every change in this respect, is to be 
 ascribed to some external influence. 
 
 Again : though the motion, and consequently the 
 velocity acquired by a body, is the effect of the force 
 applied, and of that exclusively, yet the same thing may 
 be asserted of the square, the cube, or of any other func- 
 tion of the velocity; and which of these functions is to 
 be regarded as the measure of the force, is a question 
 which experience alone can determine. 
 
 Further : in estimating the energy of causes by 
 their effects, all the circumstances which influence the 
 results, should be taken into account. If then force is 
 to be measured by the quantity of motion produced by 
 it, we must consider what circumstances, besides the in- 
 tensity of the force, are concerned in this effect ; and 
 it will immediately occur, that time and space are so 
 concerned. A force, however great, must continue to 
 act during some portion of time, however short, and 
 through some part of space, however small, in order to 
 produce any motion whatever. Wherefore, in estimat- 
 ing the energy of the force by the motion produced, it 
 would seem necessary to take into account, the time or 
 
XXVlll INTRODUCTION. 
 
 space througli which the action is continued. Whether 
 then, for the measure of the force, are we to take the 
 motion generated by its action through a given space, 
 or during a given time ; and in comparing the energies 
 of two forces, how are we to ascertain which is the cir- 
 cumstance that fixes the parity of condition with respect 
 to their actions ? 
 
 Such questions may be further multiplied, and they 
 must all be solved, before we attempt to measure the 
 force by the motion produced. In the mean time, we 
 know that two forces are equal, which by acting on the 
 same material point, in opposite directions, are in equi- 
 librio : and that two equal forces, acting in the same di- 
 rection, constitute a double, three a triple force, &c. : 
 and this is quite sufficient for our purpose, when forces 
 only are contemplated, and until we come to treat of the 
 motions, which they are fitted to produce. 
 
 And here it is to be observed, that whatever method 
 is taken for the measurement of force, all our estimates 
 are comparative. This observation is applicable to 
 -quantities of whatever kind : our notion of the mag- 
 nitude of any quantity is merely the notion of the 
 relation which it bears to some other of the same kind : 
 for which reason, forces may be represented by quan- 
 tities of any other kind, the designation being under- 
 stood to be only of relative magnitudes. By this 
 commutation, much is gained, when the quantities 
 employed are more distinctly conceived, more clearly 
 expressed, or more easily exhibited, than those for 
 which they are substituted: and it is evident that 
 
INTRODUCTION. X,X1X 
 
 numbers and lines are recommended by these advan- 
 tages. Further; this comparative estimate is most 
 readily carried on, by referring the quantities of each 
 kind, to some fixed standard. Such a standard of 
 comparison is unity among numbers : and if in the 
 same way, we fix on some certain force as the unit of 
 forces, then any force whatsoever, shall be expressed 
 simply by the number, which is to unity, in the same 
 ratio, as the force in question to the unit of forces. 
 Li the like manner, by fixing on the linear unit, to 
 which all other lines are to be compared, any force 
 shall be represented simply by the line, which bears to 
 the linear unit, the same ratio, as the force in question 
 bears to the unit of force. 
 
 It is by this substitution of the more abstract quan- 
 tities for forces, that the latter are brought within the 
 province of mathematics. 
 
 2. The point of application being a point of space, 
 may be immediately exhibited to the eye, and it has 
 its algebraical designation, by reference to three co- 
 ordinate axes or planes, as in analytic geometry. 
 
 3. The direction of the force is the same with that 
 of the line which the material point would be made to 
 describe, if it were free to obey the impression. Ac- 
 cordingly, a line will serve to exhibit the direction of 
 a force, as well as to represent its magnitude. Now 
 when a right line passes through a given point, its 
 direction may be determined by means of the angles 
 which it makes with three coordinate axes : and as this 
 is the method most frequently adopted in the following 
 
XXX INTRODUCTION. 
 
 pages, it seems proper, in this place, to offer some ex- 
 planation of the principle, and of the manner in which 
 it is applied. 
 
 Let Aor, Ay, a^, (Fig. 1.) be three rectangular 
 axes, meeting at A, a point in the right line whose 
 direction is to be determined. Let this line be a??z, 
 and let «, p, 7, denote the angles mkx, mky, mAz, 
 which the line Am makes with these axes. Now if the 
 angle a is given, the position of the line Am, is limited 
 to the surface of a cone whose axis is a^t, and whose 
 vertical angle is 2«. If |3 is given, it is in like manner, 
 limited to another conical surface, whose axis is a^, 
 and whose vertical angle 2|3. Wherefore, if both a, 
 and (3 are given, the line Am is formed by the inter- 
 section of these two surfaces. But there are two such 
 lines of intersection lying in a plane which contains az, 
 and making equal angles with the perpendicular plane 
 2/A.v, and therefore making with az, angles whose sum 
 is 180°. The angle y, when given, determines the 
 line to one of these intersections. 
 
 Hence it is evident, that we are not at liberty to 
 assign to these angles any magnitudes we please, and 
 then require the position of the line from such data. 
 In fact, the angle a being assumed, the angle (3 must 
 be assigned between the limits 90° -fa and 90° — a; 
 and this condition being observed, with respect to two 
 angles, the third angle 7 is restricted to a certain angle, 
 or its supplement. 
 
 In order to perceive more clearly, the relations by 
 which the angles «, (3, y are connected, and by which 
 
INTRODUCTION. XXXI 
 
 we are, therefore, restricted in the choice of them, 
 when from such data we proceed to ascertain the po- 
 sition of the line ; it will be convenient to reverse the 
 process, by regarding the line Am as given. Then if 
 a parallelopiped is constructed, having this line for its 
 diagonal, and for its angle, the solid angle a, made by 
 the three axes, it is evident that the three linear sides, 
 AB, AC, AD, are determined by the perpendiculars let 
 fall from the point m on each of the axes, and that their 
 magnitudes may be expressed by means of the angles as 
 follows : 
 
 AB = Am.COS.a. ACZT A?;i.C0S.p. ADZZ A7>l.C0S.y. 
 
 Further ; the square of the diagonal being equal 
 to the sum of the squares of the sides, we have 
 Am*(ii:AB'-{-Ac'-|-AD*)=Am*(cos'.a+cos*.|3-|-cos''.y) ; 
 giving, 
 
 cos*, a-f-cos''. i3-|-cos*.y=r 1, 
 
 which expresses the condition to be observed, in assign- 
 ing the magnitudes of the angles, when the position of 
 the line is sought from these data. 
 
 With respect to the first angle «, there is no limi- 
 tation, inasmuch as the square of the cosine of an angle 
 cannot exceed unity ; but the angle a being assigned, 
 (3 must be such, that cos*.a-|-cos*.|3 shall not exceed 
 unity ; i. e. (3 must not be greater than 90°-f-«> or less 
 than 90° — a, its supplement. 
 
 When j3 is taken at either of these limits, we have 
 cos*.a-j-cos*.(3— 1 : and therefore, cos*.y — 0, or yi= 
 90°. When a and |3 are both given, the value of 
 
XXXll INTRODUCTION. 
 
 cos*.y, is 1 — (cos*.a-f-cos*. j3). and y is one or other 
 of the supplemental angles, whose cosines are 
 
 -f- V(l — cos*, a — cos*.|3), and — \/(l — cos*. a — cos*.j3). 
 
 If the cosine of any of the three angles is negative, 
 that angle is obtuse, and this circumstance renders the 
 designation by cosines, peculiarly explicit, the signs 
 prefixed, serving to identify, among the eight solid 
 angles contained by the three coordinate planes, that 
 which contains the line Am. Thus if cos. a is negative, 
 the line Am. falls at the side of the plane za?/ opposite 
 to that in which it should have been found, had the 
 cosine of that angle been affirmative, i. e. the abscissae 
 taken in the direction of a^, being deemed affirmative, 
 the lines Am, ajt, in the case here supposed, lie at 
 opposite sides of the plane zAy. The same is to be 
 observed with respect to the cosine of the angle j3 and 
 the plane zAx^ and of the cosine of the angle y and the 
 plane ,vAy. 
 
 If the line is in a given plane, two axes will be 
 sufficient, by taking them in the given plane. Thus 
 if the plane were XAy, we should have yzz90° and 
 cos.yizO ; and then, the equation of condition, to be 
 fulfilled in the choice of the angles, would be cos*, a 
 -f-cos*.(3=L. 
 
 If the line is parallel to a given right line as ab, 
 the latter may be taken for an axis, and the condition 
 of parallelism is expressed by the equation, cos*. « z=l : 
 in which case the angle a is either cipher or 180°. 
 
 If the line does not pass through a, the origin of 
 
INTRODUCTION. XXXIU 
 
 the coordinates, the angles by which its direction is 
 determined, are those which it makes with three lines 
 drawn through any point of the same, parallel to the 
 three axes ; and by these angles together with the 
 coordinates of the assumed point, the position of the 
 line is completely determined. 
 
 It has been already stated, that forces may be com- 
 pared together without recurring to the motions which 
 they are fitted to produce. Indeed, in numberless cases, 
 these effects do not follow, the forces being either 
 wholly or in part counteracted : and in this latter case, 
 the motion produced, is that due to the unbalanced 
 force, i. e. to the force which remains, after deducting 
 those forces or parts of forces which are in equilibrio. 
 Thus when a body is supported by the hand^ and 
 thereby prevented from descending towards the earth's 
 surface, a pressure is felt, which is then the entire 
 weight of the body. If the hand were to descend under 
 its load, the pressure would be suitably diminished; 
 but the hand is not entirely relieved from the pressure 
 of the body in contact, unless it descends with the cele- 
 rity with which the body falls, when free to obey 
 the force of gravity. In these instances, the pressure 
 is the part of the weight counteracted, and the mo- 
 tion of the body downwards is the effect of the re- 
 mainder. 
 
 From this simple instance, it will appear requisite 
 to ascertain the forces which are so counteracted, before 
 we can properly proceed to reason about the motions 
 produced. 
 
XXXIV INTRODUCTION. 
 
 Hence tlie science of mechanics naturally resolves 
 itself into two branches ; that which treats of forces 
 independently of motion ; and that wherein the motions 
 are deduced from the exciting forces, and vice versa. 
 The one branch, relating chiefly to cases of equili- 
 brium, is denominated Statics. The other, relating 
 to the effects of unbalanced forces, is named Dynamics. 
 
 The theory of statics naturally precedes that of 
 dynamics because of its greater simplicity. It is alto- 
 gether independent of the consideration of motion, 
 and therefore of time and space ; as also of the mass of 
 the body moved and its inertia. In this branch of 
 mechanics, the body acted on, is regarded, merely as 
 an assemblage of points to which certain forces are 
 applied; and its properties are considered, no farther 
 than as it is more or less fitted to transmit these forces 
 from one point to another. Further : though gravity 
 is a force by which every particle of matter is affected, 
 it will be found convenient to establish the more general 
 theorems without reference to this force, leaving it to 
 be regarded as part of the system of forces concerned 
 in the particular cases, to which these theorems are to 
 be applied. 
 
 The science of dynamics, having for its end, the 
 establishment of the connexion between forces and mo- 
 tions, under all the circumstances in which they can be 
 offered to our thoughts, must rest upon a foundation of 
 certain physical principles, denominated the laws of 
 motion : which as they seem not to be established by an 
 inherent necessity, are to be collected from observation 
 and experiment. 
 
INTRODUCTION. XXXV 
 
 Because of the peculiar modification of meclianical 
 action, which belongs to the constitution of fluids, it is 
 found convenient to separate the mechanics of fluid, 
 from those of solid substances : and adopting a like 
 distinction in this part of the subject, we give the name 
 of Hydrostatics, to that which treats of the equilibrium 
 of fluids ; and of Hydrodynamics to that which treats of 
 their motions. 
 
 Gaseous substances are distinguished from other 
 fluids, by the properties of compressibility and elas- 
 ticity ; and accordingly, these become the subject of 
 another branch of mechanical philosophy, denominated 
 Pneumatics. 
 
Tlie Reader is requested to make tke following Corrections. 
 
 207, five lines from bottom, after the word " curve" insert the words " of the 
 intrados ;" and in the next line, in place of the word "it" insert "that 
 of the extrados." 
 
 — 349, at the beginning of the third paragraph insert the words " If the motion 
 is uniform." 
 
 293, 296, 297, insert 7, 8, 9, the numbers of the Articles. 
 
 309, 313, 317, for 3, 4, 5, read 5, 6, 7, do. 
 
STATICS. 
 
 SECTION I. 
 
 OF FORCES APPLIED TO A MATERIAL POINT. 
 
 1. As all reasoning consists in connecting certain propo- 
 sitions with others previously received as true, it is requisite 
 that we should begin with those which lie nearest to first 
 principles : and such are the propositions which immediately 
 follow. 
 
 " As forces in equilibrio destroy, each the effect of the 
 rest ; the body, by means of which they are opposed to each 
 other, is in the same condition, as to rest or motion, as if 
 they had not been applied." 
 
 Hence, we are warranted, in introducing, or suppressing 
 any system of forces in equilibrio : and in the course of our 
 demonstrations, it will frequently be found convenient to re- 
 sort to this artifice. 
 
 " If a system of forces is in equilibrio, the equilibrium 
 shall not be disturbed, by fixing a point in the body to which 
 they are applied." 
 
 For if the forces engaged are in equilibrio, there is no 
 tendency to motion thence resulting, and therefore no pres- 
 sure on the fixed point, and consequently, no reaction ; i. e. 
 
 B 
 
2 STATICS. — SECT I. 
 
 there is no new force introduced into the sj'stem, by fixing 
 one or more points of the body. 
 
 " Two equal forces, applied to the same material point, in 
 opposite directions, are in equilibrio." 
 
 For no reason can be assigned, why either should pre- 
 vail: i. e. why the material point should be moved in the di- 
 rection of one of the equal forces, rather than in that of the 
 other. 
 
 Also, " Two equal and opposite forces are in equihbrio, 
 though they should not be immediately applied to the same 
 point, provided, that the line connecting the points of appli- 
 cation is of invariable magnitude." For then neither of the 
 points can be moved in the direction of the force applied to 
 it, without drawing the other along with it, and there is no 
 reason, why this common motion should be in the direction 
 of one of the forces, rather than in that of the force equal 
 and opposite. 
 
 Hence it follows, that when the system of points is in- 
 variable, such as those of a perfectly rigid body, we are at 
 liberty to change the point of application of a force, for any 
 other in the line of its direction. 
 
 Thus let the force /; applied at the point m, act in the di- 
 rection BA (Fig. 2.) this force may be transferred to any other 
 point as m' in the same line. For, if at this latter point, we 
 apply two forces, acting in the same line, but in opposite di- 
 rections, and each of them equal to p, this will have no effect. 
 There are then three equal forces engaged, viz. one acting at 
 m, in the direction of mx ; a second at w^' in the direction of 
 m'B, and a third at the same point in the direction of ?nA. 
 Of these, the first and second are in equilibrio, and may 
 therefore be suppressed ; after which, there remains but the 
 third, which is the original force p, whose point of applica- 
 tion is transferred from m to 7n, a new point in the line of its 
 direction. 
 
 Two forces not immediately opposite in direction, cannot 
 be in equilibrio. 
 
OF FORCES APPLIED TO A MATERIAL POINT. O 
 
 Let the two forces act in the directions of the different 
 lines AB, CD, (Fig. 3.) If these forces could be supposed to 
 be in equihbrio, that equihbrium would not be disturbed, by 
 making the system of points invariable, or by fixing one of 
 those points as m taken in the line ab, the direction of one of 
 the forces : this being done, the force which acts in the direc- 
 tion of this hne, will evidently be destroyed by the reaction 
 of the fixed point ; wherefore, the force in the direction of cd 
 will then remain alone, and will turn the system round the 
 fixed point m. There is then, no equilibrium when this point 
 is fixed ; and therefore, none when it is free. Hence it fol- 
 lows that 
 
 " If two forces are in equilibrio, they must be directly op- 
 posed, and therefore, they must also be equal." 
 
 When two or more forces not in equilibrio, are applied 
 to the same material point, they must give to that point, a 
 tendency to move with a certain velocity, and in a certain di- 
 rection. The effect then, is the same, as that of some one 
 force. This last is named the resultant of the forces actually 
 applied, and they its components. Generally, the resultant 
 of any forces is that force., which would produce alone the 
 same motion which results from the combined action of the 
 original forces : and the components of any force, are those 
 from whose combined actions the same motion would result : 
 wherefore, in all statical inquiries, the components may be 
 replaced by their resultant, and vice versa. 
 
 Accordingly, when a system of forces is applied to a 
 point, and not in equihbrio, the equilibrium shall be es- 
 tablished, by applying a new force, equal and opposite to 
 their resultant. 
 
 Conversely, if a system of forces is in equilibrio, any one 
 of them, its direction being changed into the opposite, is the 
 resultant of the rest. For any one of them is in equilibrio 
 with the resultant of the rest, and therefore equal and oppa- 
 site to that resultant. 
 
 b2 
 
4 
 
 STATICS. — SECT. I. 
 
 2. To find the resultant of two or more forces applied to 
 the same material point, is the fundamental problem of Me- 
 chanics. The most simple case of this problem, is that in 
 which the forces act in the same line. 
 
 If two forces p and p are applied to the same material 
 point, and in the same direction, their resultant is equal to 
 their sum, i. e. r =^3 -^-p • 
 
 This is manifest, without looking for the measures of the 
 forces in the motions they are fitted to produce. 
 
 If two forces p and jp are applied to the same material 
 point, but in opposite directions, their resultant is equal to 
 their difference, and acts in the direction of the greater 
 force, I. e. supposing p to be the greater force, we shall have 
 n:z. p — ^'. 
 
 This will be evident, by resolving the greater force ^, 
 into two, one part equal to p', and the other to 2> — 2^'' ^^^ 
 then we have three foi'ces, viz. p, p — p , acting in the same 
 direction ; and^', acting in the opposite direction: of these, 
 the first and last are in equilibrio ; and these being sup- 
 pressed, there remains only the force ^J — p, which is there- 
 fore the value of r. 
 
 These things are equally true offerees acting in the same 
 line, though not immediately applied to the same point ; the 
 line being supposed to belong to an invariable system ; and 
 they are extended to any number of forces acting according 
 to such a line, by stating, that the general resultant of the 
 forces is equal to the difference between two sums, viz. the 
 sum of those which act in one direction, and the sum of those 
 which act in the opposite direction, the direction of the re- 
 sultant being that of the greater sum. 
 
 This proposition may be announced, in a manner still 
 more compendious, if having prefixed positive signs to the 
 symbols of the forces acting in one direction, we mark with 
 negative signs, those which act in the opposite direction : for 
 then we may state, that the resultant of all the forces acting 
 
OF FORCES APPLIED TO A MATERIAL POINT. 5 
 
 in the same line, is equal to their sum ; using this term in the 
 same sense as in common algebra. Thus if three forces 
 -\-p, -\-p',-\-p" act in the same direction, and two forces 
 — p'", — /?"" in the opposite direction, we may assert, that the 
 general resultant is equal to their sum, paying attention to 
 their signs; t. e. K— p -{-p -\-p —p — p . 
 
 To find the resultant of two forces applied to the same 
 material point, when their directions are oblique to each 
 other, is a more difficult problem. We shall proceed to its 
 solution by degrees, first ascertaining the direction of the 
 resultant, and then its quantity. 
 
 The direction of the resultant must be in the plane of the 
 components. For, if it is supposed to be that of a line on 
 one side of this plane ; another line may be assigned, sym- 
 metrically situated, on the other side of the same plane ; 
 and whatever can be supposed to determine the resultant 
 to one of these directions, must equally serve to give it the 
 other. But the resultant cannot take two different direc- 
 tions ; therefore it cannot lie at either side of the plane of the 
 components. 
 
 The direction of the resultant of two forces applied to 
 the same material point, must lie within the angle contained 
 by their directions. 
 
 Let the forces act on the point a in the directions ai/, ax. 
 (Fig. 4.) 
 
 The former would draw the point out of the line ax on 
 the side of a?/ ; and the latter would draw it out of the line 
 Ay on the side of ax : wherefore the effect of both con- 
 jointly, is to give to the point a tendency to move in a di- 
 rection intermediate between the lines Ay, ax : i. e. the di- 
 rection of the resultant must lie within the angle yAx. 
 
 If the components are equal, it can be shown, by reason- 
 ing in the same way as in the first part of this article, that 
 the angle made by their directions is bisected by that of the 
 resultant. 
 
6 STATICS. — SECT. I. 
 
 If without any change in the directions of the compo- 
 nents, one of them is increased, the other remaining un- 
 varied ; the effect on the direction of the resuUant is to di- 
 minish the angle which it makes with the component so in- 
 creased. 
 
 For let the forces p and p' act in the directions \x, a?/, 
 and let a^ be the direction of their resultant r. (Fig. 5.) 
 Then, if p becomes p -f-p, the new resultant r' is that of r 
 and pt and must therefore lie within the angle contained by 
 the directions of these last forces ; but the direction of 7^ is 
 that of p or ax, wherefore the direction of r' must be a line, 
 such as A^', lying within the angle s:\x. 
 
 Knowing the direction of the resultant of two equal forces, 
 applied to the same material point, and making with each 
 other any angle whatsoever, we may proceed to ascertain the 
 direction of the resultant, when one of the equal forces is 
 multiplied by any integer number. 
 
 For A, being the point at which the equal forces are ap- 
 plied, (Fig. 6.) and Ay, ax, their directions, if two equal por- 
 tions of these lines measured from the point a, as ae, ab, 
 are taken to represent these forces, and the parallelogram 
 EB is completed, its diagonal af, which bisects the angle at 
 a, shall be the direction of the resultant. This resultant, 
 therefore, may be transferred from a to f, at which point, if 
 it is resolved in directions parallel to Ay, ax, it shall re- 
 produce its components, viz. fq, fg, each equal to ae or ab. 
 Now, if the force applied at a in the direction ax, were 
 made ac zi 2ab, this would be adding the force ab or bc to 
 the forces fq, fg ; and if the force fq were transferred to b, 
 and there compounded with bc, the resultant of these two 
 forces would take the direction of bg parallel to af ; and 
 the point g, where it meets the direction of the force fg, 
 would be a point in the direction of the resultant of the 
 forces AE and ac ; a being necessarily another : wherefore 
 the direction is that of the connecting right hne, which is 
 
OF FORCES APPLIED TO A MATERIAL POINT. 7 
 
 the diagonal of the parallelogram, whose sides are ae and 
 
 AC. 
 
 In the same manner, if at the point g, the resultant of the 
 forces AE and 2ab, is resolved in directions parallel to ky 
 AX, it shall reproduce these components, viz. gr — ae and 
 Gi n 2ab : and if the former of these is transferred to the 
 point c, and there compounded with another force equal to 
 AB acting in the direction kx or ex, the resultant shall be 
 parallel to af, and the point h, where it meets the line gi, 
 shall be a second point in the direction of the resultant of 
 the forces ae and 3ab : moreover,^ the line eh being equal 
 to 3ef = Sab, this direction shall be that of the diagonal of 
 a parallelogram whose sides are ae and 3ab. This method 
 of investigation will serve for wab any multiple of the force 
 
 AB. 
 
 In like manner, if the resultant of the forces ae and 
 ni.KB, is transferred to the further extremity of the paral- 
 lelogram formed by these lines, and is there resolved in di- 
 rections parallel to a?/, ao;, it shall reproduce these compo- 
 nents : and if w^.ab is transferred from this point to e, and 
 there compounded with another force ep z: ae, acting in the 
 direction e//, it shall give a resultant parallel to the diagonal 
 of the parallelogram whose sides are ae and m.AB: and the 
 point where it meets the side parallel to Ay, being a second 
 point in the direction of the resultant of 2ae and w?.ab, this 
 direction is that of the diagonal of the parallelogram formed 
 by these lines. Thus by treating the force w2.ab, as the 
 force ae was treated in the former part of this investigation, 
 the force ae may be multiplied by any integer number n, and 
 it may be shown, that the direction of the resultant of the 
 forces represented by w.ae and w.ab, is that of the diagonal 
 of the parallelogram formed by these lines. 
 
 From this it appears, that when two commensurable forces 
 are applied to a material point, making with each other any 
 angle whatsoever, the direction of their resultant is that of 
 
O STATICS. — SECT, I. 
 
 the diagonal of a parallelogram, whose sides are taken in the 
 directions of the forces, and in the same ratio. For if two 
 forces, each equal to the common measure, were to act at 
 the same point and in the same lines, the direction of their 
 resultant would be that of the diagonal of a parallelogram, 
 formed by equal portions of those lines ; and when the forces 
 are multiplied by any integer numbers, the direction of the 
 resultant has been found to be that of the diagonal of a pa- 
 rallelogram, whose sides, taken in the same directions, are 
 had from the former by the same multiplications. 
 
 Finally : as the unit of force may be taken of any magni- 
 tude, however small, the proposition may be extended to the 
 case where the forces are incommensurable. For let two 
 such forces applied to the point a, be represented in quan- 
 tity and direction by the lines ab ac, and completing the 
 parallelogram ae, (Fig. 7.) let a portion ef, be taken on the 
 line EB, measured from the point e ; and however small this 
 portion, it is evident that a submultiple of ab, may be 
 found less than it ; and that this submultiple, taken repeat- 
 edly from the line be, shall have one of its terminations at 
 some point as o, intermediate between f and e ; then draw- 
 ing OG parallel to ba, we have a parallelogram bg, whose 
 sides are commensurable ; and therefore, the resultant of 
 the forces, represented by ab, ag, shall have the direction 
 Ao, which makes with ac, an angle less than the angle fac. 
 But the resultant of the forces ab, ac, makes with ac, an 
 angle less than oac, and therefore a fortiori, less than 
 FAC. Again : taking from the point e, on the line eg, a 
 portion eii however small, it can be shewn in the same way, 
 that the resultant of the forces ab, ac, makes with ab, an 
 angle less than the angle hab. This resultant therefore, 
 falls within the angle fah, however small, which can be true 
 of no line, but the diagonal ae. 
 
 It is now proved, that when two forces of any magni- 
 tudes, are applied to the same material point in directions 
 
OF FORCES APPLIED TO A MATERIAL POINT. 9 
 
 making any angle wliatsoever, the direction of their re- 
 sultant is that of the diagonal of a parallelogram, whose 
 sides are taken in the directions of the component forces, 
 and proportional to them in magnitude. 
 
 But further: the resultant of two such forces, is repre- 
 sented by the same line, in magnitude also : For let the 
 forces be represented in magnitude and direction, by the 
 lines AB, AC ; (Fig. 8.) and let r denote the resultant whose 
 magnitude is to be determined. Completing the parallelo- 
 gram contained by the lines ab, ac, the direction of r is that 
 of the diagonal ae. And as the resultant, its direction 
 being changed into the opposite, is in equilibrio with its 
 components, it follows, that if ea is produced beyond the 
 point A, as to g, a force equal to r, acting in the direction 
 AG, shall be in equilibrio with ab, ac. Again : any one of 
 the equilibrating forces, its direction being changed into the 
 opposite, is the resultant of the remaining forces; accord- 
 ingly, producing ca in the direction am, a certain force 
 acting in the direction am, is the resultant of the force ab, 
 and of the force r acting in the direction ag. Wherefore, 
 am coincides with the diagonal of a certain parallelogram, 
 one of whose sides is ab, and the ether is in the direction of 
 AG. This is sufficient to determine the parallelogram ; for 
 if from B, a line is drawn parallel to ag, and from the point 
 H, where it meets the diagonal, another line, m, is drawn 
 parallel to ba, the parallelogram is completed. Now the 
 force equal to R, and acting in the direction ag, must be 
 represented by ai : for a force in this direction, if greater or 
 less than ai, would not compound with ab, a force in the 
 direction of ah, the diagonal of the parallelogram. But 
 lA rz HB zz AE, which completes the proof that " the re- 
 sultant of two forces, acting on a material point, in direc- 
 tions making any angle whatsoever, is represented in quan- 
 tity and direction by the diagonal of a parallelogram, whose 
 
10 STATICS. — SECT. I. 
 
 sides represent the component forces, in quantity and di- 
 rection."'* 
 
 3. By substituting for one side of the parallelogram of 
 forces, the opposite side which is equal and parallel, it will 
 follow, that any two forces meeting at a point, and their 
 resultant represented by the diagonal, are proportional to 
 the sides of one of the triangles, into which the parallelo- 
 gram is divided by this diagonal; and consequently, to the 
 sides of any triangle, whose sides are parallel to the direc- 
 tions of the three forces. 
 
 When the triangle is used in place of the parallelogram, 
 
 * If it be admitted that force is measured by the velocity which it generates in 
 a given body, by its action during a given time, this theorem may be demonstrated 
 in the following manner. 
 
 If a body moving in the right line AC (Fig. 9.) receive an impulse in a direction 
 mo perpendicular to that line, such an impulse does not alter the velocity with which 
 it recedes from the line ab also perpendicular to AC. For if it did, an equal and 
 opposite impulse mo should double the change, since they are symmetrical with 
 respect to AC ; but they are in equilibrio, and therefore produce no joint effect. 
 
 Now let the body receive, at the same moment, two impulses, which acting 
 separately, would carry it in the same given time over the lines oc, OD, perpen- 
 dicular one to the other. (Fig. 10.) At the end of that time, the body shall be 
 found at E, the extremity of the diagonal of the rectangle formed by the lines oc, 
 OD. For the impulse od perpendicular to oc, does not alter the velocity with 
 which the body recedes from the perpendicular OD, by virtue of the impulse 
 oc : therefore, at the end of the given time, the body shall be found at the same 
 distance from od, i. e. somewhere in the perpendicular CE, whether od acts or not. 
 For the same reason, by virtue of the impulse od, at the end of the given time, 
 the body shall be found at the same distance from oc, whether the force in oc acts 
 or not, i. e. somewhere in the perpendicular de. Wherefore, since at the same 
 moment, it arrives at both lines CE, de, it must arrive at E, the only point common 
 to both. Moreover, as the body after leaving the point o, is not supposed to re- 
 ceive any other impulse, its motion from o to E must be uniform and rectilinear : 
 accordingly in the given time, it must describe the right line oe, which therefore 
 represents the compound force both in quantity and direction. 
 
 Now when the impulses ab, ad make any other angle, (Fig. 11.) the impulse 
 AB is resolvable into the two at right angles, ao and A??;, .*. we have two impulses, 
 Ao, and AD-j- A>n=: AD-)-D« = A«, which, being perpendicular one to the other, 
 compound the impulse represented by AC, the diagonal of the parallelogram on, 
 and which is also the diagonal of the parallelogram bd. 
 
OF FORCES APPLIED TO A MATERIAL POINT. 11 
 
 it is to be observed, that the angle contained by the sides 
 which represent the component forces, is the supplement of 
 that contained by the forces themselves. And this observa- 
 tion is to be attended to, when from the component forces 
 and the angle contained by them, the resultant is to be de- 
 termined, either graphically or by computation. 
 
 The component forces and their resultant, being thus 
 represented in quantity and direction, it follows, that all 
 questions relative to the magnitudes and directions of these 
 forces, are reducible to those which relate to the sides and 
 angles of a triangle. 
 
 Thus, the sides of a triangle, being proportional to the 
 sines of the opposite angles, and the sine of an angle and of 
 its supplement being the same, these analogies will subsist 
 among the forces and the sines of the angles contained by 
 their directions. These analogies may be thus expressed, 
 p _ p' _ R 
 
 sin. 6' ~ sin. ~ sin.{d -\-B~)' 
 
 wherein, the component forces, and their resultant, are de- 
 noted by p, p', R, and the angles contained between each of 
 the former and r, by 6, 6' ; and, therefore, that which they 
 make with each other by + 0'« 
 
 The foregoing equations express the proportionality of 
 the forces to the sines of the angles, each force being analo- 
 gous to the sine of the angle contained between the direc- 
 tions of the remaining two. From which it appears, that 
 any two of the three forces, are reciprocally proportional 
 to the sines of the angles which they make with the direction 
 of the third force ; or that the products are equal, which 
 are obtained by multiplying each of the two forces into the 
 sine of the angle which it makes with the third force. 
 
 As the perpendiculars let fall on the directions of two 
 forces, from any one point in the direction of the third, are 
 proportional to the sines of the angles they subtend, the 
 same things may be expressed, by stating, that any two of 
 
12 STATICS. — SECT. I. 
 
 the three forces, are to each other, reciprocally as the per- 
 pendiculars on their directions, let fall from any one point in 
 the direction of the third force ; or that the products are 
 equal, which are obtained by multiplying each of the two 
 forces into the perpendiculars on their directions, let fall 
 from any one point in the direction of the third force. 
 
 Accordingly, the angles being given, with any one of the 
 three forces ; or two of the forces and the angle contained 
 between one of them and the third, the remaining force and 
 angles are found by these analogies. The second of these 
 cases is, however, subject to the same ambiguity, as in the 
 solution of a triangle : for the angle is sought by its sine ; 
 and the sine of an angle is also that of its supplement. 
 
 When the components are given, together with the 
 angle contained by them, the value of the resultant is known 
 from the equation 
 
 r2 _ p2 ^ p'2 _,„ 2pp'. cos. {d + 6'). 
 
 The last term of this equation is affected with a positive 
 sign, because the angle {6 + 0'), is the angle of the paral- 
 lelogram at the point where the forces are applied, and this 
 angle is the supplement of the angle of the triangle opposed 
 to the side representing r. 
 
 4. The parallelogram of forces, serves equally for the 
 resolution, as for the composition of forces ; but the pro- 
 blem is indefinite, when nothing is given but the force to be 
 resolved ; as appears from the foregoing equations, each of 
 which, contains four quantities, and therefore determines 
 the value of one of them, only when the three others are 
 given. And without resorting to these equations, it is evi- 
 dent, that any line, by which the force is represented, may 
 be made the diagonal of numberless parallelograms ; and 
 therefore, that any given force may be resolved into num- 
 berless pairs of forces. 
 
 The data most commonly given in the resolution of forces, 
 is the force to be resolved, and the directions of the compo- 
 
OF FORCES APPLIED TO A MATEUIAL POIN'T. 13 
 
 nents. It is often required to resolve a force into two at right 
 angles : i. e. into one in a given line and another perpendi- 
 cular to same. The force in the given line shall be p.cos.0; 
 6 being the angle made by p with that line ; and the force 
 perpendicular to given line is p.sin.0, as may appear by 
 completing the rectangle. 
 
 5. By applying to the same point, a third force, equal 
 and opposite to the resultant of two forces, the system 
 is reduced to equilibrium. Wherefore, " if to a material 
 point, three forces are applied, proportional to the three 
 sides of the triangle to which they are parallel, they shall be 
 in equilibrio." 
 
 In estimating the directions of the three equilibrating 
 forces by those of the sides of the triangle, the sides must 
 be taken consecutively, i. e. each is to be considered as di- 
 rected from the last named point; as when they are taken 
 in the order «6, 6c, ca, (Fig. 12.) or in the contrary order, 
 ac, c6, ha. Such forces are in equilibrio ; for ah, and a 
 force, equal and parallel to he, applied at the point a, give 
 the resultant ac, to which the third force ca is equal and 
 opposite : also ah with a force equal to ca, taken in this line 
 produced beyond a, compound a resultant equal and oppo- 
 site to be, considered as acting at the same point. 
 
 Hence all that has been demonstrated from the proper- 
 ties 'of a triangle, relative to two forces applied to a point 
 and their resultant, is equally applicable to three forces in 
 equilibrio about the same point. 
 
 6. A system of forces applied to a material point, and whose 
 directions, measured from that point, lie all at the same side 
 of a plane passing through the same point, must have a re- 
 sultant : and this will be true, whatever be the number of 
 such forces in the system. 
 
 Let the forces be op, op , op", op", &c. (Fig. 13.) all 
 directed towards the same side of the plane ah, passing 
 through the point o. The forces op, op', have a resultant 
 
14 STATICS. — SECT. I. 
 
 which hes in the plane of these forces, within the angle 
 contained by them, and therefore, at the same side of the 
 plane ah. The same thing is true of their resultant r and 
 another of the forces as op" \ of their resultant r' and another 
 of the forces op" \ and so of any number of forces acting in 
 this manner. 
 
 If three forces act on a material point, but not all in the 
 same plane, and are represented by three lines measured 
 from the common point and in the same directions, the 
 resultant shall be in the direction of the diagonal of the pa- 
 rallelepiped having these lines for its sides, and it shall also 
 be represented in magnitude by the same line. 
 
 For let AB, AD, AF, represent the component forces, 
 (Fig. 14.) and let abcdefgh, be the parallelopiped ; whose 
 diagonal passing through the point a is ah. The resultant 
 of the forces ab, ad is ac, the diagonal of the parallelogram 
 whose sides are ab, ad ; and the resultant of ac, af, is ah, 
 the diagonal of the parallelogram whose sides are ac, af. 
 
 Conversely ; a force represented in quantity and direc- 
 tion, by the diagonal of a parallelopiped, is resolvable into 
 three forces, represented in quantity and direction by the 
 sides of that parallelopiped, 
 
 7. The method of composition and resolution by means 
 of the parallelogram of forces, may be extended to any num- 
 ber of forces applied to the same material point, and in any 
 given directions, whether in the same plane or not. For as 
 any number of forces may be replaced by their resultant, it 
 follows, that the resultant of two forces being compounded 
 with a third, gives the resultant of the three forces, and that 
 this resultant, being compounded with a fourth of the given 
 forces, gives the resultant of these four ; and that the re- 
 sultant of all but one, compounded with this one, gives the 
 general resultant of the whole system of forces. These suc- 
 cessive compositions may be effected by the construction of 
 so many parallelograms. 
 
OF FORCES APPLIED TO A MATERIAL POINT. 15 
 
 Similarly, a given force may be resolved into any number, 
 acting at the same point in its direction : first, by resolving it 
 into two; then either or each of these into two others; and 
 so on without limit. And it is evident, that each of these re- 
 solutions may be effected in any plane containing the direc- 
 tion of the force to be resolved. 
 
 8. The resultant of a number of forces applied to the 
 same material point, may be found more expeditiously by a 
 method founded on the triangular scheme. This is done, 
 simply by describing a polygonal figure, whose sides are 
 proportional to the forces, and parallel to their directions. 
 This being done, the line which closes the polygon gives the 
 general resultant in quantity and direction. Thus let op, 
 op\ op', op", op", represent the forces apphed at the 
 point o, (Fig. 15.) then, from ^9, drawing the line pq, equal 
 and parallel to op \ the line qr equal and parallel to op" \ 
 the line rs equal and parallel to op" ; the line st equal and 
 parallel to op"" -, the general resultant of the forces shall be 
 ot. For the force op, and the force op , to which pq is 
 equal and parallel, shall have for their resultant the force 
 oq : this last and op' , to which qr is equal and parallel, 
 shall have for their resultant the force or : this and the force 
 op" , to which rs is equal and parallel, shall have for their re- 
 sultant the force os : and finally, this and the force op"" , to 
 which st is equal and parallel, shall have for their resultant 
 the force ot ; which, accordingly, is the resultant of all the 
 forces applied at the point o. 
 
 If an additional force is applied at the point o, repre- 
 sented by to, this being equal and opposite to the resultant 
 of the rest, the whole system of forces would be in equi- 
 librio, and the point o would not be disturbed by them : 
 whence it follows, that any number of forces applied to a 
 material point are in equilibrio, when the polygon is closed, 
 by whose sides they are exhibited in quantity and direction ; 
 which indeed is evident, by considering, that as the general 
 resultant is represented by the side which closes the polygon, 
 
16 STATICS. — SECT. I. 
 
 there can be no resultant when the polygon is closed by the 
 lines representing the forces. 
 
 9. The method of graphical delineation is well fitted to 
 assist the imagination ; but when the resultant is to be made 
 out by computation, its quantity and the angles which deter- 
 mine its direction should be expressed as functions of the 
 component forces, and of the angles which determine their 
 directions. These expressions are most conveniently ob- 
 tained, by resolving each of the component forces according 
 to three rectangular axes ; then adding into one sum the 
 forces which act in the same line, and finally compounding 
 these three sums. 
 
 Thus let 2iy P) p"i p", &c. be the forces to be com- 
 pounded; A their point of application ; (Fig. 16.) hx, Ay, az, 
 the three rectangular axes; a, a, a", a", Sec. the angles 
 made by the directions of the several forces with the axis ax ; 
 (5, /3', /3", ft'", &c. the angles which they make with the axis 
 A^^; and 7, y', y", 7'", &c. those which they make with the 
 axis A^. Then any force ^j, resolved according to these 
 three axes, shall give the forces/). cos. o, p.cos. ft, /J.cos.y ; 
 the other forces are resolved in like manner. Wherefore, 
 the sum of the forces acting according to the line ax, is 
 p.COS.a 4-f/.COS.a'+/Acos.a"-|-y'.COS.a"', -f &C. 
 The sum of the forces acting according to Ay, is 
 
 ;?.cos./3+p'.cos.j3'+/'.cos./3"+/y".cos./3"', -f- &c. 
 And the sum of those acting in the line a^, is 
 
 J^.COS.y +2/.C0S.y' -{- p'.COS.y" -j- p"'.COS.y'", -f &C. 
 
 The whole system of forces is thus reduced to three 
 forces acting at right angles, which may be denoted by the 
 symbols x, y, z ; and as the relations of these three forces to 
 their resultant, are the same as those of the lines by which 
 they are represented to the diagonal of the parallelopiped 
 having these lines for its sides, the magnitude of the general 
 resultant is given by the equation 
 
 R-V{^' + ^' + z'). (1) 
 
OF FORCES APPLIED TO A MATERIAL POINT. 17 
 
 The direction of the resultant is to be known by the an- 
 gles which it makes with each of the three axes, \x, mj, kz: 
 these angles being denoted by a, b, r, are given by the 
 equations 
 
 cos. \zz — . cos. "B — — . COS. r ~ — . (») 
 
 R R R 
 
 10. The equations of the line of direction of r, are those 
 of its projections on the coordinate planes. Its projection 
 on the plane of zx makes with the axis kx, an angle whose 
 
 nr 
 
 tangent is — . Its projection on the plane of xy, makes 
 
 X 
 
 with the axis ky, an angle whose tangent is — . and its pro- 
 jection on the plane zy, makes with the axis a^, an angle 
 whose tangent is — . 
 
 Therefore, the equations of this line are the following : 
 
 Z X Y 
 
 %z=: — .X. X "::=. — .y. v — — . z. 
 
 These are to be regarded, only as two independent 
 equations ; inasmuch as any one of the three is derived from 
 the other two. 
 
 If the origin of the coordinates is not taken at the point 
 to which the forces are applied, let the coordinates of this 
 latter point be denoted by x^, y^, z^ ; and the equations of 
 the line of direction of the resultant will be any two of the 
 three following : 
 
 z — z^=Yi^ — •^\)> or x(ar — z^), - z{x - x) 
 x-x^-— {y — y), or y{x - x), = x(y - y) ) (3) 
 y-y,-^{s~ z), or z(y - yj, - y{z - z) 
 
18 STATICS. — SECT. I. 
 
 11. If the forces 2^,p',p",p"» &c. act, all in the same 
 plane, this may be taken for one of the coordinate planes, as 
 for example for ihe plane xy. In this case, we have 
 
 z r: 0. cos. r zz 0. 
 whereby equation (1) becomes 
 
 R = V (X^ + Y^). 
 
 and the direction of the resultant is determined by either of 
 
 X Y 
 
 the two equations cos. a •=. — . cos. b = — : or by the se- 
 
 Iv R, 
 
 cond of the equations (3) : and there is the same reduction 
 when the portions of the forces which act in the direction az 
 are in equilibrio ; for then, 
 
 ipcos.y -^-p'cos.y '\-p"cos.'^" -^p"'cos.'y"' + &c. — 0: 
 i. e. z zz: 0. 
 
 If z — 0, Y iz 0, the direction of the resultant is parallel 
 to the axis ax: and agreeably to this, equation (1) will be- 
 come R ;z X : and equations (2) will be cos. a = 1. cos. b z: 0. 
 COS. r z: 0. 
 
 In this case, the forces p cos.jS, ^/cos.jS', ^/'cos-jS", 
 ^/"cos,j3"', &c. must be either severally equal to cypher or 
 else they must be in equilibrio : And the same is true of the 
 forces 2i cos.y, ^/cos.y', ^/'cos.-y", p"co?,.y"', &c. 
 
 12. For equilibrium among the forces of the entire sys- 
 tem, we must have 
 
 RziO;2. e. V(x^+Y--fz2)rz0. 
 
 And as the quantities within the parenthesis are essen- 
 tially positive, this condition cannot be fulfilled, unless those 
 quantities are, separately, equal to cypher. 
 
 The same thing appears by referring to Articles 6 and 1. 
 In Article 6 it was shewn that three forces meeting at a 
 solid angle, cannot be in equilibrio ; and if one of the three 
 components is cypher, the same thing has been proved of the 
 two remaining forces in Art. 1. It appears then, by those 
 
OF FORCES APPLIED TO A MATERIAL POINT. 19 
 
 articles, that for r z= 0, we must have the three forces 
 X, Y, z, severally equal to cypher, i. e. 
 
 X = 0, Y := 0, z = 0. 
 which three equations in their more expanded form are, 
 y;.COS.a -4-73 .cos.a +2J .cos. a -f- p .COS, a + &c. iz U.>w 
 
 ;^.cos./34-/.cos./3'+/'.cos.j3"+/".cos./3'"+ &c. = 0. I (4,) 
 p.cos.7+/^'.cos.'y'+^/'.cos.7"4-i3"'.cos.7"'+ &c. =iO.J 
 
 Such are the conditions of equilibrium, among a system 
 of forces, applied to a material point, considered as being at 
 liberty to move in any direction : and if these conditions are 
 fulfilled, it is evident that the equation r = is also fulfilled, 
 i. e. that the system is in equilibrio. 
 
 If the point, to which the forces are applied, is confined 
 to a surface, one of the coordinate axes, as for example, the 
 axis A^, may be taken perpendicular to the surface at the 
 point; and consequently, the other two axes in the tangent 
 plane. Then 
 
 p.cos.7 +2?.cos.7 +/? .C0S.7 -j-p .cos. 7 -f- «^c. 
 
 shall express the perpendicular pressure on the surface ; 
 with which pressure the reaction of the surface is in equi- 
 librio. Denoting this reaction by ^j^, and including this 
 among the forces acting in the line ass, the equation 
 
 ^9^+^.cos. 7-|-p'.cos.7'4-i^".cos.7"+i?"'.cos.7"'+ &c. z=:0. 
 
 will be necessarily fulfilled. Accordingly, in the case here 
 considered, there will be equilibrium, whenever the two first 
 conditions are satisfied. 
 
 If the point is confined to a certain line, two of the three 
 coordinate axes, as for example, the axes a^, mj, may be 
 taken perpendicular to this line, i. e. perpendicular to the 
 tangent at the point. The third axis, ax, will be then this 
 tangent. The forces estimated according to the axes a^, \y, 
 and consequently the resultant of these forces shall be per- 
 
 c2 
 
20 STATICS. — SECT. I. 
 
 pendicular to the line to which the material point is restrict- 
 ed, and therefore shall be destroyed by its reaction, i. e. it 
 shall be in equilibrio with this reaction. Putting p^^ for the 
 reaction in the line Ay and /?,, as before, for that in the line 
 AS, the equations, 
 
 p^^.-\-p.cos.^ -i-7/.cos./3'+^/'.cos.|3"+y'.cos./3"'+ &c. z=. 0. 
 Pi' +^J-cos.7 4- ^/.cos.7' +i>"'Cos.7" +^/".cos.7'" + &c. =z 0. 
 shall be necessarily fulfilled ; and therefore in this case there 
 will be equilibrium whenever the remaining condition is sa- 
 tisfied, expressed by the first of the three equations. (4) 
 
PARALLEL FORCES. 
 
 21 
 
 SECTION II. 
 
 OF THE COMPOSITION OF PARALLEL FORCES. 
 
 1. When two forces act on an invariable system, in pa- 
 rallel directions, and towards the same side of the line con- 
 necting their points of application, the case may be regarded 
 as included in that of forces meeting at a point ; by sup- 
 posing this point to have receded to an infinite distance. 
 
 Thus, let m, m, (Fig. 17.) be the points of application of 
 the forces p, p', acting in the same plane, and towards the 
 same side of the line mm-, and first, let their directions be 
 niA, m\', inclined to each other in the angle h. Then, if 
 without making any change in the direction of the force p, 
 we suppose the direction of the force p' to turn round the 
 point m in the same plane, so as to assume, successively, the 
 positions w?'a', w'a", w'a'", &c. and finally, to coincide with 
 m ^"" , parallel to w2A, which is the direction of the force p, it 
 is evident, that during this revolution of the line w'a', it shall 
 meet the line a/«, successively, at the points h, h', h", h'", &c. 
 and that when the direction of the force p' coincides with 
 the Hne m^"" parallel to mk, the distance at which they meet 
 becomes infinite. 
 
 During the progress of the system of forces p, p', towards 
 this state, they had a resultant continually increasing in mag- 
 nitude: therefore, the same forces, when parallel, have a 
 resultant, whose magnitude and direction are those of the 
 limit to which the resultant of the forces had been approach- 
 ing, whilst the angle made by their directions was gradually 
 diminished. Now, the angle of a parallelogram being dimi- 
 nished to cypher, the diagonal passing through that angle 
 
22 STATICS. — SECT. II. 
 
 increases towards a limit, which is the sum of the sides : 
 wherefore, the sum of the component forces is the hmit 
 sought. Accordingly, when two parallel forces act towards 
 the same side of a line transverse to their directions, the re- 
 sultant is equal to the sum of the components. 
 
 Again: the direction of the resultant was ever inter- 
 mediate between the directions of the components, whilst 
 they met at an angle ; wherefore, the direction of the resul- 
 tant of two such parallel forces is intermediate between their 
 directions, and, therefore, also parallel to the same. 
 
 Finally : whilst the forces met at an angle, the perpen- 
 diculars on the directions of the components, let fall from 
 any point in that of the resultant, were reciprocally propor- 
 tional to those forces : and when the forces become parallel, 
 those perpendiculars lie in directum. Therefore, the per- 
 pendicular between the directions of the parallel forces is 
 divided by the resultant into segments, which are recipro- 
 cally proportional to those forces ; and the same thing is 
 consequently true of the segments, into which a line con- 
 necting any two points in the directions of the parallel forces, 
 is divided by the direction of their resultant. 
 
 2. But though these truths may be thus inferred, from 
 those already established relative to forces meeting at a point, 
 yet the great importance of the doctrine of parallel forces in 
 the theory of mechanics, may seem to warrant, if not to re- 
 quire, a demonstration more rigorously exact. This may be 
 obtained in the following manner : — Let the parallel forces 
 p, p' acting at ?n, m (Fig. 18.) and towards the same side of the 
 right line connecting those points be represented by w?a, mV. 
 To this system of forces let there be added two equal and op- 
 posite forces mg^mg acting atm,m' and in the connecting line 
 produced beyond those points. By this addition no change 
 will be made in the result of the original system of forces. 
 Now compounding the forces mg, m\, and also the forces 
 mg\ m k , the resultant of the former pair will act in some 
 
PARALLEL FORCES. 23 
 
 line within the angle gniA as mo, and that of the latter pair 
 in some line within the angle g't)iA' as mo, and these lines 
 being no longer parallel shall meet at some point h. Trans- 
 ferring diese partial resultants to this point, and resolving 
 them in directions parallel to mg, mk, and mg , m'k', it is 
 evident that they shall reproduce the original components ; 
 viz. nd, H5 equal respectively to mg, mk, and ud', us to m'g', 
 m'k'. Of these nd, ud', being equal and opposite, may be 
 suppressed; and the resultant of the remaining two is their 
 sum H* + h/, or mk + m'x' ; i. e. 
 
 R = p+p'. (1) 
 
 It is now proved, that " the resultant of two parallel 
 forces, acting towards the same side of the line connecting 
 their points of application, is parallel to their directions, and 
 equal to their sum." 
 
 To find the point c, where the line mm' is intersected by 
 the direction of the resultant, we have, from similar tri- 
 angles. 
 
 giving 
 
 mc : CH : : ns : *h, 
 m'c : CH 
 
 ca.ns 
 mc — • mc 
 
 5H sn 
 
 But cvL.ns — cii.n's, and therefore 
 
 mc : 711 c : : s'n : *h : : p' : P. 
 Shewing, that " the line which connects the points of appli- 
 cation, is divided by the resultant into segments, which are 
 reciprocally proportional to the forces." 
 
 By compounding the last proportion, we have, 
 
 mc + m'c : m'c : : P + P' ^ p« «• ^« "mm' : m'c : : r : p. 
 
 and in the same way, 
 
 mm' : mc : : r : p'. 
 
 which analogies are succinctly expressed by the following 
 equations : 
 
2^ STATICS. — SECT. II. 
 
 A = A = -"-. (2) 
 
 mm cm cm 
 
 Stating, that " any two of the forces are, one to the 
 other, reciprocally as the distances of their lines of direction 
 from that of the third force." 
 
 3, The resultant r, its direction being changed into the 
 opposite, is in equilibrio with p, p'. Thus cc representing 
 the resultant of the parallel forces mA, m'A, (Fig. 19. No. 1.) 
 becomes the equilibrating force, when its direction is changed 
 into the opposite, as in No. 2 ; wherefore, '' the conditions 
 of equilibrium, among three parallel forces, are, 1. That 
 the greatest should be equal to the sum of the other two. 
 2. That it should act in the contrary direction. 3. That it 
 should divide the line connecting the points of application 
 of the lesser forces into segments, which are reciprocally as 
 those forces." 
 
 4. Having established the conditions of equilibrium 
 among three parallel forces, the transition is easy, to the 
 resultant of two forces, acting in parallel but contrary direc- 
 tions ; for the equilibrating force, its direction being changed 
 into the opposite, is their resultant. Thus m'A' which equi- 
 librates the forces t?iA, cc", No. 2.., becomes their resultant 
 when its direction is changed into the opposite, as in No. 3. 
 Whence it follows, that " if two parallel forces act in con- 
 trary directions, with respect to the line which joins their 
 points of application, 1. The resultant is equal to their dif- 
 ference. 2. Its direction is parallel to those of the compo- 
 nents, and corresponds to that of the greater. 3. Its point 
 of application lies beyond that of the greater force, at a dis- 
 tance which, measured from that of the lesser force, is to 
 the distance between those forces, as the greater to the diffe- 
 rence : or which, measured from the point of application of 
 the greater of the component forces, is to the distance be- 
 tween them, as the lesser of those forces to their differ- 
 ence." 
 
PARALLEL FORCES. 25 
 
 In fact, the three cases of parallel forces here consider- 
 ed, differ only with respect to their names and directions ; 
 but not as to their relative magnitudes or points of applica- 
 tion. If p and p' act towards the same side of the line which 
 connects their points of application, r, equal to their sum, is 
 their resultant ; and its direction corresponds to that of the 
 forces p and p'. If the direction of r is changed into the 
 opposite, its magnitude and point of application remaining 
 vmchanged, the three forces are in equihbrio ; or any one of 
 them is in cquilibrio with the remaining two ; and finally, if 
 the direction of either of the first pair, as p', is also changed 
 into the opposite, its magnitude and point of application re- 
 maining unchanged, it becomes the resultant of p and r, which 
 are then parallel and contrary forces. Wherefore, all ques- 
 tions, respecting the relative magnitudes and points of ap- 
 plication of three parallel forces, whether one of them is in 
 equilibrio with the other two, or their resultant, are to be 
 managed by means of the same equations or analogies. 
 
 ■ Equation (1) gives any one of the three forces when the 
 other two are given : for r iz p -f- P) and therefore, 
 p' zz R — p, p r: R — p'. 
 
 The three equations (2) contain, each of them, four 
 quantities, viz. two forces, and two distances ; and any three 
 of those quantities being given, the fourth is immediately 
 obtained by these equations. 
 
 Thus, if two parallel forces are given, acting towards 
 the same side of the line which joins their points of applica- 
 tion, these points being also given ; and if it is required to 
 find their resultant and its point of application, it is evident 
 that the are data p, p', mm ; and that the quantities sought 
 are r and cm, or cm. But by equation (1), r = p -f- p', and by 
 
 p' 
 
 equations (2), cm ■=. mm'. -,. 
 
 p -f- p 
 
 If it is required to resolve a given force into two parallel 
 
 forces acting at given distances, the quantities given are 
 
 R, cm, cm', and those sought are p, p'. But 
 
26 STATICS. — SECT. II. 
 
 cm , 
 
 TZZR. r- P — R. 
 
 nwi nun 
 
 If two parallel and contrary forces are given, as also the 
 points of application, and it is required to find their resul- 
 tant and the point to which it is applied, the data are p, R, 
 and cm : and the quantities sought are p', cm' or mm'. But 
 p' := R — p, and 
 
 P , R 
 
 cm ~ cm. . 7nm zzcm.- 
 
 R— P R— p 
 
 p 
 5. Considering the equation cm'zz.cm. . it appears 
 
 R — P 
 
 that if whilst r remains unchanged, p increases from nothing 
 
 to R, cm' shall constantly increase: and that when p = R, the 
 
 P 
 value of p', which is r — p, is then cypher, and cm' zz cm, - 
 
 is then infinite. This shows, that when two parallel forces, 
 acting in contrary directions, are equal, there is no single 
 resultant ; or, which is the same thing, that there is no sin- 
 gle force by which the equilibrium can be established. In 
 fact, could we suppose a single resultant, there would be no 
 reason why it should be nearer to one than to the other of 
 the components ; it must, therefore, either be at an infinite 
 distance from both, or at equal distances between them ; and 
 if this latter case be supposed, there is no reason why it 
 should corrrespond in direction with one, rather than with 
 the other of the components, for all here is symmetrical. 
 
 p 
 
 If p still further increases, the denominator oi cm. , 
 
 R — p 
 
 which is the value of cm', changes its sign ; which denotes that 
 
 this latter quantity is then to be measured on the transverse 
 
 line, in the direction of cm : and because the absolute value of 
 
 the denominator is less than p, the absolute value of cm' is 
 
 greater than cm ; diminishing from infinity to this limit, as p 
 
 increases from r to infinite. 
 
 From the equation p' — r — p, it also appears, that in 
 
 this case p' is negative ; or that its direction, which on the 
 
PARALLEL FORCES. 27 
 
 supposition of r>p, corresponded to that of R ; now, on the 
 supposition of p>r, corresponds to that of p, all of which is 
 but the analytical expression of the truth already stated, viz. 
 that the resultant of two parallel and contrary forces is equal 
 to their difference; that its point of apphcation lies beyond 
 that of the greater of the component forces, and that it cor- 
 responds with this force in direction, with respect to the line 
 which connects the points of application. 
 
 6. The principles already established will direct us how 
 to proceed in the composition of any number of parallel 
 forces. 
 
 Thus let p, p', p", &c. be any parallel forces applied at 
 the points of an invariable system, whether in the same plane 
 or otherwise ; and let it be required to find the general re- 
 sultant, as also its point of application. 
 
 We first compound those which act at the same side of 
 a plane transverse to their directions : let p and p' be two 
 such forces, we have for their resultant r := p + p' ; again, 
 compounding r with a third of these forces p", we have the 
 resultant of r and p", i. e. r' = r -f- p" i= p -f p'H- ?"• Thus, 
 the resultant of all the parallel forces, which act towards the 
 same side of the plane transverse to their directions, is the 
 sum of all such forces. In the same way, the resultant of all 
 the forces, which act towards the other side of the same 
 plane, is equal to the sum of these latter forces : the 
 general resultant of all is the difference of these two partial 
 resultants, and corresponds in direction with the greater: 
 i. e. it is equal to the sum of those which act towards the 
 same side of the transverse plane, minus the sum of those 
 which act towards the opposite side of the same plane ; and 
 it corresponds in direction with that of the greater sum: so 
 that regarding as positive those which are directed to- 
 wards one side of the plane ; and as negative, those di- 
 rected towards the opposite side of the same plane, and un- 
 derstanding by the sign + in the formula wherein they are 
 combined, that the forces are to be connected with their 
 
28 STATICS.— SECT. 11. 
 
 proper signs, as in common algebra, it may be stated, that 
 the general resultant is equal to the sum of the parallel 
 forces, or that r zz p + p'4- p" + &c. 
 
 The point of application is found for the resultant by the 
 repeated use of the rule for finding that of the resultant of 
 two component forces. 
 
 First, the point c (Fig. 20.) is had by dividing the line mm 
 in the inverse ratio of the forces p and p'. Then conceiving 
 the force r (z= p + O ^o ^^ applied at this point, we connect 
 it with the point m", and we find c, the point of application 
 of r' by dividing the line cm" in the inverse ratio of the forces 
 R and p", i. e. of p + p' and p". Having found, in this man- 
 ner, the point of application of the resultant of all the forces 
 which act towards one side of the tranverse plane, we do the 
 same for the forces which act towards the other side of the 
 same plane, and the point of application being thus found for 
 the resultant of each of the groups of parallel forces, we find 
 that of the general resultant by the rules already delivered 
 for two parallel forces acting in contrary directions. 
 
 If these two partial resultants are equal in magnitude, as 
 well as contrary in their directions, the system offerees ad- 
 mits of no further reduction: i. e. it admits of no single ge- 
 neral resultant, and therefore cannot be equilibrated by any 
 single force. 
 
 From the method of finding the point c, it is evident that 
 its position in the line mm' is altogether independent of the 
 angle kmm \ therefore, whilst the directions of the forces p 
 and p' turn round their points of application, the intensities 
 of these forces and the parallelism still remaining, R the mag- 
 nitude of the resultant, and c its point of application, shall 
 continue unchanged. The same is to be observed of r', the 
 resultant of r and p", or of p, p', p", if these three forces, 
 their intensities and parallelism remaining, turn round their 
 respective points of application : and the same thing is true 
 of all the partial resultants, and, therefore, of the general 
 
PARALLEL FORCES. ^9 
 
 resultant when there is one : so that if the directions of all 
 the forces of the system are simultaneously changed, the pa- 
 rallelism remaining, as also the intensity of the force applied 
 at each point, the general resultant will continue of the same 
 magnitude, and will have the same point of application. 
 This point is denominated the centre of parallel forces, and 
 may be defined to be that point of the body through which 
 the general resultant always passes, whatever be the di- 
 rections of the forces ; the parallelism, as well as the in- 
 tensities of the forces applied at the several points, still re- 
 maining. 
 
 The general resultant of any number of parallel forces 
 being equal to the sum of those which act in one sense, mi- 
 nus the sum of those which act in the contrary sense, this 
 sum or difference may be supposed to be applied at the cen- 
 tre of parallel forces ; and, therefore, if a force equal to this 
 is applied at the same point, and in the opposite direction, 
 the whole system shall be maintained in equilibrio, whatever 
 be the position of the body, or of the lines connecting its 
 several points, with respect to the directions of the parallel 
 forces. 
 
 7. As the centre of parallel forces frequently offers itself 
 to our consideration in mechanical questions, it is of impor- 
 tance to show how its coordinates are determined, as func- 
 tions of those of the points of application, and of the intensi- 
 ties of the forces there applied. This is done by computing 
 its distance from three coordinate planes. 
 
 Wherefore, beginning with one of those planes, which 
 may be that oi xy. Let mo, mo, be the perpendiculars on 
 this plane, from m, m , the points of application of the forces 
 P, p': (Fig. 21.) and let cr be the perpendicular, from the 
 point at which their resultant is applied. It is evident that 
 o, r, o' , the projections of these points shall lie in the right 
 line 00 . From m, let mg be drawn parallel to this line oo' , 
 
30 STATICS. — SECT. II. 
 
 meeting cr at e. Then, from similar triangles, there is ob- 
 tained the following proportion : 
 
 ce : m'g (: : mc : mm) : : p' : r, 
 wherefore, 
 
 R X ce = p' X m'g. 
 But also, 
 
 R X er rr p X mo 4- P' X go. 
 And adding these equations, 
 
 R X cr zz p X wo + P' X m'd. (a) 
 
 In the same way, from m", the point of application of a 
 third force p", letting fall the perpendicular m"o", as also 
 from c, the point of application of r' the resultant of R and 
 p", letting fall the perpendicular cr, we have 
 
 r' X c'r'=R X cr + p" X m"o". (b) 
 
 Proceeding in the same way with m'", the point of appli- 
 cation of a fourth force p'" ; and with c", the point of appli- 
 cation of r'', the resultant of r' and p'", we have 
 
 R X e r z= R X c / - + P Xni o , (c) 
 
 Then adding together the equations (a), (b), (c), we have 
 
 r" X c'V'z= p X mo + p' X w/o'-f- P" X m"o"+ p'" X m"'o"' : 
 
 and so for any greater number of forces. 
 
 In the figure, all the forces, p, p', p", See. are represented 
 as acting towards the plane X7/, and all the points of applica- 
 tion, m, m , in' , &c. are placed on the same side of that plane ; 
 but it is easily perceived, that if any of the forces act in the 
 contrary direction, or if any of the points of application are 
 situated on the opposite side of the plane, such forces and 
 the distances of such points are to be marked with negative 
 signs; and that the theorem is still true, viz. that "The 
 sum of the products, formed by multiplying each of the com- 
 ponent parallel forces into the distance of its point of appli- 
 cation from any assumed plane, is equal to the single product 
 
PARALLEL FORCES. 31 
 
 of their resultant into the distance of its point of application 
 from the same plane." So that if z denotes the distance of 
 the general resultant r, and z, z, %" , &c. the distances of the 
 several points m, m, m", &c. we have 
 
 Rz = p.^ + pV+ pV'+ y"'z"' + &c. 
 or, putting for r its value, and dividing, 
 
 P^ + pV+ p"^"+ p' V+ &c. 
 
 z =:- 
 
 (3) 
 
 P 4_ p'_)_ p"4-p'''+ &c. 
 
 By this theorem the centre of parallel forces is limited to 
 a plane, parallel to that of xtj, at the distance z so deter- 
 mined. To fix it absolutely, its distance from each of the 
 other two coordinate planes must also be determined ; which 
 is done for them as for the plane of X7j. Let x be its dis- 
 tance from the plane zy, and y its distance from the plane 
 zx, the distances of the points of apphcation of the compo- 
 nents from these planes being in like manner denoted by 
 X, x', x", x'". Sec. ?/, y', 2/", 9/"', &c., we have 
 
 _ Yx +f'x'+f"x''+ t'"x"'-\-Scc, . 
 
 ^ — P + P'-f P''+ P'"+ &c. ^ ^ 
 
 _ py 4- pV-i- t'Y+ v"Y'+ & c. 
 
 Y - p + p'+ p"+ P'"+ &c. ^' ^ 
 
 These are the values of the coordinates of the centre of 
 parallel forces, and by them its position is completely deter- 
 mined. And it is manifest, if the three lines, z, x, y, so ob- 
 tained, are measured on the three axes, which are the inter- 
 sections of the assumed planes, that the centre of parallel 
 forces is at the remote extremity of the diagonal of the pa- 
 rallelopiped, having the same three lines for the sides of its 
 solid angle. 
 
 If the points of application of the forces p, p', p", &c. lie 
 in the same plane, this plane may be taken for one of the co- 
 ordinate planes, as for example, that of xy ; and this, whe- 
 ther the directions of the forces lie in the same plane or not. 
 
32 STATICS. — SECT. 11. 
 
 By this disposition we shall have s, z', s", &c. each equal to 
 cypher; and, therefore, also z — 0: so that the position of 
 the centre of parallel forces is then determined by the equa- 
 tions (4) and (o), which give the values of x and y. 
 
 If the points of application are all arranged in the same 
 right line, this line may be regarded as the intersection of 
 two of the coordinate planes ; as for example, of the planes 
 xy, xz. We shall then have z, z', z", &c. each equal to 
 cypher ; and, therefore, z — 0. Likewise, y, y', y", &c. each 
 equal to cypher, and therefore, also, y zzQ. Wherefore, the 
 centre of parallel forces is in the same line with their points 
 of application, viz. the axis of a.-: and its position in this line 
 is determined by the single equation (4), which expresses 
 the value of x, its distance from the plane zy, perpendicular 
 to the right line in which the points are arranged. 
 
 In this case, therefore, it will be sufficient for the deter- 
 mination of the centre of parallel forces to ascertain its dis- 
 tance from any point, assumed in the line itself, as the origin : 
 and this is done by means of any one of the three equations, 
 (3), (4), (5). 
 
 For example, let it be supposed that there are five points, 
 a, b, c, d, e, arranged along the line os, (Fig. 22.) to which 
 points parallel forces are applied, denoted by the numbers, 
 5, 4, 8, 10, 2. These points may be referred to any point in 
 the line itself, or in this line produced, as the origin. Let 
 them be referred to the point 7n, and let the distances from 
 this point be expressed by the numbers — 3, — 1, +4, •\-G, 
 + 7 : the distances on one side of this point being deemed 
 affirmative, as mc, md, me-, and those on the other side ne- 
 gative, as mb, ma. The same distinction is to be made 
 among the forces themselves, according as they tend to one 
 side or the other, Tif the line os. In the above scheme all the 
 forces are supposed to act in the same sense, and therefore 
 to be affected with the same sign. Then, for the distance of 
 the centre of parallel forces from the point m, the forces are 
 
PARALLEL FORCES. 
 
 33 
 
 to be multiplied, severally, into the distances of their points 
 of application from the point w?, and the sum of these pro- 
 ducts is to be divided by the sum of the forces. In this in- 
 stance we shall have 
 
 32_l_60-f 14— 4— 15 87 
 ^ = -¥j = ^="+"' 
 
 Wherefore, taking three units of the line from the point m, 
 and in the direction of the affirmative values, we have the 
 point g, which is the centre of the parallel forces. 
 
3i STATICS. — SECT. IIT. 
 
 SECTION III. 
 
 OF EQUAL AND PARALLEL FORCES ACTING ON AN INVARIABLE SYS- 
 TEM, TOWARDS OPPOSITE SIDES OF A LINE TRANSVERSE TO 
 THEIR DIRECTIONS, 
 
 1. When two parallel forces act towards opposite sides 
 of a line transverse to their directions, those directions, 
 though not immediately opposite, may be said to be con- 
 trary. 
 
 In last section, it was shown that two such forces, when 
 equal, are incapable of being equilibrated by a single force. 
 It is now to be shown how they are equilibrated, and how 
 transformed. To avoid circumlocution, a pair of equal pa- 
 rallel and contrary forces shall be simply named a pair ; and 
 in all transformations of a pair of such forces, it is to be un- 
 derstood that the intensity of the forces, and the perpen- 
 dicular distance between the lines of direction, remain un- 
 changed, unless when the contrary is expressly stated. 
 
 A pair is in equilibrio with another pair, equal to the for- 
 mer, and applied to the same points in opposite directions. 
 
 This is evident, inasmuch as the forces applied at each 
 point are in equilibrio, and the forces in equilibrio being 
 suppressed, there remains no force to disturb the system. 
 
 The equilibrium continues when the second pair is trans- 
 ferred to any part of the same plane, in parallel directions. 
 
 Let one pair consist of the forces -\-f, — p, acting in the 
 directions a«, b6. (Fig. 23.) These are in equilibrio with 
 the pair — p, + p, acting at the same points in the opposite 
 directions Aa, Bb'. Let the line ax be perpendicular to 
 those directions, and taking a'b' — ab, let the latter pair be 
 
PARALLEL FORCES, EQUAL AND CONTRARY. 35 
 
 transferred to the points a' and b', in the directions ^a, n'lf, 
 parallel to the former. The lines ab' ba' shall be bisected, 
 each at the same point o. The forces -\-v, 4- ?> acting at a 
 and b', shall compound a force + 2p, acting at o ; and the 
 forces — p, •— p, acting at b and a', shall compound a force 
 — 2p, acting at the same point o ; and as these resultants 
 are equal and opposite, they shall be in equilibrio. 
 
 The equilibrium continues when the second pair is 
 transferred, in parallel directions, into any parallel plane. 
 
 Let the parallel planes be mn, m'n', (Fig. 24.) and let the 
 directions of the pair, thus transferred, be Aa, b'6'. These 
 lines being parallel to a«, b6, the perpendicular distances 
 AB, a'b', shall be parallel, and therefore in the same plane ; 
 and being also equal, the lines ab', ba' shall bisect each 
 other, as at o. The two forces + p, -|- p, shall compound 
 the force + 2p, acting at o ; and the two forces — p, — p, 
 shall compound the force — 2p, acting at the same point ; 
 and as these equal resultants are also directly opposed, they 
 are in equilibrio. 
 
 The equilibrium continues, though the second pair is 
 turned round in its own plane, in an angle of any magnitude. 
 Let the two pairs, first, act at the extremities of the 
 same line ab, perpendicular to their directions; (Fig. 25.) 
 and this line being bisected at o, let the second pair turn 
 round this point in the same plane ; and let Act, ^'U be their 
 new directions, meeting b6, aci at the points n and m. Draw- 
 ing the line mo, the two triangles, Amo, B'mo, are right 
 angled ; and having ao zz: b'o, and ?no common, the angles at 
 m, as also the angles at o are equal : and the same is proved 
 in the same way of the triangles A'no, Bno : wherefore, the 
 angles aob', a'ob, which are vertically opposed, being bi- 
 sected hy the lines ?no, no, these lines lie in directum. The 
 forces -j- p, -f p, acting at m, compound a force in the direc- 
 tion ?}w, and the forces — p, — p, acting at fi, compound a 
 
 d2 
 
36 STATICS. — SECT. III.~ 
 
 force in the direction no : and these resultants, being equal 
 and opposite, are in equilibrio. 
 
 It is now proved, that " a pair is in equilibrio, not only 
 with a pair of equal forces directly opposed, but with the 
 same pair, when transferred, in parallel directions, to any 
 other part of the same plane, or to a parallel plane, or when 
 turned round in any of those planes, in any angle what- 
 soever." 
 
 If at the point which bisects the perpendicular distance 
 between the forces of a pair, a line is raised perpendicular 
 to their plane, this perpendicular may be named the axis of 
 the pair : and the same things are only differently stated, 
 when it is said, that " a pair is equilibrated, not only by a 
 pair of equal forces directly opposed, but by the same pair, 
 when transferred, in parallel directions, to any other point 
 of the same axis, or to any parallel axis ; or when turned 
 round its axis in any angle whatsoever." 
 
 Moreover : after any of these changes the pair is in 
 equilibrio with a pair of equal and opposite forces ; and as 
 all systems of forces are equivalent which are equilibrated 
 by the same system, it follows, that " a pair may be trans- 
 ferred, in parallel directions, to any part of the same plane, 
 or to any parallel plane, or be turned round in its plane, 
 in any angle whatsoever. Or, which is the same thing, it 
 may be transferred to any point of the same, or of a parallel 
 axis, or be turned round any of those axes in any angle 
 whatsoever." 
 
 2. The tendency of a pair is to give to the points of ap- 
 plication, movements in the directions of the individual 
 forces ; and, therefore, to the system to which those points 
 belong, considered as invariable, a movement about some 
 axis perpendicular to the plane of the forces. The energy 
 of the pair to produce this effect, can depend only on the 
 magnitudes of the forces, and the perpendicular distance 
 between their directions. The product of one of the equal 
 
PARALLEL FORCES, EQUAL AND CONTRARY. 37 
 
 forces, into the perpendicular distance between their direc- 
 tions, may be named the moment of the pair : and if two 
 pairs act in the same or in parallel planes ; or, which is the 
 same thing, on the same or on parallel axes, their moments 
 may be said to be of the same or of contrary directions, ac- 
 cording as they tend to turn the system in the same, or in 
 contrary directions. 
 
 A pair is in equilibrio with another pair when the mo- 
 ments of the two pairs are equal and contrary. 
 
 Let one pair consist of the forces -f- p, — p, acting in the 
 directions Aa, Bb ; (Fig. 26.) and let their perpendicular dis- 
 tance, AB, be bisected at o. Let the other pair be — p', + p'» 
 which may be supposed, first, to act in the directions aV, 
 b'6', parallel to the former, and equidistant from the same 
 point o. Then as p X ao i= p'x b'o, the resultant of the two 
 forces -f- p, +p', shall be + (p -\- p')> acting at the point o: 
 and as — PXBon — p'x a'o, their resultant shall be — (p-j-p'), 
 acting at the same point o : and these equal resultants, 
 being dii*ectly opposed, shall be in equilibrio. Now the pair 
 (-f-p', — p') may be transferred to any point of the same, or 
 of a parallel axis, or be turned round any one of those axes, 
 in any angle, and the equilibrium shall continue. 
 
 Hence it follows, that all pairs are equivalent whose mo- 
 ments are equal and of the same direction. 
 
 Two pairs, whose moments are of the same direction, may 
 
 be compounded into one pair, whose moment is equal to the 
 
 sum of their moments, and of the same direction. 
 
 • For, let one pair of forces, p, act at the distance o?; and 
 
 let another pair, p', act at the distance d' : this latter pair 
 
 may be transformed into another pair of forces, p'.-tj acting 
 
 at the distance d. The pairs, now acting at the same dis- 
 tance, may be made coincident ; and they are then reduced 
 
 to one pair of forces, (p -f' P'-rj' whose moment is 
 
88 STATICS. — SECT. III. 
 
 (If \ 
 p + p'. — T j d, or vd + p'c/'. 
 
 In the same way it is shown, that two pairs, whose mo- 
 ments are of contrary directions, may be compounded into a 
 single pair, whose moment is equal to the difference of their 
 moments, and of the same direction with that of the greater. 
 And, in general, that " whatever be the number of pairs^ 
 placed on the same or on parallel axes, they may be com- 
 pounded into a single pair, whose moment is equal to the 
 difference between the sum of the moments of one direction, 
 and the sum of the moments of the contrary direction; the 
 direction of the moment of the resulting pair being the same 
 as that of the greater sum." 
 
 Hence it follov/s, that " a pair may be resolved into any 
 number of such pairs, wdiose moments, severally, are of the 
 same direction with that of the pair to be resolved, or of the 
 contrary direction; provided, that the sum of the former 
 moments exceeds that of the latter, by a difference equal to 
 the moment of the pair to be resolved." 
 
 The planes of two pairs being inclined to each other in 
 any angle, if the moments are represented by the sides of a 
 parallelogram, whose angle is that which measures the in- 
 clination of the planes of the pairs, those pairs may be com- 
 pounded into a single pair, whose moment is represented by 
 the diagonal of the parallelogram ; and whose plane divides 
 the angle, made by the planes of the components, in the same 
 manner as the diagonal divides the angle contained by the 
 sides of the parallelogram. 
 
 Let the two pairs consist of forces of the same magni- 
 tude, p ; and let the forces of corresponding directions, as 
 for example, the forces -f- p act in the Hne ba, the intersec- 
 tion of the planes of the pairs. (Fig. 27.) Let those planes 
 be intersected by a third plane in the lines ac, ad ; and let 
 the forces — p act at the points of those lines m and n. The 
 two forces — p are equivalent to one force — 2p, applied to 
 
PARALLEL FORCES, EQUAL AND CONTRARY. 39 
 
 the point o, at which the hne mn is bisected. The two pairs 
 are then equivalent to one pair of forces 2p, parallel to the 
 forces p, and applied on the line ao, which is the half of ar, 
 the diagonal of the parallelogram, whose sides are am, an ; 
 and this is equivalent to a pair of forces p, parallel to the 
 former, and applied on the whole diagonal ar. 
 
 The two pairs of forces, p, applied on the lines am, an, 
 are then equivalent to one pair of forces p, parallel to the 
 same, and applied on the diagonal of the parallelogram con- 
 structed with those sides. 
 
 If the forces of the two pairs are not of the same magni- 
 tude, they may be brought to this condition, by transforming 
 one of the pairs into another of the same moment, whose 
 forces shall agree in magnitude with those of the other pair : 
 and if the forces of corresponding directions do not act in 
 the line ba, the pairs may be turned round, each in its own 
 plane, until the directions of the forces become parallel to 
 this line ; and then they may be transferred, each in its own 
 plane, so as to satisfy this supposition. 
 
 Were the intersecting plane perpendicular to the line ab, 
 the directions of the forces would be perpendicular to the 
 lines AM, AN, ar ; and the forces being all of the same mag- 
 nitude, the moments of the pairs would be proportional to 
 those lines. And as the angles made by the sides and dia- 
 gonal of the parallelogram measure the inclinations of the 
 planes of the parallel forces, it follows, that the angle made 
 by the planes of the component pairs is divided by that of 
 the resultant pair, in the same manner as the angle of the 
 parallelogram by its diagonal. 
 
 Since the resultant pair is equivalent to its components, 
 it follows, conversely, that the pair of forces p, applied on 
 the line ar, is resolvable into two pairs of forces, parallel to 
 the same, and of the same magnitude, applied on the lines 
 AM, AN, which are the sides of the parallelogram whose dia- 
 gonal is AR ; and that the moment of the pair is to those of 
 
40 
 
 STATICS. — SECT. III. 
 
 the pairs into which it is resolved, as the diagonal to the 
 sides of a parallelogram ; the diagonal making with its sides, 
 the angles which measure the inclinations of the plane of 
 the pair, to those of the pairs into which it is resolved. 
 
 If a direct demonstration is required, such is readily sup- 
 plied ; for the pair of forces p applied on ar, is equivalent to 
 the pair of forces 2p, parallel to the former, and applied on 
 Ao, the half of ar: and —2? acting at o, is equivalent to 
 — P, — P, parallel to the same and applied, one of them at 
 M, the other at n. The given pair is then equivalent to two 
 pairs of forces, equal and parallel to the same, and applied 
 on the sides of the parallelogram, of which ar is the dia- 
 gonal. 
 
 The angles formed by the planes being the same as those 
 made by their axes, the same relations of the resultant pair 
 to its components, as to the magnitude of their moments and 
 position of their planes, may be expressed by stating, that 
 "if on the axes of the component pairs, two portions are 
 taken, which, measured from the angle, are proportional to 
 their moments ; and if a parallelogram is constructed, . 
 having these lines for its sides, the diagonal shall be the axis 
 of the resulting pair ; and shall also represent the magnitude 
 of its moment." 
 
 In constructing the parallelogram, it is to be observed, 
 that when the corresponding forces are brought to act in the 
 same line, the sides of the parallelogram should be mea- 
 sured from a point in this line, either both towards the di- 
 rections of the remaining forces, or both from those direc- 
 tions : and that the sides thus measured, if made to describe 
 an angle of 90°, by revolving in their plane, become the sides 
 of the corresponding parallelogram, to be constructed with 
 the axes. 
 
 By proceeding according to these methods, any number 
 of pairs may be compounded into a single pair ; first com- 
 pounding two pairs into one, then the resulting pair with 
 
PARALLEL FORCES^ EQUAL AND CONTRARY. 41 
 
 another pair, and each resulting pair with a new pair, until 
 all are compounded. 
 
 As two forces, whose lines of direction meet at an angle, 
 must have a resultant, and, therefore, cannot be in equili- 
 brio ; so two pairs, whose planes meet at an angle, must have 
 a resulting pair, and therefore cannot be in equilibrio. 
 
 When the planes of the component pairs meet at a right 
 angle, the parallelogram becomes a rectangle ; and the mo- 
 ments of the component pairs being denoted by x and y, and 
 that of the resulting pair by R, we have 
 
 R= V(x2-fY^). 
 
 AlsOj denoting by a, the angle formed by the plane of the 
 pair whose moment is x, with the plane of the resulting pair, 
 we have 
 
 X , Y 
 
 cos. a n — . sm.a r: — . 
 
 R R 
 
 If the axes of three pairs are parallel to three lines meet- 
 ing at a solid angle, and if on these lines three portions are 
 measured from the point, to represent the moments of the 
 several pairs, the diagonal of the parallelopiped constructed 
 with these sides, shall represent the moment of the resulting 
 pair, and shall be parallel to its axis. For let the axes meet 
 at the point a, (Fig. 1^.) and the moments of the three pairs 
 being represented by the lines ab, ad, af, taken on the axes, 
 let the parallelopiped dg be constructed ; then, the two 
 pairs, whose axes are ab, ad, and whose moments are repre- 
 sented by these lines, shall compound a pair, whose axis is 
 AC, and whose moment is represented by the same line. 
 This resultant of the two pairs, compounded with the third 
 pair, whose axis is af, and whose moment is represented by 
 that line, shall give a pair, whose axis is ah, the diagonal of 
 the parallelogram constructed with the sides AC, af, and 
 whose moment is represented by the same line. But ah is 
 the diagonal of the parallelopiped. 
 
42 STATICS. — SECT. III. 
 
 These axes may be now transferred, each by a parallel 
 movement, to any distance, and the resulting pair shall not 
 be affected, as to the magnitude or direction of its moment. 
 
 Hence it follows, that three pairs, whose planes consti- 
 tute a solid angle, must have a resultant ; and, therefore, 
 cannot be in equilibrio. 
 
 If the three planes intersect at right angles, the paral- 
 lelopiped is rectangular ; and, in this case, the moment of 
 the resulting pair is equal to the square root of the sum of 
 the squares of the moments of the component pairs, i. e. put- 
 ting R, X, y, z, for these moments, we have 
 
 R = V (X^ + Y^ -f Z^). 
 
 If a, /3, 7, denote the angles made by the axis of the re- 
 sulting pair, with those of its components ; or by the diago- 
 nal with the sides of the rectangular parallelopiped, we have 
 
 X o Y z 
 
 cos. a — — . cos.p — — . COS.7— — . 
 
 R ' R ' R 
 
 From the four last equations, any one of the seven quan- 
 tities, R, X, Y, z, a, )3, 7, is known, if three of them are given. 
 But among the given quantities, there must be a moment : 
 for if the three angles, only, are given, we can determine, 
 only, the relative magnitudes of the moments. 
 
 3. A pair may be compounded with a single force pa- 
 rallel to its plane, into a single force, and this single resul- 
 tant shall be equal and parallel to the single component ; the 
 plane in which they are contained being parallel to the plane 
 of the pair; and the distance between them depending on 
 the moment of the pair. 
 
 For let 4- p, — p be the forces of the pair ; d their dis- 
 tance, and p' the single force. The pair may be transformed 
 into another pair of forces -f- p', — p', of equal moment; and 
 
 the distance between their directions will then be d'=: —7-. 
 
 p 
 
 This pair may be now transferred, so that the force — p' 
 
PARALLEL FORCES, EQUAL AND CONTRARY. 43 
 
 shall be directly opposed to the single force ; these two being 
 in equilibrio may be suppressed ; and there will then remain 
 the force + p', which is the force of the pair so transformed 
 and transferred, that corresponds in direction with the sin- 
 gle force ; or it may be regarded as the single force itself, 
 transferred in a plane parallel to that of the pair, to a dis- 
 
 „ p.c? 
 tance a — — ,-. 
 p 
 
 Conversely ; a single force may be resolved into another 
 equal and parallel at any distance, and a pair whose plane is 
 parallel to the plane of translation, and whose moment is the 
 product of that force into the distance to which it is so trans- 
 ferred. 
 
 Thus, if a force + p acts in the direction ba, (Fig. 28.) 
 the effect w ill not be changed by applying to any point of the 
 system, as d, two forces, each of them equal to p, and acting 
 in the line EC parallel to ba, but in opposite directions, i. e. 
 a force + Pj acting at d in the direction dc, and a force — F, 
 acting at the same point, in the direction de. We have then 
 three forces, viz. -f- p applied at d, in a direction parallel to 
 Ba, and a pair of forces p, whose moment is F.d. ; d being the 
 distance to which the given force is transferred. The pair, 
 thus generated, may afterwards be transferred or trans- 
 formed, in any of the ways already specified. 
 
 A pair and a single force, not parallel to the plane of the 
 pair, cannot be compounded into a single resultant. 
 
 Let the pair consist of the forces -{-f, — p, acting in the 
 directions a«, b6; (Fig. 29.) and let -f-p' be the single force, 
 acting in the direction cc, not parallel to the plane of the 
 pair. If there were a single resultant R ; a force — r, equal 
 and opposite, would be in equilibrio with the force + p', and 
 the pair +p, —p. This supposed equilibrating force, — r, 
 cannot be equal, parallel, and contrary to -|-p': for then 
 they would constitute a pair in a plane which, not being pa- 
 rallel, intersects the plane of the pair -{-f, — p; these two 
 
44 
 
 STATICS. — SECT. III. 
 
 pairs would then have a resultant pair, (Art. 2.) and there- 
 fore, the equilibrium would not subsist. This being ob- 
 served, let i/be the direction of — r, and let the force + p' 
 be transferred to o, a point in that line : by this translation, 
 a pair will be generated ; and we shall then have two pairs, 
 and two single forces. The pairs are + p> — -Pj acting in the 
 lines Aa, Bb, and -|- p', — p', acting in the lines cc, og: and 
 as the planes of these pairs intersect, it follows that they must 
 give a resultant pair. The single forces are — r and + p'j 
 acting in the directions og, of, and these forces, which act 
 at the same point, not being equal and opposite forces, must 
 give a single resultant. V/e have then a pair and a single 
 force, which cannot equilibrate. As the force p', and the 
 pair of forces p, cannot be equilibrated by a single force, it 
 follows that they cannot be compounded into a single force. 
 
 Hence it follows, that two forces p, p', not in the same 
 plane, cannot have a single resultant. 
 
 For, let p' be translated to a point in the direction of p, 
 and there compounded with it into a resultant r : we have 
 then a force R, and a pair of forces — p', -f p', generated by 
 the translation of + p' ; and as p is inclined to the plane of 
 this pair, R must also be inclined to the same plane ; where- 
 fore, the force r cannot be compounded with the pair into a 
 single resultant. 
 
 Any number of forces, applied to the points of an in- 
 variable system, may be reduced to a single force and a pair. 
 For all the forces may be transferred to any one point of 
 the system, and by these translations there will be generated 
 so many pairs. All the forces so transferred may be com- 
 pounded into a single force ; and all tlie pairs into a single 
 pair. 
 
 If the single forces applied at the same point are in equi- 
 librio, there remains but the resulting pair : and if the pairs 
 are in equilibrio, or if they give a resulting pair, whose plane 
 contains the direction of the single resulting force, the whole 
 
PARALLEL FORCES, EQUAL AND CONTRARY. 45 
 
 system of forces may be compounded into a single resultant. 
 And if the translated forces are in equilibrio, and also the 
 pairs, generated by the translations, the whole system of 
 forces applied to the different points of the body is in equi- 
 librio. 
 
 Conversely, if the system of forces applied to the diflferent 
 points is in equilibrio, the equilibrium must subsist separate- 
 ly, in the system of translated forces, and among the pairs 
 generated by the translations, i. e, the former cannot have a 
 resulting force, nor the latter a resulting pair: for were both 
 to result, they could not equilibrate, and if either were to 
 result exclusively, the system of forces from which it results 
 could not be in equilibrio. 
 
 It remains to show how these conditions of equilibrium 
 are expressed by the magnitudes of the forces, their direc- 
 tions, and points of application : and this is what is proposed 
 in the following section. 
 
46 EQUILIBRIUM AMONG FORCES APPLIED 
 
 SECTION IV. 
 
 CONDITIONS OF EQUILIBRIUM AMONG FORCES APPLIED TO DIFFE- 
 RENT POINTS OF AN INVARIABLE SYSTEM, AND IN ANY DIREC- 
 TIONS. 
 
 1. It has been shown in last section, that the forces ap- 
 plied to an invariable system, whatever be their number, 
 their intensities, their directions, or points of application, are 
 always reducible to a single force, and a single pair: in- 
 cluding the cases in which either or each of these resultants 
 is equal to cypher. It was, moreover, shown that a single 
 force and a single pair cannot equilibrate. Whence it fol- 
 lows, that for equilibrium among the forces applied, it is re- 
 quisite that the resulting force, and the moment of the re- 
 sulting pair, should be separately equal to cypher ; and it is 
 obvious, that when these conditions are fulfilled, the forces 
 are in equilibrio. 
 
 To reduce these conditions to formulae easily applied to 
 particular cases, it would seem requisite to express these two 
 resultants, as functions of the magnitudes of the forces, the 
 coordinates of their points of application, and the angles by 
 which their directions are determined : and then, to equate 
 to cypher, each of these resultants so expressed. To make 
 out these formulae, and then to apply them to some important 
 cases, is the business of the present section. 
 
 The process by which we arrive at these formulae is as 
 follows. Three axes being assumed, which meet at a solid 
 angle ; each of the forces is resolved into three, parallel to 
 these axes ; each component is then transferred to its pa- 
 rallel axis, and all in the same axis combined into one force, 
 
TO AN INVARIABLE SYSTEM. 47 
 
 equal to their sum. But by the translation of each compo- 
 nent force to its parallel axis, there is generated a pair. 
 Each of these pairs of forces is resolved into two, in the co- 
 ordinate planes to which they are parallel : and all the pairs 
 being thus brought into the three coordinate planes, those in 
 each plane are combined into one pair. 
 
 In this manner all the forces applied to the system are 
 reduced to three forces, directed along the axes, and three 
 pairs in the three coordinate planes. The three forces thus 
 obtained are readily compounded into one force, and the 
 three pairs into one pair. But this further step is unneces- 
 sary for the present purpose, which is not to find the expres- 
 sion for the resultant force or resultant pair, but the expres- 
 sion of the conditions to be fulfilled, in order that each re- 
 sultant should be equal to cypher. Now, the resultant of 
 the forces directed according to the three axes cannot be 
 equal to cypher, unless each of the three forces is, separately, 
 equal to cypher: (Sect. I. Art. 12.) and that the same thing 
 may be asserted of the moments of the pairs in the three co- 
 ordinate planes, will appear in the same way from (Sect. III. 
 Art. 2.) Wherefore, the two general conditions of equili- 
 brium are equivalent to six ; requiring that " the sum of the 
 forces in each of the three axes should be equal to cypher, 
 and that the sum of the moments of the pairs in each of 
 the coordinate planes should be equal to cypher." 
 
 To illustrate this reasoning, let Aiv, a?/, a;^, be the three 
 coordinate axes, (Fig. 30.) and let os represent one of the 
 forces. Then if this line is made the diagonal of a paral- 
 lelopiped, whose sides, om, on, or, are parallel to the three 
 axes, the force os, may be resolved into three, represented 
 in quantity and direction by these three lines. Any one of 
 these, as or, may be. transferred to its parallel axis az, and 
 by this translation there will be generated a pair of forces, 
 or. Let then the line ro, produced if necessary, meet the 
 plane yAx, at the point b ; it may be supposed to be applied 
 
48 EQUILIBRIUM AMONG FORCES APPLIED 
 
 at this point ; the pair of forces — or, + or, are then apphed 
 on the Hne Ab: and drawing the hnes bf, bg parallel to ax. 
 Ay, this pair may be resolved into two pairs of forces equal 
 and parallel to the former, and applied on the lines a/, a^. 
 The force om, in the same way, produces an equal force in 
 the axis ax, and two pairs in the planes zax, xAy : and the 
 force on produces an equal force in the axis Ay ; and two 
 pairs in the planes zAy, xAy. All the forces being treated in 
 the same way, if the sum of the forces in the axis ao;, the 
 sum in Ay, and the sum in a^, are denoted by x, y, z, and 
 the sums of the moments in the planes %Ay, zkx, yAx, by l, 
 M, N, we shall have for equilibrium 
 
 X = 0. Y = 0. z = 0. (1) 
 
 L z: 0. M = 0. N = 0. (2) 
 
 These expressions are now to be developed ; and if the 
 several forces in the axis ax, are denoted by x', x", x'", &c. ; 
 those in the axis Ay, by y', y", y'", &c. ; those in the axis a^, 
 by z', z", z'", &c. : the three first conditions are equivalent 
 to the three following : 
 
 x'+x"4-x"'+&c. =0. ^ 
 y'+ y"+ y'"+ &c. =0. I (3) 
 z'4.z"+z'"+&c. = 0. j 
 
 Of these forces, such as tend, by their directions, to in- 
 crease the ordinates of their points of application within the 
 same plane angle, may be deemed positive, and those which 
 act in the opposite directions, negative. The sign + with 
 which they are connected in the formvUae, denotes that they 
 are to be added with their proper signs. 
 
 Of the three equations which relate to the moments, it is 
 to be observed, first, that the coordinates of the points of 
 application of each of the forces, are those of three forces 
 into which it is immediately resolved, and that when these 
 last are transferred to the origin, and the pairs generated by 
 
TO AN INVARIABLE SYSTEM. 49 
 
 these translations resolved in the three coordinate planes, the 
 same coordinates are the portions of the axes on which these 
 component pairs are applied. Thus, a^, ajT, on which the 
 two pairs of forces, or, are applied, are two of the co- 
 ordinates of the point o, and so of the rest. Secondly, it is 
 to be observed, that the directions of the forces of the pairs 
 in each of the planes, being equally inclined to the axes on 
 which they are applied, the product of the force of each 
 pair into the distance measured on the axis on which it is 
 applied, may be taken for the relative measure of its moment 
 in the same coordinate plane. Wherefore, putting x\ x", x", 
 &c. for the coordinates of the points of application mea- 
 sured on the axis ax; y , y", y", &c. for those measured on 
 the axis a?/, and z, z" , z", &c. for those measured on the 
 axis Az, the moment of the pair of forces z', in the plane 
 zAx, is z'.x', and that of the pair of same forces in the plane 
 zAy is z.y ; and so of the moments of all the pairs in the co- 
 ordinate planes. Thirdly, the pairs in each of the coordi- 
 nate planes are divided into two sets, distinguished by the 
 axis, to which their forces are parallel. Thus, in the plane 
 yAx there are the pairs whose moments are y'.x', Y".x"y 
 y"'.x"\ &c., and the pairs whose momencs are x'.^, x".y", 
 x'".y"', &c. and of these it is to be observed, that if the forces, 
 and also the abscissse, are positive in any one of the co- 
 ordinate planes, the moments of these different sets are of 
 contrary directions. Accordingly, the equations l = 0. m =: 0. 
 N = 0. are equivalent to the three following: 
 
 (yV-zV) + (yV-zVO + (y"V"-z'V") + &c. = 0. ^ 
 (z'^'-xV) -f (^l'x"- x'V) -f (z"V" - x"V") + &c. = 0. > (4) 
 
 (xy-YV)-f (x'y- y".o+(x'V'- y"V") + &c. = o. J 
 
 The negative sisns in these formulae denote, that the 
 moments to which they are prefixed are to be subtracted 
 with their proper signs. 
 
 E 
 
50 EQUILIBRIUM AMONG FORCES APPLIED 
 
 The formulae, as already presented, contain neither the 
 forces immediately applied, nor the angles by which their di- 
 rections are ascertained. But let the axes be rectangular, 
 and let p', p", v"', &c. be the forces, making with lines pa- 
 rallel to the axis kx, the angles a, a", a'", &c, with lines pa- 
 rallel to the axis Ay, the angles j3', /3", /3"', &c., and with 
 lines parallel to the axis ass, the angles 7', -y", '^'", Sec, we 
 shall have 
 
 x'z: p'.cos.c/. y'— p'.cos./3'. 7/— p'.cos.y'. 
 
 and so of the other component forces. Wherefore, making 
 these substitutions in equations (3) and (4) they are presented 
 as follows : 
 
 p'.COS.a' -\- p".COS.a"-l- p'". COS. a'" 4- &c. — 0. 
 
 p'.cos.j3'4-p".cos./3"4-p"'.cos./3'"+ &c. =0. } (5) 
 
 p'.cos.y' + p".cos.7"+ p'".cos.7"'4- &c. = 0. 
 
 p (cos.p .:^ — cos. 7 .?/ ) + P (cos.p .^ — cos. 7 .?/ ) + 
 p (cos.p .z — C0S.7 .;/ ) -|- &c. zz. U. 
 
 P(cos.7 .a: — cos.a.^j + P (cos. 7 .x — cos. a .^ ) + 
 
 p (C0S.7 .X —COS. a .z )4-&c. 1=0. / V / 
 
 p(cos,a .y —cos.p .o^^j+P (cos. a .// — cos.p .x ) + 
 p (cos. a .y —cos.p .x j-f-etc. = U. 
 
 Such are the six conditions of equilibrium expressed by 
 the magnitudes of the forces, the coordinates of their points 
 of application, and the angles which determine their di- 
 rections. 
 
 2. To find the geometrical magnitudes denoted by l, m, n, 
 let the plane yAx pass through the point o, (Fig. 31.) and 
 retaining the representations of the force p' and its compo- 
 nents made in (Fig. 30.) let the lines mo, on, be produced to 
 meet the axes Ay, ax, at the points c and d. We shall then 
 have 
 
 p'.COS.a',?/'— "s! .y'iz. om X ac 
 
TO AN INVARIABLE SYSTEM. 51 
 
 and 
 
 T'.cos.[5'.x'—Y'.a:'zz on X Ad. 
 
 Whence, 
 
 p'.(cos.a'.?/'— cos.jS'.o;') = ot7i X AC — on X Ad. 
 
 These products may be combined into one, by the follow- 
 ing theorem : 
 
 " If from a point in the plane of a parallelogram, per- 
 pendiculars are let fall on the diagonal and contiguous sides, 
 the product of the diagonal into its perpendicular distance 
 from that point, is equal to the sum or difference of the pro- 
 ducts, had by multiplying each of the sides into its perpen- 
 dicular distance from the same point ; according as the 
 point is placed without or within the angle contained by the 
 sides." 
 
 To prove this proposition, let 07}ibn (Fig. 32.) be the pa- 
 rallelogram, and A the point from which are let fall on the 
 diagonal and sides the perpendiculars a^, ac, Ad. Also, 
 from the points m, b, n, let mr, hp, nq, be perpendicular to 
 the line ao. There will then be 
 
 oh X Ag-=. 2tv\axig.Aob — ao X bp. 
 
 am X AC — 2 triang.Ao;« n: ao X mr. 
 
 on XAd— 2triang. Aow — ao X nq. 
 
 But bp — mr ± nq. Wherefore ao X bp — ao [mr ± nq) ; 
 
 i. e. ob X Ag •=! om X ac ± on X Ad, 
 
 the positive sign being taken when the point a is placed 
 without the angle of the parallelogram; and the negative 
 sign when that point is placed within. 
 
 This theorem may be expressed as follows: "A line 
 being drawn from the angle of a parallelogram and in its 
 plane, the product of the diagonal into the sine of the angle 
 which it makes with that line, is equal to the sum of the pro- 
 ducts of the sides into the sines of the angles which they 
 
52 EQUILIBRIUM AMONG FORCES APPLIED 
 
 make with the same Hne, when that hne falls without ; and 
 to the difference of those products, when that line falls with- 
 in the angle of the parallelogram." 
 
 In the formulas, the moments of the pairs generated by 
 any force are expressed as if the coordinates were both posi- 
 tive, or both negative, i. e. as if the origin a were taken 
 within the angle ?iO)n, or the angle vertically opposite : ac- 
 cordingly, referring to (Fig. 3 1 .) 
 
 p'.(cos.a'.,;/'— cos.jS'.o;') zz: omXAc—onXAcl ^. obXAg. 
 
 The line ob is the projection of the line os on the plane 
 i/Ax ; wherefore, the moment of the pairs generated by the 
 force p', in the plane j/a.x, or round the axis a^', is the pro- 
 duct of this force projected on the plane y^x, into the per- 
 pendicular from the origin on this projection. 
 
 This may be otherwise expressed, for if the lines ao, as, 
 are drawn from the origin to the extremities of os, by Avhich 
 the force is represented, (Fig. 3].) and if the triangle aos is 
 projected on the plane t/acc, it may be stated that the mo- 
 ment of the force p' in the plane yAx, or round the axis 
 A^, is repi'esented by twice the area of the projected tri- 
 angle. 
 
 The projection of a force on a plane, is the product of 
 that force into the cosine of its inclination to that plane, i. e, 
 into the sine of the angle which it makes with a line perpen- 
 dicular to that plane. Wherefore, the projection of p' on 
 the plane zAy, is p'.sin.a': its projection on the plane zax, is 
 p'.sin./3': and that on the plane yAx, is p'.sin.y'. Denoting 
 the projections of p", p'", &c. in the same v.^ay, and putting 
 'p , p" , p" , &c. for the perpendiculars from the origin, on the 
 forces projected on the plane zaij. Also, q , q", q'", &c. for 
 the perpendiculars on the same forces projected on the plane 
 ZA* ; and r, r", r", &c. for the perpendiculars on the same 
 forces projected on the plane yhx; the equations (6) may re- 
 ceive the following more simple forms : 
 
TO AN INVARIABLE SYSTEM. 53 
 
 f'.sm.a'.p -\-.F".sm.a".p" + v"'.sm.a"\p" + &c. = 0. ^ 
 p'.sin./3'.g'+ P".sin.i3".9"+ p'".sm.^'".?'"+ &c. = 0. V (7) 
 p'.sin.y'. / 4- p".sin.7". r" + p'".sin.7'". r" + &c. =0. ) 
 In (Fig. 31.) A^ perpendicular to ob, is perpendicular to 
 the plane of projection sob ; it is then the shortest distance 
 between a;^ and that plane, and therefore the shortest dis- 
 tance between the lines a^ and os; which enables us to ex- 
 press the three last conditions, by stating that " the sum of 
 the products of the shortest distances of the several forces 
 from each axis, into the projections of the same forces on the 
 plane of the two remaining axes, is cypher." 
 
 3. The equations of equilibrium are now readily found, 
 on any particular supposition, by considering how the six 
 general formulge would be thereby affected. 
 
 Thus ; if the conditions of equilibrium were required for 
 a system of forces meeting at a point, this point may be taken 
 for the origin, and then the coordinates of the points of ap- 
 plication being all cypher, the three equations (4), or the 
 three equivalent equations, (6), would be necessarily fulfilled, 
 and the perpendiculars from the origin on the directions of 
 the forces being cypher, the same is true of equations (7). 
 There remain therefore only the three first conditions, ex- 
 pressed by the three equations (1), or the equations (3) or (5), 
 which are equivalent. 
 
 If the forces are parallel, by taking one of the axes, as az, 
 parallel to the forces, we have 
 
 X = 0. Y = 0. 
 
 and therefore the two first and the last of the six general 
 conditions are necessarily fulfilled. Accordingly, the only 
 conditions of equilibrium required in this case are those ex- 
 pressed by the last of the equations (1), (3), or (5), and the 
 two first of the equations (2), (4), (6), or (7), and in this case, 
 those equations become more simple ; the equations (4) be- 
 comiuff 
 
54 EQUILIBRIUM AMONG FORCES APPLIED 
 
 z'.y ■\-z".y" -\-7!".y"'-^ 8cc. zi 0. 
 
 z,x -^-z .X -\-z ,x -j- iKC. r: 0. 
 
 and the corresponding equations (6) and (7) then assuming 
 similar forms. 
 
 These conditions of equihbrium for parallel forces are 
 expressed by saying, that " the sum of the forces should be 
 equal to cypher, and that the sum of the moments round 
 each of the axes in the perpendicular plane should be equal 
 to cypher." 
 
 If the directions of the forces are in the same plane, this 
 may be taken for one of the coordinate planes, as for exam- 
 ple, for the plane t/ax. Then z r= 0. cos. 7', &:c. zz 0, which 
 reduces the equations (I), (3), (5) to the two first. And «', 
 &;c. being cypher, cos. 7', &c. being cypher, and the perpen- 
 diculars p', &c. q\ &c. being also cypher, the three equa- 
 tions (2), (4), (6), (7), are reduced to the last. Moreover, 
 Y, Y'i y"'> &c. being right angles, the last of the equations, 
 (7), assumes the form, 
 
 p .r + P .r +P -^ + ofc. ~ (J. 
 Wherefore, the three conditions of equilibrium, for forces 
 whose directions are in the same plane, are expressed by 
 saying, that " the sums of the forces resolved in directions 
 parallel to two axes in that plane, should be separately equal 
 to cypher, and that the sum of the moments round the third 
 axis should also be equal to cypher." 
 
 If the directions of the forces are parallel and in the same 
 plane, that plane may be taken for one of the coordinate 
 planes, as for example, for the plane of the axes of a; and y; 
 also, one of these axes, as the axis of x, may be taken pa- 
 rallel to the directions of the forces ; and then it will appear, 
 in the same way, that the conditions are reduced to the first 
 and last, which require " that the sum of the forces parallel 
 to AX, should be cypher, and the sum of the moments round 
 the axis a^ equal to cypher." 
 
TO AN INVARIABLE SYSTEM. 55 
 
 4. When the six conditions of equilibrium are not satis- 
 fied there must be a single resultant, or a resulting pair, or 
 both. 
 
 To have a resulting pair without a resulting force, it is 
 requisite, first, that the partial moments l, m, n, should not 
 all be equal to cypher: for if each of these moments is 
 cypher, there is no resulting pair. Secondly, the partial re- 
 sultant forces must be, each, equal to cypher, i. e. we must 
 have 
 
 X = 0. Y = 0. z = 0. 
 
 otherwise these forces would have a resultant, and it has 
 been shown, that a force and a pair cannot be compounded 
 into a pair. 
 
 To have a resulting force without a resulting pair, it 
 might, at first view, appear requisite, that besides having 
 some one of the quantities x, y, z, diflferent from cypher, we 
 should have 
 
 L = 0. M = 0. N = 0. 
 
 But it is to be considered, that though the forces of the sys- 
 tem are reducible to a single resulting force, yet if that re- 
 sultant does not pass through the origin, it generates a pair 
 when transferred to that point : and that it is in this complex 
 form it would be at first presented by the process for com- 
 position, above described : whence it appears, that the last 
 equations express the conditions requisite for a single resul- 
 tant passing through the origin ; and that, generally, the case 
 of a single resultant does not require this condition. 
 
 To find the conditions for a single resultant, it is therefore 
 to be supposed, that the moments l, m, n, may be different 
 from cypher; and it is to be inquired, whether by any 
 change of the origin, they may be all reduced to cypher. 
 By assuming any other point whose three coordinates are 
 X, 7/, %, and taking this point for the origin, the new coordi- 
 nates would be diminished by those quantities : whereby the 
 
5G EQUILIBRIUM AMONG FORCES APPLIED 
 
 sum of the moments of the pairs of forces y', y", y'", &c. in 
 the plane zy, would become 
 
 y'.«'+ x".z"^- y"'J"-\- &c. - (y'4- y"+ y"'+ &c.) z, 
 
 that is, 
 
 y'.^'H- y".^"+ y'".;^'"+ &c.-y.<^. 
 
 Those of the forces z', z'', z% &c. in the same plane would 
 become 
 
 z'./+ z"./ + z"'.y"+ &c. -z.y. 
 Making the corresponding changes in the moments in each 
 of the other planes, the three last of the six general condi- 
 tions of equilibrium will become 
 
 L— y;^ + zy = 0. "^ 
 
 M-za' + x^:r:0. \ (8) 
 
 N — x?/ + Ya;:=0. j 
 
 From these three equations, any two of the three indeter- 
 minate quantities being exterminated, there results the 
 equation 
 
 XL + YM + ZN zr 0. (9) 
 
 This equation expresses the relation, which should subsist, 
 among the three partial resultant forces and the three par- 
 tial resultant moments, in order that the forces of the system 
 should have a single resultant. 
 
 This condition may be otherwise investigated, from the 
 principle, that the resultant of the forces transferred to the 
 origin, can be compounded Avith the resultant moment when- 
 ever it is parallel to the plane of that moment ; and that in no 
 other case is it possible that they should be compounded into 
 a single resultant. Therefore, the condition inquired after 
 is the same as that to be fulfilled, in order that the direction 
 of the resulting force should be parallel to the plane of the 
 resulting moment, or, which is the same thing, perpendicular 
 to the axis of that moment. 
 
TO AN INVARIABLE SYSTEM. 57 
 
 Let R denote the resultant of the forces transferred to 
 the origin; a, (5, j, the angles which it makes with the axes 
 AX, Mj, Kz, we shall have 
 
 cos. a — — . cos.p ~ — . COS.7 r: — . 
 
 R ' R ' R 
 
 Putting G for the resulting moment, and A, ju, v, for the an- 
 gles made by its axis with the coordinate axes hx, Ay, hz, we 
 have 
 
 . L M N 
 
 cos.A rr — . cos.w n — . cos.v :r — . 
 
 G '^ G G 
 
 Now in order that R should be perpendicular to the axis 
 of the moment g, the cosine of the angle contained by those 
 lines must be cypher, i. e. 
 
 cos. a cos.X + C0S.j3. COS.M + COS.7. cos.v -zi 0. 
 
 which, putting for the cosines their values as above, gives 
 the equation 
 
 XL -f- YM + ZN = 0. 
 
 When this condition is not fulfilled, there is a resulting 
 force and a resulting moment : and this completes the ac- 
 count of the conditions required ; first, for equilibrium, when 
 there is neither resulting force nor resulting moment; se- 
 condly, for a resulting moment ; thirdly, for a single result- 
 ing force ; and fourthly, for both a resulting force and a re- 
 sulting moment. 
 
 The values of the resulting moment, and of the angles 
 made by its axis with the three coordinate axes, have been 
 already given: likewise, the value of the resultant force, and 
 of the angles made by its direction with the same axes, when 
 the direction passes through the origin; and these values 
 are not changed when it is compounded with a pair to whose 
 plane it is parallel. But by this composition, the line of di- 
 rection is changed for another to which it is parallel, in a 
 plane parallel to the plane of the pair ; and the distance to 
 
58 EQUILIBRIUM AMONG FORCES APPLIED 
 
 which it is tranferred is — . Making the directions of the 
 
 R "= 
 
 forces of g parallel to that of R, the side on which r is thus 
 transferred, is that of the force of g, which corresponds with 
 it in direction, as compared to the force of the contrary di- 
 rection. (Sect. III. Art. 3.) 
 
 If, however, a formula is required for the point of appli- 
 cation of the single resultant, this is readily supplied, for as 
 the point of application of a force may be any point, taken 
 indifferently in its line of direction, x, y, z, are the coordi- 
 nates of any point in the direction of this resultant ; and, 
 therefore, equations (8), which express the relations among 
 these coordinates, are the equations of this line. These, 
 constituting but two independent equations, do not give the 
 absolute, but the relative values of the coordinates. This, 
 however, is sufficient to furnish the values of two of the co- 
 ordinates, for any supposition made with respect to the mag- 
 nitude of the third. Thus, if it were proposed to find the 
 intersection of the resultant with the plane of ^o--: by making 
 2/ zz 0, in the first and third of these equations, we have this 
 point determined by the equations, 
 
 ~ Y ' Y * 
 
 And in the same way, by making ^ — 0, in the first and se- 
 cond of those equations, we have 
 
 , M z 
 
 z ^ z 
 
 for the intersection with the plane of t/x. 
 
 It was observed of equation (9), that it expresses the 
 condition to be satisfied, in order that the forces of the sys- 
 tem should have a single resultant ; but it does not follow 
 that there will be a single resultant whenever this condition 
 is satisfied : for it must be remembered, that the condition 
 was investigated on the supposition of the existence of r, the 
 
TO AN INVARIABLE SYSTEM. 59 
 
 resultant of the forces transferred to the origin, i. e. on the 
 suppositon that x, y, z, were not, each of them, equal to 
 cypher. If each of these is cypher, there will be only a re- 
 sulting pair, and equation (9) is equally satisfied. 
 
 To apply this to the cases of parallel forces, and of forces 
 in the same plane ; if the forces are parallel, and their sum 
 different from cypher, they have a single resultant, and by 
 taking one of the axes, as for example, the axis A.r, parallel 
 to their directions, we have 
 
 Y = 0. z = 0. L = 0. 
 
 which reduces the equations (8) to 
 
 M-\-xz zzO. 
 
 -ti ~x^ zzO. 
 giving 
 
 _ N _ M 
 
 •^ ~ x' ~ x' 
 
 both constant quantities. 
 
 If the forces act in the same plane, that plane may be 
 taken for the plane of yx ; and then 
 
 z = 0. L = 0. M = 0. 
 
 which reduces the equations (8) to 
 
 YSS = 0. 
 
 xss = 0. 
 
 N ■- Xl/ -{-YX — 0. 
 
 the last giving 
 
 N 4- YX. 
 
 y— . 
 
 X 
 
 If the forces are parallel and in the same plane, they may 
 be supposed to be parallel to ax, and in the plane of x?/ ; and 
 then 
 
 Y = 0. zzzO. L- 0. M = 0. 
 
 which reduces the equations (8) to 
 
x^ z= 
 
 0. 
 
 N-x?/!:: 
 
 0. 
 
 1 N 
 3/ = + Y' 
 
 
 60 EQUILIBRIUM AMONG FORCES APPLIED, ETC. 
 
 giving 
 
 a constant quantity. 
 
 To recapitulate the principal truths established in this 
 article : 
 
 When each of the six quantities, x, y, z, l, m, n, is 
 cypher, the system of forces is in equilibrio. 
 
 When each of the three quantities x, y, z, is cypher, 
 some one or more of the quantities l, m, n, being different 
 from cypher, the system is reducible to a pair without a sin- 
 gle force. 
 
 When each of the quantities l, m, n, is cypher, some one 
 or more of the quantities x, y, z, being different from cypher, 
 the system of forces is reducible to a single force passing 
 through the origin. 
 
 Some one or more of the partial resulting forces, and also, 
 some one or more of the partial resulting moments being dif- 
 ferent from cypher, if the condition expi'essed by equation 
 (9) is satisfied, there is a single resulting force, which does 
 not pass through the origin. 
 
 Finally, retaining the former part of the last supposition 
 respecting the existence of a partial resulting force, and a 
 partial resulting moment, if the condition expressed by equa- 
 tion (9) is not satisfied, there is a resulting pair, and also a 
 resulting force, and the latter not being in the plane of the 
 former, the system cannot be reduced to a single force, or to 
 a single pair. 
 
OF EQUILIBRIUM AVHEN THE BODY IS IN PART RESTRAINED. 61 
 
 SECTION V. 
 
 OF THE CONDITIONS OF EQUILIBRIUM, WHEN THE BOUV IS IN PART 
 RESTRAINED BY FIXED OBSTACLES; AND OF THE PRESSURES ON 
 THE POINTS OF CONTACT. 
 
 1. When a body is found not to be at liberty to move, or 
 when its motion is not in the direction of the force im- 
 pressed, it is plain that some new force is introduced, which 
 in the one case, is in equilibrio with the force actually applied, 
 and which in the other case, compounded with that force; 
 gives a resultant in the direction of the motion produced. In 
 the former case, the force introduced is, plainly, equal and 
 opposite to that applied ; and in every case, it were easy to 
 find, experimentally, the magnitude and direction of the new 
 force: for if a third force is applied sufficient to maintain 
 the body in a state of rest, it is evident that the three forces 
 are in equilibrio ; and of these, two are given in magnitude 
 and direction ; whereby the force sought is fully ascertained. 
 But this experimental investigation is wholly unnecessary for 
 the present purpose. 
 
 If a force is applied to a fixed point, it is destroyed; or, 
 more properly speaking, there is another force brought into 
 action, with which it is in equilibrio. 
 
 When a force is applied to a material point, whose move- 
 ments are confined to a certain line ; if the direction of the 
 force is perpendicular to that line, there is no reason why it 
 should move the point along that line, in one direction, ra- 
 ther than in the opposite direction ; whence we may con- 
 clude that no motion will ensue. But if the force is oblique, 
 it may be resolved into two ; of which, one is perpendicular 
 
62 STATICS. — SECT. V. 
 
 and the other parallel to the line : the former shall be 
 wholly counteracted, but the latter shall produce its full 
 effect. 
 
 When a force is applied to a material point in contact 
 with a plane surface ; if the force is directed in the perpen- 
 dicular and towards the surface, it cannot move the point 
 from the plane towards which it is directed ; neither can it 
 move the point through the plane considered as impassable; 
 and acting perpendicularly, there is no reason why it should 
 move the point, on the surface of the plane, in any one di- 
 rection rather than another. Hence it may be inferred, that 
 the perpendicular force cannot produce any motion in the 
 point; and, therefore, that it is effectually counteracted. 
 
 But, if the direction of the force is oblique to the plane, 
 it may be resolved into two components, one of them perpen- 
 dicular to the plane, and the other in that plane : of these, 
 the former is wholly counteracted, but the latter, which is 
 not obstructed, shall produce its full effect in driving the 
 point along the plane. The same things hold when the 
 surface is a curve of any kind, inasmuch as the surface at 
 the point of application may be taken for that of its tangent 
 plane. 
 
 Hence we derive the following conclusions relative to the 
 force thus brought into action. 
 
 The force thus excited, being never exerted but in op- 
 position to some force actually applied, and never exceed- 
 ing the measure of this force, is merely a counteracting 
 force ; for which reason it is denominated a force of re- 
 sistance. 
 
 With respect to the direction of this force of resistance, 
 it appears that a fixed point is capable of a resistance in any 
 line directed from that point. That the resistance of a line 
 is directed perpendicularly from that line ; and, therefore, 
 that the resistance, at any point of the line, is confined to a 
 plane perpendicular to the line at that point: and that the 
 
OF EQUILIBRIUM WHEN THE BODY IS IN PART RESTRAINED. 63 
 
 resistance of a surface is directed from the surface in the 
 perpendicular at the point of contact. 
 
 In the two last cases, it is supposed, that the force is not 
 immediately applied to a point in the line or plane, for then 
 it would be applied to a fixed point, and would belong to the 
 first case ; but that it is applied to a material point, whose 
 movements are restricted by the line or plane. 
 
 The magnitude of the power of resistance in such ob- 
 stacles, i. e. the magnitude of the force of resistance which 
 they are capable of exerting, is unlimited ; this force being 
 always equal to that by which it is excited, «'. e. to the force 
 applied, resolved in a direction opposite to that of the re- 
 sistance. 
 
 The force which is equilibrated by the resistance is de- 
 nominated a pressure. 
 
 To seek the conditions of equilibrium, in the case of 
 a body restrained by fixed obstacles, is to seek the con- 
 ditions to be satisfied, in order that the forces actually ap- 
 plied, may be equilibrated by the resistances : and here it 
 is evident, that as the resisting forces may be of any mag- 
 nitudes, the inquiry is limited to the conditions to be sa- 
 tisfied, in order that the forces immediately applied, or 
 their resultants, should be opposed in direction to the re- 
 sistances. In other words, the inquiry is, what forces of the 
 system are equilibrated by the resistances? and then, the 
 conditions are those to be satisfied, in order to insure the 
 equilibrium of the remainder. And as it is of the nature of 
 all resistances to produce either a total or partial equili- 
 brium, it may be expected that the general conditions of 
 equilibrium, when not altogether satisfied by the resistances, 
 shall leave those to be otherwise provided for, fewer, and 
 in form more simple than for a body altogether free. 
 
 2. If the system to which the forces are applied contains 
 a fixed point, it is requisite and it is sufficient for equili- 
 
6i> STATICS. — SECT. V. 
 
 brium, that the forces should be reducible to a single result- 
 ing force passing through the fixed point. 
 
 But when the forces are referred to three axes meeting in 
 a solid angle, the condition required, in order that they 
 should have none but a single resultant passing through the 
 origin, is that expressed by the equation g rz 0. which im- 
 ports, that the forces should have no resulting moment rela- 
 tive to the origin. Accordingly, " where there is a fixed 
 point, it is requisite and it is sufficient for equilibrium, that 
 the forces should have no resulting moment relative to that 
 point." 
 
 The condition g = 0, was shown to be equivalent to these 
 three, 
 
 L = 0. M = 0. N =: 0. 
 
 Accordingly, when there is a fixed point, it is requisite for 
 equilibrium, that *' the forces applied should have no result- 
 ing moment round any one of three axes meeting at the 
 fixed point in a solid angle ; and these conditions being sa- 
 tisfied, the equilibrium is established." 
 
 When the forces of the system are parallel, one of the 
 axes, as for example, the axis of x, may be taken parallel 
 to the directions of the forces ; and then, of necessity, the 
 value of L is cypher, which reduces the equations of con- 
 dition to 
 
 M == 0. N iz 0. 
 
 importing, that the equilibrium is established, when there is 
 no resulting moment about either of the two axes, whose 
 plane is perpendicular to the directions of the forces. 
 
 When all the forces are directed in a plane, containing 
 the fixed point, that plane may be taken for the plane Xf/; 
 there is in that case no moment relative to the axis of x, or 
 the axis of y ; i. e. of necessity, we have l ~ 0. m ~ 0. 
 wherefore, 
 
 N zz 0. 
 
OF EQUILIBRIUM WHEN THE BODY IS IN PART RESTRAINED. 65 
 
 is the only condition to be provided for: i. e. when the 
 forces act in a plane containing a fixed point, there will be 
 equilibrium, when there is no moment about an axis perpen- 
 dicular to that plane. 
 
 The resistance made by the fixed point, being in equili- 
 brio with the forces of the system transferred to that pomt, 
 in parallel directions, it follows that the pressure is the re- 
 sultant of those forces so transferred. 
 
 4. If the forces are applied to a body which contains two 
 fixed points, the line by which those points are connected, 
 and therefore, every point in that line is fixed. Each point 
 of this line is capable of resisting in every direction : but it is 
 evident, that among these forces of resistance there can be 
 no pairs of parallel forces, except those whose planes con- 
 tain the fixed line. 
 
 To investigate the conditions of equilibrium with such 
 resistances, let one of the fixed points be made the origin, 
 and the connecting line the axis of ^; and let the three axes 
 be rectangular. Then, the forces being reduced in the usual 
 way, to three forces acting in the axes themselves, denoted 
 by X, Y, z, and three pairs in the three coordinate planes, 
 whose moments are denoted by l, m, n ; the forces x, y, z, 
 being applied to a fixed point, are effectually counteracted. 
 
 The planes of the moments m, n, contain the fixed line, 
 and therefore the forces of these moments may be applied to 
 that line; whereby those forces are also effectually resisted. 
 It is otherwise evident, that these moments must be equili- 
 brated by moments of resisting forces, inasmuch as by con- 
 taining the fixed line in their planes, the motion of rotation, 
 which each is fitted to produce in its own plane, is effectually 
 resisted. There remains then only the moment l, and this 
 cannot be equilibrated but by a moment whose plane is per- 
 pendicular to the axis of x; i. e. to the fixed line. But 
 there can be no pair of resisting forces, except in a plane 
 which contains this line; wherefore, this n^.oment cannot be 
 
 F 
 
66 STATICS. — SECT. V. 
 
 equilibrated by the resistances. Accordingly, the condition 
 expressed by the equation l rr 0. remains to be satisfied in- 
 dependently of the forces of resistance ; all the other condi- 
 tions being necessarily satisfied by those forces. Where- 
 fore, *' for equilibrium in the case of two fixed points, it is 
 requisite and sufficient, that there should be no moment of 
 rotation round the line by which those points are con- 
 nected." 
 
 If the body is at liberty to slide along the axis, this axis 
 will oppose no resistances, except to the forces to whose di- 
 rections it is perpendicular ; but such are all the forces, 
 equilibrated by the resistances in the former case, except 
 the force x. Therefore, to the condition l zz 0, required in 
 the former case, there is now added the condition x =: 0, i. e. 
 *' for equihbrium when the body is at hberty to slide along 
 a fixed axis, it is requisite that there should be no resulting 
 force in the direction of that axis, and no moment of rotation 
 about it." 
 
 5. Hitherto the line, containing two fixed points, was 
 considered generally as a fixed line ; and this was fully suf- 
 ficient when the conditions of equilibrium were sought. 
 But if the line becomes fixed by the securities afforded to 
 two points of that line, it is evident that its powers of resist- 
 ance are supplied from those securities, and that by them, 
 the pressures on the various points of the line must be sus- 
 tained ; and it is desirable to ascertain the pressure sustained 
 by each ; or as the problem is usually stated, to determine 
 the pressures on each of two fixed points. 
 
 If the forces applied consist of a pair, whose plane con- 
 tains the two fixed points, the pair may be turned round in 
 its own plane, so that its forces shall be perpendicular to the 
 line which connects those points. It may then be trans- 
 formed into another, of equal moment, having the distance 
 between the fixed points for that between the directions of 
 its forces, and these forces may be then applied at those 
 
OF EQUILIBRIUM WHEN THE BODY IS IN PART RESTRAINED. 07 
 
 points. From which it appears, that the pressure on each 
 of the points is the moment itself, divided by the distance 
 between the fixed points, /. e. putting g for the moment, a 
 for the distance betw^een the fixed points, andjp for the pres- 
 sure on either, we have 
 
 G 
 
 The pressures at the two points are both in the plane of the 
 moment, and both perpendicular to the line which connects 
 the fixed points ; but in contrary directions with respect to 
 that line. 
 
 If the line connecting the fixed points is parallel to the 
 plane of the pair, the measure of the pressures is the same, 
 inasmuch as the pair may be transferred to the parallel plane 
 in which that line is contained. 
 
 But if the line connecting the fixed points is inclined to 
 the pair in the angle a, the pair being resolved in two planes, 
 of which, one contains the line, and the other is perpen- 
 dicular to the same line; the moments of these two compo- 
 nents are 
 
 G. cos.o. G. sin.ct. 
 
 of these, the latter not being equilibrated, either in the whole 
 or in part, by the resistances, makes no charge on the fixed 
 points, and the pressure made on each of them by the for- 
 mer, is 
 
 G. COS. a 
 
 To find the pressures resulting from any system of forces 
 on two fixed points, is to find the pressures made by the 
 forces X, y, z, and by the moments l, m, n. 
 
 Treating the forces as before, it is evident that the pres- 
 sure sustained at the origin, in the direction of each axis, is 
 due to the partial resultant directed along that axis, and to 
 
 r 2 
 
68 STATICS. — SECT. V. 
 
 the moments m and n ; it being already shown that the mo- 
 ment L could not charge any point of the line to which its 
 plane is perpendicular. The pressures made by these mo- 
 ments, at the origin, are, 
 
 M 
 
 in the direction of the axis of ar, 
 
 and A , 
 
 a 
 
 in the direction of the axis of y ; and at the other fixed point 
 the same, but in the contrary directions. Wherefore, the 
 pressures at the origin are expressed as follows : 
 
 In the direction of the axis of z, 
 
 M , x^— zar z(a — x)-\-xz 
 
 z = z -I — — ^ —^ . 
 
 a a a 
 
 and in the direction of the axis of y, 
 
 , N YX—Ky Y ia •\- x) — -Kit 
 Y^ — Y-\ -=. — ■ -. 
 
 a a a 
 
 The pressures at the other fixed point are these : 
 Parallel to the axis of z, 
 
 M _ ZX — XZ 
 
 a "~ a ' 
 
 parallel to the axis of y, 
 
 N __ xy—Yx 
 a ~~ a ' 
 
 We should arrive at the same conclusion by considering 
 that the force z is transferred from the origin, by the mo- 
 
 M 
 
 ment m, to a distance — . And the force y, by the moment 
 
 N 
 
 N, to the distance . For the pressures made by the 
 
 forces z and y, transferred to these distances, would be 
 
OF EQUILIBRIUM WHEN THE BODY IS IN PART RESTRAINED. 69 
 
 
 Ya 4-N 
 
 at the origin. 
 
 And H . . 
 
 a a 
 
 at the other fixed point. 
 
 Putting p and q for the pressures at the origin, in the di- 
 rections of the axes of « and y ; and ^/ and q' for those at 
 the other fixed point, in the parallel directions ; and putting 
 p, p' for the resultants of these pressures at each point, we 
 have 
 
 FzzVip'-^q') 
 for the pressure at the origin ; 
 
 for that at the other fixed point. 
 
 Moreover, putting y, for the angle made by p, with the 
 axis of % ; and j, for that made by p', with a line parallel to 
 the same axis, we have 
 
 P ' P 
 
 y/{p+q) y/{p ^q) 
 
 Thus, these pressures are completely determined in magni- 
 tude and direction. 
 
 But this does not relate to the pressure made by the 
 force x; neither is it possible to ascertain the manner in 
 which this force is divided between the fixed points ; for as 
 every force may be applied indifferently at any point in its 
 line of direction, it is evident that the force x may be applied 
 at either of the fixed points, or divided between them in any 
 ratio whatsoever. All that we can know concerning the 
 pressures produced by it is, that the sum of those pressures 
 is equal to the force by which they are made, and the same 
 thing is true of the pressures parallel to either of the other 
 
'^ STATICS. — SECT. V. 
 
 axes, i. e. their sum is equal to the partial resultant of all the 
 forces in the parallel axis, as will immediately appear by re- 
 ferring to the values of those pressures given above. 
 
 6. When there are three fixed points in the system, and 
 not in the same right line, the whole system becomes im- 
 moveable, and therefore, every point in it is to be regarded 
 as a fixed point. It is evident, therefore, that the equili- 
 brium is, in this case, necessarily estabhshed by the resis- 
 tances ; the forces, whatever their directions, being applied 
 at fixed points. 
 
 But as the system may be rendered immoveable, by se- 
 curing the positions of three points not in the same right 
 line, it is evident that the pressures, produced by the forces 
 of the system, must be ultimately sustained by those secu- 
 rities, and therefore it may be expected that it should be 
 shown, how those pressures are distributed upon three such 
 points, and how their values and directions may be ascer- 
 tained. 
 
 7. To find the pressures made by any system of forces 
 on three fixed points not in the same right line : let one of 
 the fixed points, a, be taken, as before, for the origin ; 
 (Fig. 33.) and the lines ab, ac for the axes of x and t/ ; also, 
 the axis of ^ perpendicular to the plane xi/. Then, reducing 
 the whole system of forces to three single forces, x, y, z, 
 acting in the three axes, and three moments, l, m, n, in the 
 three coordinate planes, the forces x, y, z, and the moments 
 M, N, may be distributed between the points a and b, as be- 
 fore, when these were the only fixed points ; observing, that 
 the moment n, when designated by the forces and the ordi- 
 nates of their points of application, should be multiplied into 
 the sine of the angle bac : thus, 
 
 N :3 {■S.I/ — YX) sin. ang. bac. 
 
 Putting e for the line ac, the moment l produces at a the 
 
 pressure -| , in the direction of the axis of ^ ; and the equal 
 
OF EQUILIBRIUM WHEN THE BODY IS IN PART RESTRAINED. 7 1 
 
 and contrary pressure at the point c. In this way, we may 
 arrive at expressions for the pressures at the several points ; 
 but this, in general, would be useless, as it is evident that the 
 distribution of the pressures is arbitrary : thus, the force x 
 may be indifferently applied at a or b, or divided between 
 them in any ratio ; and the same is manifest respecting the 
 force Y, and the points a and c. Also, the moment n may 
 be indifferently applied on any one of the lines ah, ac, or he. 
 It is only when the forces x, y, and the moment n are 
 cypher, i. e. it is only when there is no resulting force, ex- 
 cept in the perpendicular to the plane of the triangle, and no 
 resulting moment, whose plane is not perpendicular to the 
 same plane, that the question, relative to their bearings on 
 the three angular points, admits of a determinate answer. 
 For a force is equilibrated by the resistance of a single point 
 in its direction, and therefore, the addition of a second fixed 
 point in that direction, renders the problem indeterminate. 
 Also, a moment is equilibrated by the resistances of two 
 fixed points in its plane, and therefore the problem is ren- 
 dered indeterminate by a third fixed point in that plane. 
 
 The problem being determinate only when the direction 
 of the resulting force, or the plane of the resulting pair is 
 perpendicular to the plane of the triangle made by joining 
 the three fixed points, it may be proper to show how to find 
 the pressures made on those points, by such forces or pairs. 
 
 For the pressures made by a force z, whose direction is 
 perpendicular to the plane of the triangle ahc : let o be the 
 point where the plane is met by the direction of this force ; 
 and from one of the three angles, as a, let the line ao be 
 drawn, meeting the opposite side, as at d; the force z may be 
 distributed on the points a and d: the portion sustained by 
 the point a, being 
 
 od 
 ad 
 
72 STATICS. SECT. V. 
 
 and that by the point d, being 
 
 ao 
 zx—j. 
 act 
 
 This last may be distributed on the points b and c ; that on 
 the point b, being 
 
 ao dc 
 ad be' 
 
 and that on the point c, being 
 
 ao db 
 ad be ' 
 
 If the sum of the pressures, or z, is represented by the 
 area of the triangle abc, the portion sustained by the point a 
 shall be represented by the area of the triangle boc: and the 
 sum of the pressures on b and c, by the four-sided figure 
 aboc. Moreover, this sum is distributed on those points, in 
 the ratio of dc to db ; i. e. in the ratio of the area of the tri- 
 angle aoc to the area of the triangle aob. Whence it follows, 
 that if from the point of the plane where it is met by the di- 
 rection of the force z, three lines are drawn to the angles of 
 the triangle, the force z being represented by the area of 
 this triangle, those at the three points shall be represented 
 by the areas of the triangles, into which it is so divided, that 
 at each angle, by the area of the triangle placed on the op- 
 posite side. 
 
 The same theorem will be true, though the point a 
 should fall without the area of the triangle abc ; the pres- 
 sure on the point d being then in a direction corresponding 
 to that of the force z, and the pressure on a in a direction 
 parallel but contrary. 
 
 The pressures made by a pair on three fixed points are 
 ascertained as follows : the plane of the pair being perpen- 
 dicular to the plane of the triangle, made by joining the fixed 
 points. 
 
OF EQUILIBRIUM WHEN THE BODY IS IN PART RESTRAINED. 73 
 
 Let the pair be resolved into two pairs, whose planes 
 shall contain two sides of the triangle. The pressures made 
 by each component pair are had by dividing its moment by 
 the line on which it is applied. 
 
 Or thus : from one of the angles, as a, let the line ad be 
 drawn to the opposite side, parallel to the plane of the pair. 
 (Fig. 33.) The pressures may be supposed, in the first in- 
 stance, to be made on a and c?; and putting g for the mo- 
 ment of the pair, as before, the pressure at a, and also that 
 at d, is 
 
 G 
 
 ad' 
 
 The pressure at d is supported at i and c, and the parts sup- 
 
 dc 
 ported at those points are expressed by the fractions -j-» 
 
 and 7-. Wherefore, the pressure at b will be 
 
 G dc 
 
 and that at c, 
 
 ad be' 
 
 G db 
 ad be' 
 
 If the line ad is bisected at o, and if from this point lines 
 are drawn to the angles of the triangle, the pressures at the 
 angles shall be proportional to the three triangular areas ; 
 the pressure at each angle being as the triangle placed on the 
 opposite side. 
 
 For if the sum of the pressures, taken without regard to 
 
 their signs, is — ^, and if this is represented in magnitude by 
 
 the entire triangle ahc, its half, which is supported at a, may 
 be represented by the area of the triangle boo, and then the 
 four-sided figure baco, shall represent the sum of the pres- 
 sures on the points b and c, and this is divided between 
 
74 STATICS. — SECT. V. 
 
 them in the ratio of dc to db ; i. e. of the triangle coa to the 
 triangle boa. 
 
 If the point d lies between the points b and c, it is evi- 
 dent that the pressures at a and d shall be of contrary di- 
 rections ; and that, in this case, the pressures at b and c cor- 
 respond in direction with that at d. But if d falls without 
 the triangle, the point b, or the point c, beyond which it falls, 
 shall sustain the pressure equal and contrary to the sum of 
 those sustained by the two remaining points. 
 
 If the directions of the resultant forces, or the planes of 
 the resulting pairs, are not perpendicular to the plane of the 
 triangle, they may be resolved in the plane, and in the per- 
 pendicular to the plane ; the pressures made by the forces 
 and pairs acting in the plane are indeterminate, but those 
 made in the perpendicular directions are fully determined. 
 
 If the body touches a plane in a single point, any force of 
 resistance which may be excited at that point, is perpen- 
 dicular to the plane. Therefore, when the forces of the 
 system are not in equilibrio, they may equilibrate with the 
 resistance, provided they are reducible to a single resultant, 
 directed perpendicularly towards the plane at the point of 
 contact. To see how these conditions may be expressed by 
 means of the quantities which enter into the general formulse, 
 let the point of contact be taken for the origin, and the 
 plane for that of xy, and the axes rectangular. Then, in 
 order that there should be a single resultant passing through 
 the origin, it is requisite that each of the moments l, m, n, 
 should be cypher. And in order that this same force should 
 be perpendicular to the plane, it is requisite that each of the 
 partial resultants x, y, should be cypher. Wherefore, for 
 equilibrium, it is requisite that each of the five following 
 equations should be satisfied, 
 
 X = 0. Y = 0. L = 0. M = 0. N = 0. 
 
 and these being satisfied by the forces of the system, there 
 
OF EQUILIBRIUM WHEN THE BODY IS IN PART RESTRAINED. 75 
 
 will be a single resultant, which passes through the origin in 
 a line perpendicular to the plane. 
 
 But besides the conditions expressed by the five equa- 
 tions given above, it is further required, that the resulting 
 force should be directed towards the plane. For if it is di- 
 rected from the plane, no resisting force is excited ; and 
 then, the six conditions are to be satisfied, as for a body per- 
 fectly free. 
 
 The pressure on the point of support is the force z ; 
 which is the sum of the forces resolved in directions perpen- 
 dicular to the plane, and transferred to the origin in parallel 
 directions. When the equilibrium is established with the 
 resistances, this is the resultant of the forces of the system. 
 
 If the body touches the plane in two points, the only 
 forces of resistance which can be excited, are two such 
 forces, perpendicular to the plane at the points of contact : 
 and as these are parallel forces, of corresponding directions 
 with respect to the line which joins the points of applica- 
 tion, they have a parallel resultant, applied at some inter- 
 mediate point. In order, therefore, that the equilibrium 
 may be established by means of such resistances, it is requi- 
 site that the forces of the system should be reducible to a 
 single resultant, directed perpendicularly towards the plane, 
 and to some point intermediate between the points of con- 
 tact. And when these conditions are satisfied, the forces 
 of the system will be necessarily equilibrated by the resis- 
 tances. 
 
 To see how these conditions are expressed by means of 
 the quantities which enter into the general formulas, let one 
 of the points be taken for the origin, the connecting line for 
 the axis of x, and the plane for that of xy. Then, in order 
 that there may be a single resultant perpendicular to the 
 plane o^ xy, it is requisite, in the first place, that each of the 
 quantities x and y should be cypher; for if either of these 
 
76 STATICS. — SECT. V. 
 
 is different from cypher, r is oblique to the plane of xy : 
 then, if r can be compounded with g, the general resulting 
 moment, into a single resultant, this, which is the general 
 resultant of the system, is parallel to R, and, therefore, 
 oblique to the plane of xy : and if r cannot be so com- 
 pounded with G, the forces of the system are not reducible 
 to a single resultant. Whence it follows, that unless each 
 of the partial resultants, x and y, is cypher, the forces of the 
 system are not reducible to a single resultant perpendicular 
 to the plane of xy. Accordingly, the equations 
 X = 0. Y = 0. 
 
 are to be satisfied, z is the only partial resultant force that 
 now remains. This, being in the planes of the moments l 
 and M, may be compounded with both of these moments into 
 a single force, equal and parallel to z, and, therefore, per- 
 pendicular to the plane xy. But a force parallel to z, and 
 therefore perpendicular to the plane of n, cannot be com- 
 pounded with this moment into a single resulting force; 
 whence it follows, that the equation n =: 0. is also to be sa- 
 tisfied. 
 
 Further ; m may be compounded with z, and the resul- 
 tant is a single force equal and parallel to z, in the plane of 
 zx ; and this resultant of z and m may be compounded with 
 the moment l into a single force, which will be also equal 
 and parallel to z : but this resultant of z, m, and l, is not in 
 the plane of %x ; whence it follows, that the equation l = 0. 
 is also to be satisfied. Wherefore, the equations to be sa- 
 tisfied, are 
 
 X = 0. Y = 0. L =3 0. N r: 0. 
 
 The moment of m may remain, but under certain restric- 
 tions. For as it is requisite that the single resultant of the 
 system should be applied to the line connecting the points of 
 support, at a point not beyond those limits; it follows, that 
 the moment m should not exceed the product of z into the 
 
OF EQUILIBRIUM WHEN THE BODY IS IN PART RESTRAINED. 77 
 
 distance between the points of support, i. e. putting a for 
 that distance, m should not exceed za. For m compounds 
 with z a force equal and parallel, in the plane of zx ; and ap- 
 plied at a point of the axis of x, whose distance from the 
 
 ... M 
 
 origin IS am — . Wherefore, if m exceeds za, x shall be 
 z 
 
 greater than a, i. e. the final resultant of the system shall be 
 
 applied to the line connecting the points of support, at a 
 
 point in that line produced beyond them. 
 
 It is to be further observed of this moment, that if it is 
 applied on the axis of a;, the force which corresponds in di- 
 rection with z, must be that which has the greater positive 
 or the lesser negative abscissa. For, were it the contrary, 
 the moment m, however small, when compounded with z, 
 would remove its point of application, out of the origin, on 
 the side of the negative abscissae, i. e. beyond the limits pre- 
 scribed for the point of application of the final resultant, in 
 the case of equilibrium. 
 
 The magnitude of this final resultant is z, the sum of the 
 forces of the system resolved in directions parallel to the 
 axis of %, i. e. perpendicular to the plane of support ; and 
 the abscissa of its point of application on the axis of x, is 
 
 given by the equation x — — . 
 
 The pressures on the points of support are now imme- 
 diately found. For putting p for the pressure at the origin, 
 and q for that at the other point of support, we have 
 
 . . z.(a~x) 
 p : z : a—x : a. giving^ :i= — ^ -. 
 
 q : z '. X '. a. giving a zz — — . 
 
 a 
 
 i. e. putting for x its value, 
 
 za — M M 
 
78 STATICS. — SECT. V. 
 
 If the body touches the plane in more than two points ar- 
 ranged in the same right hne, all but the extreme points are 
 useless for the establishment of equilibrium. For, if the 
 forces of the system have a single resultant applied between 
 any two of the points, a fortiori, it will be applied between 
 the extreme points. Wherefore, the question relative to the 
 conditions of equilibrium is to be treated as if the body 
 touched only at the two extreme points. But in the esti- 
 mate of the pressures on the points of support, a great 
 change is made, as it becomes impossible to ascertain any 
 thing more than the sum of the pressures, when there are 
 more than two points of support, and these in the same right 
 line regarded as inflexible. For certain pressures may be 
 removed from two points to an intermediate point ; the pres- 
 sures so removed, being reciprocally as the distances to 
 which they are removed. And conversely, any pressure 
 sustained by a point may be distributed upon two other 
 points, between which it is placed, in portions, which are re- 
 ciprocally as the distances of the point so exonerated, from 
 those to which the pressure is transferred. 
 
 If the body touches the plane in three points not ar- 
 ranged in the same right line, there may be so many forces 
 of resistance excited : and the direction of each of these 
 forces of resistance is perpendicular to the plane. These 
 constitute a system of parallel forces of corresponding di- 
 rections : therefore they are reducible to a single force equal 
 to their sum. The point of application of this resultant is in 
 the area of the triangle made by joining the points of sup- 
 port. Hence it follows, that in order to equilibi*ate with the 
 resistances, the forces of the system must be reducible to a 
 single resultant perpendicular to the plane, and directed to- 
 wards some point, not exterior to the triangle, made by 
 joining the points of support. And when these conditions 
 are satisfied, the equilibrium shall be necessarily established 
 by the resistances: inasmuch as such a force can always 
 
OF EQUILIBRIUM WHEN THE BODY IS IN PART RESTRAINED, 79 
 
 he distributed on the points of support, and in parallel di- 
 rections. 
 
 To see how these conditions may be expressed, by means 
 of the quantities which enter into the general formulae, let 
 one of the points of contact be taken for the origin ; the line 
 connecting it with a second point for the axis of a:; the plane 
 for that of xy\ and the axes rectangular. Then, in order 
 that the forces of the system may have a single resultant 
 perpendicular to the plane, it may be proved, as in the last 
 case, that x, y, n, should be, severally, equal to cypher. 
 There will then remain the partial resultant z, and the two 
 moments l, m. It is not required that either of these mo- 
 ments should be cypher, as both may be compounded with 
 z into a force equal and parallel to z ; the effect of the mo- 
 ment M being to remove its point of application from the 
 origin, to a point of the axis of x, whose distance from the 
 origin is 
 
 M 
 
 xzz. . 
 
 z 
 
 and that of the moment l, to remove it to a point in the axis 
 of y, whose distance from the origin is 
 
 L 
 
 and the effect of both, conjointly, is to remove the point of 
 application of z, to the extremity of the diagonal of a rect- 
 angle, whose sides, measured from the origin on the axes of 
 X and y, are 
 
 M L 
 
 z * z * 
 
 This point must not be exterior to the area of the tri- 
 angle made by joining the three points of support: and on 
 this principle, the limitations by which the magnitudes of l 
 and M are restricted, are easily found. 
 
80 STATICS. — SECT. V. 
 
 The pressures on the three pohits are known, when the 
 point of application of the general resultant is known ; and 
 this is given by its coordinates, viz. 
 
 _ M _ L 
 
 ~ z * ^ ~~ z' 
 
 Let it be the point o within the area of the triangle ahc, 
 (Fig. SS) formed by joining th§ points of support. The 
 force applied at o, is the force z, and this is to be dis- 
 tributed on the points a, b, c. Therefore, drawing the 
 line ao, meeting the opposite side at d, the pressure z may 
 be distributed on the points a and d; the parts being, re- 
 spectively, 
 
 od ao 
 
 ad' 'ad' 
 
 The latter of these is distributed on the points b and c, in the 
 ratio of dc to db. Wherefore, if from o, lines are drawn to 
 the angles of the triangle, resolving the whole area into three 
 triangular areas ; the pressures on the three points shall be 
 proportional to the areas of these triangles ; that on each 
 point being represented by the area of the triangle placed 
 on the opposite side. For if the total pressure z, is repre- 
 sented by the area of the entire triangle abc, that on the 
 point a, shall be represented by the area boc ; this being to 
 the whole area, in the ratio o£ od to ad: and therefore, the 
 pressure on d, shall be represented by the sum of the two 
 remaining triangles boa, coa. But the pressure on d is di- 
 vided at b and c, in the ratio of dc to db ; i. e. of the triangle 
 coa, to the triangle boa. 
 
 When the body touches the plane in more than three 
 points, the resistances, when excited, must compound a 
 force perpendicular to the plane, at some point within the 
 polygon formed by joining the points of contact; omitting 
 those points at which the angles are re-entering. Where- 
 fore the forces of the system, to be capable of being equili- 
 
OF EQUILIBRIUM WHEN THE BODV IS IN PART RESTRAINED. 81 
 
 brated with such resistances, must have a «ingle resul- 
 tant directed perpendicularly to the plane, at some point 
 within the area of that polygon ; and when this condition 
 is satisfied, the equilibrium is necessarily established by 
 the resisting forces. For the resultant of the system may 
 be resolved into parallel forces bearing on three or more 
 of these points, where they will be necessarily equili- 
 brated. 
 
 In all cases, the sum of the pressures is the sum of the 
 forces of the system resolved in directions perpendicular to 
 the plane ; but if the points of contact are moi'e than three, 
 the distribution of the pressures is indeterminate. This is 
 no more than must have been expected : for the point of ap- 
 plication of the general resultant being within the area of 
 the polygon, it is within the area of some one of the tri- 
 angles into which the polygon is resolvable. Accordingly, 
 the pi'essure may be totally distributed on the angles of 
 this triangle. The pressures thus supported, may then be 
 variously transferred to lines drawn between the other an- 
 gles of the polygon, and intersecting the sides of that tri- 
 angle; and the pressures laid on each of these lines, divided 
 on their extreme points. 
 
 In general ; whatever be the nature of the surface or sur- 
 faces in contact with the body, and whatever the number of 
 the points of contact, the equilibrium will be established 
 by means of the resistances, whenever the forces of the 
 system are reducible to another system of forces, bearing 
 perpendicularly against any number of those points, and not 
 otherwise : and the pressure at each point will be the force 
 directed against it, unless so far as this pressure may be dis- 
 tributed, in the whole or in part, on any of the other points 
 of support. 
 
 This account of the pressures relates to a system in equi- 
 librio. To estimate the strains on the several points of a 
 
 G 
 
S2 STATICS. — SECT. V. 
 
 system in motion, the forces required to produce those mo- 
 tions are also to be considered. The actions of the several 
 points are then the resultants of those forces and of the re- 
 sistances here treated of; and the strains are equal and op- 
 posite to these resultants. 
 
OF THE CENTRE OF GRAVITY. S3 
 
 SECTION VI. 
 
 OF THE CENTRE OF GRAVITY. 
 
 1. The theorems already established, relative to forces 
 applied to a point or to a system of points, are altogether 
 independent of the particular sources from ^vhence those 
 forces are derived. Those theorems are, therefore, equally 
 true, whether the forces are those supplied by animal 
 strength, or those by which the particles of matter naturally 
 influence each other. Of this latter kind is the force of gra- 
 vity, and as this constantly offers itself to our notice, as one 
 of the conditions from which the solution of mechanical ques- 
 tions is to be derived, it becomes requisite, in an especial 
 manner, to consider the laws of its action on terrestrial bo- 
 dies, and the manner in which it is to be treated. 
 
 It is well known of all bodies near the earth's surface, 
 that they tend to descend, each in a direction perpendicular 
 to the horizon of the place ; and that when this tendency is 
 counteracted, a pressure is sustained, which is denominated 
 the weight of the body. The descent of bodies when free 
 to move, and the pressure when the motion is restrained, 
 are, both of them, manifest indications of the action of a 
 force; and this force is denominated the force of gravity. 
 
 The force of gravity accelerates all bodies equally ; i. e. 
 it generates in them equal velocities in equal times ; for it is 
 found that in vacuo, the lightest feather and the most pon- 
 derous substance descend together as if they were parts of 
 the same mass. And the same thing is yet more satisfac- 
 torily established by observations made with the pendulum, 
 as shall be shown hereafter. This being admitted, it follows 
 
 that the force of gravity acts equally on all the units of mass, 
 
 g2 
 
8-i 
 
 STATICS. — SECT. VI. 
 
 and in parallel directions. Thence arises a system of equal 
 and parallel forces applied to the several units of mass of 
 which the body consists ; and the resultant of these is equal 
 to their sum. Wherefore, denoting the force of gravity act- 
 ing on the unit of mass, i. e. its weight by g, and the mass of 
 the body, i. e. the number of units in that mass, by m, and 
 putting w for the weight of the body, there is 
 
 W = M^. (1) 
 
 Hence, g being the same for all bodies, " The weights of 
 bodies are proportional to their quantities of matter." 
 
 From the foregoing equation it appears that the " force 
 of gravity" and " the weight of a body" are not expressions 
 of the same import. By the former is meant the intensity 
 of the power as it exists in nature, whose measure is the 
 force with which it acts on the unit of mass; by the latter is 
 meant the force of gravity as applied to the particular body 
 under consideration ; and this depends not only on the in- 
 tensity of the force as it exists in nature, but also on the 
 number of such units in the mass of the body on which it is 
 exerted. 
 
 The quantities of matter in bodies being proportional 
 to their weights, and the weights of bodies of the same bulk 
 being exceedingly various, it is evident that there must exist 
 the widest differences, with respect to the condensation of 
 the matter of which they consist. The relation between the 
 quantity of matter in a body and its bulk or volume, is that 
 which is meant by its density ; and, therefore, the densities 
 of bodies are as the quantities of matter contained in a given 
 volume. But in comparing the densities of different sub- 
 stances it is found convenient to refer them all to the same 
 scale, agreeably to what has been done with respect to the 
 quantities which enter into equation (1). This is at once ef- 
 fected by fixing on some one substance, whose density is to 
 be regarded as the standard of comparison, and, therefore, 
 as the unit of the scale of densities. Various considerations 
 
or THE CENTRE OF GRAVITY, 
 
 85 
 
 concur, to render water the most eligible for this purpose. 
 But the density of water itself is known to vary, according to 
 the quantity of foreign matter which it may hold in solution ; 
 and also, according to its different degrees of temperature. 
 The uncertainty which would follow from the former source 
 is avoided, by taking distilled water; and that from the lat- 
 ter, by taking the water at its maximum of condensation, 
 which corresponds to a temperature of 40° of Fahrenheit, or 
 somewhat more than 4° of the centigrade thermometer. 
 Therefore, putting unity for the density of distilled water of 
 this temperature ; the densities of bodies, which in the same 
 volume, contain twice or thrice the quantity of matter, are 
 denoted by the numbers 2, 3, and so forth ; the quantities of 
 matter being compared, by means of the weights to which 
 they are proportional. 
 
 When, therefore, the quantities of matter are estimated 
 by the volumes, it is on the supposition that the densities are 
 equal; but if these are. unequal, the numbers which express 
 the ratio of the volumes, must be multiplied by those which 
 denote the densities : and the quantities of matter in different 
 bodies are proportional to these products. This is expressed 
 by the equation 
 
 M = VD (2); 
 
 in which m, v, and d are numbers denoting the ratios of the 
 quantities for which they stand to their several units of mea- 
 sure. Thus, if a cubic inch is taken for the unit of volume, 
 and the density of water for the unit of density, then the 
 quantity of matter in a cubic inch of water shall be the unit 
 of mass : and for any body whatsoever, in the last equation, 
 V shall denote the number of cubic inches in its bulk or vo- 
 lume, D its density with respect to that of water, and m its 
 mass, in relation to that of a cubic inch of this fluid. Sub- 
 stituting in equation (1) this value of m, there is 
 
 '>y-y'J)'g' (3) 
 wherein g denotes the weight of a cubic inch of water. 
 
86 STATICS. — SECT. VI. 
 
 This weight is converted into grains by putting for g in the 
 last product 2o2.db2, the number of grains contained in that 
 volume of water. 
 
 2. As in all positions of the body with respect to a horizon- 
 tal plane, the same forces of gravity are applied to the same 
 points, and in parallel directions, these forces must have a 
 centre, i. e. in a body, considered as an invariable system of 
 points, there is a certain point through which, in all positions 
 of the body, the resultant of those forces passes ; and to 
 which point it may therefore be supposed to be immediately 
 applied. This point was generally named the centre of pa- 
 rallel forces ; but when the forces are the gravitations of the 
 constituent parts, it is especially designated by the name of 
 the centre of gravity. 
 
 This reduction of the weights of the several molecules of 
 a body to a single resultant applied to an invariable point, is 
 of great use in the solution of all questions in mechanics, 
 wherein the weights of bodies are concerned: for instead of 
 an indefinite number of forces and their points of application, 
 we have only to consider a single force, which is the weight 
 of the body, and its point of application, which is the centre 
 of gravity ; and having added this force to the others given 
 by the conditions of the question, we can proceed to apply 
 the statical theorems already established relative to forces 
 acting on a system of material points destitute of gravity. 
 
 In seeking the centre of gravity of a body, it is to be ob- 
 served, that if the density is the same throughout the mass, 
 the eqvial forces ai'e aj^plied at equidistant points, in which 
 case the position of the centre of gravity will depend alto- 
 ther on the figure : and this supposition of homogeneity is 
 always made in the investigation of the centres of gravity of 
 geometrical figures. But if the density is not uniform, the 
 body being supposed to be divided into particles of a given 
 mass, the number of these contained within a given space 
 will be various : i. e. the points to which equal forces are ap- 
 plied will be more condensed in one part of the body than in 
 
OF THE CENTRE OF GRAVITY. 87 
 
 another ; in which case, the position of the centre of gravity 
 shall depend, not on the figure only, but also on the way in 
 which the matter is distributed throughout the volume, i. e. 
 on the relative densities of the parts. 
 
 This being premised, we shall proceed to illustrate the 
 methods by which the centre of gravity of a body or system 
 of bodies is ascertained, and then to give an account of the 
 chief properties of this point. 
 
 3. The methods of finding the centre of gravity are the 
 same as those given for finding the centre of parallel forces, 
 the symbols p, p', p", &c. by which the parallel forces had 
 been denoted, now denoting the weights or the masses to 
 which they are proportional. 
 
 According to the first of these methods, described in 
 Sect. II. Art. 6. the distance between two elements of the 
 system is divided in the inverse ratio of the masses or 
 weights : the point, by which the connecting line is so di- 
 vided, is the centre of gravity of those two elements, whose 
 sum is then to be regarded as concentred in this point. The 
 line connecting this point with a third element is, in like man- 
 ner, to be divided in the inverse ratio of the weights con- 
 nected by it. The common centre of gravity of three ele- 
 ments being thus found, and the sum of their weights being 
 supposed concentred in it, the same process is to be conti- 
 nued until all the weights are united in one point, which will 
 be the common centre of gravity of the entire system. 
 
 If the masses are equal, the rule now delivered will be 
 reduced to the following. The line connecting two of those 
 masses is to be bisected. The line connecting this point of 
 bisection with a third of those masses is to be divided in the 
 ratio of 1 : 2. That connecting this point of division with a 
 fourth of the masses is to be divided in the ratio of 1 : 3. and 
 so-forth, i. e. one-half is to be taken from the line connecting 
 the two first points; one-third from the line connecting the 
 point thus found, with a third of the given points; one-fourth 
 from the line connecting the point thus found, with a fourth 
 
OO STATICS. — SECT. VI. 
 
 of the given points, and so-forth. The points thus succes- 
 sively found are the centres of gravity of the parts so com- 
 bined; and the last, the common centre of gravity of the 
 whole system. 
 
 If the body, whose centre of gravity is sought, is of con- 
 siderable magnitude, the subdivision into parts, such as 
 might be regarded as physical points, and the treatment of 
 these, according to the method now described, might be ex- 
 ceedingly tedious. But if the centre of gravity of any por- 
 tion of the body is already known, the labour of computation 
 is diminished^ by supposing the weight of that portion con- 
 centred in its proper centre of gravity. 
 
 Thus, if an indefinite number of material points of equal 
 weight were uniformly arranged along the sides of a polygon, 
 the centre of gravity of those in any one line is the middle 
 point of that line: and by supposing the weights, in each 
 line, united at the middle point of that line, the problem is 
 reduced to that of finding the common centre of gravity of a 
 system of weights, equal in number to the sides of the poly- 
 gon ; the weights themselves being proportional to those 
 sides, and applied at their middle points. 
 
 If a physical surface is symmetrically divided by a certain 
 right line, the centre of gravity is in this line : and if it is 
 symmetrically divided by another line also, the centre of gra- 
 vity is, at once, known to be at the intersection of the lines 
 by which it is so divided. Also, if a solid is symmetrically 
 divided by a certain plane, the centre of gravity is in that 
 plane ; and if it is divided in like manner by a second plane, 
 the centre of gravity is in both planes, and therefore, in the 
 line of intersection : and if it is so divided by a third plane, 
 the centre of gravity is at once determined to the single point 
 common to the three planes. Thus the area of the triangle 
 ABC, being divided into elementary trapezia, or physical lines 
 parallel to any one of its sides bc, as in (Fig. ol-.) ; the centre 
 of gravity of each physical line is at its middle point, where 
 accordingly its weight may be supposed to be applied : and 
 
OF THE CENTRE OF GRAVITY. 
 
 89 
 
 as all such points of bisection are situated in the line drawn 
 from the angle a to d, the middle point of the opposite side 
 BC, it is plain that the common centre of gravity of all these 
 elements, or the centre of gravity of the area of the triangle 
 is in the line ad. For the same reason it is in the hue be, 
 (Fig. 35.) drawn from another of the angles b, to e, the mid- 
 dle point of the opposite side ac : and therefore it is at g, 
 the point where the lines ad, be, intersect. It remains only 
 to determine the distance of g, from any one of the angles, 
 measured on the right line drawn from that angle to the 
 middle point of the opposite side. Now the sides ac, bc, 
 being bisected at e and d, the hne ed is parallel to the third 
 side ab: wherefore, the triangles ced, cab, as likewise the 
 triangles e^d, a^b, are similar. The latter pair give the fol- 
 lowing proportion : 
 
 Afl- : ffT) : : ab : ed. 
 
 to 
 
 and the former give 
 
 ab : ED : : BC : DC : : 2 : 1. 
 wherefore, 
 
 A^ zz 2^D. i. e. kg zz f AD. 
 
 The centre of gravity of the area of a triangle is then the 
 same as that of three equal bodies, applied at its three angu- 
 lar points ; since it is found by the same rule, viz. by bisect- 
 ing one of the sides, and dividing the line drawn from the 
 point of bisection to the third angle, in the ratio of 1 : 3. 
 
 Knowing how to find the centre of gravity of a triangle, 
 that of any right lined polygon is readily found; inasmuch 
 as any such polygon may be resolved into triangles, whose 
 weights, proportional to their areas, may be supposed con- 
 centred in their respective centres of gravity. The problem 
 is then reduced to that of finding the common centre of gra- 
 vity of so many weights applied at these points ; which may 
 be done by the method of successive composition already 
 described. 
 
90 STATICS. — SECT. VI. 
 
 A triangular pyramid being resolvable into physical sur- 
 faces parallel to any one of the triangular faces, it is evident 
 that the centres of gravity of these surfaces are arranged in 
 the same right line, drawn to the centre of gravity of the pa- 
 rallel face from the opposite angle. For the like reason, it 
 must be also in the right line drawn to the centre of gravity 
 of any of the other triangular faces, from the angle opposed 
 to it ; and, therefore, it must be at the intersection of two 
 such lines. 
 
 Thus, in (Fig. 36.) bisecting dc in e, and in the line be, 
 drawn from the opposite angle of the triangle, taking Bg 
 — I BE, the point g is the centre of gravity of the triangular 
 face BDC ; and that of the pyramid is in the right line A^, 
 drawn to this point from the opposite angle. In like manner, 
 taking a^' zz | ae, and drawing b^' ; the centre of gravity of 
 the pyramid is also in this line Bg'. Now as a^, b^"', are in 
 the same plane, viz. that of the triangle aeb, they must in- 
 tersect as at g" : wherefore, this point g'\ is the centre of 
 gravity of the pyramid. 
 
 All that now remains, is to ascertain the distance of the 
 point g", from either of the angles a or b. In order to this, 
 it is to be observed, that e^' : ea : : e^ : eb ; wherefore, the 
 lines g'g and ab are parallel : and therefore the triangles 
 g'gE, ABE, as also the triangles gg'g", Ag"b, are similar: ac- 
 cordingly, 
 
 Ag" : g"g : : ab : g'g : : ae : g'E : : 3 : 1. 
 
 which gives 
 
 AS =Z 0.£f i^ — ^AST. 
 
 Showing that the centre of gravity of a triangular pyramid is 
 in the right line, drawn from any one of the angles to the 
 centre of gravity of the opposite face ; at a distance from the 
 
 angle which is f of this line. 
 
 The centre of gravity of a triangular pyramid is then the 
 common centre of gravity of four equal bodies, applied at its 
 
OF THE CENTRE OF GRAVITY. 91 
 
 four solid angles ; those centres being found by the same 
 rule. In both cases, the line connecting two of the angles is 
 to be bisected : the line drawn from this point of bisection to 
 a third angle, is to be divided in the ratio of one to two : and 
 finally, the line connecting the last point of division with the 
 fourth angle, is to be divided in the ratio of one to three. 
 
 The rule for determining the centre of gravity of a tri- 
 angular pyramid, may be extended to pyramids standing on 
 polygonal bases. For any such polygon may be resolved 
 into triangles ; and, therefore, the pyramid into triangular 
 pyramids, having the same summit, and those triangles for 
 their bases. The centre of gravity of each of the triangular 
 pyramids, and, therefore, that of the entire pyramid, shall 
 be in a plane parallel to the base, at a distance from the sum- 
 mit equal to | of the altitude. It shall also be in a right line 
 drawn from the summit to the centre of gravity of the base. 
 Wherefore, it is at the point of this line where it is inter- 
 sected by that plane, i. e. at a distance from the angle equal 
 to I of its length. 
 
 In prisms, this resolution into pyramids is unnecessary ; 
 for the sections parallel to the bases being all similar, their 
 centres of gravity are in the same right line; and these sec- 
 tions being moreover equal, their common centre of gravity 
 is evidently the middle point of the line, which joins the cen- 
 tres of gravity of the parallel bases. 
 
 What has been proved relative to the centre of gravity of 
 a pyramid, standing on a polygonal base, is evidently appli- 
 cable to a cone ; which may be regarded as the limit of the 
 inscribed or circumscribed pyramids, and in which limit they 
 end, when the sides become infinite in number. Wherefore, 
 the centre of gravity of a cone is in the axis, at a distance 
 from the summit equal to | of its length. 
 
 In like manner, what has been shown of prisms may be 
 extended to cylinders standing on any curvilinear bases 
 whatever. 
 
92 
 
 STATICS. SECT. VI. 
 
 The centre of gravity of a sphere or spheroid is obviously 
 at the centre ; as each of these figures may be divided sym- 
 metrically by three planes, all passing through this point. 
 
 4. The method of finding the centre of gravity, by re- 
 ference to three coordinate planes, is that explained in 
 Sect. II. Art. 7. 
 
 If the body, or system of bodies, whose centre of gravity 
 is sought, is divided into equal masses, we shall have 
 
 p = p'= p", &c. 
 
 wherefore, putting n for the number of these equal masses, 
 those theorems will become 
 
 X -\- x -\-x" -\- x" 4" &c. 
 
 n 
 
 n 
 
 z J^ z' + z" -\- z" + &c. 
 
 z iz: • . 
 
 n 
 
 The equations may be always presented in this form, and 
 they show that the distance of the centre of gravity from any 
 plane, is the sum of the distances of the equal masses divided 
 by their number: or that it is the mean distance of the 
 weights of the several parts, and, therefore, that the centre 
 of gravity is the centre of the mass. 
 
 If the body is homogeneous, i. e. of uniform density, the 
 equal masses, into which it is divided, will have equal vo- 
 lumes ; and the centre of gravity will be then the centre of 
 magnitude. From this and Art. 3. it appears, that the dis- 
 tance of the centre of gravity of a triangle from any plane, 
 is the mean distance of its three angles from the same. 
 And that the distance of the centre of gravity of a trian- 
 gular pyramid from any plane, is the mean distance of its 
 four angles. 
 
 The formula of Sect. II. Art. 7. may be presented in a 
 more abridged form, by putting 
 
OF THE CENTRE OF GRAVITY. 93 
 
 s (p . a:), for p . o: + P • V + P "x" + &c. 
 the character 2, denoting the sum of all quantities corres- 
 ponding to that to which it is prefixed. In the same man- 
 ner, 2 (p) is put for p +P'+ p"4- ^c., and the same notation 
 being extended to the other two formulae, they are thus pre- 
 sented : 
 
 2(P) ' S(P) ' 2(P) 
 
 It is seldom that we have occasion for all these formulae : 
 for if the centre of gravity is known to be in a certain plane, 
 its distance from the plane being cypher, the product of this 
 distance into the whole mass is also cypher : and therefore, 
 also, the sum of the products uf the partial masses into their 
 respective distances from the same plane is cypher, the sum 
 of the products of those which lie at one side into their dis- 
 tances from the plane, being equal to the sum of the pro- 
 ducts, similarly made out, for those which lie at the other 
 side of the plane, and with contrary signs. And conversely; 
 if the sum of the products be cypher, the distance of the 
 centre of gravity is also cypher. 
 
 When any such plane is known, it may be taken for one 
 of the coordinate planes, as for example, for the plane of xy, 
 and we shall then have 
 
 2 (p..^) Z= 0. Z 1= 0. 
 
 and the problem for finding the centre of gravity is solved by 
 the two first equations, which give its distance from the axes 
 of a; and ;y. Wherefore, "when the centres of gravity of 
 the bodies of a system lie all in the same plane, the common 
 centre of gravity is found by its distance from each of two 
 axes in the same plane ; the distance from each axis beinff 
 had by multiplying each body into the perpendicular dis- 
 tance of its centre of gravity from that axis, and dividing the 
 sum of those products by the sum of the masses," 
 
9I< STATICS. — SECT. VI. 
 
 If the centres of gravity of the bodies, or partial masses, 
 lie in the same right line, by taking this line for the inter- 
 section of two of the coordinate planes, as of the planes zjc, 
 yx\ those centres being in each of these planes, the common 
 centre of gravity shall be in both. Wherefore, 
 
 z r= 0. and y zz: 0. 
 
 and the third equation will suffice, which gives the distance 
 of the common centre of gravity from a plane perpendicular 
 to the axis of x ; i. e. from a certain point on that line itself. 
 Wherefore, '^ when the centres of gravity of the bodies of 
 the system, or of the partial masses, lie all in the same right 
 line ; the common centre of gravity is on the same line, and 
 its distance from an assumed point of that line is found, by 
 taking the sum of the products of the several bodies or 
 masses into their respective distances from that point, and 
 dividing by the entire mass. 
 
 Let it be proposed to find the centre of gravity of the pe- 
 rimeter of a plane polygon bcdef, (Fig. o7.) by means of the 
 coordinates of its angles. 
 
 The centres of gravity of the several sides, being at the 
 middle points of those sides, are all in the same plane : 
 wherefore, drawing two axes in this plane, viz. a?/, kx, and 
 denoting by x, y, the coordinates of the point b ; by x\ y, 
 those of the point c ; by x", y", those of the point d, &c. and 
 bisecting the sides bc, cd, de, &c. at the points o, o, o", &c. 
 the coordinates of the point o, shall be 
 
 X ■\- x' y 4- // 
 
 T} • o ' 
 
 those of the point d shall be 
 
 X -\-x y ■\-y 
 
 2 ' 2 ' 
 
 those of the point o" shall be 
 
 X -\- X y + // 
 
OF THE CENTRE OF GRAVITY. 95 
 
 and so of the rest. Then, putting p, p', p", &c. taken pro- 
 portional to the sides, for the weights apphed at the middle 
 points of those sides, and making these substitutions in the 
 general formulae, we have 
 
 __ P (.^: 4- a:') + p^ (.r> x") + p" {x"+ x'") + &c. 
 ^~ J^(p + p'+p")+&c. 
 
 _ V (;/ + ?/) + y' (;/ 4- f) + ^" (/+ f) + &c. 
 '^ ~ 2 (p + p'+ p") + &c. 
 
 Showing, that each of the coordinates of the centre of gra- 
 vity of the perimeter of a polygon is had, by multiplying 
 each side into the sum of the corresponding ordinates of its 
 extreme points, and dividing the sum of these products by 
 twice the perimeter. 
 
 To find the centres of gravity of curvilinear figures, the 
 aid of the integral calculus is generally requisite. The dif- 
 ferential of the figure is to be multiplied into each of its 
 three coordinates ; each of these products being made a 
 function of a single variable, by means of the equations of 
 the figure, is to be integrated within the proposed limits : 
 the definite integrals thus obtained, are the numerators of 
 the values of x, y, z, the coordinates of the centre of gra- 
 vity. Their common denominator is the magnitude of the 
 figure obtained by integrating its differential in like manner. 
 
 Thus, for a curve whose length is s, the general formulas, 
 by which the coordinates of the centre of gravity are ex- 
 pressed, will be 
 
 s s s 
 
 The differential ds being 
 
 V {dx^-{-d?f-{-d%''). 
 
 the product of this by its distance from the plane of x?/, i. e. 
 sds is 
 
 s!, V {dx"" + dy" + dz"). 
 
96 STATICS. — SECT. VI. 
 
 This being converted into a pure function of z, by means of 
 the equations of the curve, and integrated, the extreme va- 
 lues of z are to be substituted ; the difference of the result- 
 ing quantities is the numerator in the vahie of z. Its deno- 
 minator, s, is the definite integral of 
 
 to be obtained in the same way. The values of x and y, are 
 found by a similar process. 
 
 If the curve is of single curvature, it is evident from what 
 has been already observed, that the problem may be solved 
 by two of the equations : and if it is situated symmetrically 
 with respect to a certain right line, that one of those ecjua- 
 tions will suffice. 
 
 For example ; let it be proposed to find the centre of 
 gravity of a circular arc. 
 
 The arc being placed symmetrically with respect to the 
 radius drawn to its middle point, this radius may be taken 
 for the axis of x ; and the problem will be solved by means 
 of the equation 
 
 s 
 
 in which, 
 
 ds = V {dx'' -{- f/;/). 
 The origin being taken at the centre, the equation of the 
 
 curve IS 
 
 which gives, 
 
 909 
 y — r-- x- 
 
 dx^ zz. 'At. df/^. 
 x^ -^ 
 
 Substituting this value, and integrating, there is 
 $.r. ds ■:z r . y -{- c. 
 Let b and — b, be the values of //, at the extreme points 
 
OF THE CENTRE OF GllAVITY. 97 
 
 of the arc ; these are to be successively substituted ; and one 
 of the resulting quantities being subtracted from the other, 
 the definite integral is 2r.h. Wherefore, 
 
 2b. r 
 X rz . 
 
 s 
 
 showing, that the distance of the centre of gravity of the arc 
 
 from the centre, is a fourth proportional to the arc itself. Its 
 
 chord and radius. 
 
 For the semicircle, 
 
 2r • 
 s — irr. b r= r. g^vmg x =: — . 
 
 3r 
 
 and therefore, 
 
 n 
 
 which shows, that the radius of a circle is equal to tlie 
 quadrantal arc of another circle, whose radius is the distance 
 of the centre of gravity of half the periphery of the former 
 from its centre. 
 
 Let it be proposed to find the centre of gravity of a 
 cycloidal arc. 
 
 This curve is generated by the motion of a point taken 
 in the periphery of a circle, whilst the circle revolves in its 
 own plane, on a right line. 
 
 Let AB (Fig. i38.) be the line on which the circle revolves, 
 or the base of the cycloid ; v its vertex or middle point ; vd 
 its axis, which is also the diameter of the generating circle. 
 It is evident that the two branches, va, vb, are symmetrical 
 about the axis ; and that if equal arches are measured from 
 V, their centres of gravity shall be equidistant from the axis 
 vd, and also from the base. Wherefore, the common centre 
 of gravity of the two equal arcs shall be at the same distance 
 from the base, and in the axis vd. Accordingly, taking v 
 for the origin, and vd for the axis of cc, it will be sufficient to 
 find the centre of gravity of an arch at either side of vd. 
 
 H 
 
98 STATICS. — SECT. VI. 
 
 For any point, tn, there will be 
 
 A^j zz axc^ini zz arc Dr. 
 therefore, 
 
 pv) or mr zz arc vr. 
 and 
 
 7}iq = arc = vr -f- f'^' 
 i. e. putting a, for vd, 
 
 t/ = arc (verse sine rz x) + V {'^^~-^'}' 
 
 and 
 
 a 
 
 dx 2~^ 
 
 iJy zz ■ + -77 ^ . dx 
 
 "^ sm.arcvr ^/{ox—x'') 
 
 a a 
 
 -^-. o^.r + —-. or. dx. 
 
 that is, 
 
 -y/ {ax—X')' ' -v/ («a; — .r") 
 
 (a—x)dx \/{a — x) . 
 
 dv zz — - = -^— H . dx. 
 
 ^ V {ax—x-) ^/ X 
 
 Accordingly, 
 
 ds zz \J {dx^-\- dif) = V -. dx. and s—9.\J ax. 
 Also, 
 
 \ a:(/s zz a~ \x^. dx zz 
 
 2 1 A 
 
 In these integrations, no constant quantities are added, 
 as both members of each equation vanish together, for 
 0;= 0. 
 
 X = 
 
 S"^ 
 
 2 i 5 
 
 ds Q « • '^ 
 
 vphich shows, that if a line is drawn through the vertex, pa- 
 
OF THE CENTRE OF GRAVITY. 99 
 
 rallel to the base, the perpendicular on that line, from the 
 centre of gravity of an arch measured from the vertex, is 
 one-third of the perpendicular distance of its extreme point 
 from that same line ; and, therefore, that the centre of gra- 
 vity of a cycloidal arc bisected at the vertex, is in the axis, 
 at a distance from the vertex equal to one-third of the sa- 
 gitta of the arc: and that the distance of the centre of gra- 
 vity of the entire curve from the same point, is 77. 
 
 The differential of a plane surface is ydx\ wherefore, 
 the formulae for its centre of gravity are, 
 
 $ x.y.dx 5 y^. dx 
 
 ~ %y.dx ' ~ ^y.dx' 
 
 If the surface is symmetrical about a certain right line, it 
 is evident that it may be resolved into elements, whose cen- 
 tres of gravity are in that line : and that if the same line is 
 taken for the axis of the abscissae, the first of these equa- 
 tions will suffice. 
 
 Example. — Let it be proposed to find the centre of gra' 
 vity of a trapezium abcd, (Fig. 39.) 
 
 If the trapezium is resolved into elements by lines pa- 
 rallel to ADj it is evident that the right line ef, which bisects 
 the parallel sides ad, bc, shall pass through the centre of 
 gravity of each element : wherefore, it will be convenient to 
 make it the axis of the abscissae. Let f be the origin, 
 EF r= a ; AD = 6 ; EC 3= Z>'; ¥m =: x ; 7nn zz y. Then, 
 
 aM+{b~ b')x 
 
 2 11 zz . 
 
 a 
 
 If the angles at a and d are equal, y shall be perpen- 
 dicular to the axis oi x\ and the element of the area shall 
 be 
 
 a 
 
 and 
 
 11 2 
 
100 STATICS. — SECT. VI. 
 
 - [aM.x+{b-h')x'''\dx 
 2x.r'.dx — ^ — - — . 
 
 Wherefore^ 
 
 and 
 
 and 
 
 
 _^lx.y.dx _ 3aM.x + ^{h-b')x'^ 
 
 2lt/.clx ~ 6a.b'+3{b~f/)x ' 
 
 In these integrations, no constant quantity was added ; 
 because, for x zi 0. both members of each equation vanish 
 together. 
 
 If the angles at a and d are unequal ; or which is the 
 same thing, if the line ef is oblique to the parallel sides, ad, 
 BC, the same determination will serve ; as will appear by put- 
 ting j3 for the angle fed, contained between the axis and 
 the ordinates. Then, each element is 
 
 2sm.p.y.dx. 
 
 and 
 
 _2sm.^.lx.t/.dx _ %x.y.dx 
 2sin.j3. $2/.c/a7 ~ ^!/-(^^ 
 
 Taking the value of x for a; z= a. it is 
 
 a{b'+2b) 
 ~ S{b'+b)' 
 
 This solution may serve for a triangle or parallelogram. For 
 the former, by making b'zz 0. in which case, 
 
 2 
 and for the latter, by making b'—b; in which case, 
 
OF THE CENTRE OF GRAVITY. 101 
 
 a 
 
 Example. — Let it be proposed to find the centre of 
 gravity of the segment of a circle, whose equation, taking 
 the centre for the origin, is 
 
 2 2 2 
 
 2/ — r — X . 
 
 The centre of gravity of the segment is evidently in the 
 radius by which the arc is bisected. Wherefore, taking this 
 for the axis of the abscissae, 
 
 _2'\^x.y.dx _2^\/ {r^— x^)x.dx _ 
 ~ area area * ~~ 
 
 area 
 
 Taking the integral from x zz. a io x zz-r; in which last 
 case its value is c, the value of y being then cypher, we 
 have 
 
 X ■=. 
 
 area 
 For the centre of gravity of the area of the semicircle, 
 
 o 
 TV T 
 
 making a z: ; and putting for the area, its value, —y-\ 
 there is for this case, 
 
 4r 
 ^ = -37- 
 
 The sector cmvn, (Fig. 40.) is composed of the segment 
 mvn, and of the triangle men. The centre of gravity of the 
 triangle is in the same radius, by which the arc mn is bi- 
 sected ; and at a distance from c equal to f cd. Wherefore, 
 the distance between the centres of gravity of the triangle 
 and circular segment is known ; and if this distance is divided 
 in the inverse ratio of the areas, the point of division will be 
 
102 STATICS. — SECT. VI. 
 
 the centre of gravity of the sector. This process is, how- 
 ever, unnecessary. For if with the centre c, and a radius 
 equal to | cv, an arc is described, bounded by the hnes 
 Ci?i, en, this arc is the locus of the centres of gravity of the 
 indefinitely small and equal sectors into which the sector 
 cmvn is resolvable ; those elementary sectors being regarded 
 as right lined triangles. Moreover, these elementary sectors 
 being equal, and their centres of gravity being uniformly 
 arranged along this arc, their common centre of gravity, i. e. 
 the centre of gi'avity of the entire sector, must be that of the 
 arc: hence, it follows, that the distance of the centre of 
 gravity of a sector, from the centre of the circle, is a fourth 
 proportional to the arc, its chord, and f of its radius. 
 
 Example. — For the centre of gravity of the segment of 
 a parabola, whose equation is i/^ — px, the diameter being 
 taken for the axis of the abscissae, and the vertex for the 
 origin, we have 
 
 ^y.x.dx _ 3 
 5 //. dx ~ 5 "^ * 
 
 In these integrations, no constant quantity is added, be- 
 cause for X zz: 0. each integral is, evidently, cypher. 
 
 It is also evident from the observation already made, 
 that the same formula will serve whether the ordinate s are 
 perpendicular or oblique to the diameter, i. e. whether the 
 diameter which bisects the chord of the segment is the axis 
 or not. 
 
 In figures of revolution, the axis of rotation being taken 
 for the axis of the abscissas, it is plain that the first of the 
 three general equations of Art. 4, will serve for the deter- 
 mination of the centre of gravity. 
 
 If it is proposed to find the centre of gravity of a surface 
 of revolution, the element of the generating curve is 
 
 ds- V {dx- + elf) ; 
 and the element of the generated surface is 
 
Wherefore, 
 
 OF THE CENTRE OF GRAVITY. 103 
 
 _%xy y/{dx'^-\-df) 
 
 S y V {dx'-\-dy'^) ' 
 
 in which, substituting for y, dy, their values in x, collected 
 from the equation of the generating curve, and integrating 
 within the proposed limits, the value of x is obtained. 
 
 Example. — The surface of a truncated cone is generated 
 by the revolution of a right line. Let CD be that hne, (Fig. 
 41.) and ab the axis round which it revolves. Then, putting 
 a for AB ; b for bd ; and b' for ac, there will be 
 
 a.b' + {b-byv 
 
 2/ = 
 
 and 
 
 Therefore, 
 
 , (b — b') dx 
 dy — ^ , 
 
 a 
 
 The invariable coefficient of dx, occurring in the nume- 
 rator and denominator of the value of x, may be suppressed. 
 Wherefore, 
 
 _ ^y.x. dx 
 
 But the section of the conic frustum, in which is the axis, 
 is a trapezium, whose centre of gravity is in the axis of the 
 cone, at a distance from the lesser base expressed as above ; 
 whence it follows, that the centre of gravity of the sur- 
 face of the conic frustrum is the same as that of the sec- 
 tion of the solid, in which is the axis : and also, that the 
 centres of gravity of the surfaces of a cone and cylinder are 
 the same as those of the sections in which are their axes. 
 
104 STATICS. — SECT. VI. 
 
 Example. — Let it be proposed to find the centre of gra- 
 vity of a spheric zone, generated by the revolution of the cir- 
 cular arc mi, about the radius cv. (Fig. 40.) 
 
 Making this radius the axis of the abscissae, and the cen- 
 tre c the origin, the equation of the circle is 
 
 Wherefox'e, 
 
 and 
 
 V{d.v'+di/^= '"'^^ 
 
 V{r^-x') 
 
 1>/.V {dx-+ dif) - rx + c. 
 
 \xyyj[dx^^df)^'^^c. 
 
 Then, putting a for ct/ ; a for ce, and taking the integrals 
 between these limits, there will be 
 
 _ a — a _ a -\-a 
 
 Which shows, that the centre of gravity of the zone is the 
 
 middle point of the portion of the axis intercepted between 
 
 its two circular bases. 
 
 For the spheric segment there is a'=r; and in that 
 
 case, 
 
 r-A-a 
 x= ' . 
 
 showing, that the centre of gravity of the surface of a spheric 
 segment is at the middle point of the sagitta. 
 
 In general, the equation of the surface of any geometric 
 ficrure being of this form 
 
 dz :=p.dx + q^dy. 
 
 in which p and q are the partial differences of ^, with respect 
 to X and y, the differential of the surface is 
 
 y/{\+p''+q^)dx.dy. 
 
OF THE CENTRE OF GRAVITY. 105 
 
 wherefore, 
 
 ~ lWil + F+q')cLv.d!/- 
 _ ll^y/[\+p'+q')dx.dj/ 
 
 The double integrals are to be taken within the proposed 
 limits. 
 
 A solid of revolution is resolved, by planes perpendicular 
 to the axis, into cylinders or conic frusta, whose altitude is 
 dx. The value of one of these elements is 
 
 TTi/^dx. 
 
 and the product of this into the abscissa is 
 
 TTxy'. dx. 
 Wherefore, 
 
 __ ^y^xdx 
 
 ~ S y'd^ ' 
 
 in which, substituting for y its value in x, and integrating 
 as before within the proposed limits, the definite value of x 
 is obtained. 
 
 Example. — Let it be proposed to find the centre of gra- 
 vity of an oblong elHpsoid, terminated by two planes perpen- 
 dicular to the greater axis. 
 
 Putting a for the greater, j3 for the lesser semiaxis ; a 
 and a , as before, denoting the distances of the planes from 
 the centre, the equation of the generating elHpse is 
 
 wherefore. 
 
 s 
 
 7f-dx = -^v (^W-x — x^) -f c 
 
106 STATICS. — SECT. VI. 
 
 5 
 
 /32 
 
 Tliese integrals taken between the limits a and a, give 
 _6a^ {a' —a^) — 3 (a'^ — «^) 
 
 This expression, being totally independent of the lesser 
 axis of the ellipse, is applicable to the segment of a sphere. 
 
 If the ellipsoid were described about the lesser axis, the 
 expression for the distance of the centre of gravity would 
 differ from this only in the change of a for /3. Whence it 
 follows, that if the axis of the ellipsoid, whether oblong or 
 oblate, is made the diameter of a sphere, the segments of 
 these figures included between two planes perpendicular to 
 the common axis, shall have their centres of gravity at the 
 same point. 
 
 If the section, perpendicular to the axis of revolution, 
 passes through the centre, the distance of the centre of gra- 
 vity of the semiellipsoid, or hemisphere, will be had from the 
 foregoing, by making a = 0. And a = a for the oblong, or 
 a=j3 for the oblate ellipsoid, which gives 
 
 3 
 x = -a 
 
 for the one, and 
 
 for the other. 
 
 Example. — Let it be proposed to find the centre of gra- 
 vity of a paraboloid, generated by the revolution of a para- 
 bolic segment, whose equation is 
 
 substituting this value of y^ in the formula, there is 
 
 _ $ x^dx _ 2 
 5 xdx 3 ' * 
 
 In this equation there is no constant quantity. 
 
OF THE CENTRE OF GRAVITY. 107 
 
 If the segment, whose centre of gravity is sought, is in- 
 cluded between two parallel planes, at the distances a and a , 
 measured from the vertex ; these values of x, being succes- 
 sively substituted in each of the integrals, the difference of 
 the results must be taken for the numerator and the denomi- 
 nator of the value sought ; thus, 
 
 2 cP - a" 
 
 o a — a" 
 
 These examples may suffice to show how the integral cal- 
 culus is applied, to solve the problem for solids of revolution, 
 which figures are, evidently, capable of being resolved into 
 elements, whose centres of gravity are in a right line. The 
 centre of gravity of the entire solid being, then, on the same 
 line, its place is determined according to a single formula. 
 
 The general method to be used when such facilities do 
 not occur, is to resolve the solid into elements, by planes pa- 
 rallel to each of the three coordinate planes. The expression 
 for an element will then be 
 
 dx,dy.d%. 
 
 and that of the volume, 
 
 \Wdx.dij,dz. 
 
 and the three coordinates of the centre of gravity of the solid 
 are found, according to the three following formulae : 
 
 ^_'^^y^xAx^dyAz^ 
 \\\,dx,dy.dz ' 
 
 \\\'dx.dy.dz ' 
 
 ^_ ll^.s.dx.di/xh 
 m.dx.di/.d:2 
 
 The integral in each numerator and denominator is to be 
 taken within the proposed limits. 
 
 The centre of gravity of any one body of a given system 
 
 Y = 
 
108 STATICS. SECT. VI. 
 
 is immediately found, if the centres of gravity of the remain- 
 ing bodies, as well as the common centre of gravity of the 
 system are known. For, having 
 
 mx + mx •\- m'x" , See. = mx. 
 
 the ordinate of the centre of gravity of any one of the bodies, 
 as of m, is given by the equation 
 
 MX — (mV4- w^'V, &c.) 
 
 x= ^ -. 
 
 m 
 
 and the other two coordinates of this centre are found in the 
 same manner. 
 
 When the centre of gravity of a body, and that of any 
 part of the same are known, that of the remaining part is 
 also known. For in this case, the centres of gravity of the 
 whole and of its two parts are in the same right line ; and 
 any two of the masses multiplied into the distances of their 
 centres of gravity from that of the third, give equal products. 
 Thus, if m and m denote the partial masses, and d, cV the 
 distances of their centres of gravity from the centre of gra- 
 vity of the sum, which is their common centre of gravity, 
 
 then f/'= — ;•. for the distance of the centre of gravity of the 
 
 mass m , from the common centre of gravity. 
 
 Generally, whatever be the point to which, as to an 
 origin, these centres are referred, the coordinates of one of 
 those centres are found from those of the two others, by the 
 following equations : 
 
 mx •\- (m—?7i) x'— MX. 
 
 mi/ -\- (u—?n) yzz my. 
 
 m% + (m — w) ^'— mz. 
 
 5. The centre of gravity was shown to a centre of parallel 
 forces, particularized by this circumstance, that the body 
 being resolved into equal masses, the forces applied to those 
 masses are not only parallel, but equal. And as such a 
 
OF THE CENTRE OF GRAVITY. 109 
 
 system of forces may be compounded into one parallel force, 
 equal to their sum, which in all positions of the body passes 
 through the same point, denominated the centre of gravity ; 
 so, conversely, any single force directed through this point, ■ 
 is resolvable into a system of equal and parallel forces, dis- 
 tributed upon the equal masses, into which the body may be 
 supposed to be divided. 
 
 Hence it follows, that there is no tendency to rotatory 
 motion produced in a rigid body, by a force directed through 
 its centre of gravity. 
 
 The properties of this centre, usually noticed, relate to a 
 body containing a fixed point, or incumbent on a plane sur- 
 face. The fixed point is denominated the point of sus- 
 pension. 
 
 The weight of a body, considered as a single force, being 
 applied at the centre of gravity, and directed in the perpen- 
 dicular towards the horizon ; and the resistance or reaction 
 of the fixed point being a force acting in the line by which 
 this point is connected with the centre of gravity, it follows 
 that these two forces cannot be directly opposed, and there- 
 fore cannot equilibrate, unless these two lines coincide, i. e. 
 unless the fixed point is in the vertical, passing through the 
 centre of gravity ; and that then, as the weight of the body 
 is directed through the fixed point, whose reaction is always 
 equal and opposite to the action upon it, the equilibrium is 
 necessarily established. 
 
 Accordingly, when the body is suspended by the centre 
 of gravity, the equilibrium is necessarily established, what- 
 ever be the position of the body round this point. 
 
 But if the point of suspension is different from the centre 
 of gravity, there are but two positions of the body in which 
 the forces are directly opposed to each other, viz. when the 
 centre of gravity is vertically above, or vertically beneath 
 the point of suspension : and these are the two positions of 
 equilibrium. 
 
110 STATICS. — SECT. VI. 
 
 But equilibrium is of three kinds, viz. stable, unstable, 
 and neutral. It is of the first kind, when the body on any 
 slight change of position, returns to its former position : of 
 the second kind, when the deviation increases : and it is of 
 the third kind, when there is no tendency in the body to 
 recede further from its original position, or to recover that 
 position. It is evident that this last kind can exist, only 
 when the equilibrium continues under a continued change of 
 position ; and, therefore, that it can have no place in the 
 case now under consideration, wherein these are but two 
 positions of equilibrium. 
 
 To distinguish between the states of equilibrium when 
 the centre of gravity is vertically above or below the point 
 of suspension, it is only requisite to consider the direction of 
 the resultant of the two forces, when the centre of gravity 
 is out of the vertical passing through the point of suspension. 
 Let B be the body ; a the point of suspension ; o the centre 
 of gravity ; and ov the vertical passing through this point. 
 (Fig. 42.) The weight of the body applied at the point o, 
 acts in the direction of the line oi\ and the resistance of the 
 fixed point in the direction of ao ; towards the fixed point, 
 when the weight acts as a pull, \. e. when the angle voa is 
 obtuse, as in the figure ; and in the opposite direction, when 
 it acts as a pressure, i. e. when that angle is acute. The re- 
 sultant of these two forces must be directed within the angle 
 made by those lines ; and, therefore, must tend to carry the 
 centre of gravity downwards to the vertical passing through 
 the point of suspension. 
 
 Wherefore, when the centre of gravity is directly above 
 the point of suspension, the equilibrium is unstable : for on 
 the smallest departure from this line it will further descend ; 
 and will not rest, until it shall have attained the position 
 vertically beneath the point of suspension. In this latter 
 position, the equilibrium is stable; because on any departure 
 from this position, the centre of gravity will spontaneously 
 return to it. 
 
OF THE CENTRE OF GRAVITY. Ill 
 
 From these properties of the centre of gravity, this point 
 may be found mechanically. 
 
 The property first noticed was, that the body, if sus- 
 pended by the centre of gravity, will rest in any position 
 indifferently. The point possessed of this property may be 
 determined by trial. 
 
 The other property was, that if the body is suspended 
 from a point different from the centre of gravity, the body 
 shall not rest, until that centre attains to the vertical passing 
 through the point of suspension. Accordingly, if the body 
 is freely suspended from two points successively, and the 
 vertical passing through each point of suspension is traced 
 on the body, the intersection of these two lines shall mark 
 the place of the centre of gravity. 
 
 When the centre of gravity is in the vertical passing 
 through the point of suspension, the pressui'e or strain on 
 this point is plainly the weight of the body. And for any 
 position, the strain can be readily determined : for resolving 
 the weight of the body applied to the centre of gravity into 
 two forces : one acting in the line joining that centre with 
 the point of suspension, and the other perpendicular to the 
 same; the former is w.cos.a, and the latter w.sina; a 
 being the angle made by the line drawn to the point of sus- 
 pension and the vertical. The former of these is destroyed 
 by the resistance of the fixed point, and produces the pres- 
 sure thereon ; the latter is that part of the weight employed 
 in producing motion. 
 
 If the body is suspended from two points, it will not rest 
 until the centre of gravity is brought to the vertical ph\ne 
 which contains the two points of suspension ; and then the 
 pressures on the two fixed points are given by Sect V. Art. 5. 
 Wherein e being the angle between the vertical and the line 
 which joins the fixed points, 
 
 zrzw'.sin. £. x — w.cos.e. 
 
112 
 
 STATICS. — SECT. Vr. 
 
 and Y being cypher, the expressions there given are reduced 
 to 
 
 w [sin. t{a — x) -{- cos, e. si] , w{sm. z. x — cos.z.z) 
 
 a ' a ' 
 
 These are the pressures in lines perpendicular to that which 
 joins the fixed points. But with respect to the pressures in 
 the direction of that line, all that can be determined about 
 them, on the supposition of the perfect immobility of the 
 points of suspension, is, that the sum of those pressures is 
 w. COS. e. 
 
 If this line is horizontal, the angle e is right, and there- 
 fore, 
 
 sin. £zi 1. cos. £ = 0. 
 
 and the pressures altogether vertical; which reduces the 
 expressions for those pressures to 
 
 w{a — x) w.x 
 
 a ' a ' 
 
 agreeably to what has been shewn, Sect. II. Art, 4. relative 
 to the resolution of a force into two parallel forces. 
 
 If the line which joins the points of suspension is vertical, 
 there will be 
 
 sin. 6^:0. cos. e z: 1. x zz w. 2 — 0. 
 whereby the expressions for the pressures in the perpen- 
 dicular to the line connecting the fixed points, are 
 
 w.z — w.z 
 
 a a 
 
 and all that can be known relative to the vertical pressures, 
 is, that their sum is w, or the weight of the body. 
 
 This is the case of a door hung on two hinges, wherein 
 a denotes the distance between the hinges ; and z the per- 
 pendicular on that line from the centre of gravity. So that 
 the upper hinge is drawn out, and the lower hinge pressed 
 
OF THE CENTRE OF GRAVITV. 113 
 
 inwards, by a force w. - . the sum of the vertical pressures 
 
 being w. 
 
 For the weight acts in the vertical passing through the 
 centre of gravity, and may therefore be supposed to be ap- 
 plied to any point of this line. This being resolved in the 
 directions of the lines drawn to the hinges from the point so 
 assumed, each of the forces into which it is thus resolved, is 
 again to be resolved in directions vertical and horizontal. 
 The vertical forces will reproduce the weight, and the hori- 
 zontal forces the strains in their directions. Thus the points 
 of suspension being m and w ; and o the centre of gravity, 
 (Fig. 43.) let the lines no, mo, be drawn, and let the latter 
 be produced beyond the point o, as to c. Also, let ov be a 
 vertical line directed downwards from the point o. Ihe 
 weight which acts in the direction ov, is resolved into two 
 forces directed in the lines oc, on ; the former drawing the 
 upper hinge outwards ; and the latter pressing the lower 
 hinge inwards. These three forces are as the sides of the 
 triangle mon ; i. e. the weight being represented by the line 
 mn, parallel to its direction, the hinge at m shall be drawn 
 out by a force, regresented in magnitude and direction by 
 the line mo ; and the hinge at w, pressed inwards, by a force 
 represented in magnitude and direction by the line on. But 
 if each of these strains is to be resolved into two, one in the 
 vertical line mn, and the other in the perpendicular to this 
 line ; this is done by drawing the perpendicular od. The 
 strain mo is resolved into md, acting vertically downwards ; 
 and the strain do acting horizontally outwards : also, the 
 strain on into dn, acting vertically downwards ; and od acting 
 horizontally inwards. The sum of the vertical pressures is 
 mn, or the weight of the door ; and this may be sustained by 
 either of the points of suspension, or divided between them 
 in any ratio; but the point m is drawn horizontally outwards, 
 
 I 
 
114 STATICS. — SECT. VI. 
 
 with the force do ; and the point n is pressed horizontally 
 inwards, with an equal force. 
 
 The same things would readily follow from the principles 
 established in Sect. III. Art. 3. For the weight being trans- 
 ferred to the vertical containing the hinges, there is gene- 
 rated a pair of equal and contrary forces, whose moment is 
 w X od, which may be turned round in the same plane, so as 
 to become perpendicular to the line mn. But the moment 
 
 w X od is equal to — . iv X mn. i. e. to the moment of the 
 
 forces — .w acting at the interval mn. 
 mn 
 
 The body being placed on a horizontal plane, if the ver- 
 tical passing through the centre of gravity meets the plane 
 in a point within the base, the body shall rest, because the 
 weight applied to the plane at this point, may be distributed 
 on the angles of a triangle within which it is contained : and, 
 by the supposition, such a triangle can be made by joining 
 certain points of support. The weight being so distributed, 
 the pressures are necessarily equilibrated by the resistances 
 of the points to which they are applied. But the stability 
 of the body will depend on the excess of the shortest line 
 that can be drawn from the centre of gravity to the contour 
 of the base, above the perpendicular distance of the same 
 point from the plane. For in order to carry it over the edge 
 of the base, the centre of gravity must be raised to an eleva- 
 tion equal to this difference. Hence in general, the stability 
 of a body is greater, as the distance of the centre of gravity 
 from the plane of support is less, in relation to the extent 
 of the base. 
 
 When the base of the body is reduced to a point verti- 
 cally beneath the centre of gravity, the smallest force, ap- 
 plied in a different direction, should produce a disturbance. 
 Yet even in this case, the equilibrium may be stable, un- 
 
OF THE CENTRE OF GRAVITY. 115 
 
 stable, or neutral. Thus, when an elhptic cylinder is placed 
 on a horizontal plane, with its lesser axis in the vertical, the 
 position is that of stable equilibrium. If the same body rests 
 on the extremity of its greater axis, the equihbrium is un- 
 stable. A cylinder with a circular base, placed with its side 
 on a horizontal plane, affords an example of neutral equi- 
 librium. The sum of the pressures made at the points of 
 contact, is the weight of the body. 
 
 If the body rests on two points of support, the vertical 
 passing through the centre of gravity must meet the line 
 connecting the two pohits of support at some intermediate 
 point ; and the pressure on each point is to the weight, as 
 the distance of the point of intersection from the alternate 
 point, to the distance between the two points of support. 
 
 If the body rests on three points, the vertical passing 
 through the centre of gravity must meet the plane within the 
 triangle, formed by joining the three points of support ; and 
 then, if lines ai*e drawn to the three angles, from the point 
 vertically beneath the centre of gravity, the pressure on each 
 angular point is to the weiglit, as the area of the partial 
 triangle placed on the opposite side, to the area of the whole 
 triangle. Sect. V. Art. 7. 
 
 If the body touches the plane on which it rests in more 
 than three points, the pressures on the several points are 
 indeterminate, on the supposition of the perfect rigidity of 
 the body, and of the plane on which it rests ; and all that 
 can be then determined, with respect to those pressures, is, 
 that their sum is the weiglit of the body. 
 
 When the vertical line passing through the centre of 
 gravity falls without the base, the body must upset; because, 
 the reaction of the plane being in a vertical line different 
 from that in which the weight is directed, those forces 
 cannot equilibrate. 
 
 If the body is placed on an inclined plane, it must neces- 
 sarily descend; the surfaces being perfectly smooth, i.e. 
 
 1 2 
 
116 STATICS. — SECT. Vr. 
 
 supposing no restraint arising from friction. For, if the 
 weight of the body, which is a force apphed at the centre of 
 gravity, is resolved into two forces, one of them perpen- 
 dicular, and the other parallel to the plane, the latter, not 
 being opposed to the reaction, must produce its full effect 
 in carrying the body down the plane. 
 
 If the perpendicular from the centre of gravity on the 
 plane, falls within the base or surface of contact, the body 
 shall descend by sliding. For if the weight of the body, 
 which is a force applied at its centre of gravity, is resolved 
 into two forces, of which one is parallel, and the other per- 
 pendicular to the plane : this last being directed to a point 
 within the base, shall be equilibrated by the resistances : 
 and as the remaining force, viz. that parallel to the plane 
 passes through the centre of gravity, there will be no rota- 
 tory movement. See Art. 4. 
 
 From these considerations, it would appear that a sphere, 
 cylinder, or a regular polyhedron should descend down an 
 inclined plane by sliding, and not by rolling or tumbling ; 
 inasmuch as in these bodies, the perpendicular from the 
 centre of gravity on the plane, always falls within the base, 
 whatever be the inclination of the plane to the horizon. But 
 in this, the force of friction is not considered. The effect 
 of friction is to impede the motion of the parts of the body 
 in contact with the plane. Accordingly, even though the 
 perpendicular on the plane from the centre of gravity, 
 should fall within the base, the body shall roll or tumble, 
 whenever the force of friction becomes equal to that which 
 is required to turn the body over the edge of its base. 
 Hence, if the body is a sphere whose contact is reduced to 
 a single point, the least imaginable friction will impart rota- 
 tory motion: but a polyhedron will require more or less, 
 according to the extent of the base, measured from the foot 
 of the perpendicular from the centre of gravity on the plane, 
 and in the direction of the slope : and the same body that 
 
OF THE CENTRE OF GRAVITY. 117 
 
 slides, when opposed by a certain force of friction, may 
 tumble if the friction is increased. 
 
 If the perpendicular on the inclined plane falls without 
 the base, the body shall always descend by rolling or tum- 
 bling. But this rotatory movement may be in any direction, 
 according to the position of the foot of the perpendicular, 
 with respect to the base of the body or surface of contact. 
 For the weight of the body resolved in a direction perpen- 
 dicular to the plane, and the reaction of the plane, being pa- 
 rallel and contrary forces, must necessarily produce a rota- 
 tory movement, whose direction shall be determined by the 
 position of the point, in which the plane is met by the per- 
 pendicular let fall on it from the centre of gravity. 
 
 It appears then, that a body placed on an inclined plane 
 may tumble backwards, i. e. up the slope, even though it 
 should lean forwards with respect to the vertical. This 
 may be exemphfied by a rod of inconsiderable thickness, 
 when i=ts direction is within the angle made between the 
 perpendicular to the plane, and the perpendicular to the 
 horizon. 
 
 6. The properties of the centre of gravity hitherto de- 
 scribed, relate to a body suspended from a fixed point, or in- 
 cumbent on a plane. There is another property of this cerr- 
 tre which deserves to be noticed. It is that expressed by 
 the following theorem : 
 
 " The content of a surface or solid of revolution is equal 
 to the product of the generating line or plane, by the path 
 described by its centre of gravity." 
 
 To prove this theorem relative to a surface of revolution, 
 let DE be the line by whose motion the surface is generated, 
 (Fig. 44,) and bc the axis of revolution: let s denote the 
 whole line ; ds any one of its elementary parts, and y its 
 ordinate, or the perpendicular distance from the axis cb : 
 also, let a denote the distance of its centre of gravity from 
 
lis STATICS. — SECT. VI. 
 
 the line bc, and w the angle described by the plane cbde ; 
 observing that w is an abstract number, viz. the quotient of 
 the arc by its radius : so that the expression for the arc 
 described by any point of the curve whose ordinate is y, 
 is (i)?/. 
 
 This notation being understood, it will be seen that the 
 conical surface described by any element, ds, is 
 
 (x).i/.ds. 
 and that the surface generated by the entire line, is 
 
 or, because w is the same for all the elements, it is 
 
 ojl{i/.ds). 
 
 But the sum of the products, had by multiplying each ele- 
 ment into its distance from the axis bc, is equal to the single 
 product of their sum, s, into the distance of its centre of gra- 
 vity from the axis, ?'. e. 
 
 ^^'.ds — a,s. 
 
 wherefore, 
 
 ix)^?/.ds zz w.a.s. 
 
 which, putting s for the generated surface, is 
 
 s zz o).a.s. 
 
 i. e. the generated surface is equal to the product of the ge- 
 nerating line, into the arc or path described by its centre of 
 gravity. 
 
 To prove the proposition for a solid of revolution, let 
 CBDE be the generating plane, and cb, as before, the axis of 
 revolution. The plane being supposed to be resolved into 
 elementary rectangles, by ordinates perpendicular to the 
 axis CB ; and dx, denoting the portion of the axis between 
 any two consecutive ordinates, the expression for the ele- 
 mentary rectangle will be //.dx. 
 
 But the sum of the products had by multiplying each of 
 
OF THE CENTRE OF GRAVITY. 119 
 
 these elementary rectangles, into the distance of its centre of 
 gravity from the axis, is equal to the single product of the 
 area of the entire plane, into the distance of its centre of gra- 
 vity from the same axis, i. e. putting a for the entire area 
 CBED, and g for the distance of its centre of gravity from the 
 axis CB, it will be 
 
 ,2 
 
 S 
 
 Y ^^^ - ^•^^ 
 
 Therefore, 
 
 The first member of this equation is the sum of the pro- 
 ducts of y.dx. into -~. or the sum of the products had by 
 
 multiplying each elementary rectangle into half the arc de- 
 scribed by the extremity of its ordinate, i, e. the sum of the 
 portions of the solid generated by the several rectangles : 
 and the second member is the area of the generating plane, 
 multiplied into the arc or path described by its centre of 
 gravity. Wherefore, putting v for the entire volume of the 
 generated solid, it will be 
 
 V ^ A.(D.g. 
 
 The whole of the surface or solid of revolution is ex- 
 pressed by replacing w in these formulae with 2 vr, the ab- 
 stract number, which denotes the ratio of the periphery of a 
 circle to its radius, i. e. the quotient of the former divided 
 by the latter. Making this substitution, the surface of re- 
 volution is expressed by the equation 
 
 s z: 2TT.a.s. 
 and the solid of revolution by the equation 
 V — 27r.g.A. 
 If the revolving line or plane lies on each side of the axis, 
 it is only the difference of the surfaces or sohds generated 
 
1^0 STATICS. — SECT. VI. 
 
 by the parts, at different sides of the axis, that is so ex- 
 pressed : which will immediately appear by considering, that 
 in the statement relative to the equality of the product of an 
 entire mass into the distance of its centre of gravity from a 
 plane, and the sum of the products had by multiplying each 
 of the partial masses into the distance of its centre of gravity 
 from the same plane, those distances, if at opposite sides of 
 the plane, are to be marked with opposite signs. 
 
 These equations give the magnitude of a surface or solid 
 of revolution, when that of the generating line or plane is 
 known, together with the distance of its centre of gravity 
 from the axis of revolution. This method of quadrature or 
 cubature is called the barocentric method. A few examples 
 of its application are subjoined. 
 
 Example 1. — Let it be proposed to find the surface of a 
 truncated cone, whose side is given, together with the radii 
 of the circular bases. 
 
 This surface may be generated by the motion of the side 
 of the cone round its axis : wherefore, putting r, / for the 
 radii of the circular terminations, we have 
 
 a zz 
 
 2 
 
 which value of a, being substituted in the general formula, 
 there is 
 
 S Z2 TT (r + *• 
 
 Showing that the surface of the truncated cone is equal to 
 the area of a circle, whose radius is a mean proportional be- 
 tween s and r -\- 1', i. e. between the side and the sum of the 
 radii of the circular bases. 
 
 Example 2. — Let it be proposed to find the content of 
 the solid, generated by the revolution of an isosceles triangle 
 round a line passing through its vertex, and parallel to its 
 base. 
 
 Let CBD be the triangle, and cm its altitude, (Fig. A5.) 
 
OF THE CENTRE OF GRAVITY. 121 
 
 and putting h for this altitude, and h for the base bd, we 
 have 
 
 bh 1 _ 2 7 
 
 A = — and ^ = - h. 
 
 Substituting for a and g these values in the general formula, 
 there is 
 
 y--Tr.h'^.b. 
 o 
 
 But TT.h^ is the area of the circle whose radius is k, or cm, 
 and TT.h'^.b is the solid content of the cylinder standing on 
 that base, and whose altitude is b, or bd. Wherefore, the 
 solid is I of that cylinder. 
 
 Example o. — Let it be proposed to find the solid content 
 of a ring. 
 
 This solid may be supposed to be generated by the move- 
 ment of the circular section of the ring, its centre describing 
 the periphery of another circle, to which its plane is every 
 where perpendicular. Therefore, if r denotes the radius of 
 the generating circle, and r that of the circle described by 
 its centre, the area of the generating plane will be 
 
 and the path described by its centre of gravity will be 
 
 Accordingly, for the volume, we shall have 
 
 V =: 2 TT^. r'^. r. 
 
 If the axis, round which the generating circle revolves, 
 is a tangent drawn to a point in its periphery, we shall have 
 
 r'^i: r. 
 
 and for the solid content, 
 
 Example 4. — To find the content of the solid, generated 
 by the revolution of a parabolic segment, round its chord. 
 
1J22 statics. SECT. VI. 
 
 Let the chord of the segment be perpendicular to the 
 axis, and putting b for this chord, and h for the abscissa, the 
 area of the generating plane is 
 
 ^b.h. 
 
 Moreover, the distance of the centre of gravity from the ver- 
 tex was found to be 
 
 ^ h 
 
 5 "" 
 
 and, therefore, the distance of the same from the axis of re- 
 volution, is 
 
 Wherefore, the path described by the centre of gravity, is 
 
 f.7^./^. 
 giving for the volume 
 
 or putting for h, its value, viz. 2 V {jp-^- it is 
 
 If the chord is oblique to the diameter to which it is ap- 
 plied, putting a for the angle of inclination, the area of the 
 generating plane is 
 
 § b.h. sin. a. 
 and the path described by the centre of gravity is 
 
 J. TT . /^ . sin. a. 
 whereby the equation becomes 
 
 V = -1 f . TT .p^ . ]i^ . sin^ a. 
 The same general formulae may be applied to find the 
 centre of gravity of the generating line or plane when the 
 content of the generated surface or solid is known. Thus, 
 
 __ s _ v 
 
OF THE CENTRE OF GRAVITY. 123 
 
 Example 5. — The solid content of a sphere is 
 
 and this solid is generated by the revolution of a semicircle 
 round the diameter, by which it is terminated, whose area 
 is 
 
 ~¥' 
 
 Wherefore, substituting these values of v and a in the last 
 equation, we have for the centre of gravity of the semi- 
 circle, 
 
 TT 
 
 The same as already found by the direct method. 
 
 In finding the distance of the centre of gravity of a line 
 or surface from any assumed axis or plane, each element 
 was multiplied into its distance from that axis or plane, and 
 the sum of the products, thus obtained, was divided by the 
 whole content of the figure : and in finding the content of 
 the surface or solid generated by the revolution of that 
 figure, the distance of the centre of gravity was multiplied 
 into 2 TT, and into the content of the generating line or sur- 
 face. It is evident then, that this division and subsequent 
 multiplication, by the content of the generating figure, may 
 be omitted ; and that when the investigation is to be con- 
 ducted by the aid of the integral calculus, it will be enough 
 to take the sum of the products of each element into its 
 distance from the axis, and multiply that integral by 2 tt. 
 
124 
 
 STATICS. — SECT. VII. 
 
 SECTION VII. 
 
 OF THE MECHANIC POWERS. 
 
 1. The use of mechanic instruments, as far as they are 
 concerned in statics, is to enable us, by a force at our dis- 
 posal, to counteract another force, to which it is not op- 
 posed in direction, and to economize the force employed for 
 this purpose. The latter force is called the power, and the 
 former the resistance ; or simply the weight, when the re- 
 sistance is a weight to be equilibrated: and it is evident, 
 that the equilibrium shall be established, whenever the re- 
 sultant of the power and resistance is directed against some 
 fixed point, or immoveable obstacle : but then the equili- 
 brium is not, properly speaking, between those forces, but 
 between them, or their resultant, and the reaction of the 
 fixed point or obstacle. 
 
 Such being the way in which the equilibrium is estab- 
 lished by means of mechanic instruments, it is apparent 
 that there may be the greatest disparity between the anta- 
 gonist forces : and it is said that there is a mechanical ad- 
 vantage or disadvantage in the instrument, according as the 
 resistance is greater or less than the power by which it is 
 counteracted. 
 
 The more simple elements, into which machines are 
 resolvable, are called mechanic powers. These may be 
 classed according to their different structures and modes of 
 application : but the classification is in a great measure ar- 
 bitrary ; the precise difference which constitutes a distinc- 
 tion of class not being generally agreed on. Accordingly, 
 by some writers, they have been distributed under six, by 
 
THE LEVER. 125 
 
 some under seven, and by others, under eiglit classes or 
 heads. What renders the distribution of the mechanic 
 powers still more uncertain is, that being only different 
 means of effecting the composition or resolution of forces, 
 the operation, in all, is reducible to the same principle. By 
 each of them we are enabled to compound with the force to 
 be counteracted, a new force called the power; so that the 
 resultant of both shall be directed against the obstacle. 
 The differences, then, of mechanic instruments, relate not 
 to the principle of their operation, but to their structures 
 and modes of application ; and those who wish to treat the 
 subject in the most simple manner, will be inclined to refer 
 to the same class, such instruments as have an obvious re- 
 semblance in these respects. According to this rule, they 
 may be classed under three heads, viz. the lever, the rope, 
 the inclined plane. The first head comprising the balance, 
 and the axle in the wheel ; and the third head the wedge 
 and the screw. 
 
 THE LEVER. 
 
 2. The lever is a bar capable of angular motion round a 
 point called the fulcrum. To this, two forces are applied : 
 that which is to be counteracted is named the resistance ; 
 and the force by which it is to be equihbrated, the power. 
 The rectilinear distances of the points of application from 
 the fulcrum are called the arms of the lever. Thus in 
 (Fig. 46.) F is the fulcrum, round which the lever ab is at 
 liberty to turn ; p the power ; and w the resistance or weight 
 to be supported ; and the right lines, fa, fb, drawn from 
 the fulcrum to the points at which these forces are applied, 
 the arms of the lever. 
 
 It is evident that the weight of the instrument itself is a 
 force, which, if it does not pass on the obstacle, must come 
 in aid of one or other of the forces to be opposed to each 
 
126 STATICS. SECT. VII. 
 
 Other. But to deduce the consequences of certain condi- 
 tions, these conditions must be contemplated apart, even 
 from those with which they may be inseparably connected. 
 Wherefore, in considering the relation between the forces, 
 which are to equilibrate on a bar, with respect to angular 
 motion, the bar is regarded as a line without flexibility or 
 weight ; and the centre of angular motion as a fixed point. 
 When the weight of the instrument is to be taken into ac- 
 count, it is to be treated as a force applied at its centre of 
 gravity in a vertical direction ; and as making part of the 
 power or resistance, according as it conspires with one or 
 other of those forces. 
 
 3. For equilibrium, it is requisite and it is sufficient, that 
 the power and resistance should have a single resultant di- 
 rected through the fulcrum or fixed point. And this condi- 
 tion is equivalent to the three following: 1st. That the di- 
 rections of power and resistance should be in the same plane 
 which contains the fulcrum ; for if the power and resistance 
 did not act in the same plane, they could not have a single 
 resultant ; and if this plane did not contain the fulcrum, the 
 resultant, which is always in the plane of the components, 
 could not pass through this point. 2d. That the power and 
 resistance should tend to turn the lever round the fulcrum in 
 opposite directions ; for if the directions of the forces meet 
 at an angle, that of the resultant must lie within the same ; 
 i. e. the fulcrum must lie within this angle. And if the lines 
 of direction are parallel, that of the resultant, and therefore 
 also the fulcrum must lie between them or beyond them, ac- 
 cording as the component forces are directed to the same or 
 opposite sides of a line transverse to their directions. 
 3d. That the power and resistance should be reciprocally 
 proportional to the perpendiculars let fall from the fulcrum 
 on their lines of direction : for the components are recipro- 
 cally proportional to the perpendiculars let fall from any 
 point taken in the line of the resultant, and the fulcrum is in 
 
THE LEVER. 127 
 
 that line. Thus p, p' representing the power and resistance, 
 and^, // the perpendiculars from the fulcrum on their direc- 
 tions, this condition requires the following relation : 
 
 p : ^' wp \p. or, Yp — p//. 
 This relation may be otherwise expressed ; for I, V being the 
 lengths of the arms, and 0, 0' the angles which they make 
 with the directions of the forces, there is ^ = /. sin. 0, ^/= V. 
 sin.0'. whence 
 
 p . /. sin. == p'. V . sin. 0'. 
 The product of a force into the perpendicular from the 
 fulcrum on its direction, is called the moment of that force ; 
 and the three conditions of equilibrium may be briefly ex- 
 pressed, by saying, that " the moments of the forces should 
 be equal and opposite." 
 
 If the fulcrum is not a point in the lever itself, but only a 
 point of the surface of a body on which it may rest, it will be 
 requisite to add to the conditions of equilibrium as stated 
 above, that the resultant of the forces should be directed to 
 the fixed point, in a line perpendicular to the surface at the 
 contact. 
 
 4. Regarding the power as a force to be economized, it 
 is usual to divide the lever into three kinds, according to the 
 position of the fulcrum with respect to the power and re- 
 sistance. 
 
 The first kind of lever is that in which the fulcrum lies 
 between the power and the resistance or weight, as in 
 (Fig. 47.) 
 
 The second is that in which the resistance or weight is 
 applied between the power and fulcrum, as in (Fig. 48.) 
 
 And the third is that in which the power is applied be- 
 tween the resistance and fulcrum, as in (Fig. 49.) 
 
 In the first kind of lever, there is a mechanical advantage 
 or disadvantage, according as the perpendicular from the 
 fulcrum on the direction of the power is greater or less than 
 that on the direction of the resistance or weight. 
 
128 STATICS.— SECT. VII. 
 
 If the lever is straight, and the directions of the forces 
 parallel, the perpendiculars from the fulcrum on the direc- 
 tions of the forces are proportional to the lengths of the 
 arms. Therefore, in this case, the power acts at a me- 
 chanical advantage or disadvantage, according as it is ap- 
 plied to the longer or shorter arm of the lever. 
 
 In the other two kinds, the arms are always unequal; 
 and, therefore, the lever being supposed to be straight, and 
 the forces parallel in direction, there will be always a me- 
 chanical advantage in the second kind of lever, and a me- 
 chanical disadvantage in the third. 
 
 5. Hitherto two forces, only, were supposed to be applied 
 to the lever ; and in that case, it was required that they 
 should act in the same plane ; otherwise, they could not be 
 compounded into a single resultant. And it was further 
 required, that the plane of the forces shovdd pass through 
 the fulcrum ; otherwise, their resultant could not pass 
 through the fixed point. These things were contained in 
 the general statement that the moments should be directly 
 opposed. But when there are more than two forces ap- 
 plied to the lever, it is no longer requisite that the directions 
 of the forces should be contained in the same plane : but 
 then, it is required that the sum of the moments of those 
 forces should be cypher, in each of the three coordinate 
 planes passing through the fulcrum, i. e. that the three con- 
 ditions of equilibrium should be satisfied as stated in Sect. 
 V. Art. 2. relative to a system containing a fixed point. 
 
 If the fulcrum is a cylindrical axle, which allows the 
 lever no liberty of movement, but in a plane perpendicular 
 to that axle, the equilibrium is provided for, when the sum 
 of the moments of the forces, reduced to that plane, is cypher: 
 and if the lever only rests on a fixed point, the equilibrium 
 is established, only when the forces are reducible to a single 
 resultant directed against that point. 
 
 6. It is sometimes requisite to consider the strain on the 
 
THE LEVER. 129 
 
 fulcrum, with a view to the safety of the instrument, or in 
 estimating the effects of friction; and it is readily under- 
 stood, that when the fulcrum is a fixed point, the strain on 
 this point is the resultant of the forces applied to the arms 
 of the lever. This resultant passes through the fulcrum, 
 which may therefore be taken for its point of application, 
 where it will reproduce its components. Accordingly, the 
 strain on the fulcrum may be regarded as the resultant of 
 the forces acting immediately at this point, in lines parallel 
 to their proper directions. The magnitude of this resultant, 
 or the charge on the fixed point, will depend not only on the 
 magnitudes of the forces applied to the arms of the lever, 
 but also on the angle contained between their lines of direc- 
 tion; the greater limit being the sum of those forces, which 
 is the strain, when they are applied in parallel directions to 
 a lever of the first kind : and the lesser limit being the dif- 
 ference of those forces, which is the strain, when they are 
 applied in parallel directions to a lever of the second or third 
 species. 
 
 7. All those mechanical instruments are to be regarded as 
 levers, wherein the motion that ensues on a violation of equi- 
 librium is circular. Thus, a hammer, when used for the 
 purpose of drawing a nail, is a lever of the first kind ; the 
 power being applied at the end of the handle, the resistance 
 at the claw where it grips the nail, and the fulcrum being 
 the heel round which the instrument turns. Crow-bars, 
 also, are levers of this kind, when the power and resistance 
 move in opposite directions: as when the power, apphed at 
 one end, descends ; whilst the weight, at the other end, 
 ascends. But when both together ascend, or together de- 
 scend, the weight is then between the fulcrum and the point 
 of application of the power ; and the instrument is, in this 
 use of it, a lever of the second kind. 
 
 An oar of a boat is another example of the second kind of 
 
 K 
 
130 STATICS. — SECT. VII. 
 
 lever : the fulcrum being in the water, the resistance at the 
 row-lock, and the power applied by the hand of the rower. 
 
 A ladder to be raised against a wall, whilst one end re- 
 mains on the ground, affords an example of the third kind 
 of lever : the fulcrum being the end on the ground, round 
 which it turns ; the weight that of the ladder acting at the 
 centre of gravity, and in a vertical direction ; the power 
 being applied by the hands of the labourer at some interme- 
 diate point. It is evident that the labourer applies his force 
 with most advantage in the perpendicular to the ladder; 
 and that the nearer the ladder approaches to the vertical po- 
 sition, the less is the perpendicular from the fulcrum on the 
 direction of the weight ; and, therefore, the less the force 
 required to overcome it. 
 
 The limbs of animals, which are moved by muscular 
 power, are also levers of the third kind : the fulcrum being 
 at the head of the bone ; and the muscles acting between 
 this point and the centre of gravity of the limb, where the 
 weight of its parts may be supposed to be concentred. 
 This mode of action might seem to require a needless ex- 
 penditure of animal force. But it is to be considered, how 
 much is gained by this contrivance, in the lightness of the 
 limb and the celerity of its movements : and whether these 
 advantages are not wisely secured by an expenditure of 
 force, which the enormous power of the muscles, in con- 
 tracting, may so well afford. 
 
 A pole with a weight, carried by two men, may be viewed 
 as a lever of the first kind, if the weight be regarded as the 
 resistance of the fulcrum, and the bearers as acting against 
 each other : or it may be viewed as a lever of the second 
 kind, the force exerted by either of the bearers being re- 
 garded as the reaction of the fulcrum, and that exerted by 
 the other as the power by which the weight is supported. 
 Whilst the forces exerted are vertical, the sum is only equal 
 to the weight to be carried, and the portions of this sup- 
 
THE LEVER. 131 
 
 ported by the two men, are reciprocally proportional to their 
 distances from its centre of gravity. But if the forces ex- 
 erted by the bearers are not vertical, their directions must 
 meet at some point of the vertical passing through the 
 centre of gravity of the load ; and as in the parallelogram of 
 forces, the sum of the two contiguous sides is greater than 
 the diagonal; so the sum of the forces exerted must exceed 
 the weight to be supported : and they are to each other reci- 
 procally as the perpendiculars, let fall from the centre of 
 gravity of the load upon their lines of direction. Whilst the 
 pole is horizontal, each of the bearers would suffer from this 
 mode of exertion. For, let the pole be ab, (Fig. 50. ) ; let 
 the weight w be applied at the point o ; and let p, p' be the 
 forces exerted by the bearers at a and b, in directions meet- 
 ing the vertical passing through o, at the point c ; and let Bd 
 be perpendicular to ac. Then, 
 
 BO 
 
 Bd 
 
 But w and bo, being constant quantities, the force p would 
 vary inversely as Bd; i. e. inversely as the sine of the angle 
 CAB. And in the same way it appears, that the force p' va- 
 ries inversely as the sine of the angle cba. But when the 
 pole is not horizontal, as when it is carried up or down a 
 hill, the lowermost bearer must act, partly, by pushing or 
 shoving ; and the uppermost by pulling or drawing : and 
 this mode of exertion being inconvenient to the latter, he 
 will naturally be disposed to relieve himself at the expense 
 of his fellow. To see to what extent he may diminish his 
 burden, let ab be the pole in its inclined position, (Fig. 51.) 
 and let Bd, Be be perpendicular to ca, co. Then, 
 
 Be 
 Bd 
 
 in which Be, depending on the slope, is to be considered as a 
 constant quantity. Wherefore, p shall vary inversely as Bd, 
 
132 
 
 STATICS. — SECT. VII. 
 
 and shall, therefore, be least when this perpendicular is 
 greatest, i. e. when bc? coincides with ba ; or when ac is per- 
 pendicular to ab. The force then exerted at a will be given 
 by the equation 
 
 Be 
 p = w X — . 
 
 AB 
 
 which is less than would be required in the horizontal posi- 
 tion of the pole, m the ratio of Be to bo ; i. e. in the ratio of 
 the cosine of the angle of elevation to radius. The addi- 
 tional exertion required of the lowermost bearer will be 
 greater than that from which the uppermost thus relieves 
 himself; inasmuch as, in this case, the sum of the forces ex- 
 ceeds the weight to be supported, 
 
 8. Sevei'al levers may be combined together ; and then 
 the action of any one of them on the next that follows, is the 
 power by which the latter is worked. Thus, abcd (Fig. 52.) 
 represents a combination of three levers of the first kind ; 
 f', f", f'", being the fulcra. The manner in which the action 
 is conveyed from one extremity to the other, is as follows : 
 A, the extremity of the first lever, being depressed by the 
 power, the other end of that lever is raised ; and together 
 with it b, the end of the second lever, which presses on it. 
 This elevation of the nearer end of the second lever produces 
 a depression of its remoter end ; and, therefore, of c, the 
 nearer end of the third lever, by which its remoter end, d, 
 is raised. 
 
 The mechanical advantage of this combination is readily 
 understood. The power applied at a is to the force which 
 it equilibrates at b, as f'b to f'a. The force at b is to that 
 which it equilibrates at c, as f"c to f"b. And the force at c 
 is to that which it equilibrates at d, as f'"d to f"'c. Where- 
 fore, the power applied at a is to the weight which it will 
 sustain at d, as f'b Xf"c Xf"'d, to f'a Xf"b Xf"'c ; i. e. as the 
 product of the several powers to that of the several weights. 
 
THE LEVER. 1J3 
 
 which they would balance in the levers taken separately. 
 Thus, if 
 
 f'a :f'b::3:2. f"b : f"c : : 7 : 3. f"'c : f'"d : : 5 : 2. 
 
 Then, 
 
 f'b Xf"c Xf'"d : f'a xf"b Xf"'c ::^x3x2:3x7x5. 
 
 or as 4 to 35. So that a force equivalent to a weight of four 
 pounds at a, shall sustain a weight of thirty-five pounds at d. 
 And if the power at a were made greater than this, the 
 weight at d would be lifted. 
 
 Such are the conditions of equilibrium in the lever, and 
 such, in general, is the manner in which this instrument is 
 applied. But there are certain instruments of this class, 
 which, from some peculiarity in their uses or structure, would 
 demand a more particular consideration. 
 
 9. The balance is a lever, applied to ascertain the weight 
 of a body. 
 
 There are five different ways in which the equilibrium 
 may be established : and these furnish so many different in- 
 dications of the weight of the body under examination. 
 1st, By adjusting the weight of the counterpoise acting at a 
 given distance from the fulcrum or axle. 2d, By changing 
 its distance from the fulcrum. Sd, By changing that of the 
 body to be weighed from the same point. 4th, By shifting 
 the fulcrum. The fifth indication is the inclination of the 
 beam when it composes itself. 
 
 Of these several indications, the first is most generally re- 
 sorted to ; recommended, no doubt, by its more extensive 
 application ; as also, by its superior accuracy. For this rea- 
 son, the adjustments to be attended to in a balance con- 
 structed on this principle, shall be more particularly con- 
 sidered. 
 
134 STATICS. — SECT, vir. 
 
 THE BALANCE FOR EQUAL WEIGHTS. 
 
 10. In this balance the body whose weight is to be as- 
 certained and its counterpoise are placed in dishes, or other- 
 wise suspended from the extremities of the beam, and are 
 therefore to be regarded as acting immediately at the points 
 of suspension ; and the fulcrum on which the beam turns is 
 situated between those points : the instrument is evidently a 
 lever of the first kind, whose arms are the right lines drawn 
 from the fulcrum to the points of suspension. Now, as the 
 equilibrium is to be established between equal weights, it is 
 plainly requisite that the anns should be of equal lengths ; 
 and at first view this may appear sufficient. But as this kind 
 of balance is frequently employed in cases where the most 
 scrupulous accuracy is desired, it is necessary to consider 
 more particularly the properties which are required, and the 
 adjustments to be attended to. 
 
 For the indication of the equality of the appended weights, 
 it is necessary that the beam should compose itself in some 
 certain position ; and the horizontal position is obviously the 
 most convenient. Wherefore, the property first required in 
 this balance is, that it should rest in none but the horizontal 
 position when loaded with any equal weights, and therefore 
 also when unloaded. The second is, that it should quickly 
 right itself when disturbed from that position : this is called its 
 stability. The third property required is that it should in- 
 dicate a small difference of weights by its deviation from the 
 horizontal position. This is called its sensibility. It now 
 remains to be shown how these perfections are to be at- 
 tained. 
 
 11. The balance cannot have the first of these properties 
 unless the common centre of gravity of the beam and equal 
 weights lies beneath the axle ; for were it at this point, the 
 balance would rest indifferently in any position ; and were it 
 
THE BALANCE FOR EQUAL WEIGHTS. 135 
 
 above the axle, the smallest deviation from the vertical pass- 
 ing through this point would cause the balance to upset. 
 
 It is necessary then that the common centre of gravity of 
 the beam and equal weights should lie beneath the axle ; but 
 this is not enough. In order that the equality of the weights 
 should be indicated by a certain position of the beam, it is 
 necessary that the right line drawn from the fulcrum to the 
 centre of gravity of the unloaded beam should bisect the hne 
 connecting the points of suspension. To show this, let aob 
 be the balance, (Fig. 53.) o its fulcrum, b the point at which 
 the line ab is bisected. This point is the centre of gravity 
 of the equal weights, where they may be supposed to be 
 concentred. Let c be the centre of gravity of the unloaded 
 beam, then the common centre of gravity of the beam and 
 equal weights shall be in some point of the line be, as at x, 
 dividing the line be into segments xb, ccc, which are reci- 
 procally proportional to the masses concentred at b and c, 
 and therefore changing its place in the line cb with every 
 change in the magnitude of the load. Hence it appears, 
 that if the line be does not pass through the fulcrum o, the 
 line ox shall make different angles with the line ab ; and 
 since the line ox is necessarily vertical when the balance is 
 at rest, it follows that the position of ab shall change with 
 every change in the magnitude of the load. 
 
 It is plain that when the line be passes through o, the 
 position of equilibrium for equal weights is independent of 
 their magnitude. 
 
 But it is not sufficient that the equality of the weights 
 should be indicated by the same unvaried position of the 
 beam, that position must be horizontal, and the former con- 
 dition being satisfied, this last is provided for by making the 
 arms of equal lengths. For in the last figure the line ob 
 being vertical when the balance is at rest, if ab is horizontal 
 the angles at b are equal, and therefore the lines oa, ob are 
 also equal: and conversely, if these lines are equal, the 
 
136 
 
 STATICS. — SECT. VII. 
 
 angles at b must be equal, i. e. right, and therefore the line 
 AB horizontal. 
 
 If the arms are of unequal lengths, the lines oh, hh. 
 (Fig. 54.) being respectively equal to oh. b6 as before, and 
 oA, OB being unequal, the angles at h are unequal, and oh 
 being vertical, ab shall be no longer horizontal, but in- 
 clined. 
 
 This suggests a ready mode of trying whether the condi- 
 tion relative to the lengths of the arms is fulfilled. For it is 
 requisite only to interchange the equilibrating weights ; and 
 if the equilibrium still subsists, the arms are precisely of 
 equal lengths. Were it otherwise, the weight which had 
 acted by the shorter arm, being the greater weight, now 
 that it acts by the longer arm, must necessarily prepon- 
 derate. 
 
 A balance which does not stand this test, though it may 
 rest in the horizontal position when unloaded, is a deceitful 
 balance: and the commodity to be dealt out, if placed at the 
 end of the longer arm, is deficient in weight ; bearing to that 
 of the counterpoise, the ratio of the shorter to the longer 
 distance. If placed in the dish at the end of the shorter 
 arm, and again counterpoised, the weight of the new coun- 
 terpoise shall bear to that of the commodity the same ratio, 
 as the latter to that of the first counterpoise : wherefore, 
 the true weight is a geometrical mean between those by 
 which it is counterpoised, when weighed in the opposite 
 scales, and may be ascertained accordingly. Thus, putting 
 p for the weight of the commodity ; I for the longer line bc ; 
 and /' for the shorter ac; and when p is appended by the 
 longer arm, let a be its counterpoise : we shall then have 
 
 p./=: A./'. 
 
 Again, when appended by the shorter arm, let b be its coun- 
 terpoise, and the equation will be 
 
 p.r=B./. 
 
THE BALANCE FOR EQUAL WEIGHTS. 137 
 
 To exterminate /, /', let these equations be multiplied, and 
 we get 
 
 F^ zz A.B. or p = -v/ (a . b). 
 
 A still more easy way of using such a balance would be, 
 after counterpoising the body, to take it out of the scale, 
 and restore the equilibrium by a weight. For those weights 
 are necessarily equal, M^iich equilibrate with the same coun- 
 terpoise in the same circumstances. 
 
 In order that the lengths of the arms should remain un- 
 varied, in all positions of the beam, it is requisite that the 
 bodies, weighed against each other, should not be applied 
 to the arms in any considerable part of their surface ; other- 
 wise the vertical, passing through the centre of gravity of 
 the body, shall meet the arm in a point whose distance from 
 the axle is variable with the inclination of the beam ; and the 
 effective length of the arm shall be changed accordingly. 
 To render the points of apphcation invariable, the ends of 
 the arms are formed into rings, and bent over at right 
 angles. Into these are inserted the hooks, which are to 
 carry the dishes or weights; and the bearings being reduced 
 to knife edges, the points of suspension are invariably the 
 same. 
 
 12. Another of the perfections, which were said to be 
 required in a balance, is stability. This relates to the cele- 
 rity with which it rights itself, when disturbed from the 
 horizontal position ; and, therefore, it depends on the force 
 of restitution, which may be thus estimated. 
 
 Let AB be the line connecting the points of suspension ; 
 (Fig. 65.) w the weight of tlie unloaded balance ; p one of 
 the equal weights ; g the common centre of gravity of the 
 whole ; and o the axle. Then, putting a for og, and 
 for the angular deviation of og from the vertical, or of ab 
 from the horizontal position, the force of restitution will 
 be 
 
138 STATICS. — SECT. VIT. 
 
 (2 p -f- w) a. sin. Q. 
 
 i. e. if is of given magnitude, the force of restitution will 
 be as 
 
 (2 p 4- w) «• 
 
 13. The third of the perfections required in a balance 
 is its sensibility ; which is estimated by the difference of the 
 weights required to produce a given inclination of the beam 
 in relation to their sum; the sensibility being the greater, 
 as this difference is less, in relation to the entire load. 
 
 In order to see on what this property depends, let iv be 
 the additional weight thrown into the scale appended from 
 A, (Fig. 55.) The effect of this will be, to remove the 
 common centre of gravity of the balance and weights from 
 g towards a, as to g\ The balance will now compose itself, 
 so that og' shall be vertical ; and the deviation of the beam 
 from the horizontal position is measured by the angle g'og. 
 
 Now, for the relation between this angle and the quan- 
 
 2 p-l-w 
 tity ^^^— , there is in the bent lever Aog, the weight w, 
 
 at A, in equilibrio with 2 P + w, at ^. Wherefore, putting 
 L for oA, there will be 
 
 w . L . sin. ang. Aog'— {2t -\-w) a. sin. 0. 
 
 But when ab passes through the axle, the angle Aog is right, 
 and Aog' is the complement of 0, which gives 
 
 „ W.I. 
 tan. t/ ZZ rr: ; r . 
 
 (2 p + w) a 
 
 Though the line ab should not pass through the axle, 
 yet the angle oab is always exceedingly small ; and, therefore, 
 Aog nearly a right angle. So that the error will not be, in 
 any case, considerable, if Aog' is treated as the complement 
 of 0. Accordingly, the preceding equation will serve gene- 
 rally for the straight beam. If then, is of given magnitude, 
 
THE BALANCE FOR EQUAL WEIGHTS. 139 
 
 O p _L. -yy L , 
 
 , shall vary as -: which shews that the sensibiUty is 
 
 greater, as the length of the arm is increased, and as the 
 distance of the common centre of gravity from the axle is 
 diminished. 
 
 14. If the centre of gravity of the unloaded beam is in 
 the line connecting the points of suspension, and at its middle 
 point, as required Art. 11. the length of the line a, shall be 
 independent of the magnitude of the appended weights ; and 
 
 -, which is the measure of the sensibility, will not be affected 
 
 by any change of load. 
 
 But if the line connecting the points of suspension does 
 not pass through the centre of gravity of the balance itself; 
 it is evident, that as the magnitude of the appended weights 
 is increased or diminished, the common centre of gravity of 
 beam and weights shall approach this line or recede from 
 it accordingly : and that a being consequently a variable 
 quantity, the sensibility and stability of the balance shall 
 vary with the load in a manner easily understood. 
 
 Thus, if the line connecting the points of suspension falls 
 below the centre of gravity of the unloaded balance, the 
 sensibility shall be diminished, and the stability increased, 
 as the weights are increased. And if the same line falls 
 above the centre of gravity of the unloaded beam, between 
 it and the axle, the effect will be the contrary. And finally, 
 if the same line falls above the axle, the balance will serve 
 only for weights below a certain limit. This limit of 2 p de- 
 pends on w, the weight of the balance, and on the distances 
 of the axle from the centre of gravity of the unloaded balance, 
 and from the line connecting the points of suspension. For 
 let o be the axle ; g the centre of gravity of the unloaded ba- 
 lance ; and let the line^o meet the line ab above the point o, 
 as ate, (Fig. 56.) This point c is the centre of gravity of 2p, 
 
140 STATICS. — SECT. Vll. 
 
 and it is evident that as this weight is increased in relation to 
 w, the common centre of gravity of the whole shall ascend 
 from g towards c; and that when it arrives at o, the balance 
 becomes useless. In that case we have 
 
 2 P X o<? = w X o^. 
 and therefore, 
 
 o Off 
 
 oc 
 At this limit the balance shall rest indifferently in any po- 
 sition. And if the weights are further increased, the com- 
 mon centre of gravity shall be raised above the axle ; and 
 the balance, on receiving the smallest inclination, shall 
 upset. If the point g lies above, and the point c below the 
 axle, the balance will serve only for weights above a certain 
 limit. This limit being expressed as before. 
 
 The qualities of stability and sensibility are not to be 
 combined together : and if the balance is improved in one 
 respect, it must be injured in the other ; as will more fully 
 appear from the theory of pendulums. Therefore, nice 
 balances are often provided with an adjusting strew ; by 
 which the distance of the common centre of gravity from the 
 axle may be varied at pleasure, according to the degree of 
 sensibility required, by the uses to which the balance is to 
 be applied. 
 
 Various contrivances have been adopted for lessening the 
 weight of the beam without prejudice to its strength or stiff- 
 ness. In ordinary cases, this is supposed to be sufficiently 
 provided for by the shape and material of the beam. This 
 is made of steel. Its thickness is far less than its depth ; 
 and this latter decreases from the axle to the extremities. 
 The sensibility of the balance is also affected by the friction 
 between the axle and its supports. This is diminished, by 
 attending to the material and polish of those parts, and by 
 the form of the axle, which is that of a knife edge. 
 
THE STEEL YARD. 141 
 
 THE STEEL YARD. 
 
 15. The balance already examined is, perhaps of all, the 
 most accurate. Other constructions are, however, frequently 
 employed, recommended by their simplicity; but chiefly by 
 the promptitude of their indications, which is often of more 
 value than extreme precision. 
 
 The steel yard is a beam with arms very unequal in 
 length. The commodity to be weighed is suspended by a 
 hook at the end of the shorter arm, and is counterpoised by 
 a sliding weight, hung by a steel edge on the longer arm. 
 This arm is graduated ; and the weight of the commodity is 
 indicated by the division at which the counterpoise is 
 placed, when the equilibrium is established. This, as well 
 as the balance before described, is a lever of the first kind : 
 and it is evident, that if at the first division, the counter- 
 poise is in equilibrio with a certain weight, appended at the 
 shorter arm ; it shall balance twice that weight, when re- 
 moved to the second ; thrice when removed to the third 
 division, and so forth. If, for example, the length of the 
 shorter arm is two inches, and the counterpoise a weight of 
 two pounds ; then, when placed at the distance of two 
 inches from the axle, it shall balance a weight of two pounds, 
 hanging from the shorter arm ; at the distance of four 
 inches, one of four pounds ; and at six inches, one of six 
 pounds ; every inch of the scale, in this case, correspond- 
 ing to a pound weight. These divisions may be conveni- 
 ently subdivided into parts of the eighth of an inch ; and 
 then the same balance will shew a difference of two ounces. 
 
 16. It is not requisite that the beam should compose 
 itself in the horizontal position when unloaded, and it is 
 therefore commonly made to hang vertically, as the most 
 convenient position, But in that case, the zero of the scale 
 
li'2 STATICS. — SECT. Vll. 
 
 is not at the axle : for the moment of the beam conspires 
 with that of the commodity or of the counterpoise, accord- 
 ing as it is the shorter or the longer arm of the unloaded 
 balance that preponderates : and for equilibrium, the equa- 
 tion will not be between the moments of the commodity and 
 counterpoise ; but between the moment of one of these, and 
 that of the other increased by the moment of the beam. 
 Therefore, it cannot be said that the weights of the commo- 
 dity and counterpoise are inversely as their distances from 
 the axle, i. e. the zero of the scale is not at this point. 
 
 To find the point from which the divisions should com- 
 mence, let p be the weight of the commodity ; ^j that of the 
 counterpoise ; and h the point where it is to be placed to 
 render the beam horizontal, when otherwise unloaded. 
 (Fig. 57.) It is plain that p X uo is the moment of the 
 beam. Therefore, when the equilibrium is established in 
 the loaded beam, the weight p being at a, and the counter- 
 poise at K, the equation is 
 
 P X AO ZZp X HO -\-p X KO =zp X HK, 
 
 Accordingly, the zero of the scale is at h ; and if from this 
 point, the portions hb, bc, cd, &c. are measured off, each 
 equal to ao, we shall have 
 
 p z:zp, p = 2p, P =. Sp, &c. 
 
 as the counterpoise is placed at b, c, d, &c. 
 
 17. There are two limitations to the use of this balance, 
 viz. the shortness of its range, and the coarseness of its in- 
 dications. 
 
 When the counterpoise is carried to the end of the scale, 
 it is in equilibrio with a weight greater, only, in the ratio of 
 the longer to the shorter arm : so that if the lengths are as 
 ten to one, and the counterpoise a weight of two pounds, 
 the balance shall weigh only to twenty pounds. 
 
 This imperfection may be remedied, in part, by chang- 
 ing the counterpoise, according to the magnitude of the 
 
THE STEEL YARD. 143 
 
 weight to be examined. In this way, the value of each di- 
 vision is changed with that of the counterpoise. 
 
 Another method is that of furnishing the shorter arm 
 with a second hook. If this hangs at half the distance of 
 the former, the shorter arm is reduced to one-half its former 
 length, and the value of each of the divisions on the longer 
 arm is doubled. 
 
 With respect to the second limitation, it is to be ob- 
 served, that the accuracy of weighing consists in the small- 
 ness of the difference of weights, which the balance will 
 indicate, in relation to the load it carries. If the smallest 
 divisions indicate ounces ; then, unity divided by the num- 
 ber of ounces in the weight of the body examined, shall 
 denote the degree of accuracy to which the weight is ascer- 
 tained. This is the same fraction, as the smallest division 
 of the scale, divided by the distance at which the counter- 
 poise is placed to equilibrate the body weighed. So that if 
 the counterpoise stands at fifty inches, and the smallest di- 
 visions are -^q inch, the nicest indication will be the j^^ of 
 the weight. This is far short of the nicety of the common 
 balance last described. A well-made balance of that con- 
 struction would shew a difference of the millionth part of 
 the entire weight. So much more accurate are the ordinary 
 methods of weighing than of measuring. 
 
 The want of a minuter division of the scale may be 
 partly supplied by a second and smaller sliding weight. If, 
 for example, the lighter be the y^^ of the heavier ; then, the 
 motion of the lighter through one division, produces the 
 same change of momentum, as would the motion of the hea- 
 vier, through Jq of a division : and this simple contrivance 
 answers the same purpose as a division of the scale ten 
 times as minute. In using the two counterpoises, the hea- 
 vier should be placed at the division nearest to equilibrium, 
 and the defect is shewn by the division to which the smaller 
 is applied, to render the equilibrium perfect. Thus, if the 
 
144 
 
 STATICS. — SECT. VII. 
 
 divisions with the greater counterpoise denote ounces, and 
 the weight of the body is found by it to be greater than 
 101b, 6oz. but less than 101b. 7oz. ; then, if the lighter be 
 y\j of the heavier, and that the equilibrium is made perfect 
 by advancing it to the third division, the weight of the body 
 under examination is 101b. 6.3oz. 
 
 In making a comparison of this with the ordinary ba- 
 lance, it should be observed that the load on the fulcrum 
 exceeds the weight of the body under examination only by 
 the weight of the counterpoise ; which is, ordinarily, much 
 less than it : whereas, the common balance is loaded with 
 twice the weight of the commodity. This is a trifling ad- 
 vantage, and it is far more than counterbalanced by the 
 disadvantages already mentioned. However, in cases where 
 expedition is more desirable than extreme precision, this 
 instrument is highly valuable. 
 
 THE BENT LEVER BALANCE. 
 
 18. The balance which gives the readiest indications is 
 the bent lever balance ; such as the corn balance, or the 
 yarn balance, represented in (Fig. 58.) We have only to 
 place the matter to be weighed in its scale ; the balance 
 produces its own equilibrium, and this most speedily, be- 
 cause of its great stability : the index, without any further 
 trouble, declares the weight. 
 
 To ascertain the principles on which it should be gradu- 
 ated, let the balance be that represented in (Fig. 58. No. 1.) 
 the arm which carries the scale being at right angles with 
 the index, and this index being vertical before the weight is 
 introduced ; and, therefore, passing through the common 
 centre of gravity of the balance with its scale and given 
 counterpoise. Then, if a weight is placed in the scale, the 
 common centre of gravity of the instrument and weight shall 
 take the position vertically beneath the axle ; the index as- 
 
THE BENT LEVER BALANCE. 
 
 145 
 
 cending through a certain arc 6 ; and co.9 will become 
 the angle, made between the arm which carries the scale 
 and the vertical. Let / be the length of this arm ; a the 
 distance of the centre of gravity of the instrument from the 
 axle ; p the weight of the commodity in the scale ; and p 
 that of the instrument. Then, we shall have .,._ 
 
 JO -a <\%r5uU^ 
 
 F.l.cos.u :zzp.a.sin.U. '- 
 
 and therefore, 
 
 p. a. ta.n.9 
 
 '=—1— 
 
 But ^-^ is a given quantity ; wherefore, p varies as 
 
 tan.9, i.e. as the tangent of the angle through which the 
 index has moved. If, then, by placing in the scale a weight 
 of one pound, the index is moved through a certain angle ; 
 by a weight of two pounds, it shall be moved through an 
 angle whose tangent is twice ; and by a weight of three 
 pounds, through one whose tangent is thrice as great ; and 
 so forth. Accordingly, to graduate the arc, beginning at 
 the point where the index rests when the balance is un- 
 loaded, it is only requisite to find the arc through which 
 the index is carried by a known weight ; and putting unity 
 for its tangent, to measure the other arcs from the same 
 point to the several points of division, such, that their tan- 
 gents shall be represented by the numbers 2, o, 4, &c. 
 Thus, let OA (Fig. 59.) be the arm to which the weight is to 
 be appended ; this being supposed to be horizontal when 
 unloaded, and the index in the vertical ov. Then a known 
 weight, e. g. of one pound being appended, let the arm 
 take the position oa, and the index that of oa. In the tan- 
 gent vs, taking ab, be, cd, Sec. each equal to va, the lines 
 ao, bo, CO, do, &c. drawn to the centre shall mark the divi- 
 sions on the arch. 
 
 In this explanation it was supposed that the arm which 
 
 L 
 
146 STATICS. — SECT. VII. 
 
 carries the dish was horizontal, and the index vertical, when 
 the instrument was unloaded, i. e. that it passed through 
 the centre of gravity of the unloaded instrument. These 
 conditions are not necessary. The arm and the index may 
 have any positions. It is enough that the zero of the scale 
 should be at the point marked by the index when the balance 
 is unloaded : for, being fixed to the bent lever, it shall par- 
 take of its angular motion ; and, therefore, describe the same 
 angle as its centre of gravity. Accordingly, the index itself 
 is commonly made to serve for the counterpoise : but if it is 
 desired that the weights in the scale shovdd be as the tan- 
 gents of the arches described, it will be requisite that the 
 arm which carries the dish, should be at right angles to the 
 line drawn from the axle to the centre of gravity of the in- 
 strument, i. e. that the arm should be horizontal when the 
 scale is empty. 
 
 A quadrantal arc would give an infinite range ; the tan- 
 gent of a right angle being infinite. But as the arches, 
 whose tangents are equidifFerent, increase by differences 
 which rapidly diminish, and as the weight which the instru- 
 ment would carry is limited ; it is evident, that a graduated 
 arc, much less than a quadrant, will serve for all practical 
 purposes. 
 
 THE WEIGHING MACHINE. 
 
 19. The machine, the best that has been hitherto con- 
 trived for weighing heavy burdens, is that called the weigh- 
 ing machine. The scale is a platform, on which the cart or 
 dray is driven; and the load, though of many hundreds 
 weight, is balanced by a few pounds. 
 
 The equilibrium between weights so very unequal, is 
 produced by means of a compound lever, as shewn in 
 (Fig. 60.) where akb is a lever, having two fulcra at a and 
 B. Die is another lever precisely similar, having its two 
 
THE DANISH BALANCE. 147 
 
 fulcra at d and c. At e, f, g, h, are upright pins, on 
 which the platform rests ; and by these the load on the plat- 
 form is made to press on the levers aeb, dec. These, then, 
 are levers of the second order : and the pins being equidis- 
 tant from the fulcra, the weight sustained by any one of 
 them, as by the former at e, is to the pressure produced by 
 it at K in the ratio of ak, to ae. The pressure thus re- 
 duced is communicated to a lever, ml of the second order, 
 whose fulcrum is at l, and would be counteracted by a force 
 
 at M, less in the ratio of lk to lm. So that if ae zz — , and 
 
 LM 
 
 LK = — 7 , a hundred weight on the platform would be coun- 
 
 o 
 
 terpoised by a weight of two pounds acting at m. To dimi- 
 nish, yet further, the weight requisite for equilibrium, the 
 point M is connected with the shorter arm of a balance, 
 whereby the counterpoise is lessened in the ratio of the 
 shorter to the longer arm. This balance may be a steel 
 yard ; and then the load is indicated by the division to 
 which the sliding weight is brought, in order to produce 
 equilibrium. Or else, the longer arm may be furnished 
 with a dish appended to its extreme point ; and then the 
 load is ascertained by the weight in the dish. If the longer 
 arm is only four times that of the shorter, it is evident that 
 in the example given above, every hundred weight on the 
 platform would be counterpoised by a weight of half a 
 pound. 
 
 THE DANISH BALANCE. 
 
 20. The most portable balance, as well as the most 
 simple in its structure, is that in which the equilibrium is 
 established by the movement of the fulcrum: such is the 
 balance ab, (Fig. 61.) called the Danish balance. This is 
 simply a rod with a knob at one end to serve as a counter- 
 
 L 2 
 
148 STATICS. — SECT. VII. 
 
 poise, and a hook at the other end to carry the commodity 
 to be weighed. The equiUbrium is established by shifting 
 the place of the loop, by which the whole is sustained ; and 
 the weight is indicated by the mark at which the loop is 
 then placed. 
 
 The principle of the gi'aduation is easily understood. 
 The weight of the rod, including the fixed counterpoise a, 
 may be supposed to be applied at c, the common centre 
 of gravity ; and before the load is appended, the instru- 
 ment would be supported horizontally by the loop at this 
 point. But if a weight is hung from the hook, the equili- 
 brium is established by shifting the loop to the common 
 centre of gravity of the weight and balance, i. e. to a dis- 
 tance from c towards b, depending on the magnitude of the 
 appended weight. Therefore, the part of the rod between 
 its centre of gravity, and the end which carries the weight, 
 is to be graduated, so that the divisions, reckoning from the 
 ■former point, shall correspond to weights increasing by 
 some common difference. These divisions are of unequal 
 magnitudes ; but they are readily calculated. 
 
 For let p denote the weight of the instrument, including 
 the hook and fixed counterpoise ; p the appended weight ; 
 and let e be the point to which the loop is to be shifted for 
 equilibrium. Then putting a, for cb ; x, for ce ; we have 
 
 x\a — X \\ P :^>. and a; : « : : P : p '\-p. 
 
 Wherefore, 
 
 p 
 
 X — — ; — . a. 
 
 p-fi? 
 
 Which, as a and^ are known quantities, gives the value of 
 X for every assumed value of p. Thus, if p were eight 
 ounces, and it were required to weigh to one ounce : there 
 would be, for the first division, or the distance of the first 
 mark from c, 
 
 1 a 
 
THE AXLE IN THE WHEEL. 
 
 For the second mark, 
 
 149 
 
 For the third. 
 
 For the fourth, 
 
 and so forth. 
 
 2 a 
 
 3 3 
 
 4 a 
 
 THE AXLE IN THE WHEEL. 
 
 21. The lever is evidently an instrument of great power 
 in overcoming a resistance ; or even in communicating mo^ 
 tion to large masses. But it would seem that the space 
 through which they could be moved by it must be small ; 
 and it might be supposed that its usefulness would be limited 
 by this condition. 
 
 There are, however, various methods of converting the 
 reciprocating motion of the lever, into one that shall be 
 rotatory or progressive. 
 
 The most simple contrivance for applying the principle 
 of the lever to the production of progressive motion, is that 
 of the axle in the wheel. This is an instrument consisting 
 of a wheel fixed on a cylinder, which turns along with it. 
 The moving power is made to act at the circumference of 
 the wheel by a strap or cord ; by coggs or teeth ; or some 
 other equivalent contrivance : and the weight is, in like 
 manner, applied at the surface of the cylinder or axle. 
 
 The axle in the wheel is therefore a lever of the first 
 kind ; the arm by which the power acts being the radius of 
 the wheel, and that to which the weight is applied, being the 
 radius of the axle. Moreover, as the forces act in the di- 
 
150 STATICS. — SECT. VII. 
 
 rection of the tangents to the wheel and axle, their direc- 
 tions are perpendicular to the arms of the lever : wherefore 
 putting R,r, for the radii, p for the power, and w for the 
 
 weight, there is for equilibrium pr — wr. or — = -. 
 
 But the axle in the wheel is a broken lever : the radii 
 of the wheel and cylinder, which are the arms, being con- 
 nected by the intervening portion of the axis, which meets 
 them at right angles. And it may be supposed that the 
 legitimacy of this application of the theory of the straight 
 lever, to one thus distinguished from it, would require to 
 be demonstrated. This may be done in several ways, of 
 which the following is, perhaps, as simple as can be de- 
 sired. 
 
 Let the force p, acting in the direction mn, be transferred 
 to c, the centre of the wheel, (Fig. 62.) by applying at this 
 point two forces, each equal to p, acting in the opposite di- 
 rections, CE, CF, parallel to mn. This, which produces no 
 change in the system, gives a force + p, pressing on the axis 
 of the cylinder at c ; and a pair of forces + p, — p, acting in 
 the directions mn, of, whose moment is p.r. The force, 
 which is the weight w, being treated in the same way, gives 
 a force + w pressing on the axis at c, and a pair whose mo- 
 ment is w.r. The pressures on the axis, which is a fixed 
 line, are necessarily equilibrated by its resistance : where- 
 fore, the only condition required for equilibrium is, that the 
 resultant of the moments should be cypher. And as the 
 planes of the moments are parallel, this condition requires 
 that their sum should be cypher, (Sect. III. Art. 2.) i. e. 
 
 p r 
 P.R — W.r :^ 0. or — := - as before, 
 w R 
 
 In this statement no account has been taken of the thick- 
 ness of the rope. If, for greater accuracy, this is to be con- 
 sidered, the forces may be supposed to be applied at its 
 axis ; and then, to the radius of the wheel or cylinder is to 
 
THE AXLE IN THE WHEEL. 151 
 
 be added half the thickness of the rope hy which it is enve- 
 loped. Thus, let the thickness of the rope which passes 
 round the wheel be 2t; and that which passes round the 
 cylinder 2 1' ; then for equilibrium the equation will be 
 
 w ~ K-\-t' 
 
 22. With respect to the pressure on each of the points 
 of support, it is to be observed, that this can arise only from 
 the force equilibrated by the resistance of the axis, i. e. from 
 the forces p and w transferred to this line ; and from the 
 weight of the engine itself, acting at its centre of gravity, 
 which is in the same line. The pressure made by each of 
 these forces is to be estimated as for a system secured by two 
 fixed points: i. e. each force is to be resolved into two pa- 
 rallel forces, acting at those points. The total pressure on 
 each point of support is then found, by compounding the 
 forces exerted at that point. Thus, putting a for s*, the dis- 
 tance between the points of support ; m for cs ; the pressures 
 produced by p, at s and s, will be 
 
 a — m m 
 
 p. . p. — . 
 
 a a 
 
 In the same manner, putting ti for os, the distance of the 
 
 weight from the same point of support, the pressures made 
 
 by w, at s and s, shall be 
 
 a — w n 
 
 w. . w. -. 
 
 a a 
 
 Finally, let w denote the weight of the instrument itself; d 
 the distance of its centre of gravity from s ; the pressures 
 made by this force at the same points, will be 
 
 a — d d 
 
 w. . w. -. 
 
 a a 
 
 Wherefore, the total pressure at s is the resultant of the 
 
 „ a — m ,w{a — n)-\-iv(a—d) ^ ,., ,, „ 
 iorces, p. , and —^ ^^ , of which, the former 
 
152 STATICS. — SECT. VII. 
 
 acts in a direction parallel to p, and the latter in the vertical. 
 
 Ill 
 And the pressure at s is the resultant of the forces p. — , 
 
 a 
 
 and , acting m du'ections respectively parallel to 
 
 the former. 
 
 If the direction of the force p is vertically downwards, 
 the pressure at s will be 
 
 a 
 
 and that at s, will be 
 
 F.ni + w.« 4- IV. d 
 a 
 
 The power, instead of being applied at the circumference 
 of a wheel, is frequently applied to a lever inserted in the 
 cylinder, as in the case of the windlass or capstan of a ship ; 
 and then the moment of the power is the product of this 
 force, into the radius of the circle described by the point to 
 which it is applied. When the resistance is to be overcome 
 by manual force, this is most commonly applied to a winch 
 handle, which is a lever consisting of three parts, whereof 
 two are parallel; and these are joined by another part at 
 right angles. One of the parallel divisions is a prolongation 
 of the axis of the cylinder, and to the other the hand is ap- 
 plied. The leverage of the power is to be estimated by the 
 length of the intermediate division. This is exhibited by 
 CDEF, (Fig. 63.) The part ef, to which the hand is applied, 
 is usually cased in a hollow cylinder which turns on it ; and by 
 this contrivance the friction is transferred from the hand to 
 the interior surface of the hollow cylinder. 
 
 23. If several weights are applied to the axle, the equi- 
 librium is established, when the product of the power into 
 the radius of the wheel is equal to the sum of the products, 
 had by multiplying each of the weights into half the thick- 
 
THE AXLE IN THE WHEEL. 
 
 153 
 
 ness of the axle, taken at the point where it acts. If, how- 
 ever, any of the weights acts in the same direction as the 
 power, with respect to rotatory motion, its moment must be 
 added to that of the power, or subducted from those of the 
 remaining weights. Thus, if the weights are w, w', w", act- 
 ing by the radii r, r, r" \ and if w" tends to turn the instru- 
 ment in the same direction as the power, w, w' tending to 
 turn it in the opposite direction, the equation for equilibrium 
 will be 
 
 p.R + w". r"-=. w.r + w'. r. 
 or, 
 
 p.R — W.r + w . r — w .r . 
 
 In this way the labour is diminished, when the work to be 
 performed consists of two opposite motions. Thus, in draw- 
 ing up ore out of mine-shafts, two buckets are attached to 
 the same axle or cylinder : the ropes, by which they hang, 
 being wound in opposite directions. As the loaded bucket 
 ascends, the empty one descends ; whereby its moment is 
 added to that of the power, or subducted from that of the 
 resistance to be overcome. 
 
 Even when the work to be performed is of one kind only, 
 the same principle has been applied to diminish the labour. 
 To understand the value of the contrivance by which this is 
 effected, it is to be observed, that the efficacy of the instru- 
 ment is increased by enlarging the diameter of the wheel, or 
 by diminishing that of the axle : that the former of these me- 
 thods is limited by the size, which the engine cannot con- 
 veniently exceed ; and the latter, by the necessity of leaving 
 sufficient strength in the parts ; and also, by the waste of 
 power occasioned by the rigidity of the rope, when the cur- 
 vature which it is to receive is too quick, i. e. when the axle 
 round which it is to be coiled is too slender. But without 
 incurring either of these inconveniences, the power of the 
 instrument is increased by the following contrivance. The 
 
154 
 
 STATICS. — SECT, VII. 
 
 axle consists of two parts, of different thickness; and the 
 rope, which carries the weight, is attached by its ends to 
 these parts, being wound round them in opposite directions ; 
 so that whilst it coils on one of them, it winds off the other; 
 and the weight is suspended by a ring or pvdly from the rope 
 where it hangs double. Each part of the rope is then 
 strained by half the weight ; and the moment of this strain is 
 had by multiplying it into the radius of the axle to which 
 that part of the rope is applied. The moment ef the weight 
 is therefore the difference of these moments, or ^ w (r—r'), 
 and the equation for equihbrium is 
 
 (>— /) 
 
 F.R— W — --. 
 
 2 
 
 This is evidently the same thing as if the axle were re- 
 duced to a radius equal to half the difference of the radii of 
 the parts, whilst the strength in those parts is that due to r 
 and r. Thus, if the radius of the winch handle is supposed 
 to be 20 inches ; and the radii of the axle in its two parts, 
 5 and 6 inches ; the power of the instrument, estimated by 
 the ratio of the equilibrating forces, is, 
 
 w 
 
 — = 40. 
 p 
 
 So that an exertion of muscular strength, equal to half a 
 hundred weight, would be put in equilibrio with a weight of 
 one ton. To propose obtaining the same mechanical advan- 
 tage, by reducing the diameter of a single axle or cylinder, 
 its diameter should not exceed one inch, and a rope, suffi- 
 cient to carry a ton, could never be coiled round such a 
 cylinder. 
 
 24. But the usual method of increasing the mechanical 
 effect in instruments of this kind, is by combining two or 
 more of them in the same engine. One wheel may be made 
 to drive another, by a band or strap passing round the axle 
 of the driving wheel, and round the circumference of that 
 
THE AXLE IN THE WHEEL. 155 
 
 to be driven by it. But the method practised where much 
 force is employed, is by furnishing the wheels to be driven 
 with cogs or teeth ; and the axle by which the motion is to 
 be communicated, with a smaller wheel, similarly indented. 
 The smaller wheels are called pinions ; and their teeth, 
 leaves : and as any one leaf of the pinion parts from a tooth 
 of the wheel it drives, the next leaf of the pinion comes into 
 contact with the next tooth of the wheel. Accordingly, the 
 intervals between the teeth that so w^ork together, must be 
 equal ; and the number of those in the pinion must be to 
 the number in the wheel it drives, as their circumferences, 
 i. e. as their radii. This combination is represented in 
 (Fig. 64.) 
 
 To estimate the advantage to be gained by such a com- 
 bination, let r, p', p". . . . p„ be the forces applied at the several 
 wheels ; w, iv, w". . . .Wn , the weights which they would equi- 
 librate at their respective pinions : then the radii of the 
 wheels and pinions, being denoted by r, r', r". . . . r„, and 
 r, r , r\ .... r„. There are the following equations, 
 
 p _ r p' _ r' v" _ r' p„ _r„ 
 
 tV K IV R W R R,, 
 
 Multiplying these equations together, and remarking that, 
 in the combination, the force at the first pinion is that ap- 
 plied to the second wheel ; the force at the second pinion is 
 that applied to the third wheel, &;c., or ivz^ p', w'zr: p'^, .... 
 M'«_i = Pn . there is 
 
 p r. r'.r', . . . r„ 
 
 li\ R . R . R . . . . R„ 
 
 or, since the force acting at the last pinion is the weight, 
 it is 
 
 p r . r. r". ...?•„ 
 
 w R . R . R . . . . R„ 
 This theorem may be expressed by means of the revo- 
 lutions performed in a given time. For the teeth which 
 
156 STATICS. — SECT. VII. 
 
 work in each other, being necessarily set at equal distances, 
 the radius of a pinion is to that of the wheel it drives, as the 
 number of teeth in the former to that of the teeth in the 
 latter, i. e. as the number of revolutions performed by the 
 wheel to that performed in the same time by the pinion. 
 Thus, if the pinion has six leaves, and the wheel driven by 
 it sixty teeth ; the pinion must perform ten revolutions, 
 whilst the wheel makes only one. And universally, the 
 number of revolutions performed in a given time by the 
 pinion and the wheel it drives, are inversely as the number 
 of their teeth, i. e. inversely as their radii. Wherefore put- 
 ting V, v', v" , &c. for the number of revolutions performed 
 by the first, second, third wheel, &;c. in the same time, we 
 shall have 
 
 Z-— — — — iL A !!^ — Jil. 
 
 R' V ' r" v'' R„ Vn-\ 
 
 Making these substitutions in equation (a), it becomes 
 
 w Rv 
 
 shewing that for equilibrium, the ratio of the power to the 
 weight is had, by multiplying the radius of the last axle and 
 that of the first wheel, each by the number of revolutions 
 performed by it in a given time. So that if r = 10. r„. and 
 V rr 40. v„. we should have 
 
 w = 400. p. 
 
 and a man working such an engine with a force equal to ^^ 
 pounds, would sustain a weight of ten tons. 
 
 25. The teeth of the pinions and wheels afford the 
 means not only of communicating motion from one to the 
 other, but also of changing the direction of that motion ; as 
 may be seen by figures 65 and 66. In the former of which, 
 the pinion drives a crown wheel in a plane perpendicular to 
 its own plane ; and in the latter, the same thing is done by 
 bevelled work. 
 
THE PULLY. 157 
 
 In the construction of bevelled work, the particulars 
 chiefly to be attended to, are the inclination of the axes, and 
 the tapering form of the teeth. The angle contained by the 
 axes is equal to that which measures the change to be made 
 in the direction of the motion ; as may be seen by (Fig, 67.), 
 where the pinion n drives the wheel m. The angle formed 
 by the axes is bad : and the four-sided figure badc being 
 right angled at b and d, the angle bad is equal to ecb, 
 which measures the change of direction. 
 
 The manner in which the teeth should taper in bevelled 
 work, will be understood by conceiving two cones to roll on 
 each other, whilst turning round their respective axes. 
 These cones are exhibited in (Fig. 68.) Whilst the sur- 
 faces are smooth, the motion cannot be communicated from 
 one to the other, without considerable pressure ; and this 
 would be attended with a violent strain on the axle. To 
 prevent this inconvenience, and consequent waste of force, 
 the surfaces are fluted as in (Fig. 69.) and it is evident, that 
 the flutings should converge to the common vertex. More- 
 over, as the more dehcate parts of the flutings, near the 
 angles of the cones, are of httle use, those parts may be dis- 
 pensed with ; and then, nothing will remain but the bevelled 
 wheels befc, cfgd. 
 
 THE PULLY. 
 
 26. If a cord is used simply as the means of acting at a 
 distance from the point to which the force is to be applied, 
 the service it performs is often of considerable value ; but 
 as the cord, by its tension, acts with equal energy in oppo- 
 site directions, it is plain that in this use of it, the power 
 to be applied must be equal and opposite to the resistance 
 to be encountered ; and, therefore, that no change is made 
 either in its quantity or direction. It is otherwise, when 
 one of those forces being applied to strain the cord in the 
 
158 STATICS. SECT. VII. 
 
 direction of its length, the other is applied transversely, 
 to inflect it. Let the cord acb (Fig. TO.) be drawn by two 
 forces, T, t', acting at its extreme points a and b ; and let it 
 be inflected by the force f, acting at any intermediate point, 
 c, in the direction ce. Then, if any line, dc, in the direc- 
 tion of F, is taken to represent this force, and if this is made 
 the diagonal of a parallelogram hk, whose sides are in the 
 directions of the lines ca, cb, the forces acting in those direc- 
 tions shall be represented by en, ck or hd, and there will be 
 the proportions expressed by the following equations : 
 
 F T T 
 
 or. 
 
 CD CH HD* ' sin. {6 + B') sin. 0' sin. B ' 
 
 6, 0' being the angles made by the directions of the forces 
 T and t', with that of f. 
 
 If A and B are fixed points, to which the cord is at- 
 tached by its extremities, the reactions of these points will 
 take the places of the forces t and t'; and the same analogy 
 will express the relation of the inflecting force f, to the 
 strains on those points. And in either case, the forces 
 T and t' are the tensions of the two parts of the cord ca, 
 
 CB. 
 
 27. But if the inflecting force f, instead of acting at the 
 same invariable point of the cord, is at liberty to change 
 its point of application ; as when it is applied to a running 
 knot, or to a ring, through which the cord is passed, then 
 the two parts of the string communicating freely, their ten- 
 sions must be equal, i. e. T — t', and therefore, zz 9'. 
 Making these substitutions in the former theorem, there is 
 
 F : T : : sin. 2 6 : sin. 6::2 cos 6:1. or 
 F n St. cos 6. 
 
 If = 90". or ACB is one right line, there shall be 
 COS. = 0, and therefore f, infinitely less than t, for equi- 
 librium. Hence it appears, that if an inextensible cord lies 
 in a right line between its extreme points, a transverse force, 
 
THE FULLY. 159 
 
 however small, shall overcome any force, however great, by 
 which it is strained longitudinally : and, therefore, that by 
 the most inconsiderable transverse force, there may be pro- 
 duced on the fastenings, a strain which is only limited by 
 the strength of the cord itself. But if the cord suffers ex- 
 tension, cos. 9. is different from cypher, and therefore, f bears 
 some assignable ratio to t. As the angle at c diminishes, 
 COS. 0. increases ; and therefore the ratio of f to t increases. 
 When that angle becomes 120°, or 9 2= 60°, 2 cos. 9=1, and 
 .*. F— T. and when the angle at c vanishes, i. e. when the 
 two parts of the cord are parallel, cos. 0—1, and .'. f — 2 t. 
 
 Wherefore, if one end of a cord is fixed, and a weight 
 or resistance is applied at the other end, and the power 
 transversely at some intermediate point, there will be a me- 
 chanical advantage, so long as the angle contained by the 
 two parts of the cord exceeds 1 20° ; and this advantage will 
 be greater, as the angle is increased ; being infinite, for an 
 angle of 180°, i. e. when the two parts of the cord lie in 
 directum. But if the angle made by the two parts of the 
 cord is less than 120°, the mechanical advantage is to be 
 gained by applying the power as a tending, and the resis- 
 tance as an inflecting force. And as the angle farther di- 
 minishes, the ratio of the power to the resistance diminishes, 
 becoming that of 1 : 2 when the angle vanishes, i. e. when 
 the two parts of the cord are parallel. 
 
 28. The loss of force which would be occasioned by the 
 rigidity of the rope, if bent at a sharp angle, is avoided by 
 passing it over or under a grooved wheel. And to lessen 
 the friction, the wheel is made to turn on an axle ; or the 
 axle on the points of support. Thus, the friction is trans- 
 ferred from the rope and circumference of the wheel to the 
 axle, where its leverage is so much less. 
 
 Such is the pully, consisting of a grooved wheel or sheave, 
 turning on an axle fixed in a block. The rope passing over 
 or under the sheave, two of the three equihbrating forces 
 
160 STATICS. — SECT. VIT. 
 
 are applied at its extremities ; and the third at the block 
 which carries the sheave. 
 
 The use of this instrument is merely to lessen the effects 
 of rigidity and friction ; so that if these were not to be con- 
 sidered, the rope might be supposed to pass round a pin, or 
 through a ring. Wherefore, the theorem stating the rela- 
 tion between the power and resistance already delivered, 
 may be immediately applied to this instrument. The power 
 always acts at one extremity of the rope ; and as the resistance 
 or weight may be applied either to the other extremity of 
 the rope, or to the block in which the sheave revolves, the 
 instrument is divided into two kinds, applicable to different 
 purposes. These two kinds are called the. fixed and the 
 moveable pully. 
 
 29. The fixed pully is represented by (Fig. 71.) where 
 ABD is the wheel or sheave turning on an axle at c, and in 
 the block ce. The rope is mabn ; to one end of which is 
 applied the power p, and to the other, the resistance w. 
 The block is attached to a point, either absolutely fixed, or 
 to be so considered. 
 
 From Art. 26. it appears, that for equilibrium, p must be 
 equal to w ; and, therefore, that no mechanical advantage is 
 gained. The only use of the fixed pully is, to change the 
 direction of the power ; the force p, acting in the direction 
 AM, being thereby made to equilibrate the force w, acting in 
 the line bn. The change made in the direction of the force 
 is measured by the angle gem. And if the two parts of the 
 rope, AM, BN, were parallel, a power acting vertically down- 
 wards would draw a weight vertically upwards, i. e. the 
 change of direction would be measured by an angle of 180°. 
 29. If the resistance or weight is applied to the block, as 
 in (Fig. 72.) the pully ascends or descends with the weight it 
 supports ; and is, therefore, called the moveable pully : and 
 for equilibrium, we have 
 
THE PULLY. IGl 
 
 This relation between p and w may be expressed by 
 means of the circle adb. For if radii are drawn to the 
 points A and b, where the rope parts from the sheave; the 
 lines CA, cb, ab, are perpendicular to the lines am, bn, cf, the 
 dii'ections of the three equilibrating forces: and, therefore, 
 the triangle cab is similar to a triangle, whose three sides 
 are in those directions : wherefore, 
 
 p: w: : ca: ab. 
 
 i. e. the power shall be to the resistance, as the radius of the 
 wheel to the chord of the arch embraced by the rope ; so 
 that putting c for this chord, and r for the radius of the 
 sheave, it will be 
 
 p : w : : r : c. 
 
 If the two parts of the rope are parallel, we shall have 
 
 c zz 2r. 
 
 and therefore, 
 
 w 
 
 Accordingly, the moveable pully is used with most advantage 
 when the parts of the rope are parallel : the energy of the 
 power being then doubled at the working point. 
 
 SO. By combining the fixed and moveable pullies, the 
 twofold advantage is gained, of reducing the power by which 
 the weight is to be equilibrated ; and of changing the direc- 
 tion in which it is to be exerted. This is represented in 
 (Fig. 73.) where the rope, attached to a fixed point, is 
 passed under a moveable pully, and then over a fixed pully. 
 The power, applied to the end which hangs from the fixed 
 pully, shall sustain a weight of twice its magnitude applied 
 at the block which contains the moveable pully. And if the 
 first end of the rope, instead of being secured by a fixed 
 point, is passed over a second fixed pully, and then attached 
 to the block which carries the weight, as in (Fig. 74.) the 
 
 M 
 
162 STATICS. — SECT. VII. 
 
 power shall sustain a weight of thrice its magnitude. For 
 the rope being at perfect hberty to move over cr under the 
 pulhesj must be equally strained in every part. One of those 
 strains is measured by the power, and if the ropes are verti- 
 cal, the remainder by the weight. 
 
 31. Several moveable pullies may be combined in the 
 same block, and as many fixed pullies in another block. In 
 these combinations there is one rope which passes alternately 
 over the fixed, and under the moveable pullies. The power 
 is applied to one end of this rope, whilst the other end is 
 fixed ; or else, attached to either block. Such are the com- 
 binations represented in (Figures 75 and 76.) 
 
 The power is to the weight as unity to the number of 
 strains by which the weight is supported : and this number 
 is, in one case, twice the number of moveable pullies ; and 
 in the other case, greater by unity. Wherefore putting n 
 for the number of moveable pullies, it will be 
 
 p : w : : 1 : 2 ??. or p : w : : 1 : 2 w + 1 . 
 
 according as the end of the rope is attached to a fixed point, 
 or to the block which carries the weight. 
 
 32. Single pullies or systems of pullies, such as have 
 been described, may become the constituent parts of other 
 systems ; and hence arises considerable variety. Thus, to 
 a system, such as that described in the last article, a move- 
 able pully may be added, having a distinct rope, one end 
 of which is attached to the block containing the system of 
 moveable pullies. The other end may be attached to a fixed 
 point, as in (Fig. 77.) Or it may be carried over another 
 fixed pully, and then connected with the block of the pully 
 which it carries, as in (Fig. 78.) The weight is appended 
 from the block of this moveable pully; and it shall be twice 
 as great in the one case, and in the other thrice as great, 
 as the power would sustain, without this addition to the 
 system. For the appended weight it equal to the sum of 
 the strains of the parts of the last rope ; and that sustained 
 
THE PULLY. 163 
 
 by the block above it, is equivalent only to one of those 
 strains. 
 
 If to each moveable pully there is a distinct rope, one 
 end of which is attached to the block of the next superior 
 pully, and the other end to a fixed point, the block of the 
 lowest pully carrying the weight, as in (Fig. 79.) ; the strain 
 on each succeeding rope, beginning with that to which the 
 power is applied, is twice as great as the strain of the pre- 
 ceding rope, and the weight is equivalent to twice the strain 
 of the last rope. Wherefore, n denoting the number of 
 moveable pullies, the relation between the power and weight 
 shall be 
 
 p : w 1 : 1 : 2". 
 
 If the end of each of these ropes, instead of being at- 
 tached to a fixed point, is carried over a fixed pully, and 
 then attached to the block which it carries, as in (Fig. 81.) 
 the ratio of the power and weight will be given by the 
 analogy. 
 
 p : w : : 1 : 3«. 
 
 The energy is yet greater, in either of these systems 
 when inverted : the ends of the ropes, or the pullies which 
 before were fixed, being now attached to the weight, as in 
 (Figures 81 and 82.) Thus, in (Fig. 81.) the strain of the 
 rope attached to the weight at b, is p ; that of the rope at 
 c, is 2 p ; that of the rope at d, is 4 p ; and that of the rope 
 E, is 8 p. Wherefore, the weight w, being equal to the sum 
 of these strains, is p, multiplied into the sum of the terms 
 of a geometrical series whose first term is unity, and whose 
 ratio is 2, the number of terms being one more than the 
 number of moveable pullies. Consequently the equilibrating 
 
 n + l 
 
 weight is expressed by the equation w — (2 — 1) p. In the 
 arrangement represented in (Fig. 82.) the strain at b is 2 p; 
 that at c is 2. 3 p ; that at d, is 2. 3. 3 p ; and that at e is 
 2. 3. 3. 3 p. Wherefore, the weight is p, multiplied by the 
 
 M 2 
 
164 STATICS. — SECT. VIT. 
 
 sum of a geometrical series whose first term is 2, and whose 
 ratio is 3 ; the number of terms being, as before, one more 
 than the number of falhng pullies. Accordingly, the equi- 
 
 re+l 
 
 librating weight is expressed by the equation w r= (3 — 1)p. 
 3S. These several arrangements may be variously com- 
 bined ; and from what has been delivered, the weight which 
 the power will carry in each, is easily computed. This may 
 be exemplified in the combinations known by the name of 
 the Spanish Burtons, represented in (Fig. 83.) In the first 
 of these, a rope, fixed at one end, passes under a moveable 
 pully which carries the weight ; then over another pully ; 
 and to this rope the power is applied. But the second 
 pully, instead of being fixed, is carried by a distinct rope, 
 which, passing over a fixed pully, is attached by its other 
 end to the lower block. Here the strain of the second rope 
 is double of that to which the power is applied, or 2 p: and 
 this strain is applied in supporting the weight. Moreover, 
 the rope to which the power is applied, being doubled about 
 the lower pully, exerts the same force of 2 ? in sustaining 
 the weight. Wherefore, 
 
 w rz 4 p. 
 
 In the second Burton, a rope applied at one end to the 
 weight, passes over a fixed pully. The other end carries a 
 moveable pully having a distinct rope, which, being applied 
 to it at one end, passes under a second moveable pully, 
 which is attached to the weight, then over the former, and 
 the power is applied to the end of this rope. The strain of 
 the former rope is evidently thrice that of the latter, or 3 P ; 
 and the second carries a portion of the weight equal to 2 p. 
 Wherefore, 
 
 ^v — 5p. 
 
 34. In this account of the pully, it has been supposed 
 that the several ropes are strained, in directions parallel to 
 that of the weight or resistance which they support. If any 
 
THE PULLY. 165 
 
 part of a rope encompassing a moveable pvilly, is inclined to 
 the direction of the rope it carries, the energy with which it 
 acts against the resistance is to be estimated, by multiplying 
 its strain into the cosine of that angle of inclination. For by 
 resolving the oblique strain into two, one of them parallel, 
 and the other perpendicular to the direction of the strain 
 on the pully, the former is the energy with which it acts in 
 sustaining that pully ; and this is the strain of the rope, 
 multiplied into the cosine of the angle of inclination. The 
 latter is the strain multiplied into the sine of the same angle ; 
 and this is counteracted by the opposite strain of the other 
 part of the same rope. 
 
 If one end of the rope that carries the pully is fixed, the 
 strain on the pully is supported by two strains of the rope, 
 and these are inclined at equal angles. In this case it was 
 shewn, that the strain of the rope was to that supported by 
 the pully, as the radius of the sheave to the chord of the 
 arch embraced by the rope. Art. 29. This affords a ready 
 method of computing the ratio of the power and weight, in 
 a system, consisting entirely of such pullies. Thus, in a 
 system of fixed and moveable pullies, with a single cord fixed 
 at one end, putting r, r, r", 8cc. for the radii of the move- 
 able pullies, and c, c, c" , &c. for the chords of the arches 
 embraced by the ropes, we shall have for any obliquity of 
 the directions, 
 
 p _ r, r . r" . &c. 
 w ~ c.c . c". &c. ' 
 
 S5. When the power and weight act in the same direc- 
 tion, it is evident that the pressure on the fixed point is 
 equal to \v -f- p ; and if the power acts in the direction op- 
 posite to that of the weight, that this pressure is w — p. If 
 there are several such fixed points, this strain is distributed 
 among them, as in the system represented in (Fig. 79.) where 
 the points b, c, d, e sustain the several pressures, p, 2 p, 4 p, 
 8 P, and the point a the pressure 2 p. 
 
1G6 STATICS. — SECT. VII. 
 
 THE INCLINED PLANE. 
 
 36. If a material point is urged obliquely against a plane, 
 it may be maintained in a state of rest, by the application of 
 another force ; provided that the resultant of the two forces 
 is directed perpendicularly against the plane. For such a 
 resultant would be completely counteracted by the resistance 
 or reaction of the plane. Thus, let the plane be cd, (Fig. 
 84.) against which, a material point at o, is virged by the 
 oblique force fo ; and let o// be perpendicular to the plane, 
 at the side opposite to that on which is placed the material 
 point: the equilibrium shall be established by any force, 
 which, compounded with fo, gives a resultant in the direc- 
 tion of Of/. 
 
 To find the force or forces which will satisfy this con- 
 dition, it is to be observed, that the direction of the sustain- 
 ing force must be in the plane in which are fo, oy. Where- 
 fore, if the line ab be the intersection of this plane with the 
 given plane cd, it may be taken for this latter plane. If 
 then FE be drawn perpendicular to ab, i. e. parallel to oy, 
 the only condition is, that one of the components being re- 
 presented in quantity and direction by Fo, the direction of 
 the resultant must be parallel to fe, and it is plain that this 
 condition will be satisfied by a force represented in quantity 
 and direction by a line drawn from the point o, to any point 
 such as M, in the perpendicular fe, produced indefinitely 
 beyond the line ab. 
 
 Hence it appears, that the sustaining force, when di- 
 rectly opposed to FO, is equal to this force : that it is least, 
 when directed in the plane itself, being then represented 
 by OE : that at equal angles above and below the plane, 
 the sustaining forces are equal ; and that no force, of what- 
 ever magnitude, acting in the direction oy, perpendicular 
 
THE INCLINED PLANE. 167 
 
 to the plane, would suffice to sustain the material point 
 against it. 
 
 The force fo, is resolvable into the forces fe perpen- 
 dicular to the plane, and eo in that plane ; and in like man- 
 ner, the force om into the forces em and oe. Of these, the 
 forces EO, oe, being equal and opposite, destroy each other, 
 and FE, EM, being in the same line coalesce into one force 
 FM, which is the pressure on the plane. This pressure is the 
 sum or difference of the forces fe, em. 
 
 To express these results algebraically, let p designate 
 the force fo, and s the sustaining force om, then the sides 
 of the triangle fom, being as the sines of the opposite angles, 
 the forces p, s, are reciprocally proportional to the sines 
 made by their directions with the perpendicular to the plane, 
 i. e. to the cosines of the angles made by those lines with 
 the plane. Wherefore, a, a, denoting those angles, there 
 
 p. COS. a. z=. s. cos. a'. (1) 
 
 which is the condition of equilibrium. 
 
 Also, denoting the resultant or pressure on the plane by 
 R, there is 
 
 R. COS. a =. p. sin. (a + «')• (^) 
 
 The force p remaining unaltered in magnitude or direc- 
 tion, it appears from equation (1), that s. cos. a is constant, 
 or that the sustaining force raises inversely as the cosine of 
 the angle, which its direction makes with the plane. It is 
 therefore least when cos. a — 1 . or a — o. The sustaining 
 force is, therefore, most economised when directed in the 
 plane itself, its magnitude being then s z= p. cos. a. 
 
 37. Such, in general, is the theory relative to a material 
 point supported against a plane. But there is a particular 
 case which deserves to be considered distinctly. It is, that 
 wherein the plane is inclined to the horizon, and the force 
 Avith which the point is urged is its weight : such is denomi- 
 nated the inclined plane. 
 
168 STATICS. — SECT. VII. 
 
 The whole theory of the in dined plane flows at once 
 from what has been thus generally established : for in this 
 case, the force p is the weight of the body whose direction 
 is vertical ; and therefore, the angle formed by it with the 
 plane, is the complement of that made by the plane with the 
 horizon. Denoting this latter angle, or the elevation of the 
 plane by £ ; and the weight of the body by w, there will be 
 p = w. a z: 90 — £. Also, patting 7 for the angle made by 
 the direction of the sustaining force and plane, there will be 
 cos. a z=: sin.f . cos. a' = cos. 7. sin. (o + a) = sin. (90° — £ + 7) 
 = cos. (7 — e). Wherefore, substituting these values in (1) 
 and {2), there will be 
 
 sin.E COS. (7 — f) 
 
 s = w . R = w. ~ — — ^. 
 
 COS. 7 COS. 7 
 
 giving the values of the sustaining force and pressure. 
 
 When the sustaining force acts in the direction of the 
 plane, there is 7 = o. and therefore, 
 
 s = w. sin.E. R =: w. cos.e. 
 
 Or putting /, h, b for the length, height, and base of the 
 plane, there shall be 
 
 7t b 
 
 S = W. -y. R = W. -J. 
 
 When the direction of the sustaining force is parallel to 
 the base of the plane, or 7 = e. there shall be 
 
 s = w. tan.E. R = 
 
 COS.E 
 
 or, 
 
 h I 
 
 S = W. -. R = W. V 
 
 b b 
 
 On account of the importance of this subject it may not 
 be amiss to establish this theory by a separate investigation. 
 Let AC be the inchned plane (Fig. 85.), bc its height, and 
 AB its base. Then if from the right angle b, the line bo is 
 
THE INCLINED PLANE. 
 
 169 
 
 let fall perpendicular to the plane, and from the point c, the 
 line cm is drawn parallel to the direction of the sustaining 
 force, meeting the line bo at m, the three sides of the ti'i- 
 angle CBin, being parallel to the directions of the three 
 forces, shall be proportional to them in magnitude. Where- 
 fore s, R, w denoting, as before, the sustaining force, the 
 pressure and the weight, there shall be 
 
 s : w : : 77ic : cb. 
 R : w : : B7n : cb. 
 
 When the sustaining force is parallel to the plane, ?nc coin- 
 cides with oc. 
 
 But 
 and 
 
 Therefore, 
 
 oc : CB : : cb : CA : : /i : /. 
 
 OB : cb : : BA : ca. b : I. 
 
 h b 
 
 W. -. R = W. J. 
 
 mc : CB : 
 
 : cb ; 
 
 : BA : 
 
 :h:b. 
 
 mn : cb : 
 
 : CA ; 
 
 ; BA : 
 
 : I: b. 
 
 h 
 
 s = \v. -. 
 b 
 
 
 R = 
 
 I 
 
 When the direction of the sustaining force is parallel to 
 the horizon, wc is perpendicular to cb, and 
 
 and 
 
 These results are readily put into an analytical form, for 
 the sides of the triangle Bmc being as the sines of the oppo- 
 site angles, if t and y denote as before the elevation of the 
 plane and the angle made by it with the direction of the 
 sustaining force, and Z is put for the angle made by this 
 same line of direction with the base, there will be 
 
 mc : CB : : sin.wiBC : sin.B?«c : : sin.e : cos.y. 
 mB : CB : : sin.wiCB : sin.Bwc : : cos.^ : cos.y. 
 
170 STATICS. SECT. VII. 
 
 sin.c cos.^ 
 
 . *. S = W, . R = W. . 
 
 COS.y COS.y 
 
 which results agree with those ah'eady given. 
 
 oS. If instead of a material point, it is a body of any 
 definite magnitude, whose weight is to be supported on the 
 inclined plane, certain other conditions are to be satisfied 
 respecting the position of the body, and the direction of the 
 force to be applied. 
 
 With respect to the position of the body, it is to be ob- 
 served, that its weight and the reaction of the plane are to 
 be equilibrated by the sustaining force. But the forces of 
 resistance being directed in lines perpendicular to the in- 
 clined plane at the points of contact, the direction of their 
 resultant is perpendicular to that plane, at some point of the 
 surface of the polygon, formed by connecting the points of 
 contact. Wherefore, if the plane passing through the cen- 
 tre of gravity of the body, perpendicular to the inclined and 
 horizontal planes, i. e. perpendicular to the intersection of 
 those planes, does not pass through the surface of contact, 
 the forces to be equilibrated are not in the same plane ; they 
 have then no single resultant, (Sect. III. Art. 3.) and conse- 
 quently, they cannot be equilibrated by a single force. 
 Wherefore, to render the body capable of being sustained 
 by the application of a single force, the plane passing through 
 its centre of gravity, and perpendicular to the intersection of 
 the inclined and horizontal planes, must pass through the 
 surface of contact. 
 
 Such is the condition relative to the position of the body. 
 If this condition is not satisfied, it will be requisite to apply 
 two forces, which must be such, that the resultant of the 
 weight and two sustaining forces shall be directed towards 
 the plane, in a perpendicular at some point of the surface of 
 contact. 
 
 The condition relative to the position of the body, being 
 fulfilled, the body may be sustained by the application of a 
 single force directed in the plane passing through the centre 
 
THE INCLINED PLANE. 171 
 
 of gravity, and perpendicular to the intersection of the in- 
 clined and horizontal planes. But the force applied for this 
 purpose is restricted by a condition relative to its point of 
 application. 
 
 To see tlie nature of this restriction, let cd be the in- 
 clined plane, (Fig. 86.) ; gv the vertical line passing through 
 the centre of gravity ; abc the surface of contact, intersected 
 by the plane of the forces in the line ab. Then, if from a 
 and b, the extremities of this line, perpendiculars are raised 
 to the plane cd, they shall meet the vertical gv, as at d and e. 
 And as the perpendiculars to the plane at the several points 
 of the line ab must necessarily pass through the points of the 
 line de, it foUow^s, that the direction of the sustaining force 
 must pass through some point of the line de. 
 
 39. If two planes of equal heights are placed back to 
 back, as in (Fig. 87.), and if the weights w, w' are laid on 
 them, connected by a cord passing over a pully at the 
 highest point, so that the parts of the cord shall be parallel 
 to the planes ; it will be requisite for equilibrium that the 
 cord should be drawn equally by the two weights, /'. e. that 
 the parts of those v/eights, which act in the directions of the 
 planes, should be equal. Wherefore, putting l, l' for the 
 lengths of the planes, and h for the common height, the 
 
 weights must be such as to satisfy the equation w. — r: 
 
 H W w' 
 
 w'.— ^. or — zz — ^. I. e. the weights which equilibrate in this 
 
 L L Li 
 
 manner, must be to each other, as the lengths of the planes 
 on which they are placed. 
 
 40. If the body is to be supported by two planes which 
 meet at an angle, its weight is to be equilibrated by their re- 
 actions : accordingly, the directions of those three forces 
 must meet at a point. And from hence are derived the con- 
 ditions relative to the planes themselves, and to the position 
 of the body to be supported. 
 
172 STATICS. — SECT. VII. 
 
 The plane of the forces, being perpendicular to each of 
 the inclined planes, must be perpendicular to their inter- 
 section. The same plane is also vertical ; inasmuch as it 
 contains the vertical line passing through the centre of gra- 
 vity of the body. Therefore, the intersection of the inclined 
 planes must be perpendicular to a vertical plane, i. e. it must 
 be horizontal. This condition, relative to the positions of 
 the planes, being satisfied, the body is to be placed so that 
 the plane passing through its centre of gravity, and perpen- 
 dicular to the intersection of the inclined planes, shall pass 
 through each of the surfaces of contact. If the body is not 
 so placed, it will not compose itself until this condition is sa- 
 tisfied; and then, the pressure on each of the planes shall 
 be to the weight, as the sine of the elevation of the alternate 
 plane, to the sine of the angle made by the planes. And 
 these, which are the total pressures on the planes, are to be 
 distributed on the points of contact, as already stated. 
 
 THE WEDGE. 
 
 41. The wedge is a triangular prism, i.e. a solid, bounded 
 by two equal and parallel triangular planes, and three rect- 
 angles contained between their parallel sides. Two of these 
 inclined faces are to be introduced between the bodies to be 
 separated ; and the angle, made by these faces, is called the 
 angle of the wedge. The rectangular surface, opposite to 
 this angle, is the back of the wedge, and to this the power is 
 applied. 
 
 Of all mechanical instruments, this may appear to be the 
 most simple : yet in accounting for its operation some variety 
 will be found, arising from the diversity in the nature of the 
 forces, both of those to be encountered, and of those by 
 which they are to be surmounted. In each of the instru- 
 ments already treated of, the force to be encountered was 
 ever applied in the same manner. But the resistances to be 
 
THE WEDGE. 173 
 
 overcome by the wedge, may be forces applied immediately 
 to its faces, as in separating two distinct bodies ; or to its 
 edge, as in cutting ; or they may proceed from parts situated 
 beyond the edge, as in splitting or cleaving. And according 
 to the way in which these forces are applied, the several 
 parts of the instrument will be more or less effective. Also, 
 the power applied may be either simple pressure, or that 
 arising from percussion. But though the force brought into 
 action by percussion is, in general, that which is most effec- 
 tively applied to the wedge, yet as the treatment of percus- 
 sion belongs to Dynamics, the force employed as a power 
 must, for the present, be regarded as a pressvu'e or weight. 
 
 As the wedge is forced between the resistances, these 
 may be supposed to slide along its faces; and if this is done 
 without attrition, the faces will transmit only those forces 
 which act in lines perpendicular to those surfaces. Where- 
 fore, the resistances, when applied to the faces of the wedge, 
 can be equilibrated by the power, only when they act in di- 
 rections perpendicular to those faces. The same thing is to 
 be imderstood of the faces of a cleft, into which the wedge 
 has fully entered : these surfaces, being regarded as without 
 friction, can receive an impression from the wedge, i. e. from 
 the power, only in lines perpendicular. Wherefore, in all 
 cases, the power is to be resolved in directions perpendicular 
 to the surfaces through which the forces are to be trans- 
 mitted ; and the resistances are supposed to act in the oppo- 
 site directions. 
 
 The resistances being applied to the faces of the wedge, 
 let this be represented by its triangular end abc, and let df, 
 perpendicular to the back of the wedge, represent the power, 
 (Fig. 88.) Then if this line is made the diagonal of a paral- 
 lelogram GE, whose sides, dg, de, are perpendicular to the 
 faces AC, BC, it is evident that the force df will be in equili- 
 brio with the forces of resistance represented by gd, and ed 
 or FG. But the lines df, fg, gd, being perpendicular to ab, 
 
174 STATICS. — SECT. VIT. 
 
 BC, AC, the triangle dfg is similar to the triangle abc: 
 wherefore, the power is to the sum of the forces which it 
 will equilihrate, as ab to ac + bc. 
 
 If the triangular end of the wedge is isosceles, as repre- 
 sented in (Fig. 89.) the power is to the sum of the resis- 
 tances, as half the back of the wedge to one of the sides, 
 i. e. as the sine of half the angle of the wedge to radius. 
 
 Hence it follows, that the more acute the angle of the 
 wedge, the greater the resistance which it will overcome by 
 the application of a given power ; and that the efficacy of the 
 instrument, estimated in this way, is inversely as the sine of 
 the semiangle. 
 
 42. In cleaving, the motion is along the faces of the cleft, 
 and therefore, the power is to be resolved in directions per- 
 pendicular to those surfaces, and the resistances act in the 
 opposite directions. Let the angle of the cleft be adb, 
 (Fig. 90.) ; and putting b for ab, the base of the triangular 
 end of the wedge ; and l for bd, the depth of the cleft mea- 
 sured from the points a or b, along its faces ; the energy 
 
 with which the power acts against the resistance, is p. -. 
 
 B 
 
 But as the resistance yields, not in a right, but in a curved 
 line, whose centre of curvature is at the angle of the cleft, it 
 is evident, that to estimate the efficiency of the instrument in 
 this use of it, we must take into account the leverage by 
 which the forces act. Now, the resistance to be overcome is 
 the cohesive force of the fibres, which are more and more 
 extended from the angle d, to the point of fracture. The 
 point D is, therefore, the fulcrum : and putting I for the dis- 
 tance from this to the point where the strains of the fibres 
 may be supposed to be concentred, r.^ shall be the moment 
 
 P.L 
 
 of the resistance : and — — , is the moment of the force to 
 
 B 
 
 which it is opposed. Wherefore, for equilibrium. 
 
THE WEDGE. 115 
 
 T.ir , R.l.B 
 
 — R.l. or, p :z — 7—. 
 
 B L- 
 
 which shows in what way the efficacy of the wedge depends 
 on the constitution of the substance to be divided, as also, on 
 the size of the instrument. Being at present concerned in 
 estimating the power of the wedge, we shall suppose the ma- 
 terial to be given. In this case, r is given: also, the angle of 
 
 the cleft, or -, is constant ; and, therefore, I is likewise con- 
 
 L 
 
 stant. Wherefore, the power to be applied varies inversely 
 as L, i. e. inversely as b, the back of the wedge ; and the 
 efficiency of the instrument varies directly as that quantity. 
 
 Such is the advantage gained by the size of the wedge 
 when applied to a substance that yields in this manner. 
 When once the faces of the cleft part from those of the 
 wedge, the accuracy of the angle of the instrument becomes 
 of no value. And the same may be said of the magnitude of 
 that angle after that the v^edge has fully entered the cleft. 
 But in cutting, its efficacy mainly depends on this part. The 
 smaller the angle at which the faces are inclined, and the 
 more accurately they are brought to an edge, the less is the 
 number of parts whose cohesive strength is to be encoun- 
 tered. And if it is a soft or flexible substance that is to be 
 divided, the effect is to be ascribed to the angle exclusively. 
 For then, the faces of the incision, by reason of their flexibi- 
 lity, are incapable of transmitting the forces applied to them 
 by the power, to the part of the substance beyond the edge, 
 which remains to be divided. Those faces being then inca- 
 pable of acting as levers, the effect is altogether independent 
 of the depth of the wedge. The advantage arising from the 
 sharpness of the angle, in dividing a flexible substance, will 
 further appeir in the next section. 
 
176 STATICS. — SECT. VII. 
 
 THE SCREW. 
 
 43. The screw is an instrument consisting of two parts: 
 one of them is a sohd cyhnder ; the other, a hollow cylinder 
 of the same diameter. The former of these is encompassed 
 with a salient thread, proceeding round it in the form of a 
 spiral : and in the latter is a spiral groove, in which it may 
 be lodged. The solid cylinder is more especially called the 
 screw, and the hollow cylinder the nut. It is evident that 
 the screw cannot advance within the nut, or the nut upon 
 the screw, hut in the direction of the spires or threads, i. e. 
 by a motion compounded of two motions ; one of them circu- 
 lar, the other progressive. When one of these parts is 
 moved on the other, in this manner, it will press against any 
 obstacle by which its further progress is impeded. To as- 
 certain the force thus exerted by the moveable part, in rela- 
 tion to the power applied to move it, the form of the spiral 
 thread must be more particularly considered. 
 
 Let abc be a right angled triangle, whose base he is equal 
 to the circumference of the base of the cylinder, (Fig. 91.) 
 Then, if the line ah is applied to one of the sides of the 
 cylinder, and the triangle wrapped round it, the point c shall 
 reach the point h, and the hypothenuse ac shall become one 
 circumvolution of such a spiral. If the same thing is done 
 wnth a right angled triangle, whose base is 2ch, and whose 
 altitude is ^ah, its hypothenuse shall mark out two circum- 
 volutions of the same spiral ; and if the base and altitude 
 are och, 3ah, three circumvolutions shall be marked on the 
 surface of the cylinder, and so forth. 
 
 The spiral formed by the hypothenuse of a right angled 
 triangle, when its surface is thus wrapped round a cylinder, 
 is by geometricians called a helix. The base of the triangle 
 being taken equal to the circumference of the base of the 
 cylinder, its altitude is the interval between the spires mea- 
 
THE SCREW. 177 
 
 sured on the side of the cylinder, i. e. in a direction parallel 
 to its axis. 
 
 This may be represented in a manner somewhat different, 
 thus : let a rectangular parallelogram be described, whose 
 altitude is the side of the cylinder, and whose base is equal 
 to the circumference of its circular base, as in (Fig. 92.) Let 
 this rectangle be divided by the lines mn , rn'n", &c. parallel 
 to the base, and at the equal distances 7m\ nn , &c. and let 
 the transverse lines mn , mn , &c. be drawn. Then, if the 
 side of this rectangle is applied to the side of the cylinder 
 and the surface wrapped round it, the line mri shall become 
 the first spire ; and the point n falling on the point m , the 
 line inn' shall become the second spire, and the point n fall- 
 ing on the point m\ the line m'n" shall become the third 
 spire, and so forth : the several lines mn , mn, m"n" , m"n"\ 
 &c. becoming one continued spiral, proceeding from one end 
 of the cylinder to the other. 
 
 44. The thread of the screw is then an inclined plane, 
 carried round the surface of a cylinder : for which reason, 
 the ratio of the power to the resistance in equilibrio, is fur- 
 nished by the theory of the inclined plane. The direction 
 of the resisting force is plainly that of the axis of the cylin- 
 der or height of the plane. With respect to the relative mo- 
 tion of the parts, the moving power may be applied indiffe- 
 rently to the screw to move it within the nut, or to the nut 
 in the opposite direction to advance it on the screw : and the 
 resistances equilibrated by the same power, applied in these 
 two ways, shall be equal, but in opposite directions. Where- 
 fore, to assimilate this to the case of an inclined plane, on 
 which a weight is to be supported, the power may be sup- 
 posed to be applied to the nut, to carry it up a vertical screw, 
 against a resistance acting downwards. This power is ap- 
 plied in a direction parallel to cb, the base of the inclined 
 plane, (Fig. 91.) And if it were applied immediately at the 
 surface of the cylinder, the power would be to the weight as 
 
 N 
 
178 STATICS. SECT. VII. 
 
 the height of the plane to its base. Putting h for ab, and c 
 for the circumference of the base of the cyhnder, 
 
 p : w : : A : c. or, p = -. w. 
 c 
 
 But the power is usually applied to the arm of a lever 
 inserted in the moveable part, as in (Fig. 93.) The ful- 
 crum being at the centre of the circular motion, ^. e. at the 
 axis of the cylinder, if r is put for the length of the arm to 
 which the power is applied, the enei'gy of that power at the 
 
 surface of the cylinder shall be p. -. giving 
 
 or, 
 
 
 R, 
 
 
 h 
 
 p. 
 
 — 
 
 — 
 
 - w. 
 
 
 r 
 
 
 c 
 
 PZ 
 
 -h 
 
 
 r 
 — . w 
 
 
 
 ■ R.C 
 
 
 
 h 
 
 
 
 — 
 
 c 
 
 , w. 
 
 c being the circumference of circle whose radius — r. i. e. 
 the power is to the resistance which it equilibrates by means 
 of the screw, as the interval between the spires, to the 
 circumference of the circle described by the power. It 
 will be observed, that this proportion is altogether inde- 
 pendent of r, the radius of the cylinder. 
 
 The screw, therefore, as it is commonly used, combines the 
 advantages of the lever and inclined plane. In reasoning on 
 that part of the instrument which works on the latter principle, 
 it has been supposed that the power and resistance are both 
 apphed at a single point of the plane ; or, which is the same 
 thing, that the screw and nut touch in a single point. In fact, 
 they touch throughout the entire extent of the thread of the 
 nut, or, at least, in many separate points. But however 
 great the number of the points of contact, the relation be- 
 tween p and w is the same. For the whole of the resist- 
 ance \v may be distributed into the parts m, m\ m\ &c. en- 
 
THE SCREW. 179 
 
 countered at those points : and if the power P is, in like 
 manner, distributed into parts n, n, n' , &c. such that 
 
 //, 
 
 n — -. m, 
 c 
 
 ,_h , 
 
 w — -. m, 
 
 c 
 
 71 — -. m . cfec. 
 c 
 
 the equilibrium shall subsist at each point ; and therefore, 
 throughout the whole extent of the plane. And by adding, 
 we have 
 
 that is. 
 
 « ■+• w' + w"+ &c. = -. (m + m-\- m + &;c.) 
 
 h 
 
 p = -w. 
 c 
 
 4j. The mechanical advantage afforded by this instru- 
 ment is increased, either by augmenting the length of the 
 lever handle, or by diminishing the interval between the 
 spires. But the augmentation of the length of the lever 
 handle is an increase of bulk, which it is desirable to avoid. 
 And to the diminution of the interval between the spires, 
 there is a limit, arising from the necessity of leaving suffi- 
 cient strength in the parts to endure the enormous pressure, 
 and consequent friction, without breaking. The device of 
 Mr. Hunter, published in the Philosophical Transactions 
 of the year 1781, seems well fitted to give any power to the 
 instrument, without either of these disadvantages. In this 
 instrument, represented by (Fig. 94.) the screw which turns 
 within the nut, is not solid as usual, but is itself the nut of a 
 somewhat finer screw, on which it turns. 
 
 To shew the power of this contrivance, let h be the in- 
 terval between the threads of the exterior, and h' that be- 
 tween the threads of the interior screw. Now by one revo- 
 
 n2 
 
180 STATICS. SECT. VII. 
 
 lution of the exterior screw it would be carried forward 
 through the space //, and the interior screw, if firmly at- 
 tached to it, would be carried through the same space. But 
 the revolution of the interior screw is prevented by a cross 
 bar by which it is perforated, and which bears against the 
 frame work, wherefore that screw will perform one revolu- 
 tion relatively to the exterior screw, and in the opposite 
 direction, and therefore shall move upwards through the 
 space h'. The space actually described by the interior screw 
 
 is the difference of these, or h — //. Thus, when hzzz-— of an 
 inch and // = — - the progressive motion is ( — — — Jofan 
 
 inch ■=. T-jTjth of an inch. 
 
 It appears then, that the instrument, in this example, 
 combines the mechanical advantage of a screw, the interval 
 between whose threads is but the ixoth of an inch, with 
 the strength of one whose threads are ten times as gross : 
 and this, without increase of bulk, or any disadvantage, ex- 
 cept what may arise from increase of friction. 
 
 It would seem that this contrivance might be benefi- 
 cially adopted in the construction of micrometer screws, and 
 a degree of accuracy thereby given to our observations, 
 altogether unattainable by the most delicate instruments 
 now in use. Thus, if the exterior screw has 50, and the 
 interior screw 51 spires to an inch, we should obtain the 
 advantage of a screw of the ordinary construction, having 
 2550 spires to an inch. In general, the number of spires to 
 an inch, in the equivalent screw of the common construc- 
 tion, is the product of the numbers of the spires to an inch 
 in the two members divided by their difference. 
 
 46. There is another modification of this instrument, 
 called the endless screw : so named, because by turning, it 
 never comes to an end. It is evident that this advantage 
 
THE SCREW. 181 
 
 can be obtained, only when the motion of one of the parts 
 is circular. The screw, in this instrument, consists of a few 
 spires ; and these work in the teeth of a wheel, which, 
 therefore, serves in place of a nut; as represented by 
 (Fig. 95.) The screw has a circular, but not a progressive 
 motion. By the action of the spire engaged, the tooth of 
 the wheel would proceed forward : but the wheel is limited 
 to a circular motion round its axis ; and by this motion the 
 tooth is soon disengaged from the spire, and then its place 
 is supplied by another tooth, which is acted on as the 
 former ; and this tooth is succeeded by a third in the same 
 manner ; and so on perpetually. 
 
 The mechanical effect of this combination is thus esti- 
 mated. 
 
 Let c, as before, be the circumference of the circle de- 
 scribed by the winch handle, r its radius ; /^, the interval 
 between the spires. Also let r' be the radius of the toothed 
 wheel, and r that of its axle. Then p' being the force equi- 
 librated by the screw alone, there is 
 
 p = - p. 
 c 
 
 But the force p' is immediately applied at the circumference 
 of the toothed wheel, and equilibrates the weight applied at 
 its axle. Wherefore 
 
 r 
 
 p =: —7. w. 
 
 r' 
 
 Multiplying these equations, there is 
 
 Ji r 
 
 p = -. -7-. w. 
 
 c r 
 
 Now h, the interval between the spires of the screw, is 
 also the interval between the teeth of the wheel : wherefore 
 if 71 denotes the number of teeth, and c' the circumference 
 
 of the wheel, there is nh zz. c, and h — — . This value of h 
 
 n 
 
182 STATICS. — SECT. VII. 
 
 being substituted in the last equation, it becomes 
 
 n c R' 
 or smee 
 
 it is 
 
 i. e. the power is to the weight, as the radius of the axle 
 which carries the weight, to the radius of the circle de- 
 scribed by the winch handle, multiplied by the number of 
 teeth in the wheel. 
 
 _1 
 
 ~' n 
 
 c' r 
 c * r' 
 
 c' _ 
 
 r' 
 
 c 
 
 ' r' 
 
 1 
 p = -. 
 
 r 
 -. w. 
 
 R 
 
THE FUNICULAR POLYGON AND CATENARY. 183 
 
 SECTION VIII. 
 
 THE FUNICULAR POLYGON AND CATENARY. 
 
 1. If a system of forces is in equilibrio when applied to 
 a body of variable form, it is evident that the equilibrium 
 shall not be violated should the body become rigid. Hence 
 it follows, that the conditions to be satisfied for the equili- 
 bration of forces applied to a rigid body, should also be 
 satisfied in the case of a body of variable form. But it is 
 equally evident that, in this latter case, these conditions are 
 not sufficient, i. e. that other conditions are to be fulfilled in 
 order that the body may become capable of opposing to each 
 other the forces of the system. These conditions relate both 
 to the nature and the form of the body. With respect to 
 the former, it is plainly requisite, that the body should be 
 capable of transmitting the applied force from one point to 
 another, and, therefore, that the parts between the points of 
 application should be either inextensible or incompressible, 
 at least to a degree sufficient to resist the action of those 
 forces. 
 
 If the body is simply inextensible, it shall be capable 
 only of transmitting divellent forces, i. e, forces which tend 
 to increase the distances between the points of application : 
 and if it is simply incompressible, it shall serve only for the 
 transmission of compressing forces. A flexible but inexten- 
 sible cord affords a famihar example of a body of the former 
 kind. Let o, o, o", o'", o"", be the points of such a cord 
 (Fig. 96.) to which the forces p, j/, p", p",j)"", are applied; 
 these forces must be divellent : therefore, the forces 2h p"", 
 applied at the extreme points, must act in the directions o' o, 
 
IS'! STATICS. — SECT. VIII. 
 
 d" o"", and the forces ^/, p", p", applied at the intermediate 
 points, must be directed to the exterior side of the polygon. 
 If the lines od, dd' , d'd", d"d"' , represent a system of rigid 
 bars barely apposited, they will serve only for the transmis- 
 sion of compressing forces, i. e. of forces directed towards 
 the interior of the polygon, or contrary to the former. But 
 if the bars are connected by hinges, the system may serve 
 indifferently for the transmission of distending or compres- 
 sing forces. 
 
 2. When two forces are applied at the ends of a cord, 
 and others transversely at so many intermediate points, as 
 described in the preceding article, the figure assumed by the 
 cord is that denominated the funicular polygon. Let the 
 cord be that exhibited in the figure ; and o, d, d', d", d'", 
 the points to which are applied the forces p, p' , p" ^ p'", p"" , 
 acting in the same or in diflferent planes. To investigate the 
 conditions to be satisfied in order that these forces should 
 equilibrate, we shall first suppose the equilibrium to subsist. 
 The conditions which follow from that supposition will be 
 necessary. 
 
 Because of the flexibility of the cord, the forces can act 
 against each other only in the lines od , dd', d'd" , d"d"' . 
 Let these forces be represented by the lines ko, ds, d's', 
 d"s", r'd'". and let all between the first and last be resolved, 
 each in the directions of the branches of the cord contiguous 
 to its point of application, as os into dm, o'k ; d's into o"w, 
 d'ni ; d"s" into d"n, d"r. and as the forces applied at the 
 ends of the entire cord are necessarily in the directions of 
 the first and last branches, the whole system of forces jt?, jt?', 
 p",p"',p"", shall be replaced by another system of forces 
 directed along the sides of the polygon: and if the equili- 
 brium exists among the forces of the former, it must continue 
 to subsist among those of the latter system. 
 
 But among these it cannot subsist unless each part of the 
 
THE FUNICULAR POLYGON AND CATENARr. 185 
 
 cord is drawn equally in opposite directions. This condition 
 gives 
 
 ok •\-ok'zzO. o'ni'-{'o"m' = 0. o"H-\-o"'nzzO. o"'r -\-o""r' — Q. 
 
 Because these forces are the tensions of the parts, they 
 may be denoted by t, t', t" , &:c. and the system shall be 
 t — t-\-t'—t'-^t"—t"-{-t"' — t'". whose sum is cypher. These 
 tensions may be different for the different branches of the 
 cord, the tension of any branch being the resultant of all the 
 forces acting at either side of that branch : for the force ^j or 
 t, acting at o, and in the direction of the line oo', may be sup- 
 posed to be immediately applied at the point o', where, if 
 compounded with the force // or f—t, the resultant shall 
 be + i'> arid this resultant of the forces p, p , being in the 
 direction of o'o'\ may be supposed to be immediately applied 
 ato", where, if compounded with the force ^", or t"~t', the 
 resultant, which is that of the three forces^, ^', />", shall be 
 t". In like manner, beginning at the other end of the en- 
 tire cord, the force //'" or — f may be supposed to be im- 
 mediately applied at the point o", where, if compounded 
 with the force/)'" or t"'—t", the resultant shall be — /". Thus 
 it appears that the equal forces by which the branch o"d" is 
 strained in opposite directions, are the resultants of the 
 groups of forces acting at each side of that branch ; or that 
 the resultants of the forces p, &c. taken from either end of 
 the entire cord to any angle of the polygon, is in the direc- 
 tion of the portion next ensuing, and that it constitutes its 
 tension. 
 
 Further : because the equilibrium is supposed to exist in 
 the entire cord, it must exist among the forces acting at each 
 angular point. But these consist of one of the forces p, ap- 
 phed immediately at that point, and of the tensions of the 
 adjacent branches of the cord ; and those two tensions being 
 the resultants of the groups preceding and following, may be 
 replaced by those forces ; whence it follows, that the whole 
 
186 STATICS. — SECT. VIII. 
 
 system of forces ^;, p' , p", p", p"", must equilibrate at the 
 same point. Thus the lorces acting at the point o" are 
 t\ — /"and^y, and if these three are in equiUbrio at that 
 point, tlie equiUbrium shall continue when the forces t' and 
 — /" are replaced by their components; but t' is the resul- 
 tant of the forces p, p \ and —f of the forces p" ^ p"'\ 
 Hence it follows, that any system of forces in equilibrio when 
 acting against each other by means of a funicular polygon, 
 would also equilibrate about any material point to which 
 they may be transferred in parallel directions. 
 
 The same thing appears by the tensions into which those 
 forces have been resolved ; these tensions being equal and 
 opposite in pairs. 
 
 It now appears that a system of forces cannot equilibrate 
 by means of a funicular polygon, unless the following condi- 
 tions are satisfied. 1st, The forces must be such as would 
 equilibrate about a material point to which they may be 
 transferred in parallel directions. 2d, Each branch of the 
 cord must be parallel to the direction of the resultant of the 
 forces acting at either side of that branch. Further, it is 
 evident, that when these conditions are satisfied by the forces 
 applied, and the form of the polygon, the equilibrium must 
 necessarily subsist. For the first branch being taken in the 
 direction of the force p, that force may be supposed to have 
 been immediately applied at the point d \ and the second 
 branch being taken in the direction of the resultant of the 
 forces p and p acting at d , this resultant may be supposed 
 to be immediately applied at the point o", and so forth. In 
 the same manner it may be shewn, that the resultant of all 
 the remaining forces acts at the same point o" ; and the same 
 may be shown of any other angle of the polygon. 
 
 It appears, then, that when this form is given to the poly- 
 gon, the whole system of forces acts at each angle ; and that 
 if these forces are such as would equihbrate about a material 
 
THE FUNICULAR POLYGON AND CATENARY. 187 
 
 point, the equilibrium shall necessarily subsist at each angle 
 of the polygon. 
 
 It is now proved, that for equilibrium among forces 
 acting against each other by means of a funicular polygon, 
 it is requisite and sufficient that the forces should be such 
 as would equilibrate about a material point to which they 
 may be transferred in parallel directions, and that each 
 branch of the polygon should be taken parallel to the direc- 
 tion of the resultant of the group of forces from either end 
 of the entire cord, to the nearer extremity of that branch 
 inclusively. 
 
 3. Seeing that when the applied forces, taken from either 
 end of the entire cord to any of its angles inclusively, are trans- 
 ferred in parallel directions to a material point, and there com- 
 pounded, the resultant is equal to the tension and parallel to 
 the direction of the succeeding branch ; it follows that the 
 tension and direction of any branch are given by the same 
 equations which determine the magnitude and direction of the 
 resultant of a system of forces acting on a material point. 
 Thus let there be three rectangular axes meeting at a point, 
 
 and let a, a', a" a„. be the angles made by the several 
 
 branches of the cord with the axis of a; ; b, h\ h" i„. 
 
 those which they make with the axis of y; and e, c', 
 
 c" c„. those made with the axis of^. Likewise let 
 
 a, a, a! a„; b, j3, [5' b,,; c, y, y' c„, 
 
 be the angles made by the directions of the forces ^;, //, 
 
 p" -Pn-]-!) with the same axes. Then as the tension of 
 
 any member, as t", is the resultant of the forces p, p , p" . 
 there shall be 
 
 t" .0,0%. a zi p. 0,0%. a 4-^'.cos.a +/* '-cos-a'. \ 
 Acos.6"=:^j.cos.6+ij'-cos./3-|-p"-cos./3'. v (1) 
 ^".cos.c"^z:^.cos.c -\- p .dQ^.y •\- p" .oo%.y' . I 
 
 Or, putting x", v", z", for the second members, i. c. for 
 
1S8 STATICS. — SECT. VIII. 
 
 the sums of the forces /?, p , p" , reduced to the directions 
 of the axes of x, y, %, the equations shall be 
 
 r.cos.a"= x". \ 
 
 r.cos.b"=r\ I {2) 
 
 r.cos.c"=:z". j 
 Whence by squaring and adding, there are 
 
 r='v/x"^-hY"^+z'^ (3) 
 
 and 
 
 cos. a zz ■ ■ cos. 6 = 
 
 V x"^+ y"^+z"^ V x"2+ y"H z'" 
 
 z" 
 
 cos.c'^:= ' . (4) 
 
 Vx"^'+y"^4-z"' 
 These equations applied to the last branch give the va- 
 lue and direction of tn, which are already known ; yet these 
 equations are not therefore useless, for replacing t„ with its 
 known value, —pn + i, in equations (1), they become 
 
 p. COS. a -{-p'.cos.a-^ p'\cos ex," -{-pn+i.cos.an :=■ 0. 
 
 p.cos.b -\- p\cos. (5' +p". cos.fi" -|-/?„^-i.cos.6„ = 0. 
 
 p.cos.c -{-p'.cos.y'-^-p'.cos.y" -\-p„^.^.cos.Cn — 0. 
 
 Which express the condition to be satisfied in order that 
 the equilibrium may be possible. 
 
 If the cord is attached to a fixed point at one end, the 
 equations give the tension and direction of the last branch ; 
 and as in this case, the number of equations is only equal to 
 that of the unknown quantities, the problem is not restricted 
 by any condition, and the equilibrium is therefore always 
 possible. 
 
 If the cord is attached to fixed points at both ends, the 
 equilibrium is possible a fortiori. In this case it is not 
 solved by these equations, which are fewer than the number 
 of unknown quantities, of which there are eight, viz. ji and 
 
THE FUNICULAR POLYGON AND CATENARY. 189 
 
 p„ + i. And the angles a, b, c, a„, b„, Cn. But the problem 
 is not therefore indeterminate, for beside the three equa- 
 tions (1) there are the two following, viz. 
 
 cos.^a + cos.^b + cos.Vrz 1. cos.^a„+ cos.'^b„-{- cos.^(?„= 1. 
 
 with three others arising from this, that the fixed points 
 are given in position. For taking one of those points for 
 the origin of the coordinates, those of the other fixed point 
 may be expressed as functions of the quantities which enter 
 into the other equations, and of the lengths of the several 
 branches of the cord, which last are given. The expres- 
 sions to which this computation would lead are exceedingly 
 complicated ; and as their value would not reward the labour 
 of the analyst, it seems sufficient thus briefly to point out 
 the method by which the figure of the polygon may be com- 
 pletely ascertained. 
 
 4. The extreme points being fixed, if the forces applied 
 at the intermediate points are parallel, the sides of the po- 
 lygon, and therefore, also, the directions of the forces must 
 all be in the same plane. For the plane of any two conti- 
 guous sides contains the direction of the force applied at the 
 angle made by those sides, and the plain of any two succes- 
 sive parallel forces contains the intermediate side of the 
 polygon. Thus, the plane oo'p' is the same with the plane 
 p'o'o" (Fig. 99.) And the forces p\ p'' being parallel, the 
 plane p'o'o" is the same with the plane p"o"o. The same 
 reasoning may be continued through the whole system of 
 lines. 
 
 If the forces are appended weights, their resultant acts 
 in the vertical passing through their common centre of 
 gravity. And as the reactions of the fixed points are in 
 equilibrio with this resultant, it follows, that the directions 
 of the two extreme portions of the cord shall meet at some 
 point of the vertical passing through the common centre of 
 gravity of the weights. And similarly, the tensions of any 
 
190 STATICS. — SECT. VIII. 
 
 two parts of the cord, regarded as divellent forces, being in 
 equilibrio with the weights appHed at the intermediate points, 
 those parts, if produced, shall meet at some point of the 
 vertical passing through their common centre of gravity. 
 Thus, oo', o""o"' produced, shall meet at the same point of 
 the vertical passing through the common centre of gravity 
 of the weights ]j',p", p'", and oo' , o'^'o", produced, shall meet 
 at some point of the vertical passing through the common 
 centre of gravity of the weights p, p". Wherefore, the 
 theorem which expresses the relations of the three forces 
 engaged at any one angle of the polygon may, in this case, 
 be extended to the strains of any two parts, however remote, 
 and the sum of the weights applied at the intermediate an- 
 gular points, i. e. 
 
 t\t"'\p' -\-p" -^p'" 
 
 : : sin. c' : sin. a : sin. (a + c'). 
 
 The case of a cord loaded with weights being that which 
 offers itself most to our notice in this theory, it may not be 
 amiss to shew how the tensions and weights may be repre- 
 sented by a simple construction. 
 
 From any point, o, let there be drawn the lines oo', oo'', 
 oo'", &c. (Fig. 100.) respectively parallel to the sides of the 
 polygon, and meeting the vertical line in the points o', o" , 
 o'" , &c. then the three forces t, p', — t', which are in equi- 
 librio about the point o', (Fig. 99.) shall be represented by 
 the lines oo', o'o", and o"o. Likewise, the three forces 
 /', p", and — t", in equilibrio about the point o", shall be 
 represented by the lines oo", o"o"', and o"'o : and so of the 
 three forces which are in equilibrio at any other angular 
 point. 
 
 Moreover, + 1' and — t', which are of the same magni- 
 tude, being represented by the same line oo" or o"o, it fol- 
 lows, that the sides of the triangles oo'o" , od'd" , shall serve, 
 not only to exhibit the relations of the three forces acting at 
 
THE FUNICULAR POLYGON AND CATENARY. 191 
 
 one and the same point, d or o", but also to compare the 
 forces acting at one of those points with those which act at 
 the other : and the same thing being observed of all the tri- 
 angles, it follows that there is this proportionality. 
 
 t 
 
 ■.p' 
 
 '.t' ; 
 
 '.f ; 
 
 -A" 
 
 :p" 
 
 : t"'> 
 
 &c. 
 
 '.OO' 
 
 : o'o" 
 
 : oo" : 
 
 :o'V"; 
 
 ; 00'" 
 
 : o"'o"" ; 
 
 ; oo"", 
 
 &c. 
 
 Wherefore, if the directions of the sides of the polygon are 
 known, together with any one of the appended weights, all 
 the other forces, both weights and tensions, are absolutely 
 determined. 
 
 Drawing the horizontal line oh, it appears that the ten- 
 sions of the several parts of the cord are to one another as 
 the secants of the angles of their inclinations to the horizon : 
 and that the weights appended at the several angular points 
 are as the differences of the tangents of those angles. Also, 
 that the horizontal strain, which in each part of the cord 
 acts equally in opposite directions, is represented in magni- 
 tude by oh, and is therefore the same throughout the entire 
 system. 
 
 5. Hitherto the cord has been considered, only, as the 
 means of conveying the actions of the forces, to points re- 
 mote from those to which they are immediately applied ; and 
 for this reason it was regarded as being, itself, without 
 weight. But the cord and its minutest parts having weight, 
 it is plain, that if it is fastened at its two ends, those points 
 not being in the same vertical, it must be deflected from a 
 right line by the weights of its several parts ; which, in this 
 position of the cord, are so many forces applied transversely 
 to its length. The number of these forces being infinite, 
 that of the sides of the polygon must also be infinite, i. e. 
 the figure assumed by such a cord is a curve. 
 
 This curve is known by the name of the catenary; and 
 its properties follow immediately from those of the funicular 
 
192 STATICS. — SECT. VIII. 
 
 polygon loaded with weights, merely by supposing the num- 
 ber of those weights to be infinite. 
 
 As it will be sufficient to consider one branch of this 
 curve, its lowest point may be supposed to be fixed ; and, 
 therefore, the other branch may be cut away. This being 
 premised, let ov be one branch, terminating in the lowest 
 point, V, (Fig. 101.) in which, taking any point, 7n, and draw- 
 ing the tangent tns, and the horizontal line ?«/?, let these lines 
 meet the vertical passing through the point v, at s and // ; 
 then putting e for the angle hms ; w for the weight of the 
 cord between m and v ; t for the tension at the point m ; and 
 A for the tension at v, the relative magnitudes of t, a, w shall 
 be represented by ms, mh, hs. Wherefore, 
 
 hs , , 
 
 w^ 1= A. — -, =: A. tan. e. (a) 
 
 ?)ih 
 
 ms A , , . 
 
 T = A. — = {b) 
 
 mh cos.e 
 
 Showing that the weight between any point m, and the 
 lowest point, varies as the tangent of the inclination to the 
 horizon at the former point ; and that the tension varies in- 
 versely as the cosine, i. e. directly as the secant of the same 
 angle. 
 
 If the cord is uniform, the curve is called the homo- 
 geneous catenary, in which w r= X's ; s denoting the length 
 measured to the lowest point ; and h the weight in a unit of 
 length. Wherefore, 
 
 A. tan. e z= lis. (c) 
 
 From which it appears, that the two branches into which the 
 homogeneous catenary is divided by the lowest point, are 
 precisely similar. 
 
 Taking the point v for the origin, and the horizontal line 
 through this point, vr, for the axis of the abscissae ; and the 
 vertical vh for the axis of the ordinates, there will be for any 
 point of the curve, as tn, 
 
THE FUNICULAR POLYGON AND CATENARY. 193 
 
 vq ~ X, mq zz y, and tan. e zr. — . 
 
 Whereby equation c becomes 
 
 A.dy=.k.s.dx. {d) 
 
 And as ds — \/{dy^-\- dx^), any of the three quantities, y, x,s, 
 may be eliminated, and an equation obtained between the 
 remaining two. 
 
 - , ,_ A 
 To render the expressions more smiple, let a _ t ; 
 
 whereby equation (d) becomes 
 
 A'dy — s.dx. 
 which, being differenced, gives 
 
 A'cP2jzzV{df+dx^).dx. 
 
 And multiplying both sides by ■ . .^ , 2 \> i* ^^^^^ ^^ 
 
 h'.dy.d'^y _ 
 
 that is, 
 
 a', d. V {di/-\- dx'') - dy.dx. 
 
 Wherefore, by integration, 
 
 a'. ^/ {dy~-\- dx^) = ydx + cdx. 
 
 For the lowest point, v, there will be 
 
 'y=0. di/ = 0. 
 
 which gives 
 
 c r: a'. 
 
 and therefore, 
 
 a' V (^/ + dx"") - (a' + y) dx. {e) 
 
 for the difFereniial equation of the homogeneous catenary. 
 This being integrated, gives 
 
 2. = dy.dx. 
 
 X — a', log. 
 
 / A^+y+V(gAV/ + r) \ 
 
194 STATICS. — SECT. VIII. 
 
 From which it appears, that the homogeneous catenary is a 
 transcendental curve. 
 
 To find equation between y and s, let dx be eliminated 
 between the equations A'dy zz s.dx, oxvdi. ds zz. yj {dif -\-dx^)» 
 and we shall have 
 
 ^^-^\- ds" - dy\ 
 
 or 
 
 {p^''J^s^)dy''-s',ds'. 
 and 
 
 s. ds . 
 
 VW+7) = '■"■ 
 
 And by integration, 
 
 VW' + s')zzy + c. 
 
 But for y = 0. there will be * = 0. Wherefore, 
 
 c =: A' ; and V (a'^ ^ s-) = y + a'. 
 
 so that 
 
 s'' = y(2A' + y). 
 
 which shews that the homogeneous catenary is a rectifiable 
 curve ; its length from any point to the lowest, being a 
 mean proportional between y and 2a' + y- Wherein y is 
 the ordinate of the highest point of the portion to be rec- 
 tified. 
 
 6. If any of the forces, as //", is applied to a ring, or 
 loop running on the cord, its direction must bisect the angle 
 made by the adjacent branches o"o"', o"'o"". For if the 
 equilibrium subsists, it shall not be violated by fixing the 
 points o", o"" : and then the ring is confined to the surface 
 of an ellipsoid whose foci are those points, and whose major 
 axis is equal to o"o"' + o"'d"'. On this surface the ring 
 cannot rest unless the direction of the force is normal : i. e. 
 unless it bisects the angle made bylines drawn from the point 
 of apphcation to the foci. These angles being equal, the 
 
THE FUNICULAR POLYGON AND CATENARY. 195 
 
 tensions of the branches o<'o"' , o"'o"" , also must be equal : 
 and if all the forces j) are applied at rings, the tension must 
 be the same throughout the entire cord. This indeed is 
 evident, inasmuch as the whole is then one cord ; and from 
 this consideration, what has been proved relative to the 
 directions of the transverse forces would have immediately 
 followed : for the tensions of o"o'" and o"'o"" being equal, 
 they must make equal angles with the direction of the force 
 p" by which those tensions are equilibrated. This is the con- 
 dition of a cord strained round a polygon. The transverse 
 forces are the reactions of the angular points, the reaction at 
 any point being equal to the tension multiplied by twice the 
 cosine of half the angle. Thus, putting w for the angle, 
 
 there i^ p — %t. cos. -. 
 
 If the sides of the polygon are indefinitely small, and 
 their number indefinitely great, the polygon becomes a curve. 
 The tension of the cord is in every part the same, and there- 
 fore equal to the force by which it is strained at each end ; 
 but the transverse force acting at any point, /. e. the reac- 
 tion of that point of the curve, and, therefore, the pressure 
 upon it is incomparably smaller than the tension. This 
 appears from the preceding equation, for considering the 
 curve as a polygon, with an indefinite number of sides, the 
 
 angle -. becomes a right angle whose cosine is cypher. 
 
 Wherefore, in the case of a curve, the transverse force, to 
 be comparable with the tension, must be taken, not for 
 a mathematical point, but for some definite portion of the 
 curve. 
 
 To obtain the pressure on an element of the curve, which 
 shall be supposed to be of single curvature, let n be the 
 pressure made by a unit of length. This may vary from one 
 element of the cord to another, but may be considered as 
 unvaried in the same element ds. Now, referring this ele- 
 
 o2 
 
196 
 
 STATICS. — SECT. VIIF. 
 
 ment to the axes of x and //, as the tension at any point is 
 the resultant of all the forces applied to the cord from the 
 beginning up to that point, it follows that the tension, re- 
 duced to any one direction, is the sum of all the applied 
 forces reduced to the same direction. Wherefore, a being 
 the angle formed by the tangent at the first point of ds, any 
 element of the cord, with the axis of x, and a being the angle 
 made by the direction of the tangent at the beginning of the 
 
 curve, with the same axis there is — ^.cos.a = ^cos.« -f- \n. 
 sm.a.ds. and differencing, t.da. — n.J*. Now, w being the 
 angle made between the perpendiculars to the tangents at 
 ds, and the beginning of the curve, there is da,:=zdto. Where- 
 
 ds 
 fore, by last equation, t.dio = Nds. Butf?w — — . wherein r 
 
 is the radius of curvature. Wherefore, 
 
 t 
 
 N Z= - . 
 R 
 
 which shows that the pressin'e on the curve is every where 
 inversely proportional to the radius of curvature. 
 
 If the cord is strained round a cylinder with a circular 
 base, there is r = constant, and n r= constant. In this case 
 the equation ^ds — t.dw, by integration, gives ns ~ /w -|- c. 
 and as at the beginning of the arch there is szz o, u) = o. 
 there is also c =: o . wherefore the equation is 
 
 N* = fu). 
 
 which shov/s that the pressure on any arc s, is the product 
 of the tension into the angle of incurvation ; viz. into the 
 angle made by two perpendiculars to the arc, raised at its 
 two extreme points ; and that the pressure made on the cy- 
 linder by each circumvolution of the cord, is 2Tri. viz. the 
 tension multiplied by the number expressing the ratio of 
 the circumference of a circle to its radius. 
 
HOOFS, MICHES, AND DOMES. 197 
 
 Library, 
 
 SECTION IX. ^ — ^-- — -'•'• 
 
 OF THE EQUILIBRIUM OF ROOFS, ARCHES, AND DOMES. 
 
 1. To support a beam by two forces applied at its ends, 
 the forces, if not vertical, should be directed to some point 
 of the vertical passing through the centre of gravity of the 
 beam: and further, the two forces and weight should be 
 proportional to the sines of the angles contained by their 
 directions ; each force being as the sine of the angle con- 
 tained by the directions of the other two forces. Therefore, 
 if the beam is given, together with the direction of one of 
 the sustaining forces, the whole is given : for if the given 
 line of direction is produced to meet the vertical passing 
 through the centre of gravity of the beam, the line, drawn 
 from the other end to this intersection, gives the direction 
 of the other sustaining force; and each is to the weight, as 
 the sine of the angle made by the other sustaining force and 
 vertical, to the sine of the angle contained by the directions 
 of the sustaining forces. 
 
 The two following examples are given, as of much im- 
 portance in practice. 
 
 2. A beam rests with one end against a wall: the pres- 
 sures on the points of support are required. 
 
 Let the beam be ab, (Fig. 102.) resting against the wall 
 at A ; and let o be its centre of gravity. The wall being 
 supposed to be perfectly smooth, its reaction is according 
 to the perpendicular to its surface, at tlie point A. Let this 
 perpendicular be ag, meeting the vertical passuig through 
 o, at G. Now if the end b were to rest on a horizontal plane, 
 and if this plane were also perfectly smooth, its reaction 
 
198 STATICS. — SECT. IX. 
 
 would be vertical ; and the beam could not possibly be sup- 
 ported by these reactions. To support the beam, the force 
 applied at b must be in the direction bg : and then, if the 
 horizontal line, Bn, is drawn, meeting the vertical passing 
 through o, at w, the three equilibrating forces shall be as 
 the sides of the triangle bgw, parallel to their directions. 
 And if the force bg is resolved into two, one of them vertical 
 and the other horizontal, these will be no, Bit, the former 
 equal and opposite to the weight, and the latter to the re- 
 action of the wall, 
 
 Putting F, w, T for the force eg, the weight and the 
 horizontal thrust, we shall have 
 
 BG Bn 
 
 F = w. — . Tr:w. — . 
 
 AD AD 
 
 Moreover, making ab : ob : : a : 1. we shall have 
 BD , BD w 
 
 Bniz — and t — w. 
 
 a a. AD a. tan. abd 
 
 Whereby it appears, that in a given beam, the horizontal 
 thrust varies inversely as the tangent of its inclination to 
 the horizon ; and accordingly, that for the horizontal posi- 
 tion, the thrust is infinite. 
 
 If the span bd is given, the thrust shall vary as the 
 weight divided by the vertical height of the point a, above 
 the point b; i. e. if the weight of the beam is as its length, 
 the thrust shall be inversely as the sine of the inclination to 
 the horizon, 
 
 3. If the beam rests on a prop at the upper end, as on 
 the edge of a wall, the reaction of the prop shall be per- 
 pendicular to the beam. Wherefore, drawing the line AG 
 perpendicular to ae, meeting at g the vertical on, passing 
 through o, the centre of gravity, i^Iig. 103.) the direction 
 of the support at b must be bg. And the line Bm being 
 drawn parallel to ag, the three equilibrating forces shall 
 be as the sides of the triangle GBtn, parallel to their direc- 
 
ROOFS, ARCHES, AND DOMES. 199 
 
 tions ; i. e. putting p for the reaction of the prop, we shall 
 
 have 
 
 mn BG , Bn 
 
 p = W. . F = W. . andTrrw. . 
 
 Gm Gin Gill 
 
 Making ab — a. ob. we shall have 
 
 Gjn = a. of/i. 
 
 and 
 
 niB COS. ABD 
 
 p — w. = w. . 
 
 a. om a 
 
 BW W 
 
 T = W. 
 
 a.om a(tan.+ cot.) ABD ' 
 
 and T shall be greatest when (tan. + cot.) abd is least ; i. e. 
 
 w 
 when ABD = 45°. Its value being then, — . from which it 
 
 diminishes to cypher, as the beam approaches either to the 
 
 vertical or the horizontal position. 
 
 If the beam is bisected by its centre of gravity, « = 2. 
 
 w 
 and at the elevation of 45°, t = -—. 
 
 4 
 
 If the span, b« = — , is given ; and if w varies as the 
 
 length of the beam, the thrust shall vary as — ; i. e. as the 
 
 om 
 
 sine of the horizontal elevation of the beam. 
 
 The case, considered in the last article, is that of a rafter 
 abutting against another, the top or ridge being at liberty 
 to descend in the vertical. The present article relates to a 
 rafter loosely supported on a prop at the upper end, and at 
 liberty to slide from it, in the direction of its length. 
 
 4. If the funicular polygon loaded with weights is in- 
 verted, it is evident that the weights, decomposed according 
 to the directions of the sides, shall become thrusts instead 
 of tensions, i. e. compressing in place of divellent forces ; 
 and that these thrusts are to be communicated from one 
 point to another, by incompressible bars, instead of inex- 
 tensible cords. 
 
200 STATICS. — SECT. IX. 
 
 The weights applied at the angles, when resolved in the 
 directions of the bars, yield components depending on the 
 angles of inclination, as in the funicular polygon : and when 
 the equilibrium is established, the thrusts in each separate 
 bar are equal and opposite. Wherefore, putting thrusts for 
 tensions, the theory of the equilibration of apposited bars 
 is the same as that of the funicular polygon loaded with 
 weights ; and established by the same reasoning. The 
 weights, it is true, are never placed at the angles exclu- 
 sively: but the vertical pressure on any bar or beam, in- 
 cluding the weight of the beam itself, may be referred to the 
 extreme points ; i. e. to the angles of the polygon. For the 
 portion of the weight bearing on either extremity, is to the 
 entire weight, as the distance of the centre of gravity from 
 the other extremity to the entire length of the beam. The 
 vertical pressure made on the same point, by the beam 
 which constitutes the other side of the angle, is found in the 
 same manner ; and the sum of these is the pressure on the 
 intermediate angular point. In a roof covered with mate- 
 rials of uniform thickness and density, the weight on each 
 angular point is half the sum of the weights on the contain- 
 ing sides. 
 
 Thus, let o, o', o", o'", o"" be the polygonal roof (Fig. 104.) ; 
 P) P'> v" i p"'i P " *he weights incumbent on the angular 
 points, or suspended from those points. Then, the weights 
 at each angle being resolved in the directions of the rafters 
 which meet at that point ; it is requisite and sufficient for 
 equilibrium, that the thrusts in each bar should be equal : 
 t, t', t", t"' being those thrusts, we have 
 
 , sin. a , ,, sin.i' 
 
 P' ~ — T'"T — T\ —t— P ' 
 
 ' sin.(« -f a') ^ *sin.(6 + 6')' 
 
 or, 
 
 , sin. a „ sin. 6' 
 
 sin.a.cos.ff -j-sin.a'.cos.a sin.6 .cos.i+sin.6.cos.6' 
 
that is, 
 
 ROOFS, ARCHES, AND DOMES. 201 
 
 P £ 
 
 cos.tt'+sin.a'.cot.a cos.6 -\-^\x\.b.co\..b' 
 Or dividing the denominators by sin. a', sin.Z>, which are 
 equal, it will be 
 
 P' _ P" 
 
 cot. a + cot. a' cot.6 + cot. 6' ' 
 
 and so of the rest. 
 
 It is to be observed of the angles of the polygon, except 
 that at the highest point, that the parts into which they are 
 divided by the vertical lines, are, one of them acute and the 
 other obtuse ; and, therefore, that the co-tangents of those 
 parts'^ are affected with opposite signs. The theorem, 
 therefore, agrees, as it should, with that for the funicular 
 polygon ; where we had 
 
 p' _ p 
 
 tan.e — tan.e' tan.e' — tan.e 
 
 -, &c. 
 
 -^ — — J (= f . sin. a') is the horizontal thrust; 
 
 cot.« + cot. a' 
 
 whereby it appears, that the horizontal thrusts are equal 
 throughout the system. 
 
 The number of equations thus supplied, is one less than 
 the number of angles in the polygon, i. e. n being the num- 
 ber of rafters, it is n — 2. Putting w-, m', m", &c. for the 
 lengths of the rafters ; s for the span ; and k for the height 
 or pitch, we have also the equations 
 
 s = in. sin.a+m'.sin.6+ Sec. h= tn.cos.a 4-???/.cos.6 + &c. 
 
 the rafters, concerned in the second of these equations, 
 being those on one side of the highest point. If the poly- 
 gon is symmetrical, the number of distinct equations is less 
 than that of the unknown quantities. For example, if it is 
 proposed to construct an equilibrated roof of four equal raf- 
 ters, m, (Fig. 105.) the structure being symmetrical about 
 
202 STATICS. — SECT. IX. 
 
 the vertical passing through the highest point o", it will be 
 
 sufficient to consider one half; and for this we have, from 
 
 the conditions of equilibrium, 
 
 cot.a + cot.a' = 2 cot.b. 
 but 
 
 wherefore, 
 
 cot.a' = — cot.b. 
 cot.a = 3 cot. 6. 
 
 Moreover, 
 
 h = m{cos.a + cos.b), s = 2m{sm.a + sin. 5). 
 
 The quantities here concerned are 7n, s, k, and the angles a 
 and b ; and as there are but three equations, it is requisite 
 that two of these five quantities should be given: one of 
 them being a line. 
 
 5. In constructing an arch of masonry, a piece of frame 
 work, called a centre, is first erected, whose circumference 
 corresponds to the figure of the intended arch. On this are 
 placed truncated wedges, called voussoirs, beginning at the 
 piers or abutments, and finishing at the top or crown. On the 
 voussoirs are laid other materials suitable to the particular 
 purpose ; whether to support a road way, an aqueduct, or a 
 building. The centering is then removed, and the whole 
 abandoned to the mutual pressures of its parts. The vous- 
 soir at the crown is called the key-stone ; the interior curve 
 the intrados ; and the exterior the extrados. 
 
 In treating of the equilibration of such a structure, the 
 faces of the voussoirs are supposed to be perfectly smooth, 
 and therefore incapable of transmitting lateral pressure, eX' 
 cept in directions perpendicular to the joints. 
 
 The tendency to descend, in each voussoir, by its weight 
 and that of the load it carries, is to be equilibrated by the la- 
 teral thrusts of the voussoirs on each side : or, which is the 
 same thing, those weights, resolved in directions perpen- 
 dicular to the joints, should yield components equal and op- 
 posite in pairs, i. e. the weight of each voussoir, including 
 that of its load, should be resolvable into forces, equal and 
 
ROOFS, ARCHES, AND DOMES. 203 
 
 opposite to the pressures against its faces. Accordingly, the 
 theory of the arch may be derived from that of the wedge, 
 or of the catenary. Because of the importance of the pre- 
 sent subject, it is proposed to consider it without reference 
 to those theories. 
 
 Let the arch be that represented in (Fig. 106.) a, b, c, d, 
 &c. the voussoirs; ab, ab\ &c. the joints; and cd, c'd', &c. 
 the vertical lines passing through the centres of gravity of 
 the several voussoirs, and the weights with which they are 
 loaded. 
 
 The pressure made by the key stone, against the face of 
 the contiguous voussoir, is the resultant of the pressures 
 made by all the points of its face ab ; and this resultant may 
 be applied at any point of that face. If m is that point, let 
 mo, perpendicular to ab, be continued to meet the line c'd' 
 at o' ; and let oW, perpendicular to a'b', meet the line c"d" 
 at o" ; and o"m" , perpendicular to a"b" , meet the line c"'d"' 
 at o'" ; and let o"'m"' be perpendicular to a"'b"', &c. The 
 broken hne, ooo"o"''m"', shall be the line of pressure for one 
 side of the arch ; and it is similarly made out for the other 
 side. As a voussoir can support a pressure, only when di- 
 rected perpendicularly against its face, it follows, that if the 
 line of pressure passes above the lower voussoirs, the upper 
 part of the arch must spread over them ; the arch descend- 
 ing at the crown, and rising at the haunches : and that if the 
 line of pi-essure passes below the lower voussoirs, the arch 
 shall fail in the opposite way, i. e. by rising at the crown, 
 and descending at the haunches. 
 
 The line of pressure may vary through a space, depend- 
 ing on the depth of the voussoirs ; and this affords the 
 means of providing against the failure of the structure in 
 this way. Thus let two curve lines be described perpendi- 
 cular to the joints ; one of them passing through the upper, 
 and the other through the lower extremity of the key stone. 
 The condition, relative to the line of pressure, requires that 
 
204 STATICS. — SECT. IX. 
 
 every joint should be contained, in the whole or in part, be- 
 tween those curves ; and this condition is the more easily 
 secured, the greater the depth of the voussoirs. 
 
 If this condition is not satisfied, the forces cannot be 
 opposed to each other, and therefore the equilibrium can- 
 not subsist; but this being satisfied, it may be established : 
 and for this it is requisite, that the resolved forces should 
 be equal to those by which they are to be severally sup- 
 ported. 
 
 To develope this second condition, let the line CD be 
 vertical, and dd„, horizontal : and let the lines cd', cd'', 
 cd'", &c. be parallel to the joints ab, a'b' , a"h", &c. Then, 
 in order to consider one-half of the arch, let the voussoir, a, 
 be supposed to be divided into two equal parts, by the line 
 CD. The weight of the half voussoir, cabd, is perpendicular 
 to dd'; and the forces into which it is resolved, are perpen- 
 dicular to CD, cd'. Wherefore, this weight and the forces, 
 into which it resolved, shall be represented by dd', cd, cd'. 
 In the same way, if the weight of the voussoir b, is re- 
 presented by d'd'', the forces into which it is resolved shall 
 be cd', cd". And conversely, if these are the resolved 
 forces, dd" shall be the weight of b ; and so of the rest. 
 
 This statement is altogether independent of the sup- 
 position of equilibi'ium among the several voussoirs. But 
 for equilibrium, it is requisite that the thrusts should be 
 equal, as well as opposite. Wherefoi'e, if the weight of 
 half the voussoir a, is represented by dd', and accordingly 
 the forces into which it is resolved by cd, cd', one of those 
 into which the weight of b is resolved, must be likewise re- 
 presented by cd' ; and, therefore, its weight, and the other 
 of the forces into which it is resolved, by d'd" and cd". And 
 the thrust from above, against the joint a'6', being cd", this 
 must also represent the opposite thrust from below ; and 
 accordingly, the weight of c, and its resolved forces shall be 
 d"d'", cd", cd'". Thus the lines of the figure shall serve 
 
ROOFSj ARCHES, AND DOMES. 205 
 
 to compare, not only the weight of each voussoir with the 
 forces into which it is resolved, but all the, forces promis- 
 cuously : the weight of the half voussoir a, and those of the 
 rest being represented by dd', d'd", d"d"', &c. and the 
 thrusts at the several joints by the lines cd', cd", cd'", &c. 
 Wherefore, if w\ w", w" , &c. are the weights ; and «', a\ 
 a", the angles made by the joints with the vertical, v/e shall 
 have 
 
 10 m w 
 
 tan. a tan. a — tan. a tan. a"' — tan. a" 
 
 w' + to" + iv" 
 tan. a 
 
 i: e. the weights of the several voussoirs are as the differences 
 of the tangents of the angular inclinations of their faces to 
 the vertical ; and the weight taken from the crown to any 
 joint, as the tangent of the angle made by that joint with 
 the vertical. 
 
 Putting t' , t" , f, &c. for the thrusts perpendicular to 
 the joints, these are as the secants of the same angles, i. e, 
 
 t' _ t" _ t'" __ 
 sec. a' sec. a" sec. a!" 
 
 And if the weight of the arch from the crown to any joint is 
 known, the thrust is given by the equation 
 
 sm. a 
 wherein a and t relate to any joint indifferently; and w 
 denotes the weight of the arch, from the crown to that 
 joint. 
 
 The thrust perpendicular to a joint, multiplied by the 
 cosine, or divided by the secant of the inchnation of that 
 joint to the vertical, is the horizontal thrust : and as the 
 thrusts perpendicular to the joints vary as those secants, the 
 horizontal thrust is the same throughout the entire system ; 
 
206 
 
 STATICS. — SECT. IX. 
 
 and, therefore, equal to the thrust at the crown. Its value 
 is given by the equation 
 
 , , , COS. a IV 
 
 n.t zz. f.cos.a := w. —. =z . 
 
 sin. a tan. a 
 
 The horizontal thrust is therefore equal to the weight of 
 the arch, taken from the crown to the joint inclined to the 
 vertical in an angle of 45°. 
 
 All this is agreeable to what has been found relative to 
 the funicular polygon or catenary ; the horizontal elevations 
 of the cord, in its several parts, being the inclinations of the 
 perpendiculars on the cord to the vertical. 
 
 6. On these principles the curve of the extrados might 
 be assigned, when that of the intrados is given, together 
 with the depth of the work on the crown ; and vice versa. 
 But the application of these principles to practice is some- 
 what precarious, owing to the general uncertainty of the real 
 conditions of the problem to be solved. The three following 
 are given as extreme cases, of which all others are more or 
 less compounded. 
 
 7. Let it be proposed to find the extrados ; the voussoirs 
 extending to this line, i. e, filling the space between the 
 upper and lower curves. 
 
 The joints may be supposed to be every where perpen- 
 dicular to the curve of the intrados ; and the voussoirs them- 
 selves indefinitely thin wedges ; in which case, their faces 
 shall converge to the centres of curvature, as represented in 
 (Fig. 107.), where o, o', are centres of curvature; the angles 
 at those points being indefinitely small and equal. 
 
 Putting w for the weight of the arch, measured from any 
 joint to the crown, as before, and B for the angle made by 
 the same joint with the vertical, we shall have 
 
 d.w d.tan.6 „„ ,,. 
 
 ■7^- = ^--^^=^^-^^^-^- ^'^ 
 
 Moreover, if r is put for the radius of curvature at any point 
 
ROOFS, ARCHES, AND DOMES. 207 
 
 I, of the intrados ; and I for the distance of the extrados 
 from the centre of curvature, we shall have the sector 
 
 r'^.M , ^ r-.iie 
 
 OIK — — -— , and the sector ogh = —-: — . 
 2 2 
 
 and the vousson* ighk — ^^ — -^ — . wherefore, 
 
 P - r"" = Asec.^e. 
 
 omitting the multiplication by 2, as a is a constant quantity 
 as yet to be determined. 
 
 At the crown, sec. dz^l. wherefore, putting I' and r for 
 the analogous quantities belonging to this point, we have 
 
 A-P- r\ 
 and therefore, 
 
 giving 
 
 P zz r^ + {P - O sec.20. (2) 
 
 If ^ denotes the perpendicular from the extremity of the 
 line I, upon the horizontal line drawn through the centre of 
 curvature, we shall have 
 
 whereby 
 
 f. sec.2 = r^ + {P- i''^)sec.W. 
 or, 
 
 ^^ = ^e + ('^-'")- 
 
 which, for — 90°. is 
 
 7f = P-r'\ 
 
 shewing, that when the curve springs at right angles from a 
 horizontal plane, it has an asymptote parallel to that plane, 
 at a distance equal to Vp — r''^, below which it does not 
 descend. 
 
 If the curve of the intrados is a circle, r' = r. and equa- 
 
5^08 STATICS. — srcT. IX. 
 
 tion (2) may receive this form, 
 
 /2 = r- + (r- r") tan.'-^e. 
 
 In flat arciies, such as are formed of brickwork over the 
 windows of dweUing houses, the joints all converge to a 
 common centre ; and therefore, the differences of the tan- 
 gents are as the breadths of the voussoirs themselves, mea- 
 sured either on the upper or lower edges. Accordingly, 
 the weights of the voussoirs being proportional to their 
 breadths, the vertical heights must be equal, i. e. the upper 
 line of the arch must be also right. 
 
 The extreme joints, at the angles of the opening, are 
 usually inclined to each other in an angle of 60° ; therefore, 
 the distance from the angles to the centre, is equal to the 
 v/idth of the opening; and this, for windows, is generally 
 about five feet. Accordingly, the circle passing through 
 the angles, and perpendicular to the joints, rises above the 
 reveal, by a space equal to 5^.vers.sin.30°, which is some- 
 what more than 8 inches ; and the line of pressure shall not 
 be contained within the voussoirs, unless they are at least 
 equal to this in depth. 
 
 8. Let the voussoirs be supposed to be of evanescent 
 depth, i. e. let the series of wedges be reduced to a line, 
 which is the curve of the intrados ; and let the pressure of 
 the incumbent materials be vertical. 
 
 This is the supposition on which the theory of the 
 arch is commonly founded. It differs most widely from 
 the cases which present themselves to the engineer : yet 
 the conclusions, derived from this supposition, may afford 
 him some elements for his calculations, in what relates to the 
 pressures of the loose materials, with which the arch is fre- 
 quently loaded. For this reason it is here briefly developed. 
 
 Let s be the arch measured from any point to the crown ; 
 
 ds shall be its element: and having d.B zz -^. equation (1) 
 may be changed into 
 
ROOFS, ARCHES, AND DOMES. 209 
 
 ^"-=i.sec.'9. (3) 
 
 d.s r 
 
 Moreover, taking the span at the abutments for the axis of 
 Xi we shall have 
 
 ds = dx. sec. Q. 
 
 Wlierefore, 
 
 d.iv A 3/j 
 
 -7 — = -. sec.'e'. (4) 
 
 d.x r 
 
 If h is put for the height of the work, i. e. for the vertical 
 distance between the intrados and extrados, we have 
 
 d.io = h.dx. and h zz -. sec.'^O. 
 
 r 
 
 To find the value of A, we have at the crown, sec. 0=1. 
 Wherefore, putting // for the height of the work, and r' for 
 the radius of curvature at this point, we shall have, for the 
 same point, 
 
 h' = - . and a = h'.r'. 
 
 r' 
 
 Using this value of a, the equation is 
 
 h'r' 
 h = -— . sec.^'a. (5) 
 
 r 
 
 If the height of the extrados is required, above the hori- 
 zontal line on which the span is measured ; putting ?/ for the 
 ordinate of the intrados, and y for that of the extrados for 
 the same abscissa, the equation will be 
 
 h'r' 
 Y=i/-\-h = 7/-\ . sec.''^. (6) 
 
 If the intrados is a semicircle, we have 
 
 r = r', and sec. = -. 
 
 y 
 
 whereby the value of h becomes 
 
210 STATICS. SECT. IX. 
 
 and the equation of the extrados, 
 
 This curve may descend from tlie crown to a distance 
 greater or less, depending on the vahie of W ; but it has a 
 vertical asymptote at the extremity of the horizontal radius, 
 in approaching which, it rises to an indefinite height. The 
 same consequence follows from the supposition made in this 
 article, for every arch which springs from the horizon at 
 right angles. 
 
 To find the point where the curve of the extrados cuts 
 its horizontal tangent, we have only to make y = r -f- ^*' in 
 the last equation ; which then becomes, 
 
 '■ + ''' = 2/ + -^- 
 
 and this gives the value of //, the depth of work at the crown, 
 for any proposed extent of horizontal road way. 
 
 Thus, for y — -^, i. e. for an arch of 120°, the equation 
 gives /* = — r. 
 
 ^ 14 
 
 To find the curve of the extrados, that of the intrados 
 being a semiellipse having the lesser axis vertical. 
 
 Let a be the greater, and b the lesser semiaxis. Then, 
 
 The radius of curvature varies as the cube of the normal, 
 which is 
 
 y sec. 0. i. e. r: ~j- :: y^.sec^O : b^, 
 
 which gives 
 
ROOFS, ARCHES, AND DOMES. 211 
 
 Substituting these values of r' and r, in equation (5,) it is 
 
 h- 
 
 y' 
 
 Were the greater semiaxis vertical, the equation would 
 be 
 
 h- 
 
 f 
 
 The expressions, in both cases, being similar to that already 
 obtained for the circular arch. 
 
 9. The depth of the voussoirs being still supposed eva- 
 nescent, if the curve of the intrados were loaded with a fluid, 
 the perpendicular pressures on the equal elements of the 
 curve would be as the vertical heights of the fluid; and, 
 therefore, the distances of the several points of the intrados 
 below the fluid surface would be, every where, reciprocally 
 proportional to the radii of curvature, (Sect. VIII. Art. 6.) 
 And from this principle the intrados can be determined by 
 two successive integrations. 
 
 10. The principle of equilibration in a dome, except in 
 what relates to the horizontal thrusts, is contained in equa- 
 tion (3), viz. 
 
 dlG A g^ 
 
 -J- — -.sec^'t/. 
 as r 
 
 To apply this principle, let yABC, (Fig. 108.) be a portion 
 included between two planes intersecting at the axis to, at an 
 indefinitely small angle ; and let this portion be supposed to 
 be insulated, i. e. supported merely at the two extremities, 
 viz. at V, by the thrust of the equal and opposite portion ; 
 and at the lower surface abc, by the reaction of the work 
 from which it springs. 
 
 Let ahcfed be an element, whose upper and lower sur- 
 faces are perpendicular to the interior surface of the dome. 
 
 p '^ 
 
212 STATICS. — SECT. IX. 
 
 The weight of this element is proportional to its solid con- 
 tent, i. e. to the continued product of its three linear dimen- 
 sions. Wherefore, putting t, for hc\ ds, for af\ and x, the 
 distance from the axis, for ah, to which it is proportional ; 
 we shall have 
 
 c c 
 
 t,x.ds ■=. -. sec.'^O.ds. or, / = . sec.W. 
 
 r T.x 
 
 At the crown, x is cypher ; and at the base, sec. is infi- 
 nite : wherefore, at the crown and at the base the thickness 
 is infinite, unless r is also infinite at those points. 
 
 If the dome is a segment of a sphere, r is constant ; and 
 
 sec.0 is - ; ?/ being the ordinate on the horizontal diameter 
 
 of the sphere. Wherefore, 
 
 c.r 
 t- 2- 
 
 The thickness Ms a minimum, when x.ij^ is a maximum, 
 i. e. when x -zz——. which corresponds to a distance from 
 
 the summit of 36 degrees, nearly. 
 
 11. From the thickness required at the top, it appears 
 that an equilibrated arch consisting of two sectors of a dome, 
 meeting at their highest points, cannot be constructed, un- 
 less by making the curvature at the highest point infinitely 
 small : and it might be supposed, that the same limitations 
 belong to a dome consisting of apposited sectors. But in a 
 dome there are other means of security, arising from the 
 thrusts in each horizontal course. The nature of these ho- 
 rizontal thrusts will be understood, by conceiving the dome 
 to be open at the top. The opening being a horizontal 
 circle, the highest course is an arch, and the tendency of 
 any part to fall inwards, being equable throughout, is effec- 
 tually counteracted by the mutual pressures of the parts. 
 Moreover, whilst the parts of each course are prevented 
 
ROOFS, ARCHES, AND DO^MES. 213 
 
 from falling inwards on the principle of the arch, the inward 
 pressures, whilst permitted to subsist, are fitted to resist any 
 accidental violence tending to push them outwards ; and 
 being of the nature of compressing forces, to bind all toge- 
 ther into one compact mass. 
 
 Such is the advantage to be derived from the inward 
 pressures in the horizontal courses. But in a dome con- 
 sisting of equilibrated sectors, these sectors being merely 
 apposited, do not press against each other. The pressures 
 inward, in each horizontal course, are therefore counter- 
 balanced by the thrusts from above and below. Accord- 
 ingly, a dome consisting of such sectors, having no such 
 security against any violence tending to push any part out- 
 wards, is the weakest of all domes, which can be made to 
 stand independently of the adventitious aid of hoops, cramps, 
 cement, and friction. 
 
 In a dome which swells outwardly beyond that composed 
 of equilibrated arches, not only is the inward pressure in 
 the horizontal courses removed by the thrusts directed 
 from above and below, but there is moreover a tendency 
 to fly outwards ; and as there remains nothing by which 
 this is resisted, the dome must necessarily fall, by bursting 
 outwards. 
 
 But in a dome, which falls within that composed of equi- 
 librated sectors, the tendency inward in the parts of the 
 horizontal courses is never fully removed ; and in a structure 
 of this kind, outwardly concave, this tendency is even in- 
 ci'eased by the jjressures from above and below : for which 
 reason, such domes are much stronger than those consisting 
 of equilibrated sectors. 
 
 From hence it would appear, that a convex dome may 
 be strengthened by removing part of the crown : for the out- 
 ward thrusts from above and below are thereby diminished ; 
 whilst the upper course, by the tendency of its parts inward, 
 constitutes an arch which serves as an abutment for the 
 
214 STATICS. — SECT. IX. 
 
 portions of the sectors beneath. A dome open at top may, 
 
 therefore, carry a cupola or lantern ; provided that the 
 weight of the superstructure does not exceed that by which 
 the opening should be filled up, to render it a dome of 
 equihbration. 
 
 It is true, that the tendency to burst outwards, may be 
 counteracted in a dome of any figure, by hooping, &c. and 
 the horizontal thrusts, moreover, taken from the walls of 
 the building over which it rises, by the same means. \Mien 
 this method is not taken, the massiveness of the walls must 
 be suited to the thrust they have to sustain : and it appears 
 from Art. 5. that, in an equilibrated dome, this is equal to 
 the weight of a portion of the dome, measured from the 
 summit to the course, whose upper surface is inchned to the 
 vertical in an angle of 45°. 
 
STRENGTH AND STRAINS. 215 
 
 SECTION X. 
 
 OF STRENGTH AND STRAINS, DIRECT AND TRANSVERSE ; AND OF 
 THE PRINCIPLES OF FRAME WORK. 
 
 1. When a soft body, of a uniform substance, is com- 
 pressed by a force, equally applied to every part of its sur- 
 face, it is not broken. Hence it follows, that fracture is pro- 
 duced by a compressing force, in consequence of the in- 
 equality of its actions, or of the forces of cohesion, by which 
 it is resisted. 
 
 Bodies are crushed by the lateral detrusion of the parts 
 from their places : and this is variously effected, as the bo- 
 dies are malleable or rigid, of a uniform, fibrous, or granular 
 structure. 
 
 If the body is malleable or ductile, it is easily conceived, 
 that of the particles situated in the direction of the com- 
 pressing force, some shall be detruded, laterally, from be- 
 tween others, without breaking the continuity of the body : 
 and if this operation of the compressing force were equable 
 throughout, the length of the body would be diminished, 
 and the breadth uniformly increased. But this is not the 
 case ; for if the compressing force is evenly applied to the 
 circular ends of a column of such a substance, the detrusion 
 or swell shall be greatest where tlie resistance is least, /. e. 
 at the ends, as is represented by the toruses of architectural 
 columns ; the force of resistance being there least, in relation 
 to the compressing force. If the body is rigid, and still 
 homogeneous, it shall begin to fail, by splintering off in the 
 same parts. 
 
 To find the angle of the splinter, let ae^be a pillar of a 
 
216 
 
 STATICS. — SECT. X. 
 
 homogeneous substance, loaded with a weight w, (Fig. 109.) 
 and let aoh be the angle of the fracture, to be now deter- 
 mined. The cohesive force in the section ob, or od, varies 
 as the lengths of these lines, i. e. inversely as the cosine of 
 the angle aob, or cod, o being the middle point of the line ac. 
 The incumbent weight, w, is a force applied in the vertical ; 
 and, therefore, its energy in forcing the portion aob along 
 the inclined plane ob, is proportional to the sine of the same 
 angle : wherefore, the resistance divided by that energy, va- 
 ries inversely as the product of the sine and cosine of the 
 
 same anffle : i. e. as — : — -: — —. and as the body will fail 
 *= sm.2aob ^ 
 
 where the resistance divided by the energy of the weight is 
 
 a minimum, it will fail where sin. 2aob is a maximum, i. e. at 
 
 an angle ao6 = 45". Wherefore, ^0^=90°. and a pillar of 
 
 iron is said to fail in this way, under enormous pressure ; 
 
 the surfaces of fracture making a right angle with each other, 
 
 as represented in the figure. 
 
 A fibrous substance will not always follow this rule \ for 
 if the direction of the compressing force is parallel to that 
 of the fibres, and these slightly agglutinated, the fibres may 
 give way by bulging in the middle rather than at the ends, 
 where the lateral motions are resisted by enormous friction. 
 In this case the outer fibres, having least support against 
 the force by which they are bent, will be most detruded, 
 whilst those nearer the axis are crippled, as is represented 
 in (Fig. 110). This change continues until the mutual co- 
 hesion among the fibres is broken, and then the body is 
 crushed. A pillar of such material under a compressing 
 force, is greatly strengthened by hooping. 
 
 In a uniform substance, the strength to resist compres- 
 sion is generally far greater than the strength to support an 
 extending force. But in fibrous substances, it is otherwise : 
 for the extending force, having the effect of uniting the 
 fibres, cannot break the cohesion without overcoming their 
 
STRENGTH AND STRAINS. 
 
 217 
 
 united strength. However, in the relations of their strength 
 to bear a load suspended or incumbent, there is the greatest 
 diversity. Thus, a beam of oak will suspend twice as much 
 as an equal beam of fir ; whilst it will support but one-half. 
 The cause of this diversity is probably to be found in the 
 curvature of the fibres of the oak, which renders it less fit 
 to support than to suspend. 
 
 2. But the cohesive forces, which constitute the strength 
 of a beam or rod, are more easily surmounted by a force 
 applied transversely ; especially, if the straining force is ap- 
 plied at a considerable distance from the point of fracture. 
 This is plainly to be attributed to the advantage of leverage 
 then given to the straining force ; as also, perhaps, to the 
 circumstance that the full force of cohesion in the section of 
 fracture is not at once encountered by it. 
 
 The theory of transverse strains is not as yet well esta- 
 blished, owing to the difficulty of ascertaining the physical 
 conditions of the bodies subjected to those actions. Galileo 
 was the first who attempted to bring this matter within the 
 province of theoretic mechanics ; and little has been added 
 to what he had delivered on the subject. Some improve- 
 ments, however, have been made in this theory, by Mari- 
 otte, Varignon, and Leibnitz. 
 
 The operation of a transverse strain is to separate two 
 surfaces, before in contact, by an angular motion round a cer- 
 tain line, which then becomes an axle or fulcrum ; and it was 
 supposed by Galileo, that this line was in that surface which 
 is rendered concave by the action of the straining force. It 
 was also supposed by him, that the force by which the 
 motion round this line is resisted, is the same in all the fibres 
 by which the divided surfaces had been connected. On this 
 supposition, the sum of the forces of the fibres might be 
 supposed to be applied at the centre of gravity of the trans- 
 verse section, which is the surface of fracture ; and the le- 
 verage of this force of resistance would, therefore, be the 
 
218 STATICS. — SECT. X. 
 
 distance of this point from the axis of rotation. According 
 to this account, the moment of the resistances in a rectan- 
 gular beam, would be the sum of the cohesive forces of the 
 fibres in the section of fracture, multiplied by half the depth 
 of the beam, i. e. f denoting the cohesive strength in a 
 superficial unit of the transverse section ; b the breadth ; 
 and h the depth of the beam ; the sum of the cohesive forces 
 
 would hef.bJi. and its moment —^ — . 
 
 Hence it appears, that the strength of a rectangular 
 beam to resist a transverse strain, varies as the breadth 
 multiplied by the square of the depth : for which reason, 
 such a beam placed horizontally is stronger to resist a verti- 
 cal strain when its narrow face is uppermost, than when 
 it is placed with its broader surface in the same position, 
 in the inverse ratio of these dimensions : so that if the 
 breadth is double of the thickness, the strength in the two 
 positions will be as two to one. 
 
 The strongest rectangular beam that can be cut out of a 
 given cylinder is that wherein b.Ji^ is a maximum. This 
 gives 
 
 2b.h.dh + Ir.db = 0. 
 or, 
 
 2b.dh -\- h.d.b = 0. 
 Also, r being the radius of the cylinder, we have 
 
 b'' + h^ = 4^r\ 
 whence 
 
 b.db + h,dh = 0. 
 
 and db being exterminated by these two equations, we have 
 
 /i' = 2b\ i. e. W — 4r2. 
 
 whence we derive the following simple construction. 
 
 Let AsrBo be the circular section of the cyUnder ; and 
 AB its diameter, (Fig. HI.) Let this diameter be divided 
 
STRENGTH AND STRAINS. 219 
 
 into three equal parts by the points m and n : then erecting 
 the perpendiculars m%, no, and drawing the lines ^a, ^b, oa, 
 OB, we have the rectangular section required. 
 
 o. The supposition relative to the equality of the forces 
 exerted by the fibres in the section of fracture, was first 
 contested by Leibnitz. It was properly observed by him, 
 that as all beams v^ere more or less deflected before the 
 instant of fracture, the fibres could not be equally strained ; 
 and that as the forces exerted are known to be proportional 
 to the extensions, those forces of resistance in the several 
 fibres must be proportional to their distances from the ful- 
 crum, or centre of angular motion. Thus, let al), ac, be 
 two lines drawn in the two faces separated by fracture, 
 (Fig. 112.) those lines being perpendicular to the axis of ro- 
 tation passing through the point a ; we may then confine 
 our attention to the strains in the triangular surface abc. 
 The strength of a fibre to resist a longitudinal pull be- 
 ing denoted, as before, by/, this will be the force exerted 
 by the fibre 6c, when about to break : and that of any other 
 fibre will be less in the ratio of its distance from a to the 
 depth of the beam. Accordingly, the sum of the forces of 
 resistance, exerted at the moment of fracture, wovdd be only 
 
 ■PI 1 
 
 ' , instead oi f.b.h. which would have been the measure 
 
 of the strength if all the fibres acted alike, according to the 
 Galilean hypothesis. Moreover, the sum of these strains 
 would be applied at the centre of gravity of the triangle abc. 
 whose distance from a, the centre of motion, is f A. and ac- 
 cordingly, the moment of the forces of resistance would be 
 
 , , f.b.h'' . , ^f.b.P 
 
 measured by - — ; — . mstead oi ' — - — . 
 ■^ 3 2 
 
 It is be observed, that the conclusions obtained from the 
 Galilean hypothesis as to the comparative strength of a rec- 
 tangular beam, in different positions, would have equally 
 
220 STATICS. — SECT. X. 
 
 followed from this of Leibnitz ; but this agreement does not 
 extend to all figures. 
 
 Thus, if a triangular beam, with its end fixed in a wall, 
 were to support a weight suspended from the other end ; ac- 
 cording to the Galilean hypothesis, the weight which it would 
 support, with its base uppermost, would be twice as great as 
 that which it would support with its edge uppermost; the 
 distances of the centre of gravity of the triangular section 
 from the fulcrum being, in the one case f A, and in the other 
 
 -. According to the hypothesis of Leibnitz, the beam, with 
 o 
 
 the base of the triangle uppermost, would have a force of 
 
 /" 7 7 - 
 
 resistance whose moment is ^^-^ — . i. e. the area of the tri- 
 
 4 
 
 angular section being -^ , the moment of the resistance 
 
 would be that of the full cohesive force of all the fibres in 
 
 this section, acting at the distance - from the fulcrum. 
 
 Whilst the moment of resistance in the same beam having 
 
 the edge uppermost, would be " ' ' -. or that of the direct 
 
 cohesive strength of all the fibres in the section, acting at 
 
 the distance -, from the fulcrum. So that according to this 
 o ° 
 
 hypothesis, the beam in the former position would be thrice 
 
 as strong as in the latter. 
 
 This diversity, in the conclusions as to the relative 
 strength of a triangular beam, in these two positions, might 
 seem to offer a test of the truth of the hypotheses from 
 whence they follow. But there are other effects of inflec- 
 tion, which are not to be overlooked. 
 
 4. James Bernouilli was the first who proposed to consi- 
 der the compression produced in the concave surface. For, 
 
STRENGTH AND STRAINS. 221 
 
 as he remarked, the whole of the fibres m the section of 
 fracture had not been in a state of tension during the ope- 
 ration ; but whilst on one side they were in this state, those 
 on the other side were compressed. 'These portions of the 
 sections of fracture are separated by a line transverse to the 
 length of the beam ; in which line, the fibres are neither 
 compressed nor extended ; and this line is, therefore, to be 
 regarded as the real axis of rotation. The moment of the 
 forces which resist extension on one side of this line, must 
 be equal to the moment of those which resist compression 
 on the other ; for it is this equality of the moments which 
 determines the position of the neutral axis. Were the mo- 
 ment of the resistance to compression the lesser of the two, 
 the body would yield by compression, whereby the neutral 
 axis would be further removed from the concave side, and 
 this motion would continue until the equality is restored : 
 and we are to reason in the same way, if the moment of the 
 resistances to extension were the lesser. Hence it follows, 
 that the entire of the moment of resistance is twice that of 
 the forces which resist extension ; and therefore, that the 
 more compressible the body, the less the force of resistance 
 which it is capable of opposing to a transverse strain. For 
 the moment of this force is twice the moment of the forces 
 which resist extension ; and this latter moment varies as the 
 square of the distance of the convex surface from the axis 
 of rotation, which is the neutral axis. Hence it is that a 
 beam of soft timber is much strengthened by a cut on the 
 side, which, by yielding, would become concave, the cut 
 being filled with a wedge of hard wood. For the fulcrum, 
 or axis of rotation, is thereby removed to the surface, and 
 what is lost by the absence of the forces which resist com- 
 pression, is more than supplied by the increased leverage 
 given to the forces which resist extension. Thus, if the 
 neutral axis is supposed to be in the middle of the section of 
 fracture, and the saw-cut to be half the depth of the beam, 
 
222 STATICS. — SECT. X. 
 
 the distance of the centre of gravity of the uncut half, from 
 the axis of rotation, becomes thrice as great, and therefore, 
 the moment of the forces, excited by extension, is increased 
 in the same ratio : wherefore, according to the Gahlean 
 hypothesis of the equahty of the forces excited in the fibres, 
 the moment of resistance would be increased in the ratio of 
 3:2. 
 
 To see what would be the increase of strength according 
 to the hypothesis of Leibnitz, we have only to integrate the 
 
 expression y -7 a;^c?^ from a;= - to x = 7i, and this would give 
 
 ft /V 
 
 J^f.b.h^ for the moment of the resisting force; whereas the 
 sum of the moments of the forces resisting extension and 
 compression before the introduction of the wedge is -^:^f.bJi\ 
 so that according to this hypothesis, the strength would be 
 increased in the ratio of 3^ : 2. This is on the supposition, 
 that the compression can be altogether prevented in this 
 way. But, as this is only in part prevented, we are not to 
 expect an increase of strength to the amount of what is here 
 inferred. 
 
 5. Two important changes in the theory, as delivered by 
 Galileo, have been already noticed ; and it does not appear 
 that, with these emendations, the theory comprises all the 
 physical conditions which influence the result. It has been 
 judiciously remarked by Mr. Barlow, that the fibres do not 
 break immediately after being strained to the utmost of their 
 strength ; but that after this, the force of resistance still con- 
 tinues to act, though with decreasing energy ; whilst other 
 fibres, nearer to the axis of rotation, exert their greatest 
 force : and, accordingly, that fracture does not occur until 
 the aggregate of the moments of the forces of resistance 
 shall have passed its maximum. But when this occurs, i. e. 
 how far the exterior fibre is stretched, before that the whole 
 begins to give way, cannot possibly be known, unless the law 
 of the decreasing energy of a fibre, after exerting its greatest 
 
STRENGTH AND STRAINS. 223 
 
 force, were ascertained. To this may be added another ob- 
 servation, which is a consequence of the preceding ; that af- 
 ter the exterior fibres on either side have exerted their maxi- 
 mum of force, whether in resisting extension or compression, 
 the neutral axis must recede from that towards the other 
 side, where the fibres have not as yet been strained or com- 
 pressed to the Hmit of their strength. This motion of the 
 neutral axis will be towards the side, to which it had been 
 originally nearest, the weaker forces being those first over- 
 come. But it does not follow, that the beam shall be broken 
 by the weight which weakens the strength on one side only, 
 for this may occur before that the fibres at the other side 
 have arrived at the limit of the forces which they are capable 
 of exerting. Thus a beam is often crippled at one side, by 
 a force which will not break it. These observations have 
 been made for the purpose of showing that it is in vain to 
 expect a complete theory, or one that shall perfectly accord 
 with experiment, until the mechanical process towards frac- 
 ture is better understood. In the mean time, the theory, as 
 far as it has been carried, suggests many useful hints to the 
 practical mechanic. Thus it was shown, why a beam of a 
 compressible substance is strengthened by the insertion of a 
 hard wedge into a cut made in the concave side ; and the 
 same theory suggests the best methods of fishing and 
 splicing. It appears also, that the strength of a joist may 
 be increased, whilst the material is economized, by in- 
 creasing the depth in relation to the breadth : and that a si- 
 milar advantage may be attained in the construction of metal 
 pillars, by casting them hollow. For a hollow cylinder is 
 stronger than a solid one of the same substance ; the centre 
 of forces opposed to fracture, being further removed from 
 the fulcrum in the hollow than in the solid cylinder, whilst 
 the surface of fracture is the same in both. There is how- 
 ever a limit to the extent to which this principle should be 
 carried : for as all the parts should act together, they should 
 
22'i STATICS. — SECT. X, ' 
 
 be left sufficiently strong to resist such impressions as would 
 otherwise cause a partial violation of figure. 
 
 All this is seen exemplified in the woi'ks of nature, where 
 lightness is not less requisite than strength ; as in the bones 
 of most land animals, the quills of birds, the stems of reeds 
 and grasses : and suitably to the purposes of the Allwise 
 Contriver, we find that the substances of such bodies are of 
 a more than ordinary degree of hardness and tenacity. 
 
 On the whole, it is to be observed, that the equilibrium 
 between the straining and resisting forces continues until the 
 moment of fracture. For as the former is increased, the re- 
 sisting forces are also more intensely excited until their mo- 
 ment becomes a maximum ; and it is only when the straining 
 force is increased beyond this limit, that the equilibrium is 
 violated. 
 
 6. Hitherto the strength of materials to resist fracture 
 has been considered, chiefly according to the different 
 modes in which this strength may be supposed to be exerted: 
 but it is no less necessary to form an estimate of the efficacy 
 of the straining forces, according to the different modes in 
 which they may be applied. In treating of this part of the 
 subject, the energy of the resisting forces may be regarded 
 as unvaried. 
 
 If a beam, having one end fixed in a wall, carries a 
 weight on the other end, the whole may be regarded as a 
 bent lever of the first kind : the fulcrum being: at the axis of 
 rotation, in the plane of the surface in which the beam would 
 break when overloaded. The arms are the distances from 
 this fulcrum to the point at which the weight is applied on 
 the one hand, and to the centre of the resisting forces on the 
 other. And in the case of equilibrium, these contending 
 forces are reciprocally as the lengths of the arms by which 
 they act; or, more properly, they are reciprocally as the 
 perpendiculars let fall from the axis of rotation on the direc- 
 tions of the forces. In what follows, the flexure of the beam 
 
STRENGTH AND STRAINS. 225 
 
 being supposed to be inconsiderable, the length of the arm 
 shall be taken as the measure of leverage. 
 
 Whilst the beam is secured at one end, the energy of a 
 weight, w, to produce a strain in a transverse section at the 
 distance x, from the weight, is w.x. For the equilibrium 
 subsisting between the straining force, and that by which it 
 is resisted in any section, will not be disturbed if the beam 
 is fixed at that section : the weight will then act, by the arm 
 of a rigid lever, against the resistances in the same section. 
 The energy of the weight, being expressed as above, is 
 greatest in the section which coincides with the plane of the 
 wall : and putting I for the length of the beam, or the dis- 
 tance of the weight from the surface of the wall; its value at 
 that distance is iv.l. 
 
 7. If a beam, supported loosely at its two ends by props, 
 carries a weight at any intermediate point, the strain on the 
 section made at the part where the weight is applied, is 
 evidently the same as if it were there placed on a wall, and 
 loaded at each end with a weight equal to the reaction of 
 the prop which supports that end. Therefore, to find the 
 energy of the strain, we have only to find the pressure on 
 each of the props, and to multiply it into the distance of that 
 prop from the weight. Accordingly, p, p denoting the 
 pressures on the props; a and b (Fig. 113.) d, d' their dis- 
 tances from the weight at c, we have 
 
 w.d 
 
 and for the moment of the straining force, 
 
 iv.d'.d 
 
 I 
 For any other point between a and c, whose distance 
 
 7/ 
 
 from a is x, the moment of the straining force is ■ ' "^ ■■. 
 
 This moment is, therefore, greatest in the section where the 
 
 Q 
 
226 STATICS, — SECT. X. 
 
 weight is applied. And if the place of the weight is varied, 
 
 it is greatest when the weight is placed at the middle point 
 
 w I 
 of the beam, its value being then -^. The reaction of the 
 
 prop B has the same effect ; wherefore, -— , the energy of 
 
 the weight, applied at the middle of a beam loosely sup- 
 ported, is equal to that of the same weight applied at the 
 end of a beam of one-half of the length, the other end being 
 secured. Or, it is the same as that of one-half of the weight 
 ajDplied at the end of the entire beam, the other end being 
 fixed. Whence it follows, that a beam supported loosely 
 on props, would carry on its middle point, twice the weight 
 which it could support at one end, the other end being 
 fixed : and it may, therefore, be said to be twice as strong 
 when treated in the one way, as it would be in the other. 
 
 8. If a beam were secured at both ends, it becomes still 
 stronger to support a weight in the middle, in the ratio of 
 2:1. For the weight required to produce the strain at the 
 middle, and those at the two ends, is equal to the sum of 
 those which would be requisite to produce them separately, 
 i. e. it is the sum of the weight, which would produce the 
 strain in the middle, the ends being loosely supported, and 
 of the two weights which would produce the strains at the 
 wall, the beam being sawn through the middle. Now if the 
 three strains were equal, the moment of the weights which 
 by acting on the half lengths produce those at the wall would 
 
 be, each of them, equal to — ; and, therefore, to produce 
 
 the triple fracture, the moment of the weight in the middle 
 
 n 
 
 should be - z/;^ . But the deflections at the ends near the 
 
 wall are, each of them, only half of that at the middle. 
 Wherefore, the weight supported by the sum of the re- 
 sistances at the ends, is only equal to the weight supporteol 
 by the resistance of the middle section. Consequently, by 
 
STRENGTH AND STRAINS. 227 
 
 fixing the ends of the beam in the wall, the weight which it 
 will support at the middle is doubled. 
 
 If equal weights are uniformly distributed over a beam 
 fixed at one end, the strain on that end shall be half of what 
 it would have been, had the sum of the weights been applied 
 at the remote end. 
 
 For, let div be the weight resting on dx, a portion of the 
 length, we shall have 
 
 w.dx 
 
 d.w zz 
 
 I 
 
 and putting x for its distance from the fixed end, the energy 
 with which it acts on that end is 
 
 w.x.dx 
 
 I 
 and the sum of these moments, or 
 
 w C 1 w.x^. 
 
 which needs no correction. This, for x:=zl, is — . which 
 
 is half the moment of the weight tv, when applied at the 
 outer end. 
 
 If the beam is supported loosely on its ends, the strain 
 at the middle point, on the same supposition of the equal 
 distribution of the load, is the half of what it would have 
 been, had the sum of the weights been applied at that 
 point. 
 
 For the force on dx is — j — ; and its moment to pro- 
 
 V 
 
 duce fracture at o, the middle point, is had by multiplying 
 
 ... . T n TT71 r • • tv.x.dx 
 this mto X, its distance Irom o. VVhereiore, it is j — : 
 
 and the sum of these from o to a, is -^ : and the same for 
 
 o 
 
 the weights between o and b. Accordingly, the whole mo- 
 
 Q 2 
 
228 STATICS. — SECT. X. 
 
 ment at o, of the weight thus uniformly distributed is -^ : 
 
 which is half the strain produced by w placed at that point. 
 
 Since the energy of the uniformly distributed weights, 
 to produce fracture, at the middle point of the beam when 
 loosely supported, is half of that, which would belong to 
 those weights if concentred at that point ; and since, when 
 the beam is fixed at the ends and sawn in the middle, the 
 energy of the same weights to produce fracture at the fixed 
 ends, is one-half of that which would have been exerted, 
 had they been concentred at the same point ; it follows, that 
 the energy to produce both of these effects conjointly, must 
 be one-half of that, with which the weights would have 
 acted, if concentred at the middle point, i. e. if a beam 
 is secured at both ends, and uniformly loaded, it will bear 
 double the weight which it could have borne at its middle 
 point. 
 
 Therefore a beam, whether fixed at one end or loosely 
 supported at both ends, or fixed at those points, is capable 
 of bearing twice the load, uniformly distributed, that it 
 could have borne if placed altogether at the unsupported 
 end of the beam in the first case, or at the middle in each 
 of the two other cases ; i. e. in all cases, the strength of 
 the beam is doubled by the uniform distribution of the 
 load. 
 
 9. If the force by which a beam is in danger of breaking 
 is its own weight, the same theorems will apply. And it is 
 worthy of note, that as the energy of the uniformly distri- 
 buted weights varies as w.l, and that of the force of resis- 
 tance as the breadth of the beam, multiplied by the square 
 of its depth, i. e. as h.d^. so xiw is the limit of the weight 
 
 which it will carry in this manner, it shall vary as -^ — . 
 
 which, for similar beams, is as P-. i. e. the strength of 
 
STRENGTH AND STRAINS. 229 
 
 similar beams, estimated by the load which they will carry 
 in this manner, varies as the squares of their lengths. 
 
 But though the strength of a beam to support an extra- 
 neous load, increases as the square of its linear dimensions, 
 those dimensions being increased in the same proportion, 
 yet its strength to support its own weight, so far from being 
 increased, is diminished. The reason is, obviously, that 
 its own weight increases as the cube ; and the moment of 
 that weight, as the fourth power of its linear dimensions, and 
 therefore, faster than its strength to support it. Thus if 
 there are two similar beams, whose lengths are / and /', se- 
 cured each at one end in a wall: their weights are as l^ and 
 /'^, and the energies of those weights to produce fracture, 
 as ^^, l'^, whereas, the energies of the forces of resistance 
 are as h .tP : h' .d"^ , i. e. as P'.V^. 
 
 If the weight is known which a beam, fixed at one end, 
 will just carry at the other end without breaking, we may 
 find the length of a similar beam which shall just support its 
 own weight. 
 
 For, let I be the length of the smaller beam, m its mass, 
 and w the weight which it will just carry at the end. And 
 designating the length and mass of the larger beam by the 
 capitals, the energies of their loads are 
 
 (1 + ..)/. and|,L. 
 
 And the energies of the forces of resistance are as P^ and 
 L^. And the former must be proportional to the latter, or 
 
 m -\- 2w _ M 
 
 Moreover, 
 and therefore, 
 
 M '.m:: L^-.f 
 
 M = —rr 
 
230 STATICS. — SECT. X. 
 
 which value of m, being substituted in the preceding equa- 
 tion, it becomes 
 
 (m -\-2w) I— m. L. 
 
 giving 
 
 , 2w.l 
 
 -LZZ.I -^ . 
 
 m 
 
 Hence it appears, that of all similar beams of the same 
 material, and fixed in the same manner, there is but one 
 which can barely support itself without breaking. A beam 
 larger than that here estimated, will be broken by its own 
 weight. And hence also it appears, that small animals are 
 much stronger than large ones to carry their own weights ; 
 for which reason, they are far more active. Accordingly, 
 a fly will carry ten times its own weight, perhaps with more 
 facility than a horse can carry a weight merely equal to his 
 own. There is, therefore, a limit to the size of animals and 
 plants, at which they would be oppressed by their own 
 weight, and fall to pieces. This is a natural limit to the 
 magnitude of trees and land animals, which is further re- 
 duced, by the necessity of leaving them strength for other 
 purposes, beyond what would be required for the support 
 of their own weights. This limit is not so much straitened 
 for sea animals, the weight of whose bodies is in a great 
 measure supported, and whose strength to bear it much 
 assisted by the weight and compression of the medium 
 in which they live. Accordingly, we find that some of 
 these are much larger than any of the inhabitants of the 
 dry land. 
 
 The same principle which limits the size of the produc- 
 tions of nature, is equally applicable to the structures of art. 
 Thus, what appears strong in the model, may be very de- 
 ficient in strength when executed: and this must be allowed 
 for by engineers and architects, who would, otherwise, make 
 the most ruinous mistakes. 
 
STRENGTH AND STRAINS. 2Sl 
 
 10. Such are the principles on which the strength of 
 beams is to be computed, with regard to the loads they are 
 intended to bear. In treating of the equilibrium of struc- 
 tures consisting of such materials, these strains were not 
 considered. The component parts were supposed to be 
 rigid and unbending : in which case, the loads supported by 
 them might be supposed to be transferred to the angles : 
 and the subject of investigation was the equilibrium of the 
 longitudinal thrusts. When this equilibrium is secured, a 
 great object is attained : the stability of the structure being 
 then easily provided for. But this is not sufficient ; for the 
 parts of the structure are strained transversely by their own 
 weights, and by the loads they carry: and it has been shewn, 
 that the strength of beams to support such strains is far less 
 than to support a longitudinal pull or thrust ; the absolute 
 force of resistance being much less in the former case, 
 and acting ordinarily at a great mechanical disadvantage 
 in relation to the straining force. Accordingly, after the 
 equihbration of the component parts of a structure, the 
 next object, especially in carpentry, is to convert the trans- 
 verse strains into longitudinal pulls or thrusts. How this 
 is to be accomplished may appear by one or two familiar 
 examples. 
 
 Let AB be a horizontal beam attached to an upright at 
 B, destined to support a weight w, appended at a, (Fig. 114). 
 Putting I for ab, the measure of the energy of w, to strain 
 the beam at b, is iv.l : and unless the beam is very strong, 
 and well secured at b, it will be broken by an inconsiderable 
 weight, or forced out of its birth at b. This is prevented 
 by changing the straining force to a thrust in the direction 
 of AB : which is accomplished by combining with the force 
 w, another force, such that the resultant of the two shall be 
 directed according to ab : and this being effected, the po- 
 sition of the beam is secured, though it should be connected 
 with the upright only by a hinge at b. Let ai\i be a cord 
 
232 STATICS. — SECT, X. 
 
 attached to the beam at a, and to the upright at m ; if this is 
 inextensible, and sufficiently strong, it must necessarily sup- 
 ply the force requisite to change the vertical strain into a 
 horizontal thrust : for the point a cannot descend without 
 stretching or breaking the cord, and if this does not occur, 
 the three forces acting at a must be in equilibrio. These 
 forces are, 1st, the weight w, acting in the vertical ; 2d, the 
 reaction of the cord in the direction am ; and 3d, the reac- 
 tion of the upright in the direction ba. Accordingly, the 
 two first must compound a force equal and opposite to the 
 last, i. e. a force acting in the direction ab. In this case, 
 the resultant in the direction of ab is a compressing force ; 
 and the piece ab is called a strut. We have, therefore, only 
 to connect the point a, by a tie, to a point in the upright, 
 above b, and the lateral strain on ab shall be converted into 
 a longitudinal compression. 
 
 The quantity of the strain endured by the cord, and of 
 the thrust against the upright at b, are compared with the 
 weight IV, by the triangle amb, whose sides are in the direc- 
 tions of these forces. Thus, putting s for the strain on the 
 cord, and t for the thrust against the upright, we have 
 
 T z:: 
 
 sm.ang.A tan. ang. a 
 
 Similarly, by a force acting in the direction am', the 
 transverse strain may be converted into a longitudinal pull. 
 This force may be supplied by a brace an, which, if it is 
 incompressible, will prevent the point a from descending. 
 The forces acting at a are then in equilibrio, i. e. the weight 
 tv, and the reaction of the brace in the direction of nam', 
 will be a compound force in the direction of ba, which, if 
 the hinge or fastening at b is sufficiently strong, will be 
 counteracted by the reaction of that point. The relative 
 magnitudes of the three forces are determined as in the for- 
 mer case. 
 
STRENGTH AND STRAINS. 233 
 
 In the same way a beam may be strengthened, by what, 
 at first view, might appear to be an addition to its load. 
 Thus, let the beam be ab, resting on supports at a and b, 
 (Fig. 115). This shall be greatly strengthened by the piece 
 DC, descending from its middle point, and connected by its 
 remote end c, with the extremities of the beam ab. For 
 the point d cannot descend without crushing the piece dc, 
 or breaking the connexions ac, bc. Therefore, if these 
 parts are sufficiently strong, the strain on the beam at d, 
 where it is most dangerous, is transferred to the points A 
 and B whei'e it is firmly supported. In this case, the con- 
 necting pieces ac, bc, serve to thrust the piece cd against 
 the middle of the beam : wherefore, this last is a strut, and 
 the two former ties ; for which reason, these may be inex- 
 tensible cords. In this manner, the transverse strains are 
 removed by ties and struts. 
 
 11. To take away the lateral thrusts from the walls of a 
 house, occasioned by the weights of the rafters and their in- 
 cumbent load, the heads of the rafters are connected by a 
 tie beam ; which, if it had no other office, needs not be 
 gross. But, commonly, it has a ceiling to support, and 
 perhaps a floor with all the furniture of an apartment. In 
 this case, the tie beam, miless supported, will be apt to 
 yield to the transverse strain, occasioned by its own weight 
 and that of the load it carries. This support is given to it 
 by a tie, or king post, connecting it in the middle with the 
 summit of the rafters. But the rafters also, if long, will 
 require support against the transverse strains, to which they 
 are subjected. This is supplied by braces, descending from 
 the middle of the rafters, and abutting against the lower 
 end of the king post, on which they are joggled. 
 
 In this way, the middle of each rafter is supported ; and 
 the load and its bearing being reduced each to one-half, the 
 strain upon the half rafter is reduced to one-fourth of that 
 on the entire rafter supported only at the ends. All this is 
 
234 STATICS. — SECT. X. 
 
 exhibited in (Fig. 1 16.) where ab, ac are the rafters ; bc the 
 tie beam ; ad the king post ; and ed, fd the braces. This 
 is the most common kind of roof, and it is sufficient for all 
 ordinary purposes. 
 
 12. But in trusses and frame work of all kinds, besides 
 the equilibration of the longitudinal thrusts, and the re- 
 moval of the transverse strains, it is also requisite to provide 
 for the inviolability of the figure, against the various influ- 
 ences to which the work may be exposed. When this is 
 secured, all the parts shall act together as one solid mass ; 
 each contributing to support the strains on the rest. This 
 end is attained chiefly by resolving the whole into triangles. 
 For in such figures, whilst the lengths of the sides continue 
 unchanged, their relative positions must also be unchanged. 
 This is not the case in figures consisting of any greater 
 number of sides. For example, nothing could be worse 
 than a trapezium roof consisting only of three rafters and 
 a tie beam, such as that represented in (Fig. 117). For 
 though the thrusts are taken from the walls, by the tie beam, 
 and the points a and b thereby fixed, yet the angles being 
 changeable, the strain on each piece from its own load, and 
 also, the thrusts against it from the contiguous members, 
 are, consequently, changeable. 
 
 Were the middle points of the horizontal beams con- 
 nected together by the piece mn, (Fig. 1 1 8.) those beams 
 would be constrained to act together ; but the deflections to 
 which they are liable being in the same direction, little is 
 thereby gained, in proportion to the quantity of material 
 added. Were the middle points of the rafters connected 
 together by the piece op, the deflection in those parts 
 would be resisted somewhat more eflTectually, for should the 
 rafters bend, it must be in the same direction; but the 
 strains from their loads being both inwards, i. e. in opposite 
 directions, they would yield only by the difference of those 
 strains. 
 
STRENGTH AND STRAINS. 235 
 
 By means of the two uprights, ce, df, the roof becomes 
 considerably stronger : (Fig. 119.) for the angles of the tri- 
 angles, ACE, BDF, are unchangeable as long as the distances 
 of their angular points remain unchanged. And so are also 
 the angles of the four-sided figure cefd, as long as the 
 lengths of its sides, and those of the lines ac, bd, are un- 
 changed. But the chief recommendation of this construc- 
 tion is the space cefd, which it allows for an apartment. For 
 the rafters ac, bd, the truss beam cd, and the tie beam ab, 
 are all liable to deflection from the transverse strains. 
 
 ■ These deflections are effectually prevented by continuing 
 the rafters above the truss beam, so as to meet at a point ; 
 or should this roof be deemed too high, the pitch of the raf- 
 ters above the truss beam may be diminished, as represented 
 in (Fig. 120). For thereby a new fixed point is gained, by 
 means of which, the truss beam may be prevented from 
 bending by the king post hm. This gives another fixed 
 point, M, which may serve for abutments to the braces mn, 
 MP, by which the stiffness of the upper rafters is secured. 
 The triangle chd, being then a firm piece of frame work, c 
 and d are fixed points, from which the queen posts, ce, df, 
 may descend, to carry the tie beam ab. Two more fixed 
 points, E and f, are thus obtained, which serve as abutments 
 for the braces ei, fk, by which the rafters ac, bd are sup- 
 ported. The braces eg, fl may be added to counteract 
 the push of the queen posts, which might be occasioned by 
 the thrusts of the braces ie, kf. They also afford addi- 
 tional security against any change in the angles c and d ; 
 and also, further support to the truss beam cd. 
 
 A roof consisting of four rafters of this external form is 
 called a Mansard or kirb-roof. 
 
 13. Where firmness is the main object, the frame work 
 shoidd consist of triangles. For the angles being unchange- 
 able whilst the lengths of the sides are unchanged, the 
 figure cannot be violated except by the deflections of the 
 
236 STATICS. — SECT. X. 
 
 beams : and the same angles afford the means of security 
 against these deflections, being so many fixed points, which 
 may serve for abutments to the braces, by which the trans- 
 verse strains are supported. But where it is particularly 
 desirable to gain internal space, the angles of a kirb-roof 
 may be secured by braces dividing each of the rafters into 
 three parts. In this way the sides of the polygon are dou- 
 bled, as represented in (Fig. 121). And by a continuance of 
 the same method, they may be further multiplied, until the 
 interior figure approaches to the form of a vault ; it may 
 then be lined so as to perfect that figure. 
 
 But a most ingenious method of framing a roof, without 
 encroaching considerably on the space within, is that pi-ac- 
 tised by the Normans, in what are called Gothic Buildings. 
 The construction is that represented in (Fig. 122.) where 
 BE, CF, DG, are pieces descending from the angles made by 
 the rafters, the ends of those pieces being connected with 
 the angles at each side. Thus each four-sided figure, as 
 ABCE, is fixed, being resolved into two triangles by its diago- 
 nal BE. The whole being treated in this manner, the rafter 
 AB cannot move independently of bc; nor bc without cd; 
 nor this last without de. All must, therefore, act together 
 in resisting a force applied to any one part. Each of the 
 braces, ea, ec, &c. is alternately a strut and a tie ; a strut in 
 upholding the angle from whence it proceeds ; and a tie in 
 holding up the piece descending from the neighbouring an- 
 gle, by which means that angle is supported. Thus, ce, cg 
 are struts, rising from the abutments e and g, to support the 
 angle c; and cg, ge are ties upholding the piece gd against 
 the angle d. 
 
 The same method is extended to support the rafters also 
 against the strains occasioned by their respective loads, and 
 this without encroaching so much on the space within ; sim- 
 ply by connecting the pieces be, cf, dg with the middle 
 points of the rafters, as represented in (Fig. 123). In this 
 
STRENGTH AND STRAINS. 237 
 
 way the braces if, kf are struts abutting on f, and support- 
 ing the rafters cb, cd at their middle points i and k. And 
 the same pieces are ties pressing up the piece fc against the 
 angle c. 
 
 The architects who used these methods, instead of con- 
 cealing the roof by a ceiling, frequently exposed it to view, 
 ornamented with carved work : vain, perhaps of their skill 
 in supporting an enormous roof by that which, to a common 
 beholder, would seem an additional load, fitted only to hasten 
 its ruin. 
 
23S 
 
 STATICS. — SECT. XI. 
 
 SECTION XL 
 
 OF THE ELASTIC CURVE. 
 
 1 . In what has been hitherto delivered relative to trans- 
 verse strains, the forces considered were those requisite to 
 produce or resist fracture ; but by a force less than this, a 
 beam or rod is sensibly inflected. Sometimes this change of 
 figure is permanent ; in which cases, the rod is said to have 
 taken a set : but often, also, it regains its primitive figure, 
 when released from the straining force, in which case, it is 
 said to be elastic ; and the force, by which it is brought back 
 to its primitive position, is called its elasticity. 
 
 When an elastic plate, fixed at one end, is bent by a force 
 applied at the other, the figure which it assumes is called the 
 elastic curve ; and the purpose in the present section is, to 
 point out the mode of investigating that figure. 
 
 Let OB be the plate fixed at o, (Fig. 124.) and let this be 
 bent into the curve ob', by a force p, acting at b', in the di- 
 rection b'g. The equilibrium at any point, m, shall not be 
 disturbed, by rendering the plate inflexible at every other 
 point. To conceive the nature of the action of the elastic 
 force, the curve formed by the convex surface of the plate 
 may be regarded as a polygon with an infinite number of 
 equal sides, mm, mm being two contiguous sides of this 
 polygon, the tendency of the elastic force at m, is to bring 
 the line mm to the same direction with urn by a circular 
 movement round a certain line in the section made at m, 
 perpendicular to the plane of the curve ; which line is, there- 
 fore, to be regarded as the axis of circular motion. This 
 tendency being counteracted by p, the straining force acting 
 
OF THE ELASTIC CURVE. 239 
 
 at b', it follows that these two forces are in equihbrio ; or 
 that their moments are equal. Wherefore, putting e for the 
 moment of the elastic force acting at m, and drawing the line 
 M« perpendicular to b'g the direction of the inflecting force, 
 we have 
 
 EZZPXMW. (1) 
 
 In deriving the equation of the curve from this funda- 
 mental proposition, we shall confine ourselves to the case of 
 a plate of uniform breadth and depth, in which the flexure 
 is inconsiderable. Moreover, as the inflecting force when 
 oblique, may always be conceived to be compounded of two 
 forces, of which one is perpendicular to the face of the plate, 
 and the other in the direction of its length, it will be suffi- 
 cient to consider these two cases apart, which is what is pro- 
 posed in the two following articles. 
 
 2. Let cIs be any element of the interior curve ; i. e. of 
 the curve formed by the concave surface of the plate, and 
 let ds' be the corresponding element of the exterior curve. 
 In the plate before inflection these were equal ; but after in- 
 flection ds' is greater than ds. Now ds', ds being regarded 
 as similar arcs of two concentric circles, whose radii are /, r, 
 we have the following proportion : 
 
 ds' : ds : : r : r, 
 
 and 
 
 ds' — ds :ds:: r — r : r. 
 
 giving 
 
 ds —ds = . ds. 
 
 But r'—r is the thickness of the plate which is supposed to 
 be unvaried : wherefore ds' — ds which is the distention of 
 
 1 
 
 the elementary portion ds varies as -. i. e. inversely as the 
 
 radius of curvature. Now e varies as t//— J* multiplied by 
 
240 STATICS. SECT. XI. 
 
 the breadth into the square of the depth, which in the case 
 now considered are constant: it therefore varies inversely as 
 the radius of curvature ; a fact which had been previously 
 established by experiment. Wherefore nrzzA, a constant 
 quantity. Accordingly, taking the point o for the origin, 
 the right line ob, or the length of the plate before inflection, 
 for the axis of ^, and the perpendicular oc for that of?/, and 
 putting x' for od, equation (1) becomes 
 
 v(x — x)=-. 
 r 
 
 If the flexure is small, x' may be deemed equal to ob ; which 
 being denoted by I the equation becomes 
 
 v(l—x)y — A. 
 
 This is brought to an equation between x and y, simply by 
 
 (dx^-\-ch/^)^ 
 
 substituting for r its value, which is — ; — jg . Where- 
 
 ° ax. cry 
 
 fore the differential equation of the curve will be 
 
 3 
 
 But 
 
 V[l—X) -^ = A. 
 
 ax. ay 
 
 {dx--^dy'')^ ' -'"2x1 
 
 dx.d^y 
 
 
 dx^ 
 whose approximate value is -^- the flexure being inconside- 
 rable : in which case the equation becomes 
 
 f{1—X)-zj- = A. 
 
 or 
 
 v{l—x)dx-=.\.-~-. 
 
OF THE ELASTIC CURVE. 241 
 
 And by integration, 
 
 -(ai-x)=..£. 
 
 which requires no correction, inasmuch as for x=0, there is 
 
 ^ = 0. 
 dx 
 
 Multiplying the last equation by dx, and again integrat- 
 ing, there is 
 
 which as a^ and y simultaneously vanish, requires no correc- 
 tion. 
 
 For x:=zl, this becomes 
 
 And A being constant, this equation shows that the deflec- 
 tion at the point to which the inflecting force is applied, 
 varies as that force multiplied by the cube of the distance of 
 the point of application from the fixed extremity of the 
 plate. 
 
 3. If the direction of the force p coincides with the line 
 cormecting the two extreme points o and b, the point o 
 needs not to be fixed ; it will suffice if the rod rests against 
 an immoveable obstacle at that point: and the case will be 
 the same with that of a vertical rod resting on a horizontal 
 plane, and carrying a weight on its summit. 
 
 Let BO be the elastic plate resting on the horizontal plane 
 at o, (Fig. 125.) and bent into the figure bmo, by a weight 
 p, applied at b. Also, mn being a perpendicular let fall 
 from any point m, on the right line bo, the direction of the 
 compressing force, let mnh::?/, 0N=:a:. then we shall have 
 
 A 
 
242 STATICS. — SECT. XI. 
 
 in which a is constant, being the moment of the elastic force 
 at any point muUipHed into the radius of curvature at the 
 same point. The approximate value of r, for small inflec- 
 
 tions, being -72", and (Py being negative, the equation be- 
 comes, 
 
 Multiplying both members of this equation by 2dy, and in- 
 tegrating, we have 
 
 To find the value of c, let the greatest ordinate of the curve 
 
 be b, and for this ordinate we shall have -f^ziO. and there- 
 
 ax 
 
 fore, c=p.6^ which gives 
 
 or, 
 
 _^l K dy 
 
 dxzz- 
 
 WF\Vb'-.yy 
 
 Again integrating, 
 
 V A / , i/\ ■ 
 
 x-=.—r-' arc sm. zz. ^ )+c. 
 \/p \ b) 
 
 But for a;z:0, there is^zzO, and therefore, arc ( sin,=r^ j — 0, 
 
 Wherefore, c'=0. and accordingly the equation of the 
 curve is 
 
 Va ( • y\ I . Vp 
 
 a7 = -7-. arc (sin.^T-). or, v = ^' sin.—— .a;. 
 
 'Jt^ \ bt ^ ^J ^ 
 
 For « = 0B=:/, we have ^=0. and therefore. 
 
 ■v/p 
 &.sin.-V-.^=0, 
 
 Va 
 
OF THE ELASTIC CURVE. 243 
 
 l/P 
 
 Whence, either 6 — 0, or sin. -y-./ziO. and to satisfy this 
 
 V A 
 
 last equation, we must have 
 
 Vp , 
 
 — —. I zn mir. 
 
 v/a 
 
 in which m is some integer number. Accordingly, if the 
 
 force p does not satisfy this last equation, we shall have 
 
 6 = 0. i.e. there can be no inflection. Hence it appears, 
 
 that the least value of p, which can bend the rod or plate, 
 
 \/p 
 is that obtained by making w«i=I, in the equation-— ./=wz7r. 
 
 which then is 
 
 p — 
 
 I' 
 
 and the same is the greatest force which the plate can sus- 
 tain without bendinsr. 
 
 R^ 
 
244 STATICS. — SECT. XII. 
 
 SECTION XII. 
 
 OF THE FUNDAMENTAL PRINCIPLES OF STATICS. 
 
 1. The first who attempted to raise Mechanics to the 
 rank of a demonstrative science, was Archimedes ; and by 
 him it was founded on the principle of the lever, which he 
 established in the following manner. 
 
 Let AB be the lever, and c its fulcrum, (Fig. 126). 
 Taking in this line produced, bd z: ca, af =■ cb, the whole 
 line fd shall be bisected at c, and if fd were a cylinder, it 
 would be supported by a prop at c. Moreover, taking 
 AE = AF, the line ed is bisected at b. Wherefore, if the 
 cylinder were divided into two distinct cylinders at e, the 
 part FE would be supported by a prop at a ; and the part 
 ED by a prop at b, i. e. a force equal to the weight of fe, 
 acting vertically upwards at a, and a force equal to the 
 ^veight of ED, acting vertically upwards at b, would equili- 
 brate the cylinder as effectually as a force equal to its wdiole 
 weight, acting vertically upwards at its middle point c. But 
 the force, equilibrated by this last, is the entire weight 
 acting vertically downwards at c. Whence it appears, that 
 two forces acting vertically at a and b towards the same side 
 of the line ab, shall equilibrate a third force acting at c in 
 a parallel and contrary direction, provided those forces are 
 as FE, ED, fd; or as the halves of those lines, viz. cb, ca, 
 ab, each of the forces being as the distance between the 
 points of application of the other two forces. 
 
 Various improvements in this demonstration have been 
 since proposed, by those who have regarded the lever as a 
 suitable foundation of mechanical philosophy. That the 
 
OF THE FUNDAMENTAL PRINCIPLES OF STATICS. 245 
 
 whole theory of statics may be derived from thence, shall 
 be freely admitted ; but it must also be admitted, by those 
 who are most favourable to this proceeding, that the me- 
 thods of applying the principle of the lever to other matters, 
 such as the pully, the inclined plane, the composition and 
 resolution of forces, are forced, isolated, and circuitous. 
 
 2. The principle next applied to the same purpose, was 
 that of the composition and resolution of forces, first intro- 
 duced by Newton in his ' Principia Mathematica Philo- 
 sophiae Naturalis,' which had received the imprimatur of the 
 president of the Royal Society, early in July, 1GS6, though 
 the merit of the improvement has been claimed for Vai-ignon. 
 This pretension, however, does not appear to have any sup- 
 port besides his own assertion, as his ' Nouvelle Mecanique' 
 did not make its appearance until the year 1725, which was 
 after his death ; and no trace of the discovery can be found 
 in any of his other productions. It plainly appears, that it 
 had not been made by him in 1685, the year in which he 
 published his ' Memoire sur les poulies a moufles' in the 
 ' Histoire de la Republique des lettres ; wherein, unques- 
 tionably, he would have adopted this principle, had it been 
 then known to him, as it appears that he was afterwards 
 fully sensible of its immense importance; and as in that per- 
 fomance, it would have afforded him peculiar facilities in 
 calculating the efficacy of oblique strains. The principle of 
 the composition and resolution of forces pervades the whole 
 of the Principia Mathematica : and as it is impossible to 
 suppose that this performance could have been the work of 
 one or two years, it must be admitted, that it was known to 
 Newton long before the year 1685, when unquestionably it 
 was imknown to Varignon ; and, therefore, that Newton 
 was the first who made the discovery, as well as the first 
 who gave it to the public. 
 
 The demonstration by which this principle was originally 
 supported, was founded on the composition and resolution 
 
246 STATICS. — SECT. XII. 
 
 of motion. For taking the motions produced by the single 
 forces to represent these forces in quantity and direction, 
 the motion resuUing from the composition of those two mo- 
 tions was taken to represent the resulting force. This mode 
 of proof is recommended by its extreme simplicity. But it 
 has been objected, that it introduces the subject of motion, 
 which is different from that under consideration ; and that it 
 assumes the proportionality of the force to the velocity pro- 
 duced by it. That the velocity produced in a given body, 
 by a force acting during a given time, is the just measure of 
 that force, is a truth established by the most extensive ex- 
 perience ; but it is only by experience that it can be esta- 
 blished : whereas, by an independent proof of the principle 
 of the composition and resolution of forces, the whole theory 
 of statics is presented as a series of necessary truths, inde- 
 pendent of experiment or observation of any kind. These 
 objections were first urged by Daniel Bernouilli; and he was 
 also the first who supplied a demonstration of the kind re- 
 quired. Many others have been afterwards furnished, of 
 which the most simple and elegant was that of Duchalya, 
 the same which, with very little change, has been adopted in 
 Sect. I. Art. 2. of this elementary treatise. 
 
 3. But another principle could not long escape the notice 
 of writers on meclianics, being offered to their view in all 
 instances of equilibrium ; and that most obviously in the 
 several mechanic powers : this has been denominated the 
 principle of virtual velocities ; to explain which, it is to be 
 observed, that a force being applied to a material point, and 
 any small motion given to that point, by which it describes 
 the space ds in a given time, the space ds is the measure of 
 the velocity of that point. And if this line makes with the 
 direction of the force, the angle 6, then ds. cos. 6. is the ve- 
 locity estimated in the direction of the force itself. Now 
 the principle of virtual velocities is thus stated: " If any 
 number of forces are in equilibrio, whether applied to the 
 
OF THE FUNDAMENTAL PRINCIPLES OF STATICS. 247 
 
 same point, or to different points of a system, and any 
 small movements are given to the system of points, such as 
 their mutual connexions will admit, the sum of the products 
 obtained by multiplying each force into the velocity of its 
 point of application, estimated in the direction of the force, 
 is cipher ; and conversely, when for all the small movements, 
 which can possibly be given to the points of the system, the 
 sum of these products is cypher, the forces shall be in equi- 
 librio." Thus, if the points of the system are m, m, m , m'\ 
 (Fig. 127.) connected together according to any condition ; 
 and if by any small displacement they are carried to the 
 points w, 71, n", n" , respectively ; then mn, m'n, m'ti', m"n" 
 are the virtual velocities; and ns, n's, n"s", n"s" being per- 
 pendicular to the directions of the forces, the virtual ve- 
 locities, estimated in the directions of the forces, are tns, 
 ms, Ill's" , m's". Now, putting p, p', p", p'" for the forces 
 acting at those points, and jt?, p , p", p", for the lines tns, 
 m's, m's", m"'s"', then according to the principle of virtual 
 velocities, the system being in equilibrio, we shall have 
 
 v.p + p'./-F p"./'+ v"'.p"'- 0. 
 And conversely, if the condition expressed by this equation 
 is fulfilled for every small movement which may be imparted 
 to the system, the forces are in equilibrio. 
 
 In announcing this principle, it is to be noticed, that the 
 forces p, p', p", &c. are always deemed positive ; and that 
 2J, p', p", &c. are positive, when their directions are the 
 same with those of p, p', p", &;c. and negative, when their 
 directions are opposed to those of the same forces. Thus, 
 in the figure; m's', m"s", being measured from the points 
 m', m" , in the directions of the forces p', p", are positive ; 
 but ms, m"s"' , which are measured from the points m, m" , 
 in directions opposite to those of the forces p, v'", are ne- 
 gative. 
 
 4. It has been already observed, that this principle was 
 discovered by induction. Thus, for equilibrium in the lever, 
 
248 
 
 STATICS. — SECT. XII. 
 
 the power and weight acting in directions perpendicular to 
 the arms I, /', we had p./r= p'. /'. But a small angular mo- 
 tion being given to the lever, the extreme points of the arms 
 describe the spaces ds, ds. And these being similar circu- 
 lar arcs whose radii are /, /', we have 
 
 ds : ds' ::l: I'. 
 Wherefore, 
 
 T,ds = p'. ds'. 
 
 But the directions of the forces being regarded as positive, 
 whilst they tend to turn the lever about the fulcrum in op- 
 posite directions, one of the virtual velocities is positive, and 
 the other negative ; wherefore, putting ds ^.p, we shall have 
 ds zz — p'. which values being substituted in the preceding 
 equation, it becomes 
 
 P.jO + p'. p' zz 0. 
 
 More generally, the angles made by the directions of the 
 forces with the arms to which they are applied, being <p, ^', 
 
 we had for equilibrium, p./. sin. ^ = p'./'. sin. ^'. But — :z: y, 
 
 V V 
 
 wherefore, multiplying by these equals, the preceding equa- 
 tion becomes 
 
 p.f/^.sin.^ := p^ ds'. sin,^'. 
 
 Also, Q, B', being put for the angles made by the directions 
 of the forces, with the lines ds, ds', we have 
 
 sin.0rzcos.0. sin.0'z:cos.0'. 
 
 Wherefore, 
 
 T.ds.cos.0=F'. ds.cos.B'. 
 But, 
 
 ds. COS. 9=p. ds'. COS. 0'=. — J)'. 
 
 and therefore, 
 
 p./*-f p'.^rzO. 
 
 In the screw, the condition of equilibrium was found to 
 be that expressed by the equation 
 
OF THE FUNDAMENTAL PRINCIPLES OF STATICS. 249 
 
 p.^7rr = P.i. 
 
 wherein 27rr is the periphery of the circle described by the 
 power p; and I, the distance between the threads of the 
 screw, measured in a direction parallel to its axis : and these 
 being the measures of the velocities of the power and re- 
 sistance which, as before, are aiFected with opposite signs, 
 the equation is equivalent to p.^j + p'.^y = 0. 
 
 5. The principle of virtual velocities being thus verified 
 by induction, it was natural that those who treated of the 
 subject should have endeavoured to estabhsh it by a rigo- 
 rous demonstration. The proofs which have been offered 
 are, for the most part, attempts to reduce the principle in 
 question to that of the composition of forces. But to this 
 mode of proceeding it is objected, that instead of making out 
 an independent principle of mechanical science, it presents 
 a general expression of the conditions of equilibrium de- 
 duced from other principles. Lagrange has indeed at- 
 tempted to furnish a demonstration not liable to this objec- 
 tion, by resorting to the theory of the pully ; which is rea- 
 dily estabhshed, independently of the composition of forces, 
 when the parts of the cord, embracing each system of move- 
 able pullies, are parallel. Thus, if a cord passes over any 
 number of systems of fixed, and the same number of sys- 
 tems of moveable pullies, the ends of the cord being attached 
 to the blocks carrying the extreme systems, whether of fixed 
 or moveable pulhes, and all the parts of the cord being pa- 
 rallel ; there will be equihbrium in the entire system, when 
 the forces apphed to the several parts of the cord are equal, 
 i. e. when the weights appended to the several blocks con- 
 taining the moveable pullies, are proportional to the number 
 of parts of the cord reaching those blocks. Thus, in (Fig. 
 128,) where in the first system there are two, in the second, 
 three, and in the third, four moveable pullies, the extreme 
 ends of the cord being attached to the blocks which carry 
 
250 STATICS. — SECT. XII. 
 
 the systems of fixed pullies, there will be equilibrium when 
 the weights are in the following proportion: 
 
 p:p':p":: 
 4:6 :8 
 
 But if either end of the cord is carried over a fixed pully, 
 and is then attached to the lower block, the number repre- 
 senting the weight carried by that block is to be increased 
 by unity ; the equilibrium being established in all cases, when 
 the weights appended to the lower blocks are proportional 
 to the number of parts of the cord reaching those blocks ; 
 so that if «, n' , n", express the numbers of the cords which 
 
 p p' p" 
 carry the weights p, p', p'' w^e shall have — zz — = — r,. for 
 
 n n' n 
 
 equilibrium. But for vertical weights, we may substitute 
 forces equal to those weights, and acting in any different di- 
 rections, provided that the cords pertaining to each of the 
 blocks which carry the moveable pullies arc parallel, as in 
 (Fig. 129). Now if the systems of moveable pullies suffer 
 any small displacement, let ds, els', ds', See. denote the 
 spaces described by their several blocks, estimated in the 
 directions of the cords, and the length of the entire cord 
 being invariable, we have 
 
 nds -{.\'ds'-\- n"ds"+ &c. =0. 
 
 and, therefore, substituting for w, «', ?i", &c. the forces to 
 which they are proportional, 
 
 p.ds + p'. ds' + p'. ds'' -f &c. =0. 
 
 The application of this theorem is obvious ; for the blocks 
 being regarded as the points of a system to which the forces 
 p, p', p", &c. are applied, and ds, ds', ds", &c. being the vir- 
 tual velocities of those points, estimated in the directions of 
 the forces, we have 
 
 T.p + P'.;/+ T^.J}"-^ &c. =0. 
 
 Lest the demonstration, given above, should seem to be 
 limited to tending or divellent forces, let any one of the 
 
or THE FUNDAMENTAL PRINCIPLES OF STATICS. 251 
 
 moveable blocks and its corresponding fixed block be sup- 
 posed to exchange places, and the demonstration shall be 
 equally valid, i. e. the equilibrium shall still subsist, whilst 
 the direction of the force, and that of the reaction occasioned 
 by it in the fixed point, are changed into the opposite. 
 Moreover, as the equilibrium of a system is not disturbed by 
 changing the point of application of any of the forces for any 
 other point in its line of direction, it follows, that the origi- 
 nal points of application may be retained, whilst the forces 
 there applied are converted into compressing forces. 
 
 Though it has been thus proved, that in all cases of equi- 
 librium we have p./?+p'.p' + p'-p' + &c. =0. it is not to 
 be supposed that the converse of this proposition is true, in 
 the manner stated by some writers on the subject. The 
 foregoing equation is indeed a test of equilibrium, when it 
 subsists for every small motion whatever that may be sup- 
 posed to be imparted to the system, and not otherwise. For, 
 any two of the forces, p, p', p'', &c. being varied, both being 
 increased or both diminished in the same ratio, the points of 
 application of those forces may be supposed to be moved, 
 the others remaining undisturbed. In the case here sup- 
 posed, the virtual velocities of those points would be reci- 
 procally as the forces there applied ; and being, moreover, 
 affected with opposite signs, the equation p.^-j-p'.^Z-j-p''.^/' 
 -1-&C. zzO. would still subsist, though the equilibrium no 
 longer subsists in the entire system. 
 
 But the fundamental conditions of equilibrium in the 
 systems of pullies are those expressed by the equations 
 
 p p' p" 
 
 — — — ;- = -;7-, &:c. and if these conditions are satisfied, the 
 n n n 
 
 equilibrium is established. It remains then to be seen how 
 
 far these conditions are secured, when that expressed by the 
 
 equation pp.4-i'''i>'+p".i5"+&c. =0. is satisfied. Let this 
 
 equation, therefore, be supposed to subsist : from the inex- 
 
 tensibility of the cord, we have, moreover, the equation 
 
^52 STATICS. — SECT. XII. 
 
 {np-{-np -^n'p" -{- &c.) « = 0. 
 and by combining this with the preceding equation, 
 
 (p— ««)p+(i''— ^^'")/^'+(p""~*^"«)p"+ &c. =0. 
 Now, in order that this may be resolved into the equations 
 
 F~na = 0. v~n'a~^. p"— w"a=z:0. &c. 
 or, 
 
 -=:-=C,&c. -a. 
 
 it is requisite that the quantities /;, p', p" , Sec. should be so 
 many independent variables. From whence it follows, that 
 to render the condition expressed by the equation p/j + ^'p' 
 -{-p"p"-\- 8cc. = 0. sufficient for equilibrium, this equation 
 must subsist for all the small movements which can possibly 
 be imparted to the points of the system. But it is not requi- 
 site that the movements ^^,^',/>", &c. should be absolutely 
 independent, in order that the equation pjp+p'/* -|-p''i^" + &c. 
 = 0. should become a test of equilibrium. So far as the 
 motion of any point is restrained, whether by a surface to 
 which it is confined, by a fixed point, or axis, round which it 
 may revolve, or by its connexions with other points of the 
 system, so far is the equilibrium of the system necessarily se- 
 cured ; whence it follows, that we may omit those movements 
 which are impossible, and pronounce that the equilibrium is 
 provided for, when the preceding equation is satisfied in the 
 case of all the small movements which can possibly be im- 
 parted to the points of the system, consistently with the con- 
 ditions by which those movements are restricted. 
 
 6. The principle of virtual velocities being thus ex- 
 plained, it remains to show, by a few examples, how it is 
 applied. 
 
 Let there be two inclined planes, whose lengths are /, /', 
 and whose common height is //, placed back to back : it is 
 required to find the conditions of equilibrium for two weights, 
 p, p', placed on these planes, and connected by an inexten- 
 
OF THE FUNDAMENTAL PRINCIPLES OF STATICS. 253 
 
 sible cord which passes over the summit ; the parts of the 
 cord measured from the summit being respectively parallel 
 to the planes. 
 
 From the condition of the inextensibility of the cord, it 
 follows, that one of the weights cannot descend, unless the 
 other ascends through an equal space ; therefore, the virtual 
 velocities are equal and contrary. To reduce these to the 
 vertical direction, which is that of the weights, they must he 
 multiplied by the height, and divided by the lengths of the 
 planes. Wherefore, 
 
 p=.ds. -J. p — —CIS. J,. 
 
 whence the equation F.p -\-^'.p' = 0. becomes 
 
 F.ds. -J — f'.(Is. j,. 
 
 or dividing by ds.h, it is 
 
 which shows, that for equilibrium, the weights must be pro- 
 portional to the lengths of the planes on which they are 
 placed. 
 
 Let two weights, connected by an inextensible cord, be 
 placed on the surface of a horizontal cylinder; and let it be 
 required to find their positions for equilibrium. 
 
 From the condition of the inextensibility of the cord, it 
 follows, that the virtual velocities are equal and contrary; 
 and these are reduced to the vertical, when multiplied by the 
 sines of the arcs, measured from the weights to the summit. 
 Accordingly, puting Q, 0' for those arcs, we have 
 
 2)'=.cls.&in.0. p'=: — ds.smd'. 
 
 whereby the equation T.p-{-F\p'=:0. becomes 
 
 T.ds.sin,9=:F\ds.sm.$\ 
 or, 
 
254 STATICS. — SECT. XII. 
 
 p.sin.0 = p'.sin.0'. 
 which shows that the whole arc commensurate to the cord, 
 is to be divided into two parts, whose sines are as the 
 weights; and these parts being measured from the summit, 
 that the greater weight is to be placed at the extremity of 
 the lesser, and the lesser weight at the extremity of the 
 greater arc. 
 
 The same thing would appear by regarding the weights 
 as placed on the tangent planes ; for then it appears from the 
 preceding problem, that the weights must be reciprocally as 
 the sines of the horizontal elevations, i. e. reciprocally as 
 the sines of the arcs measured from the bodies to the highest 
 point. 
 
 If a number of connected weights are in equilibrio, the 
 common centre of gravity shall neither rise nor fall, in con- 
 sequence of any small movement which may be given to the 
 bodies of the system. And conversely, if such is the con- 
 dition of the common centre of gravity, the weights are in 
 equilibrio. 
 
 For the distances of the several weights from a horizontal 
 plane being denoted by z, ss' , z'\ z", &c., and that of the 
 common centre of gravity from the same plane, by z, the 
 equation v.p -f p'.//-f v".p"-\- v"'.p"-\- &c. = 0. becomes 
 
 Y.d.z + v'.d.z'-{- ^".d.z"+v".d.z'" + &cc. - 0. 
 But by Sect. II. Art. 7. we had 
 
 P.^4.p'.^'+P".^"+P'".^'" + &C. =: 
 (p + p'+p"-fp'"+&c.)z. 
 
 giving 
 
 v.d.z + v'.d.z+Y".d.z"-\-v"'.d.z"-\- &c. = 
 
 (p + p'+p"+p'"_|_&C.)(/.Z. 
 
 Wherefore, the equation vqi ■^v'.p'-\-T".p"-\~ Bcc — O. is 
 equivalent to 
 
 d.z - 0. 
 
OF THE FUNDAMENTAL PRINCIPLES OF STATICS. 255 
 
 Accordingly, when the weights equihbrate, this last equation 
 subsists ; and conversely. 
 
 When the common centre of gravity takes the highest or 
 lowest position possible, we have 
 
 dz-0. 
 
 and therefore, the system is in equilibrio. But the converse 
 of this proposition, viz. that when c?.z = 0, z is a maximum 
 or minimum, though generally true, is liable to the usual ex- 
 ceptions for maxima or minima, discovered by making the 
 differential equal to cipher. For if, as may happen in con- 
 strained motions, the common centre of gravity describes a 
 curve of contrary flexure ; at a point of that curve where the 
 tangent is horizontal, the equilibrium shall subsist, although 
 the common centre of gravity may not be then in the highest 
 or lowest position it can take. 
 
 For the wedge, the power and resistances being applied 
 in directions perpendicular to the back and faces, it was 
 found that for equilibrium, the three forces should be pro- 
 portional to the surfaces on which they act, i. e. to the three 
 sides of the triangular end of the wedge, by which those 
 surfaces may be represented. 
 
 To derive this theorem from the principle of virtual ve- 
 locities, let the wedge be represented by (Fig. 130.) where 
 B, L, l' are the back and faces, a, /3, 7, the angles to which 
 they are opposed, and p, p', p" the forces acting perpendi- 
 cularly on B, L, \!. The three forces being supposed to be 
 directed to the same point, which may be taken for their 
 common point of application, let this point describe the line 
 ds, in the direction of the power p ; the line of the motion 
 shall then make, with the direction of the force p', an angle 
 equal to the angle y, and with the direction of the force p", 
 an angle equal to the angle /3 ; and we shall have 
 
 P- V' -P" '' 
 1 : cos. 7 : C0S./3. 
 
256 STATICS. — SECT. XII. 
 
 and, therefore, by the principle of virtual velocities, 
 
 p =: p'. cos.y -f- p". C0S./3. 
 
 This equation involves the three quantities p, p', p'', and, 
 therefore, does not express their relative magnitudes. To 
 express this relation, the equation should contain but two of 
 those quantities. And as the direction of ds is arbitrary, it 
 may be supposed to be perpendicular to that of one of the 
 forces ; for then, the virtual velocity, estimated in the di- 
 rection of that force, shall be cipher ; and the correspond- 
 ing term shall vanish from the equation. Accordingly, let 
 ds be perpendicular to the direction of the force p", i. e. 
 parallel to the face l'; then 
 
 2)''- 0. 
 and therefore 
 
 p .2J = 0. 
 
 and the condition of equilibrium expressed by the principle 
 of virtual velocities is 
 
 p.^j 4- p'* P^ — 0' 
 
 But the line ds makes, with the direction of the force p, an 
 angle whose complement is /3 ; and with the direction of the 
 force p' an angle whose complement is a. Wherefore, 
 
 p :p' : : sin.|3 : sin. a : : L : B. 
 so that the proportion p : p': i^)' '-2^ is 
 
 p : p' : : B : L. 
 And in the same way, by taking ds in the direction of the 
 face L, we get 
 
 p : p" : : B : l'. 
 which two proportions may be stated as before, 
 
 P . p/ . p// 
 
 : : B : L : l'. 
 
 Ordinarily, the triangle by which the wedge is repre- 
 sented, is isosceles, i. c. l' = l, and each of the angles at 
 
OF THE FUNDAMENTAL PRINCIPLES OF STATICS. 257 
 
 a 
 
 the base is the complement of 5. in which case, the relation 
 of the forces appears by the equation 
 
 p :=: p'. cos. 7 + p''. C0S./3. 
 which then becomes 
 
 p = 2p . sm.-. 
 2 
 
 To ascertain the ratio of the power and weight for the 
 inclined plane ; let p, p' be those forces, and let their com- 
 mon point of application be supposed to be carried vip the 
 plane through the space ds. Then, p, p' are as the cosines 
 of the angles made by the plane with the directions of the 
 forces p, p'. Wherefore, if e denote the angular elevation 
 of the plane, and 6 the angle made by the plane with the 
 direction of the power, we shall have 
 
 p'.p'w COS. 9 : sin.e. 
 
 giving for equilibrium 
 
 P.COS.0 — p'. sin.e. 
 
 If the power acts in a direction parallel to the plane, 
 
 COS. 9 = 1. and p zz p'. sin.e. 
 
 7. To show what number of equations may be obtained 
 from the general theorem, and in what manner they are to 
 be apphed, let 9, 9', 9", &c. be the angles made by the di- 
 rections of the forces p, p', p", &c. with the infinitesimal 
 lines ds, ds', ds", &c. described by their several points of 
 application. The equation 
 
 T.p + p'.iy+ p".y+ &c. = 0. 
 
 is equivalent to 
 
 F.ds.cos.e + p'. ds'.cos,9' + p".ds". cos.0"-f.&c.=O. («) 
 
 Now, referring the points of application to three rectan- 
 gular axes, let the coordinates of the point of application of 
 
 s 
 
258 STATICS. SECT. XII. 
 
 the force p be x, y, ^. Those of the point of application of 
 the force p', x', y\ z', &c. Also, pvitting «, j3, 7, for the an- 
 gles made by the direction of the force p, with the axes of 
 X, y, and z respectively; a', |3', 7' for those made by the 
 direction of the force p' with the same axTes, and so forth. 
 Likewise, putting ^, x> ^ for the angles made by the line 
 ds, with the axes ; 0', x, t//' for those made by ds' with the 
 same, and so forth ; we have, by a well known theorem, 
 
 008.0= cos.^.cos.a + cos.x-cos.j3 + cos.;//. cos. 7. 
 and 
 ds. cos.B — ds (cos.0. cos. a + cos. x. cos./B + cos.i//. cos.7) = 
 dx.cos.a, + (ly- C0S./3 + ^•^- cos. 7. 
 And similarly, 
 
 ds.cos.9' = dx'. cos.a -j- dy'. COS. [5' +dz\ cos.7'. 
 and 
 
 ds". co^.B" ■=dx".cos.a"-\-dy". cos.^"-\-dz". cos, 7". 
 
 the coefficients of all the forces being expressed in like man- 
 ner. Wherefore, making these substitutions, equation («) 
 becomes 
 
 v.cos.a.dx + p'. cos.a'.dx' •{•'p". cos. a", dx" + '£'". 
 cos.c<."'.dx'"-^&LC. 
 
 p.cos.|3.(///+p'. cos./3'.f/;/+p". C0S./3". dy"-\-v"'. 
 
 C0S./3'". f//"+ &c. I -^' ^■^ 
 
 T.cos.y.dz + P. COS. 7'. dz'-\- p". cos.7". dz" + ^'"• 
 cos.j"'.dz"'+&c. 
 
 If the points of application are unconnected, so that the 
 motion of one of those points shall have no influence upon 
 another, then are the quantities p, p', p" , p'", &c. inde- 
 pendent variables ; and therefore, also dx, dy, dz, dx\ 
 dy', dz', &c. are independent variables : whereby the 
 last equation is resolvable into so many distinct equations, 
 viz. 
 
OF THE FUNDAMENTAL PRINCIPLES OF STATICS. 259 
 
 P.COS.a=iO. p'. COS.a'rrO. p". cos.a" = 0. See. 
 p.cos./3 = 0. p'. cos.^' = 0. p". cos./3"=0. &c. 
 P.cos.yz^O. p'. cos.y'zrO. p". cos.7"=:0. &c. 
 
 But if the points of application are connected in any manner, 
 so that one of them cannot move unless by moving another, 
 according to some certain law ; then, the equations express- 
 ing the conditions to which the system of points is subjected, 
 shall serve to eliminate so many variables from equation (b), 
 and the coefficients of those that remain, being put equal to 
 cipher, shall express the conditions to be satisfied for equi- 
 librium. 
 
 8. If the forces are all applied to the same material point, 
 we have 
 
 ds = ds' = ds" = ds"', &c. 
 
 wherefore, 
 
 dxzzdx' -rzdx" ^=- dx'" , &c . 
 dy-dy'-dif-dij", &c. 
 dz-dz-dz"-dz'", Sec. 
 and equation (Z>) becomes 
 
 (p.cos.a + p'. cos.a'-j-P". cos.a" + &c.) dx 
 + (p.cos.j3 + p'. cos./3' + p",cos./3" + &c.) i/y J^ = 0. {c) 
 
 -|-(p.cos.7 + p'. cos.y' + p''. cos.7"-|-&c.) dz 
 
 which, as dx, dy, dz, are independent variables, is re^ 
 ducible to three distinct equations, agreeably to Sect. I. 
 Art. 11. 
 
 If the point to which the forces are applied, is confined 
 to a surface whose equation is l = 0, the differentials dx, 
 dy, dz, are no longer absolutely independent of each other. 
 Wherefore, equation (c) is no longer resolvable into three. 
 But putting this equation in the form x.dx+Y.dy + z,dz=0. 
 
 s2 
 
260 STATICS. — SECT. XII. 
 
 and eliminating dx between this and the differential equation 
 of the surface, viz. 
 
 clh J dh , c?L , „ 
 — — . dx + -7—. dy + —;—. dz — U. 
 dx dy dz 
 
 we have 
 
 dh J </l , dh ^ di. -, f. 
 
 dx ^ dx dy -^ dz 
 
 In this equation, the variables dy, dz, are independent: 
 "wherefore, their coefficients are, each, equal to cipher, 
 giving for the conditions of equilibrium 
 
 </l c?l _, dh dh „ 
 
 Y.-. x.^- — 0. z.- x.^- := 0. 
 
 dx dy dx dz 
 
 For example, let the equation of the surface be a.dx + b.dy 
 + c.dz = 0. we have 
 
 dh _ dij dh _ 
 
 dx ' dy * dz~ ' 
 
 which values being substituted in the general formula, we 
 have the equations 
 
 Y.a—x.b=0 z.a— x. c = 0. 
 
 The elimination is more expeditiously performed by 
 adding the differential equation of the surftice, multiplied 
 by an indeterminate coefficient, to the general equation of 
 equilibrium : then putting the coefficient of each variable 
 equal to cipher, and eliminating the indeterminate coefficient. 
 Thus, 
 
 , f/L „ , dh „ dh _ 
 
 x + m.-r- = 0. Y + 7n.-r- = 0. z + m.-r- = 0. 
 dx ay dz 
 
 from which, eliminating m, we have 
 
 dh dh - dh dh 
 
 Y.- X. ^- = 0. Z.- X. -r-zzO. 
 
 dx dy dx dz 
 
 the same with those obtained by directly eliminating dx, be- 
 
OF THE FUNDAMENTAL PRINCIPLES OF STATICS. 261 
 
 tween the general equation x.da; -^Y.di/ -\-z.dz:=:0. and the 
 equation of the surface. 
 
 9. Equations (c) will serve to show the resultant of any 
 number of forces applied to a material point : for any one 
 of the equilibrating forces, its direction being changed into 
 the opposite, is the resultant of the remaining forces. 
 
 Thus, to find the resultant of two forces, there are but 
 three terms in each of those equations. And the plane of 
 the forces r, p', being taken for that of .r, ?/, we have 
 
 COS. 7 = 0. cos.7'=0. 
 
 and therefore, 
 
 cos. 7"= 0. 
 
 Wherefore, the equations by which the resultant of the 
 forces p, p', is to be determined in quantity and direction, 
 are 
 
 p". cos.a" = P. cos.a + p'. COS. a', 
 p". cos./3"=: P. C0S./3 + p'. cos.|3'. 
 The forces p, p', p", being represented in quantity and 
 direction by three right lines drawn from their common point 
 of application, the terms of the equations express the coor- 
 dinates of the extreme points of those lines : and the equa- 
 tions show that each of the coordinates of the extremity of 
 the line f", is equal to the sum of the corresponding coor- 
 dinates of the extremities of the lines p, p', i. e. that the 
 coordinates of the extremity of the line p" are those of the 
 extremity of the diagonal of the parallelogram whose sides 
 are the lines p, p', and, therefore, that the line p" is that 
 diagonal. 
 
 In the same way, and without the trouble of elimination, 
 it may be seen that the resultant of three forces applied to 
 a material point, is represented in quantity and direction 
 by the diagonal of the parallelopiped, whose three sides 
 represent the three component forces in quantity and direc- 
 tion. 
 
262 STATICS. — SECT. XII. 
 
 10. To find the conditions of equilibrium for forces ap- 
 plied to a rigid system, it is to be observed, that the system 
 cannot be moved, except by a movement of translation, by 
 which all the points describe equal and parallel lines ; or, 
 by a motion of rotation. For the first kind of movement we 
 have 
 
 dx=:dx' = dx", &c. 
 
 dy=.dy'=.dy", &c. 
 
 ds=d^'=dz", &c. 
 
 which changes the equation {b) into 
 
 (p.cos.a-f-p'. cos.a'+p". cos.a"+&c.) dx\ 
 
 +(p.cos.j3+p'. cos.j3'+p ". cos./3"-f &c.) d?/ > = 0. 
 
 +(p.cos.7 + p'. COS.y'+P". C0S.7"+&C') (f^-^ 
 giving 
 
 P.cos.a + p'. cos.a'-j- p". cos.a" + &c. = 0. 
 
 p.cos. j3 + p'. cos.j3'+ p". cos. j3 + &c. = 0. 
 p.cos.-y + p'. cos.7'-|-p".cos.7''+&c. =0. 
 
 which are the three first conditions of equilibrium among 
 a number of forces, applied to the points of a rigid system. 
 Any rotatory motion which may be given to the system, 
 is resolvable into motions round the axes of the coordinates. 
 Now if the angle dO is described round the axis of z, no 
 change is made thereby in the magnitude of any of the lines 
 z, z', z", &c. which are the coordinates of the points of ap- 
 plication parallel to that axis, therefore 
 
 dz=dz'=:dz", &c. = 0. 
 Moreover, 
 
 jn_dx _ dy _ dx' _ __dj/ „ 
 ~ y ~ X ~ y' ~ x' ' 
 
 or, 
 
 dx—y.dd, d.x'—y'.dO, &c. 
 
 dy = —x.d.B, dy' = - x'.dB, &c. 
 
OF THE FUNDAMENTAL PRINCIPLES OF STATICS. 263 
 
 and these values being substituted in equation (b), it be- 
 comes 
 
 p (y.cos.a— A\cos./3) + p' (y. cos.oc'—x'. cos./3')+&c. = 0. 
 In the same manner, for a movement of rotation round the 
 axis of I/, we shall have 
 
 p(ir.cos.'y— ^. cos.a) + p'(a;'. COS.7' — sr'. cos.a') + &c. =rO. 
 
 And for a movement of rotation round the axis of x, the 
 equation 
 
 p(.^.cos.j3— ?/.cos.7)4-P'. (^'. cos.jS — 2/'.cos.7')4-&c.=0. 
 which completes the six conditions of equilibrium for a sys- 
 tem of invariable form, agreeably to Sect, IV. Art. 1, 
 
DYNAMICS. 
 
 SECTION L 
 
 THE MEASURES OF MOTION AND FORCE. 
 
 1. That branch of mechanical science which relates to 
 unbalanced forces and their effects, is called Dynamics : and 
 as the effect of unbalanced force is motion, it seems expe- 
 dient, in the first place, to treat of motion, independently of 
 the causes by which it is produced. 
 
 When a body changes its place, it is said to be in motion ; 
 and the rate of that change is called its velocity. Now ve- 
 locity is not itself a mathematical quantity; but if it can be 
 measiu'ed by quantities of this nature, it thereby falls within 
 the province of mathematical science. 
 
 The velocity of a body is greater or lesser, according as 
 the space described by it in a given time is greater or lesser. 
 Wherefore, the space described in a given time is the mea- 
 sure of the velocity : and as in comparing the velocities of 
 different bodies, or of the same body at different periods, by 
 the spaces described, the times must be equal, it becomes 
 necessary in every case of such comparison, to fix on some 
 portion of time for this purpose. This portion is denomi- 
 nated the unit of time. The velocity is, therefore, said to 
 be measured by the space described in the unit of time. 
 
 Whatever space a body describes in a unit of time, it is 
 evident that with the same velocity, it would describe twice 
 
266 DYNAMICS. — SECT. I. 
 
 that space in two such units ; and generally, that the space 
 described with the same velocity, is obtained by multiplying 
 the space described in a unit of time, by the number of such 
 units contained in the time of the motion. Wherefore, v, 
 denoting the velocity ; t, the number of units in the time of 
 the motion ; and s, the space described, there is * z= vt. or 
 
 V — -. of which it is to be remarked, that whatever be the 
 
 magnitude of 5, it becomes, when divided by t, the space de- 
 scribed in a unit of time. 
 
 In all cases, such as this, wherein quantities of diiFerent 
 kinds seem to be compared together, the terms are rendered 
 homogeneous by regarding all, or all but one, as numbers. 
 Thus in the preceding equation not only v and t, but s also 
 may be regarded as a number ; viz. the number of units of 
 space contained in the space described. These units of 
 measure for the several quantities which are involved in the 
 same equation are, all but one, purely conventional. For 
 example, if I" is taken for the unit of time, and one foot for 
 the unit of space, then the unit of velocity is that of a body 
 which describes one foot in a second of time. So that what- 
 ever be the space described, or the time of describing it, the 
 
 s 
 quantity - shall be the number of feet described in a second, 
 
 or, which is the same thing, the number of units in the velo- 
 city of the moving body. In these estimates, the velocity is 
 supposed to be uniform ; i. e. it is supposed that the body 
 describes equal spaces in equal times during the whole con- 
 tinuance of the motion : and the same thing is always to be 
 understood, when not otherwise expressed. 
 
 For the more ready solution of questions relative to such 
 motions, it will be convenient to compute the time from the 
 moment of the passage through some one point ; and also to 
 denote the place of the body, at any instant, by its distance 
 from some point taken at pleasure in the line of its motion. 
 
THE MEASURES OF MOTION AND FOKCE. 267 
 
 Thus, let the body move in the dh-ection ab (Fig. 131.) with 
 the velocity v, and let the time be measured from the moment 
 of its passage through d ; i. e. let d be the point for which 
 t — o. Then if m is the place of the body after any time t, 
 there shall be T>m-=vt. Futher, let c be the point from 
 which the distance of the body is to be measured. Putting s 
 for this variable distance, and a for the line cd, we shall 
 have s — a {zz Dm) ^ it. and s zz a -{- vt. If the point c is 
 taken at the other side of d, it is evident that the sign of a 
 must be changed. 
 
 If a second body moves uniformly in the same line with 
 the velocity v, let d' be the point through which it passes at 
 the instant from which the time is computed, i. e. at the same 
 instant in which the former body passes through d ; and let 
 / be its distance from c, and a the distance cd'; there will be 
 in like manner for this body, s'= «'+ *''^- Now if it is re- 
 quired to find the time of the meeting of these bodies, this is 
 found by making s=:s' in the two last equations, which gives 
 
 /= -,. and by substituting this value of ^, in the expres- 
 
 1) — V 
 
 sions for s or /, there is s = —- When the value off, 
 
 v~v 
 
 given by the former of these two equations, is negative, it is 
 inferred that the meeting has occurred before the instant, 
 from which the time is computed, i. e. before the arrival of 
 the one body at d, or that of the other at d'. and when the 
 denominator of that value is cipher, /. e. when v=v, the 
 value itself is infinite: which imports that the bodies never 
 meet. This indeed is evident of itself; inasmuch as the bo- 
 dies, in that case, move in the same direction with the same 
 velocity. 
 
 2. If the spaces successively described in equal times con- 
 stitute an increasing series, the motion of the body is said to 
 be accelerated ; and if they constitute a decreasing series, 
 the motion is retarded. When this occurs for portions of 
 
268 DYNAMICS. — SECT. I. 
 
 time, however minute, the velocity is continually varied ; and 
 in such cases, it may not be so obvious how the velocity of 
 the body, at any one instant of time, or point of space, is to 
 be measured. For if the velocity is accelerated, the space 
 described in any portion of time, is greater than that which 
 would be described with the velocity at the beginning, and 
 less than that which would be described with the velocity at 
 the end of that portion of time ; and the contrary if the ve- 
 locity is retarded; i. e. v and v' denoting these velocities, 
 
 s s 
 
 there is v<C^-:<iVy for one case, and «;> - >r', for the other. 
 
 T t 
 
 This difficulty is met by supposing s, and consequently t, to 
 be indefinitely diminished : for the change of velocity, how- 
 ever rapid, is gradual. Whence it is plain, that by diminish- 
 ing the space, and consequently the time, indefinitely, v and 
 v', the velocities at the extreme points, shall approach within 
 any assignable difference ; and therefore, that the quantity 
 
 -, which is ever intermediate between them, shall be ulti- 
 mately equal to either. Accordingly, ds and dt, denoting 
 the indefinitely small increments of space and time, there 
 must be, 
 
 _dj_ 
 ^~ dt' 
 
 These things being established, it may be satisfactory to 
 see how the space, time, and velocity, as dependent, each of 
 them, on the other two, may be graphically represented. 
 
 3. Let a right line such as ab and its parts, (Fig. 132.) 
 represent the -time of the motion and the parts of that time, 
 and let a continued line hb'b"b"', &c. be drawn, such that 
 its perpendicular ordinates shall be proportional to the velo- 
 cities at the moments of time corresponding to the points 
 of the line ab on which they are raised : the spaces de- 
 scribed by the body in any portions of the time of its mo- 
 tion, shall be proportional to the areas standing on the cor- 
 
THE MEASURES OF MOTION AND FORCE. 2G9 
 
 responding portions of the base. For if the time ab is di- 
 vided into the equal portions aa, a a", d'd" , &c. and if the 
 velocity continued unvaried during each of those portions of 
 time, the spaces described in those times would be repre- 
 sented by the rectangles am, am, a"m", &c. or by the rect- 
 angles ab', a'b", a"b"', &c. according as it is the velocity at 
 the beginning or ending of each portion of time that is sup- 
 posed to continue unvaried during that time : and therefore, 
 the space described in the whole or any portion of the time 
 AB, would be represented in magnitude by the area of the in- 
 scribed or circumscribed serrated figure standing on the whole 
 or that portion of the line ab. This will hold however fre- 
 quently the velocity is supposed to change, i. e. however mi- 
 nutely the line ab is divided. It therefore holds for the 
 limit of the space so described by the body, and for that of 
 the serrated figure by whose area it is represented. But the 
 limit of the former is the space described by the body mov- 
 ing with a velocity continually varied ; and that of the latter 
 is the area of the figure wherein the serrated boundary is 
 replaced by the continued line bb'b", See. For, by produc- 
 ing each of the right lines bni, b'm , b"m", 8cc. until they in- 
 tersect the greatest ordinate, it will appear that the excess 
 of the area of the circumscribed above that of the inscribed 
 figure, is the difference between the greatest and least of the 
 elementary rectangles : /. e. putting /, and /' for the greatest 
 and least ordinates, and m, for the base of each of the ele- 
 mentary rectangles, the difference is m[l—l') which vanishes 
 with m. And since, by the continual subdivision of the line 
 AB, the circumscribed and inscribed figures shall be made 
 to approach within any assignable difference, the same must 
 hold a fortiori when either of them is compared to the figure 
 bounded by the continued line bb'b', &c. which is interme- 
 diate between them. 
 
 4. Having shown how motions are measured, we may 
 proceed to show how they are compounded and resolved. 
 
270 DYNAMICS. — SECT. I, 
 
 Perhaps there is no motion observed in nature that is not 
 the resultant of several other motions. Thus if a body is 
 moved in the hold of a ship, whilst the ship moves on the 
 surface of the sea, and the surface of the globe revolves 
 from west to east, round its axis, and the earth itself pro- 
 ceeds in its annual course round the sun, and the sun, toge- 
 ther with the whole planetary system, moves round some of 
 the nearer constellations, the motion of the body in absolute 
 space is compounded of all these motions. 
 
 It is now to be shown how the resultant motion is to be 
 collected from its components, and vice versa ; and to begin 
 with the simplest case, that of the composition of two mo- 
 tions, it may be stated, that if the two component motions are 
 represented in quantity and direction by two sides of a pa- 
 rallelogram, the resultant shall be represented in quantity and 
 direction by the diagonal of that parallelogram. This theorem 
 has been already domonstrated in the note appended to Sta- 
 tics, Sect. I. Art, 2. in addition to which the following may 
 serve as an exemplification. If a body moves from a to b in 
 the right line ab (Fig. lorS.) whilst that line, supposed rigid, 
 is carried to the position a"b" ; at the end of that time the 
 body shall be found, not at b, but at b", and if the velocity 
 of the body in the rigid line is always to that of the line it- 
 self, in the ratio of ab to aa", the line being always parallel 
 to AB, which is its first position, the absolute motion of the 
 body shall be in the right line connecting the points a and 
 b". For the velocities being constantly in the ratio of ab to 
 aa", the lines described by these motions in the same time, 
 shall be in the same ratio : and these lines being the coordi- 
 nates which mark the position of the body at any one in- 
 stant, it follows, that the body is always to be found in the 
 right line ab". 
 
 5. What has been shown respecting the composition of 
 two motions in diflferent directions may be extended to the 
 composition of any number of coexisting motions in any di- 
 
THE MEASURES OF MOTION AND FORCE. 271 
 
 rectlons, by continually compounding a new motion with 
 the partial resultant already discovered. This method of 
 successive compositions is easily practised. Thus, if a body 
 moves through a certain space in one direction, and a cer- 
 tain space in another direction during the same time ; it has 
 been shown that, at the end of that time, it shall be found 
 at the extremity of the diagonal of the parallelogram whose 
 sides are the lines it would have described by the separate 
 motions. Wherefore, the place of the body at the end of 
 the given time is found, by drawing the line it would have 
 described by one of those motions singly, and from its ex- 
 tremity a line equal and parallel to that which it would have 
 described by the other motion singly ; i. e. it v/ill be found 
 at the same point, at which it would have arrived had these 
 motions been successive instead of cotemporaneous. For 
 example, if ab is the line it would have described by one of 
 these motions, (Fig. 1 34.) and ac that which it would have 
 described by the other in the same time ; the point at which it 
 arrives, by the compound motion, is found by supposing it to 
 have been carried by the former motion to b, and to be sub- 
 sequently carried over the hne bc' equal and parallel to ac. 
 Now if a third cotemporaneous motion had been communi- 
 cated to the body, by which alone it would have been car- 
 ried over the lin-e ad in the same time, this may be supposed 
 not to have commenced, until after its arrival at c' ; and by 
 drawing from this point a line, c'd', equal and parallel to ad, 
 we have the point d', to which it arrives, in consequence of 
 the three motions simultaneously communicated ; and if 
 each of these motions were uniform and rectilinear, the risfht 
 line ad' shall be that actually described by the compound 
 motion ; and this motion shall be also uniform, bearing to 
 any of the partial motions, the ratio of the line ad' to that by 
 which the partial motion is represented. From all which it 
 is inferred, that if several motions are simultaneously com- 
 municated, the point of space at which the body is found at 
 
272 DYNAMICS. — SECT. I. 
 
 the end of any portion of time, is the same at which it would 
 arrive, if these movements were communicated successively. 
 Accordingly, the place of the body to which these move- 
 ments are simultaneously communicated, and the line de- 
 scribed by it in any given time, are found as follows. From 
 the body let the several lines be drawn which would have 
 been described by it in a given time, for example, 1'', by 
 the separate motions : then from the extremity of one of 
 those lines, let a line be drawn equal and parallel to a se- 
 cond ; and from the extremity of this, another equal and pa- 
 rallel to a third; and so on, until all the lines originally 
 drawn from the body, except the first, are thus transferred : 
 the extremity of the last line is the point where the body 
 shall be found at the end of 1", when the several movements 
 are coexistent; and the line actually described, shall be that 
 connecting the points occupied by the body at the beginning 
 and end of the motion. If the line last drawn terminates in 
 the point of departure, the polygon is closed, and the abso- 
 lute motion of the body is nothing. 
 
 It cannot escape notice, that all which has been here 
 shown, relative to the composition of motions, is strictly co- 
 incident with that before established, respecting the compo- 
 sition of forces, Sect. 1. Art. 8. 
 
 This composition of motions may be easily imagined by 
 presenting different examples. Thus, if a man climbs to the 
 topmast of a ship, whilst the ship itself is in motion, his actual 
 motion is compounded of the motion of the vessel, and of 
 that which is the consequence of his own exertions. Thus, 
 also, if a body moves on a flat surface, whilst this surface is 
 moved on another surface, and the second on a third, and so 
 forth, for any number of surfaces, the motion of the body 
 shall be compounded of all these several motions, and the 
 point at which it shall be found at the end of any given time, 
 for example, 1", shall be that at which it would have arrived, 
 if it had been directed by these motions successively, each of 
 
THE MEASURES OF MOTION AND FORCE. 273 
 
 them being continued for 1". The courses are indeed diffe- 
 rent, as are also the times. In the latter case, the figure is 
 a polygon, and each of its sides is described in 1" ; and in 
 the former case, the body describes in V the line which 
 closes the polygon : but the point to which it arrives is the 
 same, whether the motions are simultaneous or consecu- 
 tive. 
 
 6. As motions are compounded, so are they also resolved 
 in the same manner as forces : the resolved motions being 
 represented, in quantity and direction, by the sides of a tri- 
 angle or polygon constructed on the line representing the 
 motion to be resolved. But the relation between the resultant 
 motion and its components is exhibited in the way best fitted 
 for analysis, by resolving the motion according to three rect- 
 angular axes : and this also is done in the same way as for 
 forces. The motion in each axis is had by multiplying the 
 line representing that motion into the cosine of the angle 
 which its direction makes with that axis : and all the move- 
 ments, whether simultaneous or consecutive, being treated 
 in this way, the problems respecting curvilinear motion may 
 be reduced to those relative to rectilinear motion ; and the 
 whole theory of dynamics unfolded, not only with greater 
 facility, but also in a manner more fertile in general results. 
 
 7. The composition and resolution of motion being so 
 far explained, the subject which next offers itself for consi- 
 deration is the cause of motion : and for this we are com- 
 pelled to look beyond the body moved. For a body at rest 
 is observed to continue in the same condition, until it is 
 disturbed by something external to itself; and as it cannot 
 produce motion in itself, so neither can it increase the velo- 
 city of that already imparted to it. But whether a body set 
 in motion has any natural tendency to rest, whereby it is 
 disposed to return to that state, is a question not so readily 
 answei'ed. For though, like the former question, relative to 
 the generation of motion, this is to be decided by an appeal 
 
 T 
 
{274 DYNAMICS. — SECT. I. 
 
 to facts, yet the facts are such as require a more attentive 
 consideration. It is observed of all motions near the earth's 
 surface, that they require for their continuance a continued 
 renewal of the impressions by which they had been pro- 
 duced ; without whichj if not suddenly checked, they all 
 gradually decay, until at length the bodies return to a state 
 of rest. Thus, if a ball is rolled on a level plane, its motion 
 is continually retarded ; and after describing a space greater 
 or less, the motion is altogether lost. And in the same way 
 a pendulum, being let fall from a certain height, oscillates 
 through arches continually decreasing, until at length it be- 
 comes quiescent at the lowest point of the arch. 
 
 From this seemingly spontaneous relaxation of motion, 
 it might be supposed that there is in matter a natural ten- 
 dency to rest, or a preference of rest to motion. But before 
 that this conclusion can be drawn, it should be considered 
 how far the observed retardations may arise from external 
 impediments : The existence of such impediments is certain : 
 the attrition of the surface on which the body moves is one, 
 and the resistance of the medium in which it moves is 
 another. These obstructions cannot be altogether removed, 
 but they may be reduced, and then it is invariably observed, 
 that the retardations are reduced in the same proportion, 
 and the motion is continued for a period suitably longer. 
 Thus, by levigating the plane, the body which rolls on it 
 proceeds to a greater distance. By polishing the axle on 
 which the pendulum oscillates, in order to diminish the 
 attrition, the number of vibrations performed is suitably 
 increased ; and the reduction of the diameter of the axle, 
 i. e. the diminution of the leverage by which the force of 
 friction acts against the moving body, has the same conse- 
 quence. The retardation of motion in a resisting medium 
 is less, as the density of the medium is less ; and in the 
 same medium it is diminished, by increasing the weight 
 of the moving body, in relation to the surface which it 
 
THE MEASURES OF MOTION AND FORCE, 275 
 
 opposes to the reaction of the medium. By such contri- 
 vances, the resistance opposed to the motions of bodies can 
 be greatly diminished, and the result is such as to warrant 
 the inference, that if all impediment could be completely 
 removed, there would be no loss of motion sustained. In 
 this way we are obliged to reason respecting motions near 
 the earth's surface ; where, though we may diminish, we 
 cannot completely remove all manner of obstructions. But 
 the proof afforded by the heavenly bodies is decisive of the 
 question ; the motions of these bodies having continued for 
 ages, without the least sensible abatement. 
 
 The perfect inactivity of matter being then admitted, or 
 its incapacity to make any change in its own condition, as to 
 rest or motion, it follows, that every such change, whether 
 in the quantity or dh'ection of the motion, is to be regarded 
 as an indication of the action of some external force ; which 
 leads us to consider how the force exerted is measured by 
 the motion or change of motion produced by it. The ques- 
 tion will occur, whether is it to be measured, simply, by the 
 quantity of motion, or by its square, cube, or what other 
 function ? And whether by the effect produced by its ac- 
 tion, through a given space, or during a given time ? This 
 doubt as to the measure of force by motion, is easily re- 
 moved by the observation, that the motion or change of 
 motion produced in a body, by the action of a given force 
 during a given time, is the same, whether the body sub- 
 jected to this action is at rest or in motion ; and if in motion, 
 whatever be the direction of the force applied with respect 
 to that of the motion already existing. Thus, all bodies on 
 the earth's surface move from west to east round the earth's 
 axis, and are carried in the same direction by the progres- 
 sive movement of the earth round the sun: yet, if a body 
 partaking of such a motion, is acted on by a certain force 
 during a certain time, the same effect is produced, whether 
 
 t2 
 
276 
 
 DYNAMICS, — SECT. I. 
 
 the force is directed from east to west, or from west to east, 
 or in any transverse direction whatsoever. The same ob- 
 servation may be extended to a body contained in a ship in 
 motion. The relative velocity produced by a force is the 
 same, in whatever direction the force is applied. 
 
 Now, the change of motion produced in a given body, 
 by a given force, acting during a given time, being inde- 
 pendent of the state of the body to which the force is ap- 
 plied, it follows, that a double or triple force must produce 
 a double or triple change of motion ; and, generally, that 
 the force is proportional to the change of motion produced 
 by it in a given time ; the change of motion being estimated 
 in the direction of the force. For whatever be the magni- 
 tude of the force, if it is divided into a number of units, the 
 change of motion produced by each unit of force is the same, 
 whether the remainder of the force has acted or not ; and, 
 therefore, the total change of motion is the aggregate 
 of the motions, which the several units of force would 
 have produced, if separately applied to the body at rest, 
 i. e. the motion, or change of motion, produced in a given 
 body, during a given time, is proportional to the force 
 applied ; and is, therefore, the adequate measure of that 
 force. 
 
 From observations of this nature, to which there is no 
 exception, the two following laws of motion are collected : 
 
 1st, Every body continues in its state of rest, or of uni- 
 form rectilinear motion, unless influenced by some external 
 force. 
 
 2nd, Every motion or change of motion produced in a 
 given body, during a given time, is proportional to the ex- 
 citing force, and in the same direction. 
 
 The first of these laws is called the law of Inertia ; the 
 second exhibits the measure of force, or the relation which 
 subsists between the force and the velocity : and the whole 
 theory of Dynamics is but the developement of these two 
 
THE MEASURES OF MOTION AND FORCE. 277 
 
 propositions. Thus, from the second law of motion it fol- 
 lows, that if two forces are simultaneously applied to a body, 
 each of which is fitted to move it over a certain right line in 
 a given time, these lines being made the sides of a paralle- 
 logram, its diagonal shall be the space described by their 
 united actions. For the motion excited by each force, 
 being uninfluenced by the effect produced by the other, the 
 motions produced by each force acting separately, must 
 subsist together ; and, therefore, the motion of the body 
 must be that which arises from the composition of those 
 motions. It follows also, that if a force is applied to a body 
 in motion, and oblique to its direction, the effect shall be the 
 same as if it were applied to the body at rest, in conjunc- 
 tion with another force fitted to produce the previous 
 motion. 
 
 In the second law of motion the force was measured by 
 its effect produced by its action during a given time, and on 
 a given mass, and it is therefore requisite to show how 
 forces are to be compared when acting during different 
 times or on different masses. Now, it follows, from what 
 has been already established, that the motion, or change of 
 motion, produced in a given body by a given force, is pro- 
 portional to the duration of its action. For the time being 
 divided into any number of equal portions or units, the 
 change produced in any unit after the first, is uninfluenced by 
 the motion already produced : and the changes of motion in 
 all these equal portions of time, being equal and in the same 
 direction, the total change of motion must be equal to the 
 change produced in one of these moments multiplied by their 
 number ; therefore, t representing the number of units in the 
 whole time, v the velocity produced in that time, andy the 
 force, measured by the velocity produced by it in the unit of 
 
 'V 
 
 time, we have v ^::.ft, or f —~.- 
 
 To show how forces are to be compared by the effects 
 
278 DYNAMICS. — SECT. I. 
 
 produced on unequal masses of matter, it may be laid down 
 as a principle, that when two equal forces act on two equal 
 and contiguous masses of matter, during the same time, and 
 in parallel directions, there is no reason why a greater degree 
 of velocity should be produced in one of the masses, rather 
 than in the other, all influential circumstances being the 
 same. The masses, therefore, shall move with the same ve- 
 locity, and in parallel directions ; and there being no relative 
 motion of one from the other, they shall remain contiguous, 
 both during the action of the forces, and after they shall 
 have ceased to act. The bodies, therefore, may be sup- 
 posed to be connected together, so as to constitute one mass 
 equal to their sum ; and this, without inducing any change 
 in the velocity produced. Accordingly, whatever force is 
 required by its action during a given time, to produce a cer- 
 tain velocity in a certain mass, a double force will be re- 
 quired to produce, in the same time, the same velocity in a 
 double mass ; a triple force, to produce the same velocity in 
 a triple mass, &c. And generally, if m denotes the number 
 of units of mass in the body considered, and ify is the force 
 which in the time t, produces the velocity v, in the unit of 
 mass ; then the force F producing in the same time the same 
 velocity in the mass m, shall be mf. i. e. F = mf. But it has 
 
 been already shown, that/z: -. .'. f =: — . wherein the sym- 
 bols m, V, t, denote numbers. 
 
 In proceeding to connect the vari&us phenomena of mo- 
 tion with the forces, which are their efficient causes, we shall 
 begin by supposing the forces applied to a material point: or 
 if, in treating of this part of the subject, bodies are spoken 
 of, all the equal molecules of the body shall be supposed to 
 be influenced by equal forces, applied to them in parallel di- 
 rections. Therefore, in this first volume, where the subject 
 is thus restricted, our inquiries shall be confined to the con- 
 sideration of motions merely progressive. When the con- 
 
THE MEASURES OF MOTION AND FORCE. 279 
 
 nexion between the phenomena and then* efficient causes is 
 so far understoocl, the student will be prepared to enter on 
 the subject of rotatory motion, or the effect of force, when 
 unequally applied to the equal portions of the same mass ; 
 which shall be treated of in the second volume of this work. 
 
280 DYNAMICS. SECT. II. 
 
 SECTION 11. 
 
 OF FORCES PRODUCING RECTILINEAR MOTION. 
 
 1. When a material point is disturbed from a state of 
 rest by the action of a constant force, the increments of velo- 
 city generated in equal times are equal, i. e. the velocity of 
 the moving body is one uniformly accelerated. The velo- 
 cities thus generated, are as the number of those equal in- 
 crements, i. e. as the number of equal portions of time 
 counted from the beginning of the motion ; wherefore they 
 are proportional to the times in which they are acquired. 
 Accordingly, the accelerating forced*, being estimated by the 
 velocity generated by it in a unit of time, the velocity v, ge- 
 nerated in any time t, shall be expressed by the equation 
 
 v=fJ. (I) 
 
 To show the dependance of the space on the time and 
 velocity, it will be convenient to make use of the theorem 
 proved in Art. 3. of last section ; and to accommodate that 
 theorem to the present case, wherein the generated velocities 
 are as the times counted from the beginning of the motion, 
 the ordinates must be taken proportional to the abscissae. 
 Accordingly, if in (Fig. 133.) the point a corresponds to the 
 instant at which the velocity is cipher, and if the velocity ac- 
 quired in the time ab, is represented by the ordinate bc, 
 that which is generated in the time am, shall be represented 
 by an ordinate m\, which is to bc in the ratio of am to ab. 
 When such is the law of the ordinates, the line by which 
 they are terminated shall be a right line diverging from ab, 
 and the scheme a right-lined triangle. The areas of the si- 
 milar triangles amn, abc, being as the squares of the bases 
 
OF FORCES PRODUCING RECTILINEAR MOTION. 281 
 
 AM, AB, or of the altitudes mn, bc, it follows, that the spaces 
 described with a velocity uniformly accelerated, and counted 
 from the beginning of the motion, are as the squares of the 
 times in which they are described, or as the squares of the 
 velocities acquired in those times. For example, the spaces 
 described in 1", 2", 3", 4", &c., counted from the beginning 
 of the motion, are as the numbers 1, 4, 9, 16, &c. And the 
 spaces described in the several equal moments are as the 
 odd numbers 1, S, 5, 7, &c., these being the differences of 
 0, 1, 4, 9, 16, &c. the squares of the natural numbers, in- 
 cluding cipher. 
 
 If the velocity is uniformly retarded, i. e. if it suffers 
 equal decrements in equal times, the velocities are propor- 
 tional to the times, counted to the end of the motion ; and 
 the ordinates, by which they are represented, are terminated 
 by a right line converging to the base line, by which the 
 time is represented. Therefore, the scheme given in (Fig. 
 loo.) will serve to represent the theory of uniformly retarded 
 velocities, by regarding a as the point of time at which the 
 motion terminates. Thus, the spaces described in the times 
 BA, MA, being proportional to the areas bac, man, are 
 as the squares of the times ba, ma, or of the initial velocities 
 BC, mn. 
 
 If a body moves with a velocity uniformly accelerated, 
 the space described, measured from the beginning of the 
 motion, shall be one-half of that which it would describe 
 with the last acquired velocity continued uniformly for an 
 equal time: the former of these spaces being represented by 
 a triangle, and the latter by a parallelogram of the same base 
 and altitude. 
 
 The following theorems are now established relative to 
 the effects of a constant force, on a body moved by it from a 
 state of rest. 
 
 1st, The velocities generated in any times, counted from 
 the beginning of the motion, are proportional to those times. 
 
282 DYNAMICS. — SECT. 11. 
 
 2d, The spaces described, measured from the beginning 
 of the motion, are proportional to the squares of tlie times, 
 or of the acquired velocities. 
 
 3d, The space described from the beginning of the mo- 
 tion, is one-half of that which would be described with the 
 acquired velocity, continued uniform for an equal time. 
 
 The same theorems are equally applicable, when the ve- 
 locity is uniformly retarded ; the spaces and times being 
 measured to the end of the motion. 
 
 The first of these theorems is expressed by the equa- 
 tion (1), wherein^ denotes the force, or its measure, which is 
 the velocity generated in a unit of time; and t, as before, the 
 number of such units in the time of the motion. 
 
 The third theorem is expressed by the equation 2s = vt. 
 Whose terms being multiplied into those of eqviation (1), 
 there is 
 
 whereby the space is known when the time is given, and con- 
 versely. By multiplying the alternate members of the same 
 equations, there is 
 
 v''-=z2fs, (3) 
 
 whereby the velocity is known when the space is given, and 
 conversely. 
 
 If the body has an initial velocity v, that with which 
 it moves at the expiration of any time, t, is expressed by 
 the equation 
 
 v-v±ft. (4) 
 
 The force /, being taken with an affirmative or negative 
 sign, according as the velocity is accelerated or retarded. 
 The velocity, expressed by this equation, is the sura or dif- 
 ference of two velocities, of which one is the uniform ve- 
 locity Vy and the other the velocity ft, proportional to the 
 time t. 
 
OF FORCES PRODUCING RECTILINEAR MOTION. 283 
 
 The space described is the sum or difference of the 
 spaces which would have been severally described in these 
 two ways. That due to the uniform velocity v is had by 
 multiplying this velocity into the time ; and that due to the 
 uniformly accelerated velocity, by multiplyingy?, the velocity 
 generated in the time t, by half that time. The former part 
 
 is, therefore, v't, and the latter*^, which gives for the space, 
 
 the equation 
 
 s^v't±f^. (5) 
 
 Exterminating /, between equations (4) and (5), there is 
 %fs-{f — v'% (6) 
 
 Wherein f is to be taken with a negative sign, when the 
 motion is retarded. 
 
 J^. As near the surface of the earth, bodies descending 
 or ascending in the vertical, are known to be uniformly ac- 
 celerated or retarded, we have only to put for f, its proper 
 value in the foregoing formulse, in order to have the answer 
 to every inquiry, whether relating to the space of ascent or de- 
 scent, the velocity lost or gained, or the time required for this 
 purpose. Now, it is ascertained by experiments made with 
 pendulums, that in the latitude of 45°, a body, falling freely, 
 would describe sixteen feet and one inch in a second of 
 time, wherefore, the velocity acquired in the same time, is 
 that which w^ould carry the body through 32 feet 2 inches in 
 one second. In this case, the force is that of gravity at the 
 earth's surface, which it is convenient to designate by an 
 appropriate symbol. Thus /, being replaced by g, in the 
 preceding equation, they will be known to relate to bodies 
 ascending or descending, under the influence of the force of 
 terrestrial gravity. 
 
 If a body is projected vertically upwards with a cer- 
 tain velocity v\ the diminished velocity v, corresponding to 
 any time, or the time corresponding to that velocity, is given 
 
SSI- 
 
 DYNAMICS. — SECT. II. 
 
 by equation (4), which as the value of g is negative, be- 
 comes v=iv' — gt. or v' — V =. gt. The same observations 
 apply when equations (5) and (6) are used to compare the 
 time or velocity with the height of ascent. 
 
 By making v — 0. in equation (6), there is for the whole 
 
 height of ascent, *= — . which compared with equation (3) 
 
 or s =. — . shevv's, that if the spaces are equal, the velocity 
 
 destroyed in one body is equal to that generated in the other : 
 or that the body, on its return to the surface of the earth, re- 
 gains precisely the same velocity with which it had been 
 projected upwards ; and it also shows, that if a body is 
 returned upward with the velocity acquired in the descent, 
 it shall rise to the same height from which it fell. Also, by 
 comparing equations (1) and (4), it will be seen that in the 
 same case the times of ascent and descent are equal. 
 Finally, from same equations it will follow, that if one body 
 were to begin to ascend, and the other to descend at the 
 same instant, the sum of the velocities in the ascending and 
 descending bodies, shall be constantly equal to that with 
 which the former had been projected upwards. For equa- 
 tion (4) gives for the velocity of the ascending body, 
 v=.v'—gt. and equation (1) for the descending body, 
 V -n gt. and the sum of these values is v' ; which is also the 
 relative velocity of one of the bodies from or towards the 
 other. 
 
 Let the numerical value of g be 32^, viz. the number of 
 feet through which a body would be carried in a second of 
 time with the velocity generated by gravity in that time ; and 
 let this value be assigned toforg, in those equations; then, 
 t shall denote the number of seconds in the time of the mo- 
 tion ; s, the number of feet in the space described ; and v 
 shall express the number of feet which a body moving uni- 
 
OF FORCES PRODUCING RECTILINEAR MOTION. 283 
 
 furnily with that velocity would describe in one second. 
 Accordingly, the equation between the time and velocity is 
 
 V - Sn" {a) 
 
 omitting the two inches in the value of g, for sake of sim- 
 plicity. 
 
 The equation between the time and space is 
 
 And that between the space and velocity is 
 
 V^Sy/s. OV.S--^. {c) 
 
 Thus, if the body falls through a space of 100 feet, by equa- 
 
 10" 
 tion [U) there is t"z=. -— — 2J". and by equation (e) v = 80 
 
 feet per 1". 
 
 3. When a body is in part sustained by an inclined plane, 
 the force by which it is solicited is had by resolving g, the 
 force of gravity, whose direction is vertical, into two forces; 
 one of them perpendicular to the plane, and the other in 
 
 the direction of the plane. These are j.g, j.g, in w^hich, 
 
 /, h, b denote the length, height, and base of the plane. 
 
 The force -j.g, is destroyed by the resistance of the plane, 
 
 but the force -jg, produces its full effect. And as this 
 
 V 
 
 value is the same in every part of the plane, the motion of 
 the body descending by this force is uniformly accelerated. 
 Accordingly, all questions relative to the motion of such a 
 
 body, are to be solved by making y= -.g, in equations (1), 
 
 (2), (3). which then are 
 
 V = jg.t. (<0 
 
2S6 DYNAMICS. — SECT. II. 
 
 * - 27-^-^'' ^^^ 
 
 v'^=z2^^.g.s if) 
 
 Equation {d) shews that the velocity acquired in any time, 
 by a body descending down an inclined plane, is to that 
 acquired in the same time, by a body descending freely in 
 the vertical, as the height of the plane to its length. 
 
 The space described in a given time, or the time of de- 
 scent through a given space, is had from equation (e). 
 Thus, putting / for s in that equation, we have for the time 
 of descent down the entire plane. 
 
 From which it appears, that the times of descent down 
 different planes, are proportional to their lengths, divided by 
 the square roots of their heights : and that if the heights 
 are equal, the times are simply as the lengths of the planes. 
 The height itself being included among those planes, it 
 follows, that the time of descent through the length of 
 the plane, is to the time of descending through its ver- 
 tical height as the length to the height. This appears 
 also by a comparison of the values of those times, viz. 
 
 l.'y -pj , and y ~^, which are in the proportion of / to h. 
 
 If it is required to ascertain the portion of the plane 
 through which a body would descend, whilst another body 
 falls down the vertical height, we have by the same equa- 
 tion, 
 
 gh 
 and putting this equal to — , which by equation (2), ex- 
 
 O 
 
OF FORCES PRODUCING RIX'TILINEAR MOTION. 287 
 
 presses the square of the time of the fall down the vertical 
 height, we have 
 
 Is zz. It , or, s zz — . 
 
 t 
 
 showing that the portion of the inclined plane described in 
 the time of the fall down the vertical height, is a third pro- 
 portional to the length and height. Accordingly, if ab is 
 the inclined plane, and bc its height, (Fig. 135.) and CD a 
 perpendicular on the plane from the point c, the segment 
 BD shall be the portion required. For by similar triangles, 
 AB : BC : BD. 
 
 If a circle is described round the diameter bc, the several 
 chords drawn from b, such as bd, be, bf, &c. shall be de- 
 scribed by bodies descending along them in the same time ; 
 the time of descent for each being equal to that of the de- 
 scent down the vertical diameter bc. And the same thing 
 is true of the chords do, eg, fc, terminated by the other 
 extremity of the vertical diameter ; which is plain, by con- 
 sidering that these are the sam.e as to length and inclina- 
 tion, with the parallel chords bd', be', bf', drawn from 
 the upper extremity of the same diameter. The same thing 
 
 would also appear immediately from equation t =. I'y — » 
 
 which shows that the times of descent down different planes 
 are as the lengths divided by the square roots of the heights. 
 But the chords of the arches of the same circle are as the 
 square roots of their verse sines, i. e. the arches being 
 measured from either extremity of the vertical diameter, 
 the chords are proportional to the square roots of their 
 heights. 
 
 The relation between the space and velocity is given by 
 equation {/). Wherefore, if the velocity acquired by de- 
 scending down the entire length of the plane is sought, 
 this is had by putting / for s in that equation, which then 
 becomes 
 
288 DYNAMICS. — SECT. II. 
 
 TIlis is also the expression for the velocity acquired by 
 falling down h, the vertical height of the plane; from M'hich 
 it appears, that the velocities acquired by falling down dif- 
 ferent planes, are as the square roots of their heights ; and 
 that if the heights are equal, the velocities acquired are also 
 equal. 
 
 It also follows, that the velocities acquired by falling 
 through the chords of the several arches of the same circle 
 are proportional to the chords themselves, the circle being 
 placed in a vertical plane, and the chords being drawn to 
 the extremity of the vertical diameter, for the velocities are 
 proportional to the square roots of the heights, and there- 
 fore to the chords. 
 
 4. There are several other constant forces in nature, 
 wdiich render this theory of practical importance. Of this 
 kind, is the force of resistance arising from tenacity. For 
 when a projectile penetrates a solid substance, the whole 
 depth of penetration, 5, being divided into an indefinite 
 number of equal parts ds, the tenacity in each of these parts 
 is the same, and the decrement of velocity produced by its 
 resistance is proportional to the time of its action, L e. to 
 the time taken by the projectile to pass through each of 
 these spaces ; and, therefore, the retardation produced by 
 any number of such parts, i. e. the retardation suffered by 
 the projectile in passing through any portion of the whole 
 depth of penetration, is simply as the time. This is con- 
 firmed by experiment, which shows, that the depths to which 
 a projectile penetrates a bank of earth, or plank of timber, 
 are as the squares of the velocities of projection. This re- 
 lation between the spaces described, and the velocities de- 
 stroyed, is the indication of a uniformly retarded velocity ; 
 and, therefore, of a constant force. — See equation 3. 
 
 This being admitted to be the nature of the force of re- 
 sistance arising from tenacity, it is easily seen why a bullet 
 
OF FORCES PRODUCING RECTILINEAR MOTION. 289 
 
 may be sent through a plank without overturning it; or 
 through a door without moving it on its hinges. For agree- 
 ably to the notation before adopted, \etf be the constant 
 force of resistance ; s the space through which it acts, and 
 which, in this case, is the thickness of the plank; i;' the 
 velocity of projection ; and v that with which the projectile 
 escapes from the remote surface ; there is by equation (3) 
 
 a constant quantity: wherefore, (v'—v) {v-\-v) is constant ; 
 shewing, that z;'—?; varies reciprocally as v~]^v. But v — v 
 is the velocity lost, and this is proportional to the motion 
 communicated. Wherefore, the greater the sum of the ve- 
 locities with which the projectile enters and escapes, the less 
 the motion communicated by it. 
 
 5. When the action is that of a variable force, the rate 
 at which the velocity increases or diminishes, being as the 
 acting force, is no longer constant, i. e. the velocity is no 
 longer vmiformly accelerated, or uniformly retarded : and 
 the change made in that velocity, during a given time, is no 
 longer the measure of the force acting at the beginning of 
 that time. If the force increases, the change of velocity is 
 greater than that due to the uniform action of the force 
 whose expression is sought ; and if the force diminishes, the 
 change of velocity is less than that by which the same should 
 be measured. Now, to find the expression for such a vary- 
 ing force at any one instant, it is to be observed, that the 
 change made in the velocity during any time t, is interme- 
 diate between the changes that would be effected in the 
 same time, by the forces acting at the beginning and end of 
 that time, I. e. putting v , v for the velocities at the begin- 
 ning and end of the time t, j' and / for the forces ; the 
 greater velocity and greater force being denoted by the ac- 
 cented letters : there is 
 
 f.t<.{v'--v)<f.t 
 i, e. V ~v is intermediate between /if and./'/. But by di- 
 
 u 
 
290 DYNAMICS. — SECT. 11. 
 
 minishing the time indefinitely, the forcesy and/' at the 
 beginning and end of that time approach within any as- 
 signable difference ; and therefore, a fortiori, v ~v, inter- 
 mediate between them, shall differ from /if by a quantity less 
 than any assignable : therefore, denoting the indefinitely 
 small increments of the time and velocity by dt and dv, 
 there is dv :^fdt or 
 
 This theorem is demonstrated similarly for a decreasing 
 force. 
 
 (IjS cl~s 
 
 By Sect. 1. Art. 3. «;=—. and therefore, -^=:f7y=/.t//» 
 
 whence 
 
 in which the force is expressed as a function of the space 
 and time. 
 
 Also, multiplying equation (8) by the equation — r: z7» 
 
 we \\a.yef.dszzv.dv, or, 
 
 /=i^. (,0) 
 
 whereby the force is represented as a function of the space 
 and velocity. 
 
 The theorems contained in equations (8) and (10) may be 
 represented geometrically ; the former, by taking a base line 
 AB, with its subdivisions, to represent the time and its sub- 
 divisions, (Fig. lo6.) and erecting perpendiculars at indefi- 
 nitely small intervals, to represent the forces acting at the 
 several moments of time, corresponding to the points of the 
 base at which they are raised. Each areola shall then re- 
 present the increment of the velocity, or dv. gained in the 
 time represented by its base ; and, therefore, the area. 
 
OF FORCES PRODUCING RECTILINEAR MOTION. 291 
 
 standing on any portion of the base, shall represent the ve- 
 locity generated in the corresponding portion of time. 
 
 The force is never given as a function of the time, and, 
 therefore, the preceding construction is of no use but for il- 
 lustration. 
 
 The force is most commonly known as a function of the 
 space, and, therefore, equation (10) admits of a more useful 
 construction. 
 
 Let the same line ab represent the space described by 
 the moving body ; and the equidistant ordinates, the forces 
 acting at the points at which they are raised. Then, each 
 areola representing y*. J*, shall represent its equal, viz. v.dv. 
 which is half the increment on the square of the velocity. 
 Wherefore, the change on the square of the velocity, during 
 the motion through any portion of the space ab, shall be 
 represented by twice the area standing on that portion. And 
 thus, the determination of the velocity at any point of the 
 space, is reduced to the quadrature of curves. 
 
 The velocity being thus found as a function of the space, 
 
 ds 
 may be substituted for v, in equation y = -y-, whence, by in- 
 tegration, the time of describing any portion of the space is 
 found. 
 
 In general, it is to be observed, that each of the equa- 
 tions (8), (9), (10) involves three quantities : and that when 
 the force is given as a function of the space or velocity, one 
 of these may be eliminated, and an equation thus obtained 
 between the remaining two. This proceeding shall now be 
 illustrated by a few examples. 
 
 6. The force being supposed to be directed to a fixed 
 centre, and to vary as some power, n, of the distance from 
 that centre, let it be proposed to find the law of the motion 
 of a body ascending or descending in a right line passing 
 through that point. 
 
 In general, if x denote the variable distance of the body 
 
292 
 
 DYNAMICS. SECT. II, 
 
 from the centre of force, then ds= — dx ; the body being 
 supposed to descend towards that point. Equation (10), 
 therefore, becomes vdv zz —fdx, or integrating 
 
 v''=:-2^fdx+c. 
 
 Now in the case proposed, if /j. denotes the intensity of 
 the force at the unit of distance, the value of the force at any 
 distance x, is /mx": Substituting this value of/ in the equa- 
 tion just obtained, it becomes 
 
 o 2nx"+^ , 
 v^= ^^—j — he. 
 
 But v denoting the velocity at the distance x', we have 
 And subtracting, 
 
 w+1 
 
 
 This equation will equally serve in the case of a body 
 projected in a direction opposite to that of the force. For in 
 that case, the signs of dx and dv are together changed ; 
 whereby the differential equation remains unchanged. 
 
 The total height of ascent when the body is projected in 
 the direction opposite to that of the force, is had by making 
 vzzo in the last equation. The force at the distance .r'is ju.r'", 
 and if h denote the height through which the body must fall 
 under the influence of that force supposed constant, in order 
 to acquire the velocity v', we have 
 
 x"+^ - x''+^ = (« + 1 )x"h. 
 
 whence 
 
 1 
 
 x=x'[l+{?i+\)-^^. 
 
 The velocity being expressed as a function of the dis- 
 tance, the time is to be obtained from the formula 
 
OF FORCES PRODUCING RECTILINEAR MOTION. 293 
 
 To apply these results to different suppositions respect- 
 ing the law of the force. 
 
 For n— 1 . The equation between the space and velocity 
 
 IS 
 
 If the body descends from a state of rest, there is 
 
 The velocity with which the body arrives at the centre of 
 force is had by making :r=o. in the last equation, which 
 gives 
 
 The body having arrived at this point shall proceed in the 
 same direction to the distance x' with a velocity retarded in 
 the same manner as it was before accelerated : the velocities 
 at equal distances on both sides being equal. Having at- 
 tained to this distance beyond the centre of force, the body 
 shall return, describing the space 2x' with a velocity varying 
 according to the same law ; and thus is shall for ever go and 
 return through this space. 
 
 The time of descent to any distance x measured from the 
 
 centre of force is obtained by putting for v, its value in the 
 
 dx . . , 
 equation v=.~—. which gives 
 
 dx 
 
 dtzz- 
 
 y/ ix[x"~—x-) 
 and by integration 
 
 1_ 
 
 which needs no correction if the time be computed from the 
 instant when the body is at the extremity of its space of vi- 
 bration, i. e. when x=:x'. 
 
 t= — -. arc I cos. = — -, j 
 
294 DYNAMICS. — SECT. II. 
 
 The whole time of descent is had by making x=o. which 
 gives cos. = o. and the arc a quadrant. Wherefore, putting 
 T for this time its value is given by the equation 
 
 T — - 
 
 2ViU 
 
 This being independent of the distance from which the 
 body begins to descend, it follows that bodies arrive at the 
 centre of force in equal times from whatever distances they 
 may have fallen. 
 
 By reversing the general formula, we get 
 
 XZZ x'cOS.(y fjl.t). 
 
 for the distance corresponding to any time t ; and by sub- 
 stituting this value of x in the equation between the space 
 and velocity, we obtain an equation between the velocity and 
 time. 
 
 To represent these results geometrically, let a circle be 
 described (Fig. 137.) whose centre o is the centre of force, 
 and radius oa equal to .v ; and at the distance on, equal to 
 X, let the ordinate ?im be raised. It is obvious that nm = 
 y .v'^—.v^. so that 
 
 vzi'sj \x nm. 
 The velocity therefore varies as the ordinate. Also, the arc 
 
 Mn — a^'arcf cos. =— r) ; 
 
 and consequently 
 
 The denominator of this expression is the measure of 
 the velocity acquired in falling to the centre. Therefore, 
 the time is equal to that of describing the arc a;«, with that 
 velocity continued uniform. 
 
 Forces varying according to this law frequently offer 
 
OF FORCES PRODUCING RECTILINEAR MOTION. 295 
 
 themselves for consideration. Thus, if the density of the 
 earth were uniform throughout its mass, the force of gravity 
 beneath the surface would follow the law of the simple dis- 
 tance from the centre, i. e. at the distance of half the ra- 
 dius, the weight of the body would be reduced to half; at 
 the distance of a third part of the radius, to one-third, and 
 so forth. In this case, the gravity at the surface being de- 
 noted by g, there is g — /xr, and substituting in the expres- 
 sions already obtained, the velocity acquired in descending 
 to the centre is 
 
 V ^z V gr, 
 
 and the time is 
 
 rVi 
 
 The quantities so expressed are readily computed ; for 
 if in round numbers, r is 4000 miles, or 21,1^0000 feet, g 
 being 32. 
 
 
 gr — V 675,840000. feet = 26000 feet. q.p. 
 i. e. about five miles. Accordingly the body would pass 
 through the centre, moving at the rate of about five miles 
 per 1". 
 
 For the time of descent to the centre, we have tt = 
 
 22 
 
 ~-.q-p- and therefore, 
 
 ^_^./r 11. /21,12000(r ^,,, , 
 
 T--y- = — V .^^ = 21 Y nearly. 
 
 The force of elastic bodies is known to follow the law of 
 the distance from the point of rest. Thus the middle point 
 of a musical cord is accelerated or retarded by a force vary- 
 ing as the distance from the middle point of the space de- 
 scribed by its motion, or from the point of rest. The 
 theorems here established may, therefore, be applied to the 
 solution of questions relative to the motion of such bodies. 
 
29G DYNAMICS. — SECT. II. 
 
 For example, let a bow be supposed to require a weight 
 of 100 pounds applied to its string, to draw it through a 
 space of '30 inches, and that the arrow weighs one ounce ; 
 the velocity with which it is discharged is had by putting 
 their particular values for x' and fx in the equation v = a;' Vi". 
 Now the force at x', i. e. fix' is 1600 times the force of 
 gravity, or 1600x32. and x'zz2i feet. Therefore Z7 = 350 
 feet per 1". nearly ; or about the fifth part of the velocity 
 with which a ball is discharged from a well loaded musket. 
 
 If the force varies inversely as the distance from the 
 fixed point, or mzi — 1. the solution given above fails; and 
 we must have recourse to the general equation 
 
 ?;2 = - '2lfdx + c. 
 From which, by substituting for/ its value -, we find 
 
 X 
 
 v^ — — 2fl\og.X + c. 
 And 
 
 v"^ = — 2m.\og.x + c. 
 
 Wherefore, the law of the movement is given by the 
 equation 
 
 v^ — v'^ zz 2/u.log. — . 
 
 The body being projected in a direction opposite to that 
 of the force, the total height of ascent is had by making 
 V — 0. which gives 
 
 v^ :r 2u.log.— 7. 
 
 whereby it appears that the body under the influence of a 
 force acting according to this law, cannot be sent to an infi- 
 nite distance, by any finite velocity of projection. 
 
 If the body descends from a state of rest, the expression 
 
 x' 
 for the velocity is given by the equation v- — 2/u.log. — . At 
 
OF FORCES PRODUCING RECTILINEAR MOTION. 297 
 
 the centre of force, a; — 0. and, therefore, the velocity at 
 this point is infinite. 
 
 If the force varies inversely as the square of the distance, 
 then, nzz — 2, and the equation expressing the relation 
 between the velocity and space is 
 
 \X X J 
 
 If the body descends from a state of rest, v — 0. and 
 -y^ — 2|u( yj. From which it appears, that in the de- 
 scent, the velocity varies as the square root of the space 
 described, divided by the square root of the distance from 
 the centre. 
 
 For the centre, x znO. wherefore, the velocity at the 
 centre is infinite. 
 
 By substituting for v, its value in the equation v — 7-, 
 
 there is 
 
 x^.dx xdx 
 
 dt- — 
 
 X X 
 
 and integrating, 
 
 / x' ^ , , ^A , f 2x — x'\) 
 r= \/ -^ < [xx —x'^)" -^^x arc ( cos. =. j— J > . 
 
 which requires no correction. 
 
 To represent these things geometrically, let a be the 
 point from which the body begins its descent towards o, the 
 centre of force, (Fig. 138.) and let the line ao be bisected at 
 c: then a:'=: ao, and the body being at m, x—^x'=cm. 
 Moreover, if a circle is described with the centre c and ra- 
 dius CA, and an ordinate mn, is raised at /«, we shall have 
 
298 
 
 DYNAMICS. — SECT. II. 
 
 x' f X--\x \ 1 
 
 — arc ( COS. iz. — ^ ' , - 1 r: arc. aw. and {xx — otP)- =:mn, 
 
 Now, ^ Ao X arc An= sector caw ; and ^ ao X mn = triangle 
 cno. Wherefore, the time of descent to any point as 7)1, is 
 proportional to the mixed-Hned area oa«, contained by the 
 right Hnes oa, g;?, and the circular arc Afi. 
 
 This construction is given by Newton. 
 
 When the body arrives at the centre, there will be xzzo. 
 and 
 
 ' / ' 
 
 ^TT X y X 
 
 which shows that the times of descent from different dis- 
 tances to the centre of force, vary in the sesquiplicate ratio of 
 those distances. 
 
 10. To determine the law of the vertical movement of a 
 body in a fluid medium. The fluid being supposed to be 
 without tenacity, the resistance opposed to the motion of the 
 body shall be measured by the motion communicated to the 
 fluid, i. e. by the quantity displaced in a given time, multi- 
 plied by the velocity imparted to it. Wherefore, the resis- 
 tance is as the square of the velocity, and may be denoted by 
 mv^. the coefficient 7n being the resistance to the body mov- 
 ing with a unit of velocity. Its value depends on the form 
 of the body and the density of the medium, and is, therefore, 
 constant, when both of these are given. Moreover, in pro- 
 blems of this nature, the spaces described being small, in re- 
 lation to the distance from the centre of the earth, the force 
 of gravity shall be supposed to be constant. 
 
 If the body is projected vertically upwards, its motion is 
 retarded, both by the force of gravity and by the resistance 
 of the medium; and the whole force then acting on the body 
 is —{g + mv-). But if the body descends, gravity acts as an 
 accelerating force ; and this being diminished by the resis- 
 
OF FORCES PRODUCING RECTILINEAR MOTION. S99 
 
 tance of the medium, the force with which it is effectively 
 accelerated, is g — mi^. 
 
 To begin with the case of an ascending body, we have 
 fzz — (^ + mv^). and this value of the force being substi- 
 tuted in the general equations, filx rz v.d.v, fdt — dv, we 
 have 
 
 , v.dv - dv 
 
 dx — ^ -;s. dt — — 
 
 g-[-mv^' "" g-\-7nv^' 
 
 These equations being integrated, give 
 2fnx = c — log. {g-\- mv^). 
 
 V tug. t :zic—a.rc (tan. -zz ^ — .vy 
 
 The constants are determined by observing that for 
 x-ziO, or for t — O, the value of 2; is the velocity of projection. 
 Wherefore, putting v, as before, for that velocity, 
 
 c = log. {g 4- mv~). c'zi arc (tan. — / ~-^'j' 
 Wherefore, 
 
 2mx zzlos.l- 7 I. 
 
 \g + mv J 
 
 V t7igJ=i arc (tan. =: V ~.y'j — arc (^tan. = y —.v). 
 
 These two equations contain the complete solution of the 
 problem for an ascending body, since by exterminating v, 
 they give an equation between the space and time. 
 
 For the total ascent, v := 0. Accordingly putting x' for 
 
 this height, and t' for the time of total ascent, we shall 
 
 have 
 
 , 1 , /, mv~\ 
 
 x=z-~.\og. (1 H j. 
 
 2?>i \ g J 
 
300 DYNAMICS. SECT. II. 
 
 When the body arrives at the highest point of its flight, 
 
 it shall return as from a state of rest ; and the force with 
 
 which it is accelerated in its descent being g — mv^, the 
 
 equations by which this part of the problem is to be solved, 
 
 are 
 
 , v.dv J, dv 
 
 ete=r 5. ar = o^ 
 
 g—mv^ g—mv^ 
 
 The former of these, by integration, gives 
 
 ^mx —c — log. {g — mv^). (1 1) 
 
 And since x is now measured from the point where t; = 0, 
 c = log. g. Wherefore, 
 
 2mx = \og.( — ^—X 
 
 ° \g — mv^J 
 
 To prepare the second of the differential equations for 
 integration, it may be put in this form, 
 
 _ 1 t dv dv -^ 
 
 ^V g K. V g + v. \^ tti V g — vVm S 
 
 which, by integration, gives 
 
 2Vmg.t = \og.S^^I±L^l. (12) 
 
 t Vg — V. V j}i ^ 
 
 This requires no correction, since ^ := 0, gives v zz 0. 
 
 For X infinite in equation (11), or t infinite in equation 
 (12), there is 
 
 ^ m 
 
 This, therefore, is the limit to which the velocity of the de- 
 scending body always approaches, without ever attaining to 
 that value. 
 
 When the body thrown upwards with the velocity v', 
 returns to the point from which it was projected, it is evi- 
 dent that it does not regain the whole of this velocity, inas- 
 
OF FORCES PRODUCING RECTILINEAR MOTION. 301 
 
 much as in each point of the space, the force which acce- 
 lerates the motion in the descent, is less than that by which 
 the motion upwards was retarded. These velocities are 
 compared by equating the values of x and x, as given for 
 the body in its ascent and descent. This gives 
 
 €±^'=-^. and .'=-£2^,. 
 
 g g— mv g + mv^ 
 
 Wherein mv'~ is the resistance encountered by the body 
 moving with the velocity of projection. The same state- 
 ment will serve for comparing the velocities of ascent and 
 descent at any point, by putting in place of v', the velocity 
 of ascent at that point. 
 
302 DYNAMICS, — SECT. III. 
 
 CTION III. 
 
 OF FORCES PRODUCING CURVILINEAR MOTION. 
 
 1. When a body moves in a curve of any kind, the direc- 
 tion of its motion is continually changed, which cannot be, 
 unless the body is under the influence of a force or forces 
 acting in directions different from that of the motion. Now 
 it is the business of mechanical philosophy to connect these 
 forces and movements, so that each may be ascertained, as 
 to its intensity, direction, and laws, when the other is given. 
 By those who first cultivated this branch of mechanics, such 
 questions were managed by resorting to the indefinitely 
 small movements successively compounded, and the forces 
 by which they were produced. Thus, if ac (Fig. 139.) is 
 the path of the body; am the portion of the tangent at a, 
 which would have been described in a unit of time with the 
 velocity at that point ; and ao, the line actually described 
 in that time : then, an, equal and parallel to mo, is the space 
 which would have been described, had the force been ap- 
 plied to the body at rest. Moreover, the unit of lime, and 
 therefore the lines ao, an, being taken indefinitely small, 
 the force during that time is regarded as unchanged in 
 quantity and direction : and, consequently, 2an, or 2mo, is 
 the measure of the velocity due to the action of the force ; 
 and, therefore, also the measure of the force itself. In the 
 same way, am' being the portion of the tangent to the curve 
 at a' which wovdd be described in a unit of time with the 
 velocity at that point ; and a'o' the space actually described 
 in that time ; twice w^'o', or twice a'n', to which it is equal 
 and parallel, is the measure of the force at «'. xiccordinglj-, 
 
OF FORCES PRODUCING CURVILINEAR MOTION. 303 
 
 the deflecting force at «, is to that acting at a', as 7no to 
 m'o', the times of describing the elementary arcs ao, a'o, 
 being equal ; or if the times are unequal in any ratio, the 
 forces are as the same lines divided by the squares of the 
 limes, agreeably to what has been established relative to 
 right lined motions produced by the action of constant 
 forces. Now, when the curve is given, as also the law of 
 the motion in that curve, it is the business of geometry to 
 ascertain the relation of the lines mo, m'o' , and, therefore, 
 of the forces measured by those lines; likewise, when the 
 velocity and direction of the movement at any one point of 
 space are given, together with the intensity and law of the 
 force by which the body is subsequently influenced, it be- 
 longs to the same science to investigate the nature of the 
 curve or trajectory. The former of these was called the 
 direct, and the latter the inverse problem in this branch of 
 mechanics. 
 
 2. This method, which was somewhat circuitous, has 
 given place to others, according to which, the forces and 
 movements are reduced to rectangular or polar coordinates. 
 Of these, the former, as being more simple as well as more 
 extensibly applicable, is that which shall be treated of in 
 the present section. In this method, as already intimated, 
 the forces, whatever their number or directions, are reduced 
 to three forces acting in directions parallel to three rectan- 
 gular axes, after the manner explained in ' Statics,' Sect. I. 
 Art. 9, &c. Likewise, the material point being projected 
 on the same axes, those projections are regarded as so many 
 points which accompany the material point in its motion 
 through space. In this way, the movement of the material 
 point is resolved into three movements, which are those of 
 its projections : and as the movement, parallel to any one 
 axis, is not influenced by the forces parallel to the other 
 axes, it follows, that each of the decomposed motions may 
 be regarded as being influenced solely by the force in its 
 
304 DYNAMICS — SECT. III. 
 
 direction ; whereby the theory of forces producing curvi- 
 linear motion, is reduced to that of three forces producing 
 so many right hned motions. 
 
 3. To ilhistrate this method by example, let a body be 
 supposed to be projected in a line making any angle with 
 the vertical, whilst it descends in the vertical with a velocity 
 uniformly accelerated. 
 
 Let BD (Fig. 140.) be the space that would be described 
 in a given time with the velocity of projection ; and ba, taken 
 in the vertical, that described in the same time, with the ve- 
 locity uniformly increasing. Then at the end of the same 
 time, the body shall be brought by the compound motion to 
 c, the extremity of the diagonal of the parallelogram, whose 
 sides are bd, ba. Dividing bd into a number of equal parts, 
 BO, oo', do" , o"d, the times of reaching the points o, o', o" , d, 
 with the velocity of projection, are as the numbers 1, 2, 3, 4, 
 whilst the spaces on, o'n, o"n", do, described in the vertical, 
 are as squares of those numbers; i. e. as 1, 4, 9, 16. And 
 such, generally, is the relation between the lines on, o'n , 
 o"n", &c. and the lines bo, bo', bo", &c. or between b;w, bw/, 
 B»z", &;c. and mn, m'n', m'n , &c. the former being as the 
 squares of the latter ; wherefore, the curve described is the 
 parabola. Now, without adverting to the properties of this 
 curve, or embarrassing ourselves with the consideration of 
 the compound motion, all the problems respecting the time 
 of flight, the range, the height of ascent, &c. may be solved, 
 by considering the motions as subsisting separately, and de- 
 termining the question for each of these movements. 
 
 It is evident that the motion shall be in the plane of the 
 rectihnear movements, seeing that there is no reason why the 
 projectile should deviate from this plane, to one side rather 
 than the other. Wherefore, it will be sufficient to refer to 
 two axes in that plane, and for the sake of simplicity, they 
 may be drawn through b, the point of departure : one of 
 them vertical and the other horizontal. 
 
OF FORCES PRODUCING CURVILINEAR MOTION. 305 
 
 The uniformly accelerated velocity, being parallel to the 
 axis of t/, is not to be resolved : this velocity being propor- 
 tional to the time, may be denoted by gt. The velocity of 
 projection, when oblique to the horizon, is to be resolved 
 into tvro, of which one is vertical, and the other horizontal. 
 Let v' denote this velocity of projection; and e the elevation 
 of the line of direction, or the angle which it makes with the 
 horizon. Then, 2;'.cos.e, v'.sin.e, are the velocities of pro- 
 jection estimated in directions parallel to the axes of x and 
 1/ respectively. The motion parallel to the axis of x, is, 
 therefore, v'. cos. e : and the vertical movement of the pro- 
 jectile being the difference between the uniform motion of 
 projection upwards, and the uniformly accelerated motion 
 downwards, is v. sin. e—gt. 
 
 The altitude to which the projectile ascends, and the 
 time in which it returns to the axis of a:, depend solely on the 
 vertical movement. These are given by equations (4) and 
 (6), by making v — Q, using g with a negative sign, and 
 changing v' for v.sSn.e. In this way, the latter equation 
 gives for the height of ascent, 
 
 and the former, for the time, 
 
 _ v'. sin. e 
 ~ g 
 Twice this time is that of the ascent and subsequent descent, 
 or as it is commonly called, the time of flight. 
 
 To find the horizontal range, the horizontal velocity is to 
 be multiplied by the time of flight. Wherefore, putting r 
 for the horizontal range, its value is given by the equation 
 
 , St?', sin.e «j'^. sin. 2e 
 R z: r . cos.e X ■ = . 
 
 g g 
 
 Hence it appears, that when the velocity of projection is 
 
306 DYNAMICS. — SECT. III. 
 
 given, the amplitude or range varies as the sine of twice the 
 elevation: and that it is, therefore, a maximum, when the 
 elevation is 45°. Also, that for two elevations which are 
 complements, one of the other, the horizontal ranges are 
 equal. 
 
 The place of the projectile for any time /, is found by as- 
 certaining its coordinates for that time. The horizontal ve- 
 locity is uniform ; therefore, xzzt.v cos. e. But the ordinate 
 y, is the difference of the spaces described by tw'o vertical 
 motions. One of these motions, viz. t;'. sin.e, is uniform; 
 and, therefore, the space described by this motion is t.v. sin. e. 
 The other is the uniformly accelerated motion, whose value 
 for any time t, is gt, and the space described by this motion 
 
 is ^^. Accordingly, 
 
 ij-t.v.sm.e-^-. 
 
 If the problem is to find the velocity of projection, or the 
 elevation requisite for striking a given scope, it is evident 
 that the value of t must be the same in the equations 
 
 X :=: t.v\ COS. e. and y = t.v'. sin. e-——-. Wherefore, the va- 
 
 lues of t, as given by those equations, are to be put equal, 
 i. e. t is to be exterminated between them, whence 
 
 o 
 
 •^ 2y . cos.^e 
 
 This equation becomes somewhat more simple, by put- 
 ting for v'^, its value, given by the equation v^z:z2gh; in 
 which, h denotes the height from which a body should fall, 
 in order to acquire the velocity v, or, as it is called, the 
 height due to that velocity. Making this substitution, the 
 equation is 
 
 y ~ A'.tan.e -rz 5—. 
 
 4A.COS. e 
 
OF FORCES PRODUCING CURVILINEAR MOTION. 307 
 
 This is the equation of the curve described by the projectile: 
 from which it appears again, that the curve is a parabola, 
 whose axis is parallel to the axis of y. Also, that 4A.cos.V, 
 is its principal parameter ; and 4A the parameter of the dia- 
 meter passing through the point of projection. By this 
 equation, any one of the four quantities, x, y, e, h, is ascer- 
 tained when the remaining three are given. Thus, if the 
 scope is given by its coordinates x, y \ and if the angle of 
 elevation is also given, or e = e'; these values being substi- 
 tuted in the equation of the curve, we have 
 
 h - 
 
 or, because h zz -— . 
 
 4.C0S. V(.r'. tan.e—y')' 
 
 ,2 
 
 vzz 
 
 cos.e'-v/ (■^'. tan.e' — y) 
 
 If y — x'. tsiu.e, the value of v is infinite ; and when y' is 
 greater, the value of v is impossible : which is, indeed, evi- 
 dent, inasmuch as the curve must fall below its tangent. 
 
 If it is required to find the elevation, in order to reach a 
 given scope with a given velocity of projection, it will be con- 
 venient to express cos. e by means of tan. e. 
 
 Substituting in the equation of the curve for —, its 
 
 cos.'e 
 
 value, which is l-}-tan.^e. and solving for tan. e, there is 
 
 tan.e n: -, • 
 
 X 
 
 From which it appears, that the scope cannot be reached 
 with the given velocity of projection, if 4////'-^-^^ exceeds 4/i'\ 
 
 i. e. if h' is less than ■— ^^. But that when h' ex- 
 ceeds this value, there are two values of e, between whose 
 
 2/1 . 
 tangents the quantity — - is an arithmetical mean. 
 
 x2 
 
308 DYNAMICS, — SECT. III. 
 
 4. The case here adduced, when treated as above, affords 
 an exemphfication of the use of the principles stated in the 
 preceding Art. ; and as in this case, the law of the motion 
 in each of the axes was given, there could be no difficulty in 
 finding for any given time, the place of the body, its velocity, 
 or the direction of its motion. But in most cases, the direc- 
 tion of the force, with respect to the axes, is continually 
 changed; and in such cases, it is evident, that the reduction 
 of the forces and motions to the same axes must first be as- 
 sumed, in order that by the connexion between them, we may 
 proceed to the discovery of either from the other. 
 
 The relation between the force and velocity in each axis 
 is expressed as follows : 
 
 The velocities in the directions of the axes are -j-, -/» 
 
 at at 
 
 — . Wherefore, putting, as usual, x, y, z for the sum of 
 
 the forces in the axes of a:, y, and z respectively, we have 
 by equation (9) of preceding section. 
 
 d^x _ d^y__ 
 
 dt^ ~^' dt"^ ~ " dt 
 
 a-^' dF = Y. -l^-"^' (1) 
 
 When the motion is given, those into which it is resolved, 
 and, therefore, the first members of these equations, are 
 given ; whence the resolved forces are known, and, there- 
 fore, their resultant, which is the square root of the sum of 
 the squares of its rectangular components. And conversely, 
 when the force is given, those into which it is resolved, i. e. 
 the second members of those equations are given ; and, 
 therefore, the first members, which by integration give the 
 resolved movements ; and these, being compounded, give 
 the velocity in absolute space. 
 
 To find the path of the motion, equations (1) are to un- 
 dergo a second integration, and at each integration there 
 are introduced three constant quantities : those introduced 
 
OF FORCES PRODUCING CURVILINEAR MOTION. 309 
 
 by the first integration, are the resolved velocities corres- 
 ponding to certain coordinates, and expressed as at the be- 
 ginning of this article ; and the constants, introduced by the 
 second integration, are those coordinates themselves. The 
 times being eliminated from the three integrals, and the 
 forces being functions of the coordinates sc, 2/, and z, the two 
 resulting equations express the relations among those coor- 
 dinates, and are therefore the equations of the curve. The 
 three differential equations are requisite, when the path of 
 the movement is a curve of double curvature ; but when the 
 motion is limited to a plane, that plane may be taken for the 
 plane of the axes of x and //, and then 2 zizO, z — 0, and the 
 differential equations relative to the other two axes being 
 integrated, and t eliminated, there results a single equation 
 between the coordinates a: and ?/, which is that of the plane 
 figure. 
 
 3. To proceed according to this method, let the equa- 
 tions (1) be multiplied, the first by d.Vy the second by diy, 
 and the third by dss, we have 
 
 dx.d'^x , dyuPii , dz.cPz 
 
 Adding these equations, there is 
 
 -^ ., -^ ^ ' = x.dx 4- Y.df/ + z.dz. {2) 
 
 1 he first member of this equation is ^-j-^ = 
 
 d(ds^) ds.dh „„ p 
 
 ds.u^s 
 
 — 'y^- — xdx -\- Ydy 4- "idz. (3) 
 
 From equation (3) some important conclusions are de- 
 rived. For putting it in this form, 
 
 d's dx . f/?/ , dz .... 
 
310 DYNAMICS. — SECT. III. 
 
 As the coefficients -r-, -/, —r, are the cosines of the angles 
 ds' (Is ds ^ 
 
 at which ds, the element of the curve, is inclined to the 
 
 axes of X, y, and z, respectively, it follows that the second 
 
 member of this equation expresses the sum of the forces, 
 
 reduced each to the direction of the tangent. Accordingly, 
 
 d's 
 putting s for the tangential force, the equation is — ^ "=. 
 
 dx , dii , dz ^_„ . , , 
 
 X.- — f- Y. -f^ 4- z. -7- =:s. Whence it appears, that the tan- 
 ds as ds fi ' 
 
 gential force is expressed by the second differential coeffi- 
 cient of the arc with respect to the time ; an expression 
 strictly analogous to that of a force producing right lined 
 motion. 
 
 dids'^) 
 Equation (3) put in this form — Vr^ — 2.(xc?a; + \dy ■\- 
 
 zdz.) gives by integration, 
 
 t'^-2^ (yidx + Ydy + '^dz) + c. (5) 
 
 The value of the quantity within the sign of integration is 
 s.ds. and this value being substituted, the equation is 
 
 'v^-2^s.ds-^ c. 
 
 From which it follows, that if the resultant of the forces is 
 resolved into two, viz. in the direction of the tangent, and 
 in that of the normal to the curve, the velocity shall be 
 affected solely by the tangential force : and as this force can 
 have no effect in changing the direction of the motion, it 
 likewise follows, that the change of direction is effected, ex- 
 clusively by the normal force. 
 
 The last equation is particularly useful when the tan- 
 gential force is given as a function of the arc, of which a 
 remarkable example will occur when constrained motion is 
 treated of. 
 
 When the quantity xdx + ^'dy + zdz is an exact differ- 
 
OF FORCES TRODUCING CURVILINEAR MOTION. oil 
 
 ential of a function, (p{x, ij, 2;), of the three coordinates, 
 equation (5) becomes 
 
 v'-2(j>{x, y, ^) + c; 
 And if y denote the velocity at any point of the trajectory 
 whose coordinates are x', ?/, is', 
 
 Wherefore, 
 
 In this case, then, the difference between the squares of 
 the velocities in any two points of the path, is independent 
 both of the form of the curve between those points, and of 
 the time in which it is described ; or in other words, the 
 velocity being given for any point of the path, it is also 
 given for any other point ; and this, independently of the 
 figure contained between those points, and of the time taken 
 to describe it. 
 
 This will be the case when the acting force is directed 
 to a fixed centre, and is at the same time a function of the 
 distance from that centre ; or when it is the resultant of two 
 or more other forces of this nature : and then the expression 
 x.ds-{-Y.di/-\-z.ch. is reducible to another, wherein the va- 
 riables are separated. 
 
 If p denote the distance of the moving point from one of 
 
 these fixed centres, and a, j3, 7, the angles contained by 
 
 (fp 
 that line with the three axes, -4- will be the cosine of the 
 
 as 
 
 the angle contained by the same line with the element of the 
 
 curve ; and ^-, ~, -^, being the cosines of the angles made 
 ' ds' ds ds' *= ^ 
 
 by the element of the curve with the three axes, there is 
 
 dp dx , ^dii , d^ 
 
 -=cos..^+cos.|3^+cos.T,^. 
 
 Let both members of this equation be multiplied by vds, p 
 
312 DYNAMICS. — SECT. III. 
 
 denoting the intensity of the force directed to the fixed 
 centre, and it becomes 
 
 There will be a similar equation for each of the other forces, 
 so that we have by addition 
 
 li{pdj))zzdxs(pcos.a)-{-di/'^{FCOs.i^)-\-dzs(FCOS.'y). 
 
 the symbol s denoting the sum of all the products similar 
 to those to which it is prefixed. But 
 
 2(pcos.a) = x. s(Pcos.]3)=Y. s(pcos.7)=:z,» 
 
 so that 
 
 xdx+Ydy + zd%— s {vdp). 
 
 It appears then that when there are several forces di- 
 rected to fixed centres, which are moreover functions of the 
 distances from those centres, the velocity being given for 
 any one point of the path, is given for any other point ; and 
 this independently of the form of the curve described be- 
 tween those points, and of the time taken to describe it. 
 
 If there is but one such force acting on the body, the 
 proposition is true, not only for the transit between two 
 given points, but also between two given distances from the 
 centre of force. For in this case, 
 
 v'-=z2\Y.dp+c—2(l,{p) + c, 
 
 From this equation it appears, that if two bodies, influ- 
 enced by the same accelerating force directed to a fixed 
 point, have equal velocities at any equal distances from that 
 point, their velocities, at any other equal distances, shall be 
 equal, i. e. several spherical surfaces being described round 
 the centre of force, if the bodies have equal velocities at any 
 one of those surfaces, they shall arrive at any other of the 
 surfaces with equal velocities. 
 
 The geometrical proof of this proposition is exceedingly 
 simple. For let ds, ds' be the elements of the two curves 
 contained between the spheric surfaces, whose radii are r. 
 
OF FORCES PRODUCING CURVILINEAR MOTION. 313 
 
 and r + dr. The tangential forces are F-4-. andp-/-.. i.e. in- 
 ° as as 
 
 versely as the elements ds, ds. But the changes made in 
 
 the squares of the velocities by describing the spaces ds, ds' 
 
 are expressed by the double products of those forces and 
 
 the arcs, or by 2vdp, and are the same for both curves. 
 
 This is the 40th Prop, of the first Book of the Principia 
 
 Math, of Newton. 
 
 rpX n^ll Cp!S 
 
 4. Having the equations -^- = x.t/^ —;-—Y.dt. ——=z.dt, 
 ^ ^ dt dt ^ dt 
 
 if the first is multiplied by y, and the second by x, we have 
 
 by subtraction, 
 
 In the same manner we obtain two other equations 
 z.^x—x.d^s; 
 
 dt 
 y.dP^z — z.d^y 
 
 — {ssx~xz)dt. 
 
 dt -iy^-^^)dt' 
 
 The first members of these equations are the differentials 
 of the quantities 
 
 x.dy — y.dx z.dx — x.dss y.dz — %.dy 
 
 Jt * di ' dt * 
 
 Wherefore, 
 
 x.di/ — ^^.dx ^ , 
 
 z.dx — x.dz , , C , s J 
 
 ■zz c -\- \ [zx — xz)dt. 
 
 dt 
 ~di 
 
 '^- _— ^n c"+ \ {yz - zY)dt. 
 
 dt/ doc 
 
 -j~ and — being the velocities of the moving point, estimated 
 
 in the directions of the axes w and x, a: -/ — 7/ — - is the mO" 
 
 •^ dt ^ dt 
 
314 DYNAMICS. — SECT. III. 
 
 ment, with respect to the axis of z, of the quantity of motion 
 acquired at the end of the time t. The first members of the 
 other two equations, in hke manner, are the similar mo- 
 ments with respect to the axes of y and x. On the other 
 hand, since X(//, Ydt, zclt, are the quantities of motion im- 
 pressed by the acting forces during the time clt, it is obvious 
 that the quantities under the sign of integration are the mo- 
 ments of these elementary motions taken with respect to 
 the same axes ; and the preceding equations show in what 
 manner the moments of the velocity acquired at the end of 
 any time, are made up of the moments of the velocities 
 engendered during each instant by the action of the moving 
 forces. 
 
 But the first members of the preceding equations bear 
 another signification. We may suppose the elementary arc, 
 ds projected on the planes of the axes: and the reasoning 
 concerning all these projections being the same, it will be 
 sufficient to consider the projection on one of those planes, 
 which may be that of xy. Accordingly, let o be the origin, 
 (Fig. 141.) OA the axis of x, and ob the axis of //, and 7nn 
 the projection of the elementary arc on the plane of these 
 axes. Then, ma, me, nh, nf, being perpendiculars on those 
 axes, and the former being produced to meet the latter pair 
 at c and h, we shall have oa = x, oe r= y, mc z= dx, mh — dy. 
 Wherefore, xdy — ydx, is the diiference of the rectangular 
 parallelograms fm, mh. But the point o, lying within the 
 angle ema, contained by the sides of the parallelogram lie, 
 the difference of the rectangles mf, mh is eqvial to the pro- 
 duct of mn, the diagonal of the parallelogram he, into its 
 perpendicular distance from o, * Statics,' Sect. 4. Art. 2. 
 or to twice the area of the ti'iangle mon. But this triangle 
 is the projection of that whose base is ds, and whose vertical 
 angle is at o ; wherefore, xdy — ydx expresses twice the 
 projection of that area on the plane of xy. The same rea- 
 sonings apply to the other equations; and we learn that the 
 
OF FORCES PRODUCING CURVILINEAR MOTION. 315 
 
 numerators in the first members of the preceding equa- 
 tions are double of the projections, on the three coordinate 
 planes, of the elementary area described by the moving 
 point in the time dt. 
 
 Or thus : putting r for the distance of the origin o, from 
 the projection of the material point on the plane of x, y\ 
 and w for the angle made by the line r, with the axis of a:, 
 we have :r =: r.cos.w. ?/z=r.sin.w. whence, 
 
 dx zz — r.sin.(t).c?ct>. dy — r.cos.w.c/w. 
 
 and therefore, 
 
 x.dy — y.dx =. r^.du). 
 But r.d(o is the circular arc, described with the centre o 
 and radius r ; wherefore, r-.doj is twice the projection of the 
 triangular area whose base is ds, and whose vertex is at the 
 origin o. 
 
 When the forces acting on the point are reduced to a 
 single force directed to the origin, the moments of this 
 force, taken with respect to that point, vanish, and we have 
 
 XY — yxzz 0, zx — xzzz 0, yz — zy — 0. 
 
 Consequently, the equations above obtained are reduced to 
 
 xdy — ydx z=: cdt, 
 
 sdx — xdz zi cdt, 
 
 yd% — %dy zz c'dt. 
 
 The projections of the elementary areas described during 
 the time dt are therefore constant, and the areas described 
 in any finite time are proportional to that time. Recipro- 
 cally, if the areas are proportional to the times, or the latter 
 equations fulfilled, the former must hold good also, and the 
 direction of the force must pass through the origin. 
 
 Let the equations last obtained be multiplied, each by 
 the variable which it does not include, and then added ; 
 there is 
 
 cz + c'y + c"x — ^' 
 
316 DYNAMICS. — SECT. Ill, 
 
 which is the equation of a plane passing through the origin. 
 This result, therefore, shows that a body urged by a force 
 tending to a fixed point, shall move in a plane passing through 
 that point. 
 
 5. It may be satisfactory to the student to see these 
 things proved geometrically ; wherefore, let c be the centre 
 of force, (Fig. 14^) ; o the material point, whether moving in 
 the plane oi xy, or projected on that plane; and let om be 
 the line it would describe in the time dt, were it not in- 
 fluenced by a force. On this supposition, the triangle ocm 
 would be described by the radius vector in the time dt. But 
 if at o it is urged by a force directed to c, which would carry 
 it through the space on, in the time dt, the line described by 
 the compound motion in the time dt shall be oo, the diagonal 
 of the parallelogram, whose sides are otn, on ; and the area 
 described by the radius vector shall be the triangle oco'. 
 But mo' being parallel to oc, the triangles oco', ocm, standing 
 on the same base oc, are equal; If the material point were 
 to receive no other impression, it would proceed in the line 
 go' produced, describing in the time dt the space o'm' equal 
 to oo'; but if at o' it receives an impression, which in that 
 time would carry it through the space oV, the line de- 
 scribed in the time dt shall be o'o", the diagonal of the pa- 
 rallelogram whose sides are oW, oV; and the area de- 
 scribed by the radius vector shall be o'co". But the trian- 
 gles o'co", o'cm, standing on the same base, and between the 
 same parallels, are equal : and as o'cm'-zz oco', it follows, that 
 oco — oco . 
 
 The triangular areas described round the centre of force 
 in the equal moments dt, being thus proved to be equal, it 
 follows, that the areas described in any portions of time 
 whatsoever, are proportional to the times. Conversely, if 
 the areas described in the time dt are equal, the area 
 o'co"zz area oco'— area o'cm. Wherefore, the triangles o'co" 
 and o'cm' standing on the same base, being equal, they must 
 
OF FORCES PRODUCING CURVILINEAR MOTION. 317 
 
 be within the same parallels ; i. e. m'o" must be parallel to 
 o'c, and, therefore, the Une oV, in which the force is directed, 
 must pass through the point c. The equable description of 
 areas round any point is therefore the indication of a deflect- 
 ing force, whose line of direction constantly passes through 
 that point. 
 
 Whilst the force is directed to the same point c, the 
 figure of the trajectory must be a plane ; for the line od\ 
 (Fig. 142.) is in the plane of o'm' , o'n' ; i. e. in the plane of 
 oo' y o'n', or of oco'; and the same thing is shown, in the same 
 way, of the remaining sides of the polygon. Moreover, in 
 this case, the areas described in equal times, dt, being equal, 
 the velocities are inversely as the perpendiculars from the 
 centre of force on the sides of the polygon. Wherefore, 
 when the polygon terminates in a curve, the velocities at dif- 
 ferent points are inversely as the perpendiculars from the 
 centre of force on the tangents at those points. 
 
318 DYNAMICS. — SECT. IV. 
 
 SECTION IV. 
 
 OF CENTRAL FORCES, 
 
 1. When a material point is urged by a single force 
 tending to a fixed centre, which force is represented by 
 some function of the distance from that centre, the general 
 formulae, given in the preceding section, are susceptible of 
 simplifications ; which, on account of the importance of the 
 case, merit a distinct consideration. 
 
 It has been already shewn (Sect. 3. Arts. 4. and 5.) that, 
 in the case under consideration, the point will move in a 
 plane passing through the centre of force, as indeed is appa- 
 rent ; for every thing being similar on both sides of the 
 plane drawn through the direction of the initial velocity and 
 the fixed centre of force, there is no reason why the move- 
 able should deviate from that plane to one side rather than 
 the other; that plane will, therefore, contain the path of the 
 moving point. If then we take this plane for the plane of 
 xy, we may dispense with the third of our fundamental 
 equations, (1.) of last Sect, and the motion of the point will 
 be given by the two equations, 
 
 f- (•) 
 
 2. As the force by which the moving point is urged 
 varies as a function of the distance of that point from a fixed 
 centre, it is natural and convenient, that its position be re- 
 ferred to that centre by means of polar coordinates. For 
 this purpose, the equations above may be transformed as 
 
OF CENTRAL FORCES. 319 
 
 follows : if R denote the central force, and w the angle made 
 by its direction with the axis of x, we shall have the equa- 
 tions 
 
 X = — R.cos.w. Y = — R.sin.w. 
 
 in which the force r has an opposite sign to that of x and r, 
 inasmuch as its direction is toivards the origin. If now we 
 multiply the former of these equations by sin.w, and the 
 latter by cos.w, and subtract, we have 
 
 Y.cos.w — x.sin.w = 0. 
 
 Again, multiplying the former by cos.w. and the latter by 
 sin.w, and adding, we find 
 
 Y.sin.w 4- x.cos,(t> + R = 0. 
 
 It now remains to substitute for x and y their values ex- 
 pressed by polar coordinates ; these may be obtained as fol- 
 lows : if r denote the distance of the moving point from the 
 origin, or radius vector, we have 
 
 X = r.cos.w, y — r.sin.w ; 
 
 differentiating these equations twice, all the quantities being 
 considered as variable, we find 
 
 d'^x — f/V.cos.w — 2t/r.f/a>.sin.(u — rc^w^.cos.w — rdPio.sm.u), 
 d^y=. cPr.sin.w -f 2dr.do).cos.ii) — rdby^.sm.cj -f- rd^oj.cos.w. 
 
 Therefore, dividing by dt-, we have, by equations (1) and 
 
 (2), 
 
 /d^r db)S ( ^dr dw , tf^wN . 
 
 " = Kif - '■wn-'- - \^Tf7t + •••■s-J""-"- 
 
 /cZV dh)\ . , /^dr dh) , d-w\ 
 
 Y = —rrr — r.-^T] sm.w + ( ^-77.-7- + r.-7,-)cos.w. 
 V dt^ dt'J \ dt dt dt' J 
 
 Substituting now these values of x and y in the equations 
 above, we obtain the equations 
 
 2^.^4-r.^r = 0. (3) 
 
 dt dt dt^ ^ ' 
 
 dh- d(x)^ , n tA\ 
 
S20 DYNAMICS. — SECT. IV. 
 
 We have thus transformed equations (1) and (2) mto two 
 polar equations, which we shall use hereafter in determining 
 the motion of the point. 
 
 3. If we multiply equation (3) by r.df, it becomes 
 
 2rdr. — + r^. —r-- d. [ r^. i- ) = 0. 
 dt dt \ dt I 
 
 the integral of which is 
 
 ..4-=. (5) 
 
 Now, rVa> is twice the differential of the area described 
 round the centre of force ; it follows, therefore, from this 
 equation, that the increment of the area described in a given 
 time by the moving point round the centre of force is con- 
 stant ; or, that the areas described are proportional to the 
 times of their description, as has been already shown in the 
 preceding section. 
 
 Equation (5) may be thus written, 
 disi _ c 
 
 from which we conclude that the angular velocity of the 
 moving point varies inversely as the square of the distance 
 
 from the centre of force ; -j- expressing that angular ve- 
 locity. 
 
 4. The integral of equation (4) may also be obtained 
 with the aid of equation (3). For if equation (4) be multi- 
 plied by dr, and equation (3) by rdw, and the results added 
 together, we shall have 
 
 dr.—+ rdr. -^ + r^du.. -^ + ^dr = 0. 
 
 the integral of which will be 
 
 h being the arbitrary constant. 
 
OF ceKtral forces. 521 
 
 From this equation we immediately obtain the velocity of 
 the moveable at any distance from the centre of force : for, 
 observing that dr'^ + r'^dio'^ = ds^, equation (6) may be written 
 
 ds^ 
 But ^ = «)^, V representing the velocity, therefore, 
 
 v^=k- 2^Rdr. (7) 
 
 Comparing this with the expression for the velocity of a 
 point moving in a right line to or from the centre of force, 
 (Sect. II. x\rt. 6.) we see that they shall be equal, if the 
 constants are equal ; it follows, therefore, that if the velo- 
 city of the point moving in a curve by the action of a cen- 
 tral force, be equal to the velocity of the point movino- di- 
 rectly to, or from the centre, at any equal distances, they 
 shall be equal at all equal distances. We need not there- 
 fore, in what follows, dwell on the applications of this for- 
 mula for the velocity to particular laws of force, as this has 
 been sufficiently done in a preceding section, where we 
 treated of the motion of a material point in a right line pass- 
 ing through the centre of force. 
 
 As the force r is by hypothesis a function of the distance 
 
 from the centre, the integral \Rf/r is likewise a function of 
 
 the same quantity: denoting this by 0r, equation (7) may be 
 written 
 
 v^ = k—2<p{r): 
 
 The constant, k, shall be determined, if the velocity at any 
 given distance be known. For, let v be the velocity at the 
 given distance /, then, 
 
 which gives the value of k. Subtracting this from the pre- 
 ceding equation, we have 
 
 Y 
 
322 DYNAMICS. — SECT. IV. 
 
 v'^-v''- 2 
 
 (/)-0(r)^ (8) 
 
 It appears from this equation that the velocity depends solely 
 on the distance of the moving point from the centre of force, 
 and not at all on the nature of the path described. 
 
 5. The equations (5) and (6) are the integrals of the first 
 order of our fundamental equations (3) and (4), which they 
 should therefore replace ; and are evidently sufficient to de- 
 termine the coordinates of the moving point at each moment 
 of time, the law of the force being known. For if the co- 
 ordinates r and to be eliminated successively from the equa- 
 tions 
 
 the resulting equations, containing each only one of the co- 
 ordinates, and the time t, will, when integrated, express the 
 relation between the time and each of those coordinates ; 
 whereby the position of the moving point will be ascertained 
 for any given time. 
 
 Again, if the differential of the time, dt, be eliminated 
 from those equations, the resulting equation, containing only 
 the coordinates of the point, will give by integration the re- 
 lation between r and w, or the equation of the curve de- 
 scribed by the moveable. 
 
 To proceed to these applications : if we substitute in the 
 
 latter of those equations for -^ its value -^, obtained from 
 
 dt r 
 
 the former, it becomes 
 
 dr^ "2 
 
 ^f+^ + ^^^clr = L (9) 
 
 The integral of which will give us the relation between the 
 distance r and the time t. 
 
 In order to find the relation between w and t, instead of 
 eliminating r between the two differential equations above, 
 and then integrating, we shall find it more convenient to eli- 
 
OF CENTRAL FORCES. S22 
 
 minate that quantity between the integral of equation (9) and 
 the equation of the orbit. 
 
 The differential equation of the orbit is found by substi- 
 tuting the value of dt obtained from equation (5) in (6), 
 which thus becomes 
 
 C^+^)+^S-*="- 
 
 which equation will assume a more convenient form, if we 
 take for the variable ?/ = - ; it becomes by this transforma- 
 tion, 
 
 f du'^ , „\ ^C du 
 
 c 
 
 a^+"^^^v-''- (>«) 
 
 From this equation we are enabled, when the law of the 
 force is known, or r given in function of u, to obtain by inte- 
 gration the relation between u and w, or the equation of the 
 orbit : and vice versa, when the equation of the curve de- 
 scribed is given, we may obtain by differentiation the ex- 
 pression for R, or the law of the variation of the force. The 
 latter of these problems is called by English mathematicians 
 the direct, and the former the itiverse problem of central 
 forces. We shall commence with its application to the 
 former of these questions, as more naturally connected with 
 our subject, reserving the latter for a subsequent part of this 
 section. 
 
 6. The force being supposed to vary directly as the dis- 
 tance from the centre, let it be required to investigate the 
 equation of the orbit. 
 
 In this case r = mr= — , m being the force at the unit of 
 
 distance. Then, \ r.-^- = \ —~ = -—^, which being sub- 
 
 stituted in equation (10), we have 
 
 v2 
 
324 DYNAMICS. — SECT. IV. 
 
 (S + »^) 
 
 I *^ _ 7 
 
 the differential equation of the trajectory. 
 
 Before we proceed to integrate this equation, we shall 
 detei'mine the arbitrary constants by means of the maximvim 
 and minimum values of the variable u. These values are 
 
 found by making -7- = in the differential equation, which 
 
 'iTt 
 
 thus becomes ch^-\ — ^ — ^ —0- or, 
 u 
 
 C^ c 
 
 and denoting the roots of this equation by w', m", we shall 
 have 
 
 '2 I "2 '2 //2 
 
 -^■=.11 '*+ U . — 7 = M M . 
 
 Substituting now these values of the constants in the diffe- 
 rential equation, and multiplying by u^, we find 
 
 !f!lf^' + m4_^(^'2^ ^,/2^) ^^2^ ^/2^^,/2 _ Q^ 
 
 atu 
 Now, in order to integrate this equation, let us make 
 
 % being the new variable. And it becomes by this transfor- 
 mation 
 
 (ii)^-r-^TH--o- 
 
 or. 
 
 dwzz. 
 
 the integral of which is 
 
OF CENTRAL FORCES. S25 
 
 .> = -^arc.(cos.= ^^^)+a. 
 
 a being the arbitrary constant. 
 
 Now, reversing the formula, and observing that 
 COS. 2 (a — (o) = COS. 2 (w — a), we obtain 
 
 « = -1-(m'2_?/'2). cos. 2 (w-a). 
 
 whence 
 
 u'zz i (m'2+ O-i {u'^-u"^). COS. 2 (w-a). (6) 
 
 the complete integral. 
 
 This equation will assume a very elegant form, if we sub- 
 stitute for COS. 2 (w — a) its value cos.^ (tu — a) — sin.^(w — a) ; 
 we shall have by this substitution, 
 
 u^— u"^. sin.^ (ti»— a) + m"^. cos.^ (w— a). (c) 
 
 When (i) — arrO, i^:=:ii"^\ and when w — a=-, u^z=.u"^. 
 
 At 
 
 From which it appears, that u and ?*", the maximum and 
 minimum values of ^/, are those which correspond to the 
 
 TT 
 
 angles w :=--+- «> and (o=a, respectively. 
 
 Substituting for m its value -, we shall have 
 
 —7;-=. -To. sin,^(w — a) H — ^. cos.''^(w— «). 
 r^ a' 
 
 a and 6 being the gi'eatest and least values of r ; whence. 
 
 
 id) 
 
 a^. sin.^ (w — a) -j- b'^. cos."^ (w — a)' 
 
 The equation of a central ellipse, whose semiaxes are a 
 and 6, 
 
 The semiaxes a and 6, may be easily expressed in terms 
 of the constants, m^ k, and c. For, solving equation {a), 
 we shall have 
 
S26 DYNAMICS. SECT. IV. 
 
 ^,2 _ j/c+Vik'-mc' „2_ jk-Vik^-mc' 
 
 the reciprocals of which are the values of P and «^. 
 
 7. Let us now proceed to apply equation (9) to obtain 
 the relation between the coordinates of the curve and the 
 time. 
 
 R r= mr, therefore, \Rdr =■ m.—. 
 
 which being substituted in equation (9), it becomes 
 
 at r 
 or multiplying by — , 
 
 1 r^.dr^ . k ^ c^ f. 
 
 —'-^^T+r -^ + — = 0. 
 
 m at m in 
 
 The constant quantities which enter this equation have been 
 already determined in function of ti and u", the greatest 
 and least values of u ; for in the preceding article we have 
 had 
 
 
 whence, 
 
 c^ 1 
 
 m u u 
 
 and these values being substituted in the differential equa- 
 tion, it is 
 
 m dt^ 
 To integrate this equation, let us make 
 
OF CENTRAL FORCES. 327 
 
 and it is transformed into 
 
 4m' df'^'^ \ 2 J 
 
 and separating the variables 
 
 1 (h 
 dt=-— — . 
 
 whence, 
 
 t = rr-j- . arc( COS. zz —-2 — To-J + const. 
 
 Let ^=:^, when ^=|(a^— 6^), or r^ = a^; and there is, 
 t' zz const. Wherefore, observing that the cosine remains 
 the same whether the arc is positive or negative, the cor- 
 rected integral is 
 
 t—t'= ——— . arc ( cos. = ^7-9 wS- 
 
 {e) 
 
 % being equal to r^— K^^-f-^^J- 
 Reversing this formula, we have 
 
 whence, 
 
 i^=\{a^ArlP)^\{a^-})')Q,o^.2^J m{t-f). 
 or, substituting for q,o^.2^I m{t — i'^ its value cos.^v/;w(^— 
 
 r'' = a\cos.^Vm{t-t')+b\sm.''Vm(t-t') (/) 
 
 The relation between the angle w — a and the time is 
 readily found by substituting this value of r^ in the equation 
 of the orbit, 
 
 o'^.sin,^(a; — a) + ^''.cos."'^(w — a) 
 
 Multiplying the result by the denominator, and reducing we 
 find 
 
328 DYNAMICS. — SECT. IV. 
 
 whence, 
 
 tan.((o— a)= -.tan. -v/w?(^— 0- ig) 
 
 a remarkable expression for the relation between the angle 
 at the centre and the time. 
 
 It appears from this equation, that the tangents of the 
 angles w — a and \^m{t—f) vanish together, or that the an- 
 gles simultaneously arrive at those values which render the 
 tangent equal to cipher; therefore, when w— a = 27r, or the 
 radius vector performed an entire revolution, \/m(t—t^ 
 — Stt; if, therefore, the time of an entire revolution be 
 called T, we have 
 
 This result, being altogether independent of « and b, the 
 elements of the orbit, shews that all ellipses described 
 round the same centre of force, the force varying as the 
 distance, are described in the same time. 
 
 8. The force being supposed to vary inversely as the 
 square of the distance, it is proposed to investigate the 
 equation of the trajectory. 
 
 Here r= — irww^. and \ r. — ^ = \ mdu — mu. Making 
 this substitution in equation (10), we have 
 
 the differential equation of the curve. 
 
 Proceeding as in Art. 6. let u' and u" be the roots of the 
 quadratic equation, 
 
 2m k 
 
 xr 
 
 c c^ 
 
 zrO. (£) 
 
 which is obtained by making -r- =0. in the differential equa- 
 tion ; we have 
 
OF CENTRAL FORCES. 329 
 
 
 zzu'+u", —^-tiu'. 
 
 c c 
 
 and these values of the constants being substituted in the 
 differential equation, it becomes 
 
 clu) 
 In order to integrate this equation, let us make 
 
 and it becomes by this transformation 
 
 
 dz 
 
 and integrating, 
 
 (i) zz — arc( cos.= -^j-, 77- J -\-a.. 
 
 \ i{u'—u)J 
 
 or reversing the formula, 
 
 %zz^{u' — u"). cos.(aj — a). 
 
 Therefore, 
 
 tt =|(m'+m") + J(m' — e/').cos.(w - a). {k) 
 
 the equation of the curve. 
 
 This equation may be put under a form similar to that of 
 equation (c). Art. 6., and by precisely the same transforma- 
 tion. Substitute for cos. (w — a) its value cos^ — 
 
 sin^ — -TT—. and we shall have 
 2 
 
 ti = u. cos -—: h w .sm^ — :— . (I) 
 
 2 2 ^ /^ 
 
 When 10 — a:z:0, u:=u' \ and when a> — a = tt, u-=.ii'. It 
 appears, therefore, that u and ii\ the maximum and mini- 
 mum values of «, are those which correspond to the angles 
 
330 DYNAMICS. — SECT. IV. 
 
 (I) = a, and w = tt + «• They are, therefore, in the same 
 right hne. 
 
 If we substitute for u, u, and u", their values-, — r, and 
 
 r r 
 
 —r, in equation (Z:), it becomes 
 
 or, 
 
 / It 
 rr 
 r = 
 
 ^{r" + r')-{-\{r"—r'). cos. (w-a) 
 
 And making ^{r" -{-r') = «, !(/•''—/) z: «e, there shall be 
 r'r"z=. a^— a^e^ ; and the equation becomes 
 
 «(l-0 , , 
 
 1 + e.cos. (w — a) 
 
 the known equation of a conic section referred to the focus 
 as the origin. — {See Analytic Geometry, Kxt. 51.) 
 
 9. For the determination of the constants a and e, we 
 have 
 
 «=i(/+0=:i^^^^, 
 
 and 
 
 ae=i (r' — r)=: ^ , ^, \ whence e = — j, 
 
 uu" u +u 
 
 But we have had above, the equations 
 
 from which we deduce |^(m'— m")=^1 -J ; substituting 
 
 these values in the expressions for a and e, we find 
 
 m / 2 
 
 X' ^=Vl+A-.l^. (;,) 
 
OF CENTRAL FORCES. 331 
 
 It is a problem, however, of more importance to deter- 
 mine the elements of the orbit a and e by means of the 
 initial velocity, distance, and angle of projection; or, in 
 other words, a body being projected from a given point, in a 
 given direction, and with a given velocity, to determine the 
 particular conic section, which it will describe. This is done 
 by finding the values of c and Tc in terms of the velocity, dis- 
 tance, and angle of projection, and substituting them in the 
 equations («). 
 
 If be the angle formed by the tangent with the radius 
 
 vector, sin.0 = —7—, ds being the element of the curve. 
 as 
 
 Multiplying by r, r. sin. = — — =z —7-, by equation (5). 
 
 (XS do 
 
 .*. c = -j-.r.sin.0 •=. z?.r.sin.0. 
 at 
 
 Again, by equation (7), v^zz k—2 \Rc?r rz^-f- — — , when 
 
 the force varies inversely as the square of the distance, or 
 
 R = -2". Therefore, if /o be the distance of the point from 
 
 the centre at the commencement of the motion, v the velo- 
 city, and £ the angle of projection, we shall have 
 
 cnv.p.sm.E, k—\^ . (0) 
 
 P 
 
 and substituting these values in equation (w), 
 
 a = 
 
 m „ . v>^sin.2£(v2-— j 
 
 ^^=1+ ^. e_i. (p) 
 
 ^!^__v^ ??j 
 
 P 
 
 The value of a thus found, is independent of the angle of 
 projection ; from which it appears, that in whatever direc- 
 tion the body is projected, the axis major of the section will 
 be the same, if the velocity remains unaltered. 
 
332 
 
 DYNAMICS. — SECT. IV. 
 
 The curve described shall be an ellipse, hyperbola, or 
 parabola, as the excentricity e is less, greater than, or equal 
 to unity : that is, according as 
 
 v^jo^sin.^tf v^ ) 
 
 l-e- = -V 
 
 is positive, negative, or cipher. But the sign of l—e^ is 
 
 2m 
 evidently the same as that of v^. Therefore, the curve 
 
 shall be an ellipse, hyperbola, or parabola, according as v^ 
 
 is less, greater than, or equal to . 
 
 P 
 10, Let us now proceed to investigate the time of de- 
 scribing any portion of the orbit. 
 
 Since R = -2-j \Rdr= \ m.-j— . If then this value 
 
 is substituted in the equation, 
 
 and the result multiplied by r^, there is 
 
 df 
 
 — Jcr^ — 2mr-\- c^=0. 
 
 The values of the constants k and c^ have been already ob- 
 tained in function of the elements of the orbit: for from 
 equations (w). Art. 9, we have 
 
 a 
 If now we substitute these values in our differential equa- 
 tion, and multiply the result bv — , it becomes 
 ^ " m 
 
 m dr 
 
OF CENTRAL FORCES. 333 
 
 or, separating the variables, 
 
 a 
 
 V- 
 
 -.rdr 
 dt = 
 
 In order to integrate this expression, let us make a—r 
 = ae.z, whence we have 
 
 r — a{\ —ez), drzz — ae.dss. 
 
 By this substitution, the equation becomes 
 
 , a^/a {\~ez)dz 
 
 (it— — ~~. • , ~» 
 
 Vm Vl— s^ 
 
 the integral of which is 
 
 '&' 
 
 a\J a 
 
 t = —, — |arc.(cos.=^)— e. v^l — s^l +const. 
 
 y/ m * 
 
 If t' be the value of t, when «=1, ov rzz a{\— e), we 
 shall have ^'z= const. ; and the corrected integral is 
 
 t-t'-"^ |arc.(cos.=^)--eVrr^2-| , n 
 
 t' being the time elapsed when the body arrives at the lower 
 apsis. 
 
 t—t' being the time counted from the lower apsis, the 
 time of a semi-revolution will be had by making r =: « (1 +e), 
 
 the higher apsidal distance, or ^ = — 1 ; which gives — - — . -w 
 
 \/ m 
 
 for the time of half a revolution. If, therefore, the time of 
 an entire revolution be called t, we have 
 
 T=27r.-7— . (r) 
 
 From which it appears, that the periodic times in different 
 orbits vary in the sesquiplicate ratio of the major axes ; this 
 is the third law of the planetary movements discovered by 
 Kepler. 
 
 The relation between / and x or r, given by equation (5'), 
 
334 DYNAMICS. — SECT. IV. 
 
 is generally exhibited under the form of two equations, 
 which are virtually contained in that equation, and flow from 
 it by introducing a new variable, namely, the arc whose co- 
 sine z:zz: for if this be called (j), there shall be 
 
 , a^ a , . . 
 
 ^ =: cos.ri), t — t-zi — - — (0 — e. sm.ri)). 
 
 or making for conciseness ^ ^ — n, and observing that 
 
 ^J m 
 a^J a 
 r-zia{y—ez)y 
 
 r — a{l—e.cos.(p), (s) 
 
 n{f-^t') zz. <jt—e. &\n.<p. {t) 
 
 By the latter of these equations ^ may be expressed by 
 means of t — 1\ and this substituted in the former will give 
 the radius vector, r, in function of the time, or conversely. 
 
 It remains now to discover the relation between w and t. 
 For this purpose we must eliminate r between equation (s) 
 and the equation of the orbit : we thus have 
 
 1— e^ 
 
 l — e.cos.(b=. -. r-. 
 
 1+^. cos. (w — a) 
 
 whence we find 
 
 - . cos.rf) — e 
 
 cos. (w — a) = . 
 
 1— e. cos.0 
 
 from this we obtain the values of 1 — cos. (w — a) and 
 1 +C0S. (w— a), which, divided one by the other, give 
 
 1 — cos, (u) —a) _ 1 -{-g 1— cos.0 
 l+cos. ((u— a) ~" 1— e* l+cos.^' 
 
 or, 
 
 (i} — a. 
 tan. — — - 
 
 2 
 
 If, therefore, ^ be found in terms of t — t', by means of 
 equation (t), and the result substituted in {v), we shall have 
 
OF CENTRAL FORCES. S35 
 
 (o—a expressed by means of t—f, or the angle in function of 
 the time. 
 
 In order to represent the planetary movements, astrono- 
 mers have supposed a star, which sets out from perihelion at 
 the same time with the planet, and moves with a uniform an- 
 gular velocity round the sun, the angle n(t—t') formed by 
 the radius vector of this star with the axis major, at anytime 
 t, is called by them the mean anomahj ; the angle w — a formed 
 by the radius vector of the planet, the true anomaly ; and 
 the angle the excentric anomaly. 
 
 11. Let the expression for the force consist of two parts, 
 one of which varies inversely as the square, and the other 
 inversely as the cube of the distance ; and let it be required 
 to find the equation of the curve described by the body. 
 
 R rz mit^ -^ m'u^, 
 
 C du C/ , / N 7 . f u^ 
 
 .'. \ R. — 2 — \ i^m -\- m u) du := mu + m — . 
 
 making this substitution in equation (10), and arranging, we 
 have for the differential equation of the orbit 
 
 dti"^ 
 c^. -r-o + (c^—m') u^—^mu—kzzO, 
 
 du 
 If we make -7- =: in this equation, the roots of the re- 
 
 dh) 
 
 suiting equation will be the greatest and least values of w. 
 Therefore, u' and u" being those values, or the roots of the 
 equation, 
 
 2 2/» k 
 u'- 5 y.u 2 7 = 0. iw) 
 
 we shall have 
 
 9,m , . ., — k 
 
 '.." 
 
 — = ?«-f-M, -^ r'^uu 
 
 c — ni c — m 
 
 Therefore, dividing the differential equation by c^—m'i sub- 
 
3S6 
 
 DYNAMICS. — SECT. IV. 
 
 C — m 
 
 stituting these values in the result, and making - — — ^ 
 
 it becomes 
 
 1 du^ 
 
 y dh) 
 
 In order to integrate, let l{ti'-{-u") +;g be substituted for u, 
 and the equation is transformed into 
 
 or, separating the variables, 
 du)z=.- 
 
 whence, 
 
 1 ^ ^ > 
 
 (D rz .arc. < cos. =: -rr—. ;;- > -\-a. 
 
 y i i(m - M J S 
 
 or, reversing the formula, 
 
 IS — ^{u — u^).COS.y{(i) — a), 
 .*. u=:l{u' +u')-\-^{u' —u").cos.y((i) — a,). (x) 
 
 If we substitute in this equation for cos.7(w — a) its value 
 
 cos.^yf — - — j — sin.^yf— -— j, it will assume the form, 
 
 u-u\cos.'y[^^J+u''.s[n.'y\^'^J {y) 
 
 An equation which coincides with that of the focal ellipse, 
 except in this, that the variable (w — a) is multiplied by a 
 constant quantity. Hence this curve may be readily con- 
 structed; for, in the ellipse whose apsidal distances are 
 
 — r and —J}, if a line be drawn from the focus forming an 
 v! u ° 
 
 angle with the axis, which is to the angle formed by the ra- 
 dius vector of the ellipse with the same, in the ratio of 1 to 
 
OF CENTRAL FORCES. 337 
 
 7, and this line be taken equal to that radius vector, its ex- 
 tremity shall describe the curve in question. 
 
 When oj — cx^ — O, in equation (y), we shall have u — ii; 
 and when ^(w — a)=7r, there is u — u". Therefore, w and 
 w" being the values of w which correspond to m' and u" , the 
 greatest and least values of u, we shall have 
 
 , TT 
 
 7 
 Therefore, subtracting, the angle between the apsides, 
 
 ,> /_ ■^_ ^ 
 
 12. The force being represented by any function of the 
 distance ; it is required to find the equation of the orbit, 
 when it approaches very nearly to a circle. 
 
 The force r being a function of the distance from the 
 
 centre, or its reciprocal tt ; — j shall be a function of the 
 
 same quantity, which we shall denote by the character^ (m) ; 
 
 whence r = t/'./iu). Also, \ r. -j— \ f{u).du ; denoting 
 
 this integral by the character //«), and substituting in equa- 
 tion (10), we have 
 
 ^i§+>^) -m«)=^. 
 
 Now it is evident that if the body is projected in a direc- 
 tion perpendicular to the radius vector, and with a velocity 
 very nearly equal to that in a circle at the same distance, 
 the orbit will not differ much from a circle. But by equa- 
 tion (5) we have 
 
 c, d(i) ,, (Is dio . f. 
 
 c = r^. —r-^=z7-. -J-. —p- zzrv.smM. 
 at at as 
 
 Hence, a being the value of u, or the reciprocal of the dis- 
 
 z 
 
338 DYNAMICS. — SECT. IV. 
 
 tance r, at the place of projection, and v the velocity of 
 projection, we have 
 
 V 
 
 c=.-. 
 a 
 
 the angle of projection being right, and therefore sin.0=: 1. 
 Again, by equation (7), 
 
 But when a^=:a, vzzv ; wherefore, 
 
 and these values of c and k being substituted in the differ- 
 ential equation, it becomes 
 
 Now let Mzza-j-s:, a being the value of u at the point of 
 projection ; and z a very small but variable quantity ; then, 
 duzzdz, and by Taylor's theorem, 
 
 /X«+^)-/X«):=/ («). f 4-/(«). -^^ +/»• iJ^ + &^- 
 
 f being the first derived function ofy^. 
 
 Therefore, making these substitutions in the equation 
 
 above, and dividing the whole by -^, we find 
 
 Putting v' for the velocity in a circle at the point of pro- 
 jection, there is v'^zzrx radius (Equat. (/'), Art. 14.) 
 
 zza'^.fia).- = a.f{a). But by the conditions of the pro- 
 ct 
 
 blem, v^ : v'" : : 1 : 1 -f-S, S being a very small quantity ; there- 
 fore, 
 
OF CENTRAL FORCES. 339 
 
 substituting now this value in the equation just obtained, 
 neglecting the terms which contain z^, ^^8, &c. as indefi- 
 nitely small compared to the rest, and arranging, we obtain 
 
 the differential equation of the curve. 
 
 In order to integrate this equation, let us make 
 
 1 '. ; —y", and dividing the equation by y^, it will 
 
 assume the form 
 
 7 aa» \ 7 / 
 
 or, putting for ^ its equal m — a. 
 
 
 = 0, 
 
 (J 11 
 
 If we make -— zr in this equation, the roots of the re- 
 
 suiting equation will be the values of u at the apsides, or the 
 greatest and least values of u ; we thus find 
 
 = «(! + ?!). M 
 
 \ 7 
 
 These values being, as before, denoted by m" and u', the 
 above equation will become 
 
 —5-. —j—^-vu — (m -\-u )u-\-uu —V. 
 7 dis) 
 
 the same differential equation which we have had in the last 
 article, whose integral is 
 
 or, 
 
 ,,=:,/.cos.^7 (^)+,/'.sin.v(^) (^O 
 
 and if we substitute in the latter of these equations for u 
 
 z 2 
 
340 DYNAMICS. — SECT. IV. 
 
 IH — 2) ^^^ ^) ^^ shall have the 
 equation of the curve, 
 
 ?<=« ? 1 + ^.cos.V [—2~) S ^^^ 
 
 13. It appears from Art. 11. that the angle between the 
 apsides in this curve is 
 
 ,, , TT TT 
 
 tU — to = — ZZ 
 
 {dl 
 
 y a/, a-r{a) 
 ^ /(«) 
 
 When the force varies as any power of the distance, or 
 
 r:z:;wm", f{u)'=- —^"zimu"^^, andy'(?<) = »«(w — 2)?<"~^; there- 
 fore, 
 
 a.fia) , w(«-2)a»-2 ^ 
 
 1 ^tV = 1 ^ — i^^—-^ - "• 
 
 and the angle between the apsides = , 
 
 Hence, when the force varies directly as the distance, or 
 
 wz: — 1, the angle between the apsides = -, as we have al- 
 
 ready seen, Art. 6. When the force varies inversely as the 
 square of the distance, or w=2, the angle between the ap- 
 sides -zzTT, as has been already shown, Art. 8. When the 
 force varies inversely as the cube of the distance, the angle 
 between the apsides is infinite: and the force varying in- 
 versely as any higher power of the distance, the value of 
 the angle becomes imaginary ; which indicates that the body 
 leaving an apsis will never arrive at another; and, there- 
 fore, that it shall either descend to the centre or go off to 
 infinity. 
 
 Let the force consist of two parts, each varying as some 
 power of the distance, or R=:wi?i"-}-mV. In this case, 
 
OF CENTRAL FORCES. 34l 
 
 and 
 
 f'{ii) - m{n - 2)u"-^-\-m\n - 2)m"'-^. 
 
 (3— y^);?^a" -- + (3 - n)m'a'''-^ 
 and the angle between the apsides is equal to 
 
 V~ 
 
 ?na"~'' + ma" 
 
 (3 - ii)ma"-^-\-{ii - n)mar'-'^ 
 
 14. In all that has preceded, we have supposed the forces 
 given, and thence determined the nature and circumstances 
 of the motion. We shall now proceed in an inverse method, 
 by supposing the laws of the movement known, and thence 
 deducing the nature of the forces, and the laws according to 
 which they vary. 
 
 This method may be applied to the motion of the hea- 
 venly bodies, which, as it will afford the most convenient 
 illustration, will moreover lead to the most important re- 
 sults. 
 
 Let us suppose, therefore, in the first place, as was 
 judged to be the case in the infancy of astronomical know- 
 ledge, that the planets move in circular orbits round the sun, 
 and with an equable motion. In this case, it is evident, the 
 areas described round the sun are proportional to the times 
 of their description, and therefore it follows, from Art. 6. 
 and 7. of the preceding section, that the forces are always 
 directed to the sun. In investigating the nature of these 
 forces, therefore, we may employ the theorems respecting 
 central forces obtained in the preceding part of this section. 
 For the case under consideration it will be most con- 
 venient to make use of equation (4), which, since r is con- 
 stant, becomes 
 
 duP- 
 
342 DYNAMICS. — SECT. IV. 
 
 Now — expresses the angular velocity, and as this by 
 
 hypothesis is constant, we have —r — — , t being the time 
 
 at T ° 
 
 of the entire revolution ; therefore, making this substitution 
 
 R - 47r^. ^ {e') 
 
 the general expression for the force in a circle. 
 
 This formula, however, will not enable us to determine 
 the variation of the solar force dependent on the change of 
 distance, unless it should appear that there existed some re- 
 lation between the periodic times and the distances. Here 
 observation comes to our aid, by which it appears that the 
 squares of the periodic times of the planets vary as the 
 cubes of their distances from the sun ; and from this result, 
 combined with the expression for the force already found, 
 we conclude that the forces by which the different planets 
 are urged towards the sun vary inversely as the squares of 
 their distances from that body. 
 
 V being the velocity, we have v = , therefore. 
 
 'If 
 
 ^=VI7. if') 
 
 This is the general expression for the velocity in a circle de- 
 scribed under the influence of a force directed to the centre. 
 But, the force varying inversely as the square of the distance, 
 as has been proved to be the case in the planetary system, 
 
 v varies as — - ; i. e. the velocities of the different planets 
 
 vary inversely in the subduplicate ratio of the distances. 
 
 15. As the science of astronomy advanced, it was found 
 that the circular hypothesis did not exactly accord with ob- 
 
OF CENTRAL FORCES. 343 
 
 servation ; the true laws of the planetary movements were 
 first discovered by Kepler, and are as follows : 
 
 1st, Each planet describes equal areas in equal times 
 about the sun. 
 
 2d, The orbits described by the planets are ellipses, the 
 sun being in one of the foci. 
 
 3d, The periodic times of the different planets are in the 
 sesquiplicate ratio of the major axes of their orbits. 
 
 From the first of these laM^s we conclude, as before, that 
 the force, by which each of the planets is urged, is con- 
 stantly directed to the sun ; thus the nature of the force is 
 determined. 
 
 The area described in the time dt being ^r^dw, we have 
 by this law 
 
 r^dii) ■=. cdt. 
 
 Whence, the orbit being given, we can find the time of de- 
 scribing any portion of it, and also the velocity at any point. 
 
 For, in the first place, we have dt — , in which, by 
 
 means of the equation of the orbit, w may be expressed in 
 terms of r, or vice versa, and the time then found by inte- 
 grating. 
 
 ds 
 Again, the expression of the velocity is v=. — , ds being 
 
 the differential of the arc of the curve; but ds'^—dr'^-\-r'^d(jj^, 
 andc/^^= — 2— » whence v'^— c^ i ^ 2 "1"~2 )> ^^^> substitu- 
 ting - for r. 
 u 
 
 du^ 1 
 
 It is easily shown that -j-^ -\- ti^ — —^, p being the per- 
 pendicular from the origin on the tangent ; wherefore, 
 
344 
 
 DYNAMICS. — SECT. IV. 
 
 C 
 
 vzi-\ or the velocity at any point varies inversely as the 
 
 perpendicular let fall from the centre of force on the tangent 
 to the curve at that point ; as may be shown very readily 
 from geometric considerations. 
 
 16. The relation between the coordinates of the moving 
 body, and the central force by which it is urged, is given by 
 equation (10). That equation has been hitherto applied to 
 discover the relation between those coordinates, or the equa- 
 tion of the curve described by the body, when the force is 
 given. In the present case, however, the path of the body 
 is given, and it is required to deduce the law of the force. 
 For this purpose let equation (10) be differentiated, and the 
 result divided by 2du ; it becomes 
 
 ^(d'^u , N R n 
 
 whence there is 
 
 -=^'"0^^+ ') (12) 
 
 the general expression for the central force in any curve 
 described by its action combined w^ith an initial projection. 
 
 Now by the second of Kepler's laws, the curve described 
 by each of the planets is an ellipse, the sun being in the 
 focus. The polar equation of the eUipse referred to the 
 focus is 
 
 _ 1 + e. COS. (w — a) 
 '' - ^^■?) • 
 
 and differentiating twice, we have 
 
 dhi — e.cos.(w — a) , cPu . 1 
 
 ~" whence -7— t+m^ 
 
 du}^~ a(l-e*^) ' duj''^ a{\-ey 
 
 and this being substituted in the general expression for the 
 
 force, we find 
 
 2 2 
 _ c u _ m ,. 
 
OF CENTRAL FORCES. 34<5 
 
 making • ^- zz m. The force, therefore, by which each 
 
 ° rt(J — e^) 
 
 planet is solicited, varies inversely as the square of its dis- 
 tance from the sun. 
 
 17. The quantity ?u expresses the intensity of this force 
 at the unit of distance ; this is constant for any one planet, 
 but we do not as yet know whether it may change or not in 
 passing from one planet to another. The third law of Kep- 
 ler will enable us to determine this point. 
 
 Since the areas are proportional to the times of describ- 
 ing them, and c expresses the double of the area described 
 in the unit of time, ct will be double of the area described 
 in the time t ; that is, double the area of the ellipse, t being 
 the time of an entii'e revolution. Now the area of the ellipse 
 = irab. (a and b being the semi-axes,) = ira^ yj \ — e^ ; where- 
 fore, c — Stt. , and 
 
 T 
 
 a{\—e^) T^ 
 
 Now m being the absolute force for any other planet, 
 whose semi-axis major is a , and periodic time t' there is 
 
 72* 
 
 T^ 
 
 But by the third law of Kepler, — =: -75 ; wherefore, 
 
 m^m, or the absolute force is the same for all the planets. 
 If, therefore, we suppose the planets placed at equal dis- 
 tances from the sun, they shall all be urged by the same ac- 
 celerating force ; and being let fall without any initial velo- 
 city, they would descend towards the sun, describing equal 
 spaces in the same time. 
 
 Thus by Kepler's laws, which are the results of obser- 
 vation, Newton was led to the discovery of the nature of 
 those forces which animate the planetary system. From the 
 
348 DYNAMICS. — SECT. IV. 
 
 proportionality of the areas to the times of their description, 
 it has been shown that the force soHciting each planet is di- 
 rected always to the centre of the sun. From the ellipticity 
 of the planetary orbits, we have concluded, that, for each 
 planet, this force varies inversely as the square of the dis- 
 tance from the sun. And, finally, from the third law of 
 Kepler, which has been called the harmonic law, it follows, 
 that this attractive force is the same for all the planets, if 
 placed at equal distances from the sun ; and, therefore, that 
 it does not change from one planet to another, but in conse- 
 quence of the change of distance. 
 
OF CONSTRAINED MOTION IN GENERAL. 347 
 
 SECTION V. 
 
 OF CONSTRAINED MOTION IN GENERAL. 
 
 1. If a material point or body, subjected to the action of 
 a certain force, p, is constrained by a rigid line or surface 
 to take a direction different from that of p, this force natu- 
 rally resolves itself into two forces ; one of them perpendi- 
 cular, and the other parallel to the line or surface. This 
 latter, called the tangential force, shall produce its full 
 effect in accelerating or retarding the motion of the body : 
 but the normal force is equilibrated by the reaction. This 
 normal force constitutes the pressure immediately arising 
 from the force p ; and being independent of the movement, 
 it is the only pressure we have had occasion to consider in 
 unfolding the theory of Statics. 
 
 If several forces, p, p', p", &c. act on the body, what 
 has been stated relative to a single force is to be understood 
 of their resultant. 
 
 But that this is not the whole amount of the pressure 
 made by the body, moving on a curved surface, may appear 
 by supposing the action of the forces p), p', p", &c. to have 
 ceased. For the continual deflection of the body from the 
 course it would have taken if unrestrained, evinces the ex- 
 istence of a force of reaction, and, therefore, of a pressure 
 against the constraining line or surface. 
 
 The better to conceive the nature of this deflecting force, 
 the figure, to which the body is constrained to accommodate 
 its motion, may be supposed to be a polygon. Let mm'm", 
 &c. be that polygon, (Fig. 143.), and w the angle made by 
 the side nim", with the side vun produced beyond the point 
 
348 DYNAMICS SECT. V. 
 
 ni. When the body has described the hne mm with a cer- 
 tain velocity z?, instead of continuing its course in that hne 
 produced, it is constrained to move in the contiguous side 
 m'm'. The velocity with which it moves in the line mm\ 
 is had by resolving v according to the line mm' , and the 
 perpendicular to that line. The velocities in these direc- 
 tions are zj.cos.o), and z^.sin.w, respectively. The former 
 of these is the velocity retained in the side 7nm\ and the 
 latter is that destroyed by its reaction. 
 
 2. If the constraining line or surface is curved, the 
 change of direction is continuous ; and, therefore, the re- 
 action by which this change is produced, is unremitting. 
 The angle w, which measures the deflection in passing 
 from one element to another, is indeed indefinitely small; 
 but then, the time dt is also indefinitely small : and when the 
 latter becomes a definite quantity, the angle which measures 
 the change of direction also becomes of a definite magni- 
 tude. The force of reaction, in this case, being unremit- 
 ting, and moreover, requiring some definite time to produce 
 a definite effect, is comparable to an accelerating or re- 
 tarding force. Its value is expressed by the velocity de- 
 stroyed in an indefinitely small time divided by that time, 
 
 - r.sin.W V.M , , • ^ n - ^ 11 1 
 
 I.e. by — — — , or —.-'■, the arc, when indennitely small, be- 
 ing confounded with its sine. 
 
 To find a more convenient expression, let mm be the 
 element of the curve described by the body, in the time 
 dt, (Fig. 144.), and let two tangents be drawn to the points 
 m, m', intersecting at o, the former of these being produced, 
 as to n. The deflection or change of direction, which oc- 
 curs in the passage from m to m', is then measured by the 
 angle nom'. Now if mc,m'c are perpendiculars to the tan- 
 gents at the points m and m\ meeting at c, the angle nom' 
 is equal to the angle at c, which, therefore, may be denoted 
 by (X). Also, putting ds for the indefinitely small arc mm'^ 
 
ON CONSTRAINED MOTION IN GENERAL. S49 
 
 and p for mc or 7nc, which is then the radius of curvature, 
 
 ds v.dt 
 we have w = — = — — . Wherefore, substituting this value 
 
 P P ^,^^^ ^, 
 
 of ix), the deflecting force — ^ — — , The same is the mea- 
 sure of the pressure arising from the tendency of the body 
 to move in a right Hue : a tendency which is counteracted, 
 
 but not destroyed by the normal forces. The force — be- 
 ing always directed from the centre of the circle of curva- 
 ture, is denominated the centrifugal force. 
 
 3. If a body is constrained to move in a circle, the radius 
 of curvature is the radius of the circle actually described ; 
 and, r denoting that radius, andy the centrifugal force, this 
 
 force is expressed by the equation f — —zz -^- ; in which 
 
 h denotes the height due to v, the velocity of the gyrating 
 body. This equation immediately gives the relation of the 
 centrifugal force in a circle to the force of gravity ; for di- 
 viding both sides of the equation by g, there is 
 
 ff r * 
 The equation y= — may be changed for another, ex- 
 pressing the centrifugal force by the time of the revolution : 
 for, denoting this time by t, there is Z7= ; which value 
 
 being substituted, the equation becomes 
 
 47rV 
 
 The time t is found by dividing any certain portion of time 
 by the number of gyrations made in that time: and the 
 facility with which t is thus obtained, even in the case of 
 rapid movements, renders this expi'ession particularly con- 
 venient, whenever it is proposed to find the absolute value 
 of the centrifugal force. For example ; for the centrifugal 
 
350 DYNAMICS. SECT. V. 
 
 force at the equator, produced by the rotation of the earth 
 round its axis, we have 7r = 3.1415926, tz=86164, the num- 
 ber of seconds in a sidereal day ; and for the equatorial ra- 
 dius rr: 20,976876 feet. Making these substitutions in the 
 preceding formula, there is ybr.11154. 
 
 The centrifugal force at the equator is, therefore, such 
 as would generate a velocity of .11154 feet, in a second of 
 time. To compare this with the force of gravity at the 
 equator, the latter, estimated in the same way, is 32.125, 
 consequently, 
 
 •' 288 
 
 I. e. the centrifugal force at the equator is the 28Sth part of 
 the effective force of gravity. But if g denote the attraction 
 of the terrestrial spheroid 
 
 so that we have 
 
 -^ ~ 289 
 
 or the centrifugal force at the equator is the 289th part of 
 the total gravity. 
 
 Were the time of the earth's revolution diminished in 
 any ratio, the centrifugal force would be increased in the du- 
 plicate of that ratio. It is, therefore, easy to find the time in 
 which it should revolve, in order that the centrifugal force 
 at the equator should be equal to that of the attraction to- 
 wards the spheroid. Since, putting t' for this time, we have 
 
 t'^:t-:: 1 : 289. Wherefore, t'=: —, showing that if the 
 
 earth were to revolve round its axis in the 17th part of a 
 sidereal day, or in 1\ 24", 28|", the force of gravity at the 
 equator would be equilibrated by the centrifugal force ; and 
 the bodies on this part of the earth's surface would have no 
 
OF CONSTRAINED MOTION IN GENERAL. 351 
 
 weight. In this statement it is supposed that the figure of 
 the earth remains unchanged. 
 
 When a solid revolves round an axis, each point describes 
 a circle whose radius is the perpendicular let fall from that 
 point on the axis : and all those circles being described in 
 the same time, the centrifugal forces of the several points 
 are proportional to their perpendicular distances from the 
 axis. Consequently, i^f denote the centrifugal force at any 
 parallel of latitude, and r' the radius of the parallel, 
 
 /'=/^'=Aos./; 
 
 I being the latitude of the parallel. This force is directed 
 from the centre of the circle of latitude. The portion of it 
 which acts in opposition to gravity, is found by resolving it 
 in the direction of the terrestrial radius, or by multiplying it 
 by the cosine of latitude ; so that the diminution of gravity 
 arising from the centrifugal force is 
 
 /'cos./=/cos.^/. 
 
 To show this, let pepV be a great circle passing through 
 the poles p, p'; and let eg be the equatorial radius; bd that 
 of any parallel, (Fig. 145.) Then, if ce represents the cen- 
 trifugal force at e, that at b shall be represented by db ; and 
 DM being a perpendicular from d, on the radius cb, the part 
 opposed to gravity shall be mb. But cb : db : mb, i. e. ce : db : 
 MB ; wherefore, ce : mb : : ce^ : db^, or as 1 : cos.^/. Whence 
 it follows, that the decrement of gravity, immediately due to 
 the centrifugal force, varies as the square of the cosine of 
 latitude. 
 
 4. Many of the laws of unconstrained motion flow readily 
 from the principles here laid down. For when the body is 
 free, the forces acting on it must supply the place of the re- 
 action of the curve ; so that if the i*esultant of these forces 
 be resolved into two, one in the direction of the tangent, and 
 the other perpendicular to it; the latter must be equal and 
 
352 DYNAMICS. — SECT. V. 
 
 opposite to the centrifugal force. Hence it follows, that the 
 osculating plane of the curve, at any point, must contain the 
 direction of that resultant; and if be the angle which that 
 direction makes with the tangent to the curve, we must have 
 
 R.sin.(^ = — . 
 P 
 
 When the resultant, r, is directed to a fixed centre, the 
 curve will all lie in one plane. In this case the angle B is de- 
 duced from the equation of the curve itself, and the law of 
 the central force is given by the formula 
 
 V- 
 
 R = 
 
 psin.d.' 
 
 If Sy denote the chord of curvature passing through the cen- 
 tre of force, y — psin.O; and we have the formula of Newton, 
 
 tr 
 
 R=Z . 
 
 7 
 When the body moves in a circle, whose centre is the centre 
 of force, 7 ::: /*, and 
 
 r 
 
 5. To return to the problem of constrained motion. 
 Ths forces by which the movement is affected are the forces 
 p, p', p" , &c. or their resultant, r, and the reaction of the 
 line or surface. The force r is resolved into two, viz. a 
 normal and tangential force. The reaction is likewise re- 
 solvable into two forces, both of them normal. Of these, 
 one is equal and opposite to the normal force exerted by r, 
 
 and the other that expressed by — . The equal and oppo- 
 site forces being suppressed, there are only the force r re- 
 solved according to the tangent, and that part of the force of 
 
 reaction expressed by — . The former of these, being a 
 
OF CONSTRAINED MOTION IN GENERAL. 353 
 
 tangential force, changes only the velocity ; and the latter, 
 being a normal force, changes only the direction, i. e. the 
 former is merely an accelerating or retarding force ; the lat- 
 ter merely a deflecting force. 
 
 That the deflecting force makes no change in the velo- 
 city, may perhaps require demonstration, more especially as 
 it has been shown, that when the path is a polygon repre- 
 sented by mnim", &c, (Fig. 143.) the body having described 
 the line 7ti7n, with the velocity v, enters on the line m'?n", 
 with the velocity v.cos.w. Whereby it would appear that 
 at the angle ni, a portion of the velocity is lost by deflection, 
 whose value is z>(l— cos. w); or,' f.ver. sin. w. And if the 
 figure described were actually a polygon, whether right 
 lined or curved lined, the angles being then finite, those 
 losses would have a finite value, and should, therefore, be 
 estimated. But the case is different, when for the polygon 
 we substitute its limiting curve. For the velocity lost in the 
 direction of the tangent, is to that destroyed in the direction 
 
 of the normal, as ver. sin. co to sin. w, i. e. as tan. - to unity. 
 
 Now the path being curvilinear, the angle lo is infinitely 
 
 smaller than any right lined angle; and, therefore, tan. - 
 
 infinitely less than unity. The velocity destroyed at each 
 instant in the direction of the normal, has been represented 
 
 as the effect of a force, — , which must act during some de- 
 
 P 
 finite time to produce any definite effect; and the velocity 
 
 supposed to be destroyed in the direction of the tangent is 
 infinitely less : wherefore, if this latter is ascribed to the ac- 
 tion of any force, it must be to a force, such as cannot pro- 
 duce any efi'ect in any definite time. Whence it follows, 
 that there is no change made in the velocity, except that 
 which is due to the tangential force ; and that a body moving 
 in a curve in consequence of an impulse, and not affected by 
 
 2 a 
 
354 DYNAMICS. — SECT. V. 
 
 friction or the resistance of the medium, must for ever con- 
 tinue to move with the same imvariecl velocity. 
 
 6. The perpendicular action of the constraining line or 
 surface having no effect on the velocity, it might seem that 
 in developing the theory of constrained motion, it would not 
 be requisite to consider any force but that in the tangent ; 
 and undoubtedly this abbreviated process may be followed, 
 when the body moves in a given curve without friction. But 
 this is only a limited case of the general theory of constrained 
 motion: for the tangential force depends on the path ; and 
 when the body is constrained by a surface, the path itself is 
 to be investigated. In such cases, therefore, we cannot pro- 
 ceed from the tangential force to any of the objects of in- 
 quiry, viz. the time, the velocity, the path, or the pressure 
 on the constraining surface. Moreover, in many cases, the 
 tangential force depends not only on the form of the curve 
 described, but also on the pressure ; and, therefore, on the 
 normal force, as when friction is to be considered. For these 
 reasons, in developing the general theory of constrained mo- 
 tion, it seems proper to follow the usual method, which is 
 also the most elegant, by including all the forces actually 
 exerted, and then regarding the body as moving in free 
 space. In this method there are three fundamental equa- 
 tions of the movement, by which the normal force of re- 
 action will naturally be eliminated, whenever the quantity 
 sought is independent of this force ; whilst its introduction 
 into those equations aiFords the readiest means of expressing 
 its value, whenever this becomes the subject of investi- 
 gation. 
 
 Because of some peculiarities of management, as the mo- 
 tions are restrained by lines or surfaces, it seems expedient 
 to separate those parts of the general subject. 
 
OF MOTION CONSTRAINED BY A QIVEN CURVE, 355 
 
 SECTION VI. 
 
 OF MOTION CONSTRAINED BY A GIVEN CURVE. 
 
 1. Let each of the forces,^, p', ^/', &:c. be resolved ac- 
 cordmg to three rectangular axes ; and let x, y, and z, de- 
 note, as usual, the sum of the resolved forces acting m di- 
 rections parallel to the axes o£x, y, and sr, respectively. Also, 
 let N denote the force of reaction in the normal, and 0, 0', 0", 
 the angles made by this normal with the same axes. We 
 shall then have for the fundamental equations of the move- 
 ment, 
 
 3-2-=X + N.COS.O. 
 
 ^=Y + N.C0S.6I'. > (1) 
 
 ||-=:Z-fN.COS.r. 
 
 Two of these equations will suffice for a plane curve : for 
 then R, and consequently n, is in that plane. Wherefore, 
 by taking two axes in the same plane, each of the terms of 
 the third equation shall be cipher. 
 
 Further : 
 
 cos.20+cos.20'4.cos.20"= I. t2) 
 
 Moreover, as -r, -r-y ~ri are the cosines of the angles 
 as CIS as 
 
 made by the element of the curve, with the axes of x, y, 
 
 and z, respectively ; and as 0, 6', 6", are the angles made by 
 
 the normal with the same axes, it follows, that 
 
 ^.cos.0-f ~.cos.e'+ ^.cos.r=0. (.3) 
 
 as ds as 
 
 2a2 
 
356 DYNAMICS. — SECT. VI. 
 
 the first member being the expression for the cosine of the 
 angle contained between the arc and normal, i. e. of a right 
 angle. 
 
 By means of equations (1) and (3) the quantities n, 
 cos.0, cos.d', COS.0", may be at once eliminated; and the re- 
 sult shall be a differential equation of the second order 
 among the remaining quantities x, y, %, t, and the forces 
 X, Y, z ; whereby the time may be expressed as a function of 
 those forces and of the coordinates. 
 
 The equation thus obtained will also give the velocity. For, 
 
 , . . p 7 . , ds ^ dx^^-dir' + dz^ , 
 by substituting tor dt its value — , or '■ , the 
 
 equation will be between v and the same quantities. 
 
 Equation (2) points to the method of eliminating the co- 
 sines by themselves, i. e. of reducing the co-efficient of n to 
 unity ; whereby the value of this force is at once expressed 
 as a function of the remaining variables. 
 
 The coordinates in the expressions thus obtained may 
 be reduced to one, by using the equation or equations of the 
 curve for the elimination of the rest. Then, if x, y, and z 
 are given functions of the coqrdinates, those forces being 
 previously so expressed, the quantity sought, whether it be 
 V, ty or N, shall be exhibited as a function of the remaining 
 ordinate. 
 
 2. To proceed according to the method now described, 
 let the first of the equations (1) be multiplied by dx, the se- 
 cond by dy, and the third by dz, and let these equations then 
 be added ; the co-efficient of n becomes cipher, as appears 
 from equation (3), and the result is 
 
 dx.cPx-\-dy.d'^y-\-dz.d'^z , , , , 
 
 • z=.x..dx+Y.dy-\-z.dz. (4) 
 
 If we differentiate the equation 
 
 dx'^-\-dy'^-\-dz^ _ ds^ 
 d¥ ~di" 
 
OF MOTION CONSTRAINED BY A GIVEN CURVE. 357 
 
 di 
 
 the equation becomes 
 
 we find that the first number of equation (4) is ds. -j^ ; and 
 
 d^s dx dy . dz ,,. 
 
 -r-r, = X.-r- + Y.-/ + Z.— . (5) 
 
 di^ ds ds ds ^ ^ 
 
 The terms in the second member of this last equation are 
 the accelerating forces reduced to the tangent. The equa- 
 tion, therefore, shows that the tangential force is equal to 
 the second differential coefficient of the space described, 
 taken with respect to the time, as has been already proved 
 with respect to unconstrained motion. 
 
 Equation (4), by integration, gives 
 
 dx'^+dif+dz'' _ 2^(^^ja:+Y,di/ + z.dz)+c. (6) 
 
 at' *' 
 
 Using the equation or equations of the curve for elimina- 
 tion, the integral of the last equation gives the value of t, as 
 a function of one of the coordinates. And here it may be 
 observed, that as the expression for dt, to be obtained from 
 equation (6), contains \/ dx--\-dif'-\-dz^, or ds, the time is de- 
 pendent on the equations of the curve, i. e. the time taken to 
 move from one point of the path to another, depends on the 
 form of the curve described between those points ; and is 
 accordingly different for different curves described between 
 two given points. 
 
 ds^ 
 3. The first member of equation (6) being — ^j ory^, that 
 
 equation gives for the velocity, 
 
 v'-'^\^{^.dx-\-Y.dy\i.dz)^c. (7) 
 
 When mdx-irYdy-^zdz is an exact differential of a func- 
 tion, <^{x, y, z,) of the three coordinates, the preceding equa- 
 tion assumes the form 
 
S58 DYNAMICS. — SECT. Vf. 
 
 And if V denotes the velocity at any point whose coordinates 
 are x', y\ %' , we shall have v"^ — 2^{x\ y\ z)-\-c. Wherefore, 
 
 z)2-t?'2z=2^Gr, y, z)-^{x' , y' , z'). 
 
 This expression for the velocity, which is the same as 
 that found for unconstrained motion in Sect. 3. Art. 5, is in- 
 dependent of the form of the curve. Wherefore, if different 
 curves are described between the points whose coordinates 
 are x , y, z, and x", y", z", the body proceeding from one 
 of those points with a certain velocity, would arrive at the 
 other with a velocity which is the same for all those different 
 courses. And if the curve is closed, the body shall ever 
 return to the same point with the same velocity. 
 
 4. IfR = 0. e. e. ifx = 0. Y3:0. znO. then 
 
 J {^.dx + Y.dy+z.dz) = 0. and v^= c. 
 
 Wherefore, when the body moves in conssquence of an ori- 
 ginal impulse, and is not solicited by any force, but that of 
 the reaction of the constraining curve, the velocity, is unva- 
 ried, or the same at every point in the line of its motion. 
 This is agreeable to Art. 5. of last Sect, where it was shewn 
 that there is no change in the velocity, except that made 
 by the action of the tangential force. 
 
 5. If the body moving in the curve is urged by the force 
 of gravity only, taking the axis of z vertical, and measuring 
 the positive values of the parallel ordinates upwards, we 
 shall have 
 
 xz=0. Y = 0. z — —g. and v'^——2\gdz-\-c= —2gz-\-c. 
 
 Let z' be the height at which the velocity is cipher, and we 
 shall have c zz 2gz'. Wherefore, 
 
 v-' = 2g{z'-z). 
 The value of the velocity, in this case, depends exclu- 
 sively on the space z'—z; i. e. on the vertical distance from 
 
OF MOTION CONSTRAINED BT A GIVEN CURVE. 359 
 
 the point -svliere the velocity is cipher. Whence it follows, 
 that if several bodies moving in different curves under the 
 influence of the force of gravity, have equal velocities at any 
 common height above a horizontal plane, their velocities 
 shall be equal at any other given height above the same 
 plane. The time, indeed, in this, as in all other cases, de- 
 pends on the equation of the curve : but even in this respect, 
 the case now under consideration is distinguished by this 
 peculiarity, that the time depends on the relation between 
 the curve and the vertical ordinates only ; as appears by 
 
 ds 
 substituting for v, its value in the equation dt:= — which 
 
 (Is 
 then becomes df— , — Wherefore, if there are two 
 
 curves in which the relation between * and z, is the same, 
 t. e. if for equal arches, the vertical ordinates are always 
 equal, not only the velocities acquired, but also the times of 
 the descent through those equal arches shall be equal ; the 
 initial velocity being the same in both : and this is true, how- 
 ever the curves may differ as to their horizontal coordinates. 
 Two plane curves agreeing in the relation between the arc 
 and one of the coordinates must be the same. Wherefore, 
 the curves here spoken of can differ only when all but one 
 are of double curvature. This diversity may be obtained by 
 wrapping a plane curve round a vertical cylinder, whose base 
 is a continued curve of any figure; whereby an indefinite 
 variety of curves of double curvature may be produced from 
 the same plane ; and conversely, any one of the curves so 
 produced may be opened into a plane : and from the fore- 
 going equations it appears, that no change shall be made 
 thereby in the velocity acquired in the descent through any 
 arc, or in the time taken to describe it. 
 
 6. To find the pressure exerted at any point of the 
 curve; let us suppose in the first place that there is no ac- 
 
360 DYNAMICS. — SECT. VI. 
 
 celei-cating force acting on the body. Equation (1), in this 
 case becomes, 
 
 ^=Ncos.0, ^^z:Ncos.0, -J^=n<^o^^-« • 
 
 Again, ds denoting the element of the curve described in 
 
 the time dt, we have ds — vdt; in which ds is constant, since 
 
 ds 
 V is so, (Art. 4). Substituting then for dt its value — , in 
 
 the preceding equations, squaring and adding, we obtain 
 
 ^^ y'V(d'ufM^p!/f+{rI-'^r 
 ds" 
 
 But, ds being constant, 
 
 ds^ 
 
 p — 
 
 V {d^xf 4- {d'yf-\- {d'^f 
 And we find, as before, (Sect. 5. Art. 2.), that the pressure 
 arising from the inertia of the moving point is expressed by 
 the formula 
 
 v^ 
 
 Nr= — . 
 
 P 
 
 To find the total pressure we must resolve the resultant, 
 R, of the accelerating forces into two, one in the direction 
 of the tangent, and the other normal to the curve. The 
 former of these is the force which alters the velocity of the 
 moving point, and the latter produces a pressure on the 
 curve which is independent of the motion of the body. This 
 may be called the statical pressure. It is to be combined 
 with the centrifugal force, or the pressure arising from the 
 inertia of the body, and the resultant will be the actual 
 pressure sustained. 
 
 7. We may apply these results to the case of a heavy 
 body, compelled to describe a vertical circle by the tension 
 of a rod or wire, whose extremity is attached to a fixed 
 point. Let c be the fixed point, (Fig. 14G.), ca the vertical 
 
OF MOTION CONSTRAINED BY A GIVEN CURVE. 361 
 
 passing through it, and cm any other position of the con- 
 necting rod. Let the force of gravity be represented by the 
 vertical line mn, and let this be resolved into two, mq in 
 the direction of the tangent at the point m, and nq perpen- 
 dicular to it. The former of these is the force employed in 
 accelerating the body ; the latter is that portion of gravity 
 which produces the strain on the fixed point. Letting fall 
 the perpendicular M)i on the vertical radius, from the simi- 
 larity of the triangles mnq and cmw we have 
 
 MQ _ M/i nq _ c?^ 
 
 MN ~ MC' MN ~ MC* 
 
 Wherefore, if g denote the force of gravity, / the accele- 
 rating force, and p the pressure on the axle ; and if un and 
 AH the ordinate and abscissa of the arc terminating in the 
 lowest point; be denoted by y and s, and the radius of the 
 circle by r, we have 
 
 7/ r — « 
 
 The accelerating force, then, varies as the sine of the arc 
 measured from the lowest point, and the pressure as its 
 cosine. 
 
 The pressure just spoken of, however, is only the sta- 
 tical pressure, which would subsist if the force f were 
 counteracted, and the body at rest. The whole pressure is 
 
 obtained by adding to this the centrifugal force. This 
 
 .'- ' 
 
 force, we have seen, is equal to — , or to 2g; — ; so that 
 
 ;• r 
 
 if m denote the mass of the body, the total pressure will be 
 
 mg . 
 
 ^ r 
 
 If the body is let fall the rod being in the horizontal posi- 
 tion, x'zi:. r, and the tension at any point is dmg . This, 
 
362 DYNAMICS. — SECT. vr. 
 
 in every part of the curve, is thrice the tension of the rod pro- 
 duced by the weight when supported by a force equal and op- 
 posite to the tangential force. If the body is supposed to de- 
 scend from the vertical position without impulse ; i. e. if it 
 descends from b, the highest point of the circle, z=2r: 
 
 and the strain is m^ ^-. At the lowest point this is 
 
 5Mgf or five times the weight. 
 
 If it is required to find the point at which the strain be- 
 
 comes equal to the weight, we have for this, mo- ^^ 
 
 i" 
 
 2 
 
 =: M^ ; which gives z = 77^'. Accordingly, from whatever 
 
 o 
 
 height the body descends without impulse, the strain be- 
 comes equal to the weight when it has descended through 
 one-third of that height. 
 
 If it is required to find the point at which the pressure 
 vanishes, its value is to be made equal to cipher. This gives 
 
 !s = — -^ — , for the height to which the body should rise, in 
 o 
 
 order that the pressure may vanish. But it is possible that 
 
 the body may never rise to this height; for z cannot surpass 
 
 either ;^' or 2r. This points to the greater and lesser limits 
 
 of the magnitude to be assigned to s-', in order that the 
 
 body may rise to the height required by the formula. For 
 
 making ;£ successively equal to z and 2r, we have z=r, 
 
 z'z=:-^r. If z' is less than r, the force of gravity resolved in 
 
 the direction of the length of the pendulum is directed from 
 
 the centre ; and, therefore, conspires with the centrifugal 
 
 force throughout the whole curve. And if z' is greater 
 
 than 4r, the centrifugal force is, in all parts of the curve, 
 
 greater than the weight ; as will appear by putting ^' = Ar-|- 8, 
 
 in the expression for that force. Being therefore greater 
 
 than the pressure made by the weight on the centre, 
 
 the connecting rod, throughout, suffers a tension ; which is 
 
OF MOTION CONSTRAINED BY A GIVEN CURVE. 3G3 
 
 the sum of those forces in the lower half, and then* differ- 
 ence in the upper half of the circle. If z' is not less than r, 
 or greater than 4/-, the hody shall always attain to the 
 height specified in the formula for a pressure equal to ci- 
 pher: '/'. e. it shall rise to a point where the weight, re- 
 solved according to the length of the rod,hecomes equal and 
 opposite to the centrifugal force. 
 
 When the body rises above this point, the pressure due 
 to the weight is greater than the centrifugal force : and 
 being, moreover, directed towards the centre, the rod is 
 compressed by the difference of those forces. Wherefore, 
 if ;s' falls between the jjrescribed hmits, and the heavy body 
 is suspended by a flexible cord, it shall part from the circle 
 at a point, whose vertical height above a is that given by 
 
 the equation ;:;= — ^-. Being then unconfined by the 
 o 
 
 cord, it shall move in a parabola, having a common tangent 
 with the circle at the point of departure. The angle of ele- 
 vation, as well as the velocity at this point, being given, the 
 parabola is given. 
 
364 DYNAMICS. — SECT. VII. 
 
 SECTION VII. 
 
 OF THE SIMPLE PENDULUM. 
 
 1. The use of the pendulum as a regulator of those 
 motions by which time is measured, and as affording the 
 readiest and most correct measures of terrestrial gravitation, 
 entitles this instrument to particular consideration. It con- 
 sists of a heavy body, suspended by a rod from a horizontal 
 axle, round which it is at liberty to move in the arch of a 
 circle. In unfolding the theory of this instrument, it is 
 convenient to regard the rod as being without weight, and 
 the body attached to it as a material point. The instru- 
 ment, so considered, is called by writers on mechanics a 
 simple pendulum. In reality, the rod, like all other material 
 substances, is heavy, and the body suspended by it of some 
 certain bulk; but as the time of the vibration of a pendulum 
 depends not merely on its length, measured from the axle to 
 the remote extremity of the appended body, but also on the 
 form of the whole pendulum ; or more properly, on the dis- 
 tribution of the matter of which it consists, it is found ex- 
 pedient to reduce every real pendulum to the simple pendu- 
 lum : i. e. to estimate its length by that of the simple pen- 
 dulum which would perform its oscillations in the same time. 
 The mode of making this reduction shall be shown in a sub- 
 sequent part of this treatise. For the present we shall con- 
 sider the case of the simple pendulum. 
 
 2. It has been already proved, that when a body is con- 
 strained to move in a curve whose plane is vertical, by the 
 influence of gravity alone, the velocity acquired at any point 
 is that due to the height from which it has descended, mea- 
 
OF THE SIMPLE PENDULUM. 365 
 
 sured in the vertical. The elementary proof of this is ex- 
 ceedingly simple. The curve being regarded as a polygon 
 with an infinite number of sides, the body may be supposed 
 to descend down a system of planes inclined to the horizon 
 at different angles ; and the planes being infinite in number, 
 no part of the velocity acquired shall be lost in passing from 
 one to the other. Wherefore, I, I', l", Sec. denoting the 
 lengths, and //, k', li\ &c. the heights of the planes, be- 
 ginning with the uppermost, and v, v', v", &:c. the velocities 
 at the lower extremities of those planes, there will be for 
 the fall down the plane /, v^ = 9,g1i. And v being the ve- 
 locity with which the body enters on the second plane, there 
 is at the end of this plane, v''^—v^^:.2gh'. And in like man- 
 ner, at the end of the third plane, v"'^ — v'-zz2 gh"- Where- 
 fore, by addition, v"'-—2ff {h-{-h' + 7i")- The same proof ex- 
 tends to any number of planes : accordingly, if ^' denotes 
 the vertical height of the point from which the body is suf- 
 fered to descend ; and z that of the point to which it falls ; 
 both measured upwards from the same horizontal plane. 
 The velocity will be expressed by the formula 
 
 Now the condition of a body suspended by a rod is the same 
 as if its movement were constrained by a vertical circle : the 
 tension of the rod taking place of the pressure, and the re- 
 action of the fixed point that of the constraining curve. 
 Wherefore, let bmam'b be the circle whose plane is vertical, 
 (Fig. l47.); A the lowest point; and ab the vertical diameter. 
 If the body is suffered to descend from m, its velocity at any 
 point, N, is had by drawing the horizontal lines mm' and nn^ 
 meeting the vertical diameter at e and f ; for the velocity at 
 N, is that due to the height ef. The body having descended 
 to the lowest point a, where it acquires a velocity due to the 
 height EA, begins to ascend against the force of gravity, in 
 the other branch of the curve am' ; and as it appears by the 
 equation, that in the two branches the velocities are equal at 
 
366 DYNAMICS. SECT. VII. 
 
 equal heights, it follows, that the velocity in the ascending 
 branch shall not be destroyed until the body arrives at m'. 
 The velocity at this point being cipher, the body shall de- 
 scend by its weight to a ; where regaining the velocity with 
 which, in its progress, it had passed through this point, it 
 shall ascend to m : and thus the body shall go and return for 
 ever through the same arc mam', supposing no retardation 
 from friction, or from the resistance of the medium through 
 which it moves. 
 
 3, The time, di, of describing any element of the arc, ds, 
 is given by the formula 
 
 — ds 
 dt = ; 
 
 V 
 
 in which dt and ds are affected with opposite signs, be- 
 cause as the time increases, the arc measured from the 
 lowest point diminishes. Substituting in this expression for 
 V its value found in the preceding article, we have 
 
 dt= -^^ 
 
 V2g (/-z). 
 
 When the arc of vibration is indefinitely small, this for- 
 mula is readily integrated. Let s and s' be the arcs mea- 
 sured from the lowest point, corresponding to the ordinates 
 z and ^'; these arcs, being indefinitely small, are equal 
 to their chords, and we have 
 
 s'^:=2rz, s'^z^2rz\ 
 
 Substituting then for 2z, 2z', their values — , and — , the 
 
 preceding formula becomes 
 
 /~ ds 
 y g \ s — s 
 The integral of which is 
 
 t^z\/ - diVc (cos. -zz-Y 
 y g \ sJ 
 
 which needs no correction, as tzzo, when s—s'. 
 
OF THE SIMPLE PENDULUM. 367 
 
 The time of descent to the lowest point is had by making 
 5zzO; and as arc (cos.=0)=-, this time is 
 
 '^^ g 
 
 The time of an oscillation, or of the whole descent and sub- 
 sequent ascent, is double of this, and if its value be denoted 
 by T, 
 
 =V! 
 
 This expression being independent of the magnitude of 
 the arc described, it follows that the time of oscillation will 
 be the same, however the arc of vibi'ation be varied, provided 
 that that arc is indefinitely small. It is this property which 
 renders the pendulum an instrument of such value in the 
 measurement of time. 
 
 Gravity being constant at any given place, it follows from 
 the same formula, that the times of vibration of different 
 pendulums vary as the square roots of their lengths. 
 
 4. Many curious relations are at once deduced from the 
 expressions just obtained. Thus, the time of the fall in the 
 
 vertical through a space equal to |r being ^ -, it follows, 
 
 that the time of vibration is to that of the fall through half the 
 length of the pendulum, as tt : 1 ; i. e. as the circumference 
 of a circle to its diameter. 
 
 The time of the fall down the vertical diameter is 
 
 24/ -. And this being the same as the time of descent 
 
 through any chord terminating at the lowest point, it follows, 
 that the time of describing a very small circular arc ending 
 at the lowest point, is to that of the descent down its chord, 
 
SG8 DYNAMICS. — SECT. VII. 
 
 as -W - to 2 \/ -'. i. e. as tt to 4, or as the circumference 
 
 2^ g y g 
 
 of a circle to four times its diameter. 
 
 From the last equation it also appears, that if two pen- 
 dulums of different lengths are set to vibrate in small circular 
 arcs, the times shall be as the square roots of the lengths ; 
 or the lengths as the squares of the times : i. e. inversely as 
 the squares of the numbers of vibrations performed by the 
 two pendulums in the same time. Wherefore, n denoting 
 the number of vibrations made in a given time, /A r shall be 
 a constant quantity : i. e. the same for all pendulums vibrat- 
 ing in small circvdar arches, at the same part of the surface 
 of the earth. Accordingly, given the length of the pendu- 
 lum, and the number of vibrations made by it in a given 
 time, the length of a second's pendulum is thereby known. 
 For let r' be the length of the given pendulum, and r that of 
 the second's pendulum ; also, let n' be the number of the 
 vibrations made by the former in n seconds, we have n^.r=. 
 
 n'". / 
 «'^. /; whence, r = — ^. 
 n' 
 
 Hence, also, knowing the daily or weekly loss or gain of 
 
 a pendulum clock, the correction to be made in the length of 
 
 the pendulum is immediately ascertained. For let w be the 
 
 number of seconds in a day or in a week, and n±dn the 
 
 number shown by the clock; there is n\=: (n±clny.r'zz 
 
 n^r±2n.d>i.r. omitting the third term of the square of 
 
 n^dn, as being inconsiderable with respect to the second. 
 
 This equation gives w^(r — r):=z ±2}i.dn,r , or r — r'-=. ± r'; 
 
 i. e. the correction of the length of the pendulum is had by 
 multiplying that length into a fraction, whose numerator is 
 twice the error in time, and whose denominator is the time in 
 which it had accumulated. 
 
 The equation shews that /— / is positive or negative, 
 
OF THE SIMPLE PENDULUM. 369 
 
 as the clock gains or loses ; and that the pendulum is to be 
 lengthened or shortened accordingly. 
 
 5. Besides the advantages derived from the pendulum 
 for the measurement of time, we obtain from it the readiest, 
 as well as the most precise method of satisfying our in- 
 quiries relative to some of the most important subjects in 
 Physical Science. 
 
 Thus, the equation T—tt \/ -, exhibiting the depend- 
 ence of any one of the quantities t, r, and ,«• on the remain- 
 ing two, suggests the mode of finding whether the accele- 
 rating force of gravity acts alike on all terrestrial substances, 
 or with an intensity which is different for the different spe- 
 cies of substances. Now in the same part of the earth, 
 pendulums of equal lengths are found to be perfectly 
 isochronous, however different the kinds of matter of which 
 they consist. This fact is ascertained with the greatest 
 precision : for whilst two pendulums are suffered to descend 
 together, if there were the smallest difference in the times 
 of their vibrations, this difference, after a number of vibra- 
 tions, would be multiplied by that number, and would 
 thereby become plainly perceptible; the pendulums not 
 moving in the corresponding points of their paths, i. e. the 
 rods not remaining parallel. The experiment, therefore, 
 shows that when the lengths are equal, the times are pre- 
 cisely equal ; and the result being the same, however the sub- 
 stances be varied, it follows, that all are equally accelerated 
 by gravity ; or that this force acts on all bodies, in proportion 
 to their quantities of matter. 
 
 The equation T = 7r y^- gives gzz — ^. If the pendu- 
 lum vibrates seconds, t is unit ; and / denoting the length 
 of such a pendulum, the equation is g=7r-l, which expresses 
 the velocity generated by gravity in one second, by the 
 space which would be described with that velocity continued 
 
S70 DYNAMICS. — SECT. VII. 
 
 uniformly for that time. The space through which a body 
 
 falls from rest in one second is one-half of this, or jr^;^; 
 
 which shows that the space described in one second, by a 
 body falling freely from a state of rest, is to half the length 
 of the second's pendulum, in the duplicate ratio of the peri- 
 phery of a circle to its diameter. 
 
 The foregoing expressions require only the substitution 
 of their numerical values for tt and /. Now in these lati- 
 tudes, I is found to be 39.14 inches. Wherefore, substitut- 
 ing this number for /, and for tt its value, viz. 3.14159, the 
 value of ^ will be 32.19 feet : and the space described by a 
 body falling freely from rest, will be 16.09 feet, or nearly 16 
 feet and one inch. 
 
 But the force of gravity, though constant in the same place, 
 is not of the same intensity at all parts of the earth's surface. 
 In Sect. 5. Art. 3. it has been shewn, that the effective force 
 of gravity is diminished from the poles to the equator, by the 
 centrifugal force. Hence it follows, that if the earth were a 
 fluid mass, the columns extending from the centre to the 
 surface would vary in length ; those nearer to the plane of 
 the equator rising, to balance by their greater mass, those in 
 which the effective gravity is greater. There is then a fur- 
 ther change of the effective gravity arising from figure. It 
 is shewn by the writers on Physical Astronomy, that the 
 figure, which the earth would thus assume, would be an 
 oblate spheroid; and that in proceeding from the equator 
 towards the pole, the increment of gravity on the surface of 
 a homogeneous body of this figure would vary as the square 
 of the sine of latitude : L e. according to the same law as 
 the increment immediately due to the centrifugal force. 
 Wherefore, g denoting the effective gravity at the equator, 
 and mo, the whole of the excess of that at the poles, the 
 increment at any latitude, X, would be wiosin.^X; and the 
 effective gravity at that latitude would be 
 
OF THE SIMPLE PENDULUM. 
 
 371 
 
 g—G{l-\-m sin.^X). 
 From this equation the value of m is known when the 
 ratio of £r to g is found : and this is the ratio of the lengths 
 of two isochronous pendulums, of which, one vibrates at the 
 latitude X, and the other at the equator. Wherefore, I and 
 L being those lengths, the equation is 
 
 As it may not be convenient to find the length of the 
 
 pendulum at the equator, l is to be exterminated; for which 
 
 purpose, a second equation will be requisite. Wherefore, 
 
 putting I' for the length of a second's pendulum at any other 
 
 latitude A', there is, in like manner, l' — l{1 -^in sinrX) ; and 
 
 - -. . . /' l+msin.-X' x\n ^^ • 
 
 by division -7-= V-; -■ — itt"- Which gives 
 
 •^ I l+wisin.-A 
 
 I'-l 
 
 Isin.'^X' — I'&in.-X 
 
 The number of vibrations performed in a given time can 
 be observed with more ease and precision than the length of 
 the pendulum : and the lengths of isochronous pendulums 
 being as the squares of the numbers of vibrations made by the 
 same pendulum in a given time, at those different latitudes, 
 the formula may be written thus : 
 
 n"-—n' 
 
 w-sin.^A — w sin. A 
 Wherein, n and w' denote the numbers of the vibrations made 
 in a given time by the same pendulum, at the latitudes A 
 and A'. 
 
 By making 7/z:?^ + f//7, and neglecting the terms which 
 are inconsiderable with respect to the rest, the formula be- 
 comes 
 
 _ 9.dn 'Mn 
 
 ~'w(sin.''^A'— sin.^A)~wsin.(A'+A)sin.(A'— A) 
 
 Experiments conducted on these principles give w^yif 
 
372 DYNAMICS. — SECT. VII. 
 
 Shewing that the difference of gravity at the equator and 
 the poles is the 191st part of the equatorial gravity. 
 
 The pendulum by giving the value of 7n, gives also the 
 ellipticity of the earth, i. e. the difference of the equatorial 
 and polar semidiameters divided by the former. It is proved, 
 indeed, by the writers before referred to, that if the earth 
 were of uniform density, the ellipticity would be ^ of the 
 fraction expressing the ratio of the centrifugal force to gra- 
 vity at the equator : and that the same would also be the 
 value of m; i. e. q denoting the centrifugal force at the equa- 
 tor divided by gravity, and e the ellipticity, there would be 
 ezz^q::zm. Now, 5'=7r^> wherefore, ^q=-2^j, for the 
 value of (? or m. This is Newton's discovery. 
 
 The supposition of a uniform density is in the highest 
 degree improbable ; and as the value of m depends on the 
 constitution of the earth, with respect to the density of its 
 interior parts, the value of m, found on that supposition, is 
 not to be regarded even as an approximation to the truth. 
 But Clairaut has shewn, that according to whatever law the 
 density may vary from the surface to the centre, the above 
 value of e and m, found on the supposition of a uniform den- 
 sity, is an arithmetical mean between the real values of those 
 quantities : i. e, that 
 
 e + m=.lq. 
 substituting in this equation for m and q their values, we have 
 
 The value of e can, indeed, be discovered by several 
 other methods, but not with the same facility, or certainty of 
 its accuracy. The most obvious of those methods is that by 
 the measurement of the meridional ares ; and the result ob- 
 tained by the most accurate of these measurements is -^j-^. 
 The ellipticity deduced by Laplace from the observed mag- 
 nitude of two of the lunar inequalities is g^_. 
 
 Clairaut has shown, that for a density increasing towards 
 the centre, the value of e would be less, and consequeinly,. 
 
OF THE SIMPLE PENDULUM. S^fi 
 
 that of m greater than ^q, or ^Ij ; that for a uniform density 
 there would be e = ^q; and for a density infinite at the 
 
 centre, e~ ^. Accordingly, for the same limits, there would 
 
 be mr=.^q and m^z^q. The values found by observation 
 are between those limits, and, therefore, show that the den- 
 sity of the earth increases towards the centre : a conclusion 
 antecedently probable, and further confirmed by experi« 
 raents made on the attractions of mountains. 
 
 6. Hitherto the arch of descent has been regarded as in- 
 definitely small. To deduce its value generally we must sub- 
 
 rdz 
 stitute for ds its value — -, in the equation 
 
 — ds 
 
 dt — 
 
 and we find 
 
 dt 
 
 — r.dz 
 
 This equation cannot be integrated in finite terms. To 
 expand it into a series, let the quantity under the radical 
 sign be multiplied, and divided by 2>-^. This changes the 
 equation to 
 
 dt-l y -. .- X —7=. 
 
 ^ g Vz'z-z' >y/i_^ 
 
 2r 
 
 Developing the factor ( 1 — ^ ) into the series 
 l + i-|:+i-f-^+M.|.-|^+&c. we have 
 
 dt 
 
 The terms to be integrated are of the form %-■ ^ ' Z . 
 
374} DYNAMICS. SECT. VII. 
 
 the exponents m being the natural numbers beginning with 
 cipher. Wherefore, putting the several integrals, 
 
 C —z'^.dz C z^-^.dz 
 
 J Ts^fF- '"" i7sr1i= •'"-' '""'■■, 
 
 ? ~dz 
 
 \ ■ . ^ "=A^, the time of the entire vibration will be 
 
 -='/^-^.+4(;^)».+i.f(^)\+ ) 
 
 *•-* Wl ^3+ &- \ ) 
 
 Each of the terms of this series after the first, is derived 
 from the preceding term : for, 
 
 ^Vzz — z^ m *" "^ 2m ^ ^ zz~z^' 
 
 These integrals are to be taken between 2; = s', and2;=0' 
 and at each of these limits, the first term of the second 
 member vanishes. Wherefore, 
 
 e -z^.dz _ ^ (2m-l) C ~^'^-\dz 
 J y/^^-z" ~^ 2m J "^ z'z~z' '■ 
 
 This formula exhibits the dependence of each of the in- 
 tegrals on the one preceding. Wherefore, all are known 
 
 when the first, viz. Aq, or \ - , is known. But 
 
 ^ "^ z'z — z^ 
 
 — d^ ( 2z— z\ 
 , , : = arc COS. — -, — . which, taken from z =z 2', 
 
 \/zz — z^ \ z J 
 
 C •— dz 
 z = 0, is \ — —TT. Wherefore, making m suc- 
 
 S 
 
 to , .. , 
 
 cessively, 1, 2, 3, 4, &c., we have 
 
 A,= 
 
 i.' 
 
 . TT. 
 
 
 
 
 
 A2 = 
 
 1^' 
 
 A, = 
 
 1 
 2- 
 
 .3 -'2 
 
 '.TT. 
 
 
 A3 = 
 
 l~^' 
 
 ^2 = 
 
 1 
 2' 
 
 3 5 
 
 ^/3 
 
 TT. 
 
 &C. 
 
 
 &C. 
 
 
 &C. 
 
 
 
 * See Lardnei-'s Integral Calculus, Sect, 4. Art. 222. 
 
OF THE SIMPLE PENDULUM. 375 
 
 Making these substitutions in equation (1), it becomes 
 
 This series, proceeding according to the powers of 
 
 —, always converges, inasmuch as z' is less than 2r : and 
 
 it converges the more rapidly, the smaller the quantity 
 
 -7—. If the arc is very small, the first term will suffice : 
 2r 
 
 which gives T = 7r V -, the same conclusion to which we 
 
 arrived by taking the arcs for their chords. Unless the arc 
 
 of vibration is very considerable, the two first terms will 
 
 z 
 suffice : of which, the second, called the correction, is -— . 
 
 or 
 
 Denoting the semi-angle by 0, rO is the arch whose verse 
 
 sine is z' ; and this arch being confounded with its chord, 
 
 , . , r .u r.u ^ z u . . . „ , 
 
 there is z'z=. — ; — :=—--. {/.p. and -—=.-—, givmff tor the 
 
 corrected value of the time of the vibration, 
 
 7. If the curve in which the body is constrained to move 
 be the cycloid, having its axis vertical, the relation between 
 s and ~ is expressed, generally, by the equation, 
 
 s'^—4 az. 
 
 in which a is the diameter of the generating circle. Let / 
 denote the arc corresponding to the ordinate z', and let 
 
 2 '2 
 
 — , — be substituted for z and z', in the expression 
 4a 4a 
 
 V 2g{z'— z) ; we then have 
 
376 DYNAINIICS. SECT. VH 
 
 The expression for the time is derived from the equation 
 
 — ds_ A /2a ^* 
 
 ~ ~ ^ —' Jin 
 
 dt- 
 
 which, by integration, gives 
 
 t = 
 
 V — . arc cos. = -, . 
 £• V sj 
 
 This requires no correction; since for ^ = 0, there is s=.s', 
 and therefore arc (cos. = -, j r= 0. 
 
 At the lowest point of the arc, * vanishes, and arc 
 (cos.=:0) = -. Wherefore the time of the descent to this 
 
 point is expressed by the equation ^ = ^ \/ — And if t 
 denote the time of the entire vibration 
 
 g 
 
 The time of vibration, therefore, is independent of the 
 magnitude of the arc described ; and the property of 
 tautochrcr,is7n, which belongs to the circle, only when the 
 arcs of vibration are indefinitely small, holds universally for 
 the cycloid. 
 
 The velocity and time may now be represented by a geo- 
 metrical construction in the same way as in Sect. 2. Art. 7. 
 Thus in (Fig. 148.) U being the whole arc of vibration, let a 
 tangent be drawn at v the lowest point, and let a circle, 
 I'v'l'. be described, whose centre is v, and radius \l' equal 
 to the cycloidal arc \L Then, if ym' is taken from the 
 centre, equal to the arc'v?;<, the ordinate erected at w/ or 
 
OF THE SIMPLE PENDULUM. 377 
 
 = V \l''^ — Y7u- — V \l'' — \'i/r — V s'^—s'-. Wherefore, 
 
 iv S-"^ s'— A - V ^.m'q. Accordingly, the ve- 
 ^ 2a ' ^a 
 
 locity at any point, m, varies as mq, the ordinate of the 
 circle raised at the corresponding point of its diameter. 
 And wm being the cosine of the arch I'q, to the radius v/', 
 
 V — 
 
 1 • I -5 \ wccl'q _,„ /Of, 
 
 there is arc [ cos. =i - I = — ~ . V> hence, t = a/ 1— . 
 
 (s\ A / 2a arc I'q , , . , , , „ , 
 
 COS. — -\ — y — . -T-^. showing that the time of the 
 sJ ^ v^ 
 
 g 
 
 movement from I, to any point, m, varies as the circular 
 arch I'q, cut off by the ordinate mq. 
 
 Owing to the property of tautochronism, which has been 
 shown to belong to the cycloid, it has been supposed, that 
 a pendulum made to oscillate in this curve, was peculiarly 
 fitted for the regulation of the movements of clock work. 
 Induced by this supposition, the mathematicians of the se- 
 venteenth century have bestowed more than ordinary atten- 
 tion on the properties of this curve: and though their 
 labours on this subject have not been rewarded by the 
 attainment of the proposed end, it must be admitted, that 
 they have materially contributed to the advancement of ma- 
 thematical science. 
 
 The method of causing a pendulous body to move in anv 
 given curve, is naturally suggested by the generation of a 
 curve from its evolute. The evolute of a cycloid consists of 
 two semi-cycloids, equal to the halves of the given cycloid, 
 or involute, and having their vertices at the extreme points 
 of its base. Thus, (Fig. 148.) the semi-cycloids oa, ob, 
 equal to bv, av, and similarly placed, shall constitute the 
 evolute of the cycloid avb. Accordingly, a body may be 
 made to vibrate in a given cycloid, by placing two semi- 
 cycloidal plates, each equal to half the given cycloid in the 
 
378 
 
 DYNAMICS. — SECT. VII. 
 
 positions shewn in the figure ; and then suspending the 
 body by a flexible string whose length is ov, or twice the 
 axis of the cycloid. For as the string applies itself alter- 
 nately to the plates oa, ob, the body shall always be found 
 in the curve avb. 
 
 But for the regulation of the movements of clock work, 
 it is not requisite that the pendulum should vibrate in the 
 arc of a cycloid : neither is the contrivance, whereby it is 
 made to move in that curve, fitted to insure tautochronism. 
 For the point at which the string parts from the cycloidal 
 plate is the temporary centre of motion. Consequently, the 
 part which at any instant hangs freely, together with the 
 appended weight, is to be regarded as the vibrating body at 
 that instant; rejecting that part of the string whose motion 
 is prevented by the cycloidal plate. The part of the pen- 
 dulum which hangs freely, being variable both in figure and 
 weight, the point at which it may be supposed to be concen- 
 tred, or, as it is named, the centre of oscillation is not neces- 
 sarily an invariable point of the substance of the pendulum. 
 Consequently, the contrivance, however well fitted to cause 
 any given point of the body to move in a cycloid, is not 
 fitted to give this motion to the ever varying centre of oscil- 
 lation. For these and other reasons the circular arc is in- 
 variably preferred. 
 
 8. It is a problem of some interest to determine whether 
 the property of tautochronism, which has been found 
 to belong to the cycloid, is possessed exclusively by that 
 curve. 
 
 The time of descent in any curve to the lowest point is 
 
 ds 
 found by integrating the equation dtzz. i between 
 
 \ 2g{z'~ss) 
 
 the limits z = z, and ^ = 0. Wherefore, if that time be 
 denoted by t', there is 
 
 '__r~' ^* 
 
OF THE SIMPLE PENDULUM. 379 
 
 And in order that the curve may possess the property of 
 tautochronism, the value of this integral must be indepen- 
 dent of z', the ordinate of the initial arc. We are then to 
 inquire what is the relation of s and z, which will fulfil this 
 condition. 
 
 Let s be supposed to be developed according to the as- 
 cending powers of z, so that 
 
 in which the coefficients and exponents are indeterminate. 
 It is evident then, in the first place, that the exponents a, 
 €, J, &c. are all greater than cipher, since s and z vanish 
 together. Differencing this expression, and substituting 
 the value of ds in the formula given above 
 
 a—l(lr 
 
 Jo 
 
 y/2g{z'—z) 
 
 or, making z=:z'u, 
 
 •&c. 
 
 ,_ ^,a— if»l uf^—'^du , - 
 t—aaz n — hfcr. 
 
 y 
 
 in which there will be as many terms as in the development 
 of s. Now, in order that this expression should be inde- 
 pendent of -:', the coefficient of / in one of these terms must 
 be cipher, and all the other terms must vanish. We must 
 have, therefore, 
 
 — 1 
 
 ■2* 
 
 a=i. 
 
 And since the exponents S, 7, &c. are all positive, as well 
 as the definite integrals taken between the Hmits 0, 1, the 
 other terms cannot vanish unless the coefficients b, c, &c. 
 vanish ; so that there is 
 
 b-0, c=0, &c. 
 and the equation of the curve is reduced to 
 
 s — azh. 
 
380 DYNAMICS. SECT. Vll. 
 
 which is the equation of the cycloid. The cycloid therefore 
 is the only plane curve possessing the property of tautochro- 
 nism. 
 
 9. On comparing the time of descent down an indefinitely 
 small arc of a circle, terminating at the lowest point with 
 that down its chord, it was found (Art. 4.) that the latter 
 is to the former as the circumference of a circle to four times 
 its diameter. From this example it appears, that the time 
 of descent from one point to another, not in the same ver- 
 tical, is not the shortest, for the shortest line that can be 
 drawn through those points ; and it becomes a problem to 
 find the line of swiftest descent. 
 
 Let A and b be the two points, and awzb the curve of 
 swiftest descent from the one to the other, (Fig. 149.) Then, 
 if two other points are taken in this curve, as m and m , the 
 intercepted arc mhm also must be that of swiftest descent 
 from one of those points to the other. For, let it be sup- 
 posed that any other arc, as mem , is described in a lesser 
 time than the arc mhm \ the velocity at m being that due to 
 the vertical height au , is the same for both ; and, therefore, 
 the body would take the same time to describe the remaining 
 arc m'B, by whatever course it arrives at m. Wherefore, 
 the difference of the times of describing the whole arcs 
 Amhm'^, and Amcm'E, shall be that of describing the parts 
 mhm , mem , which are not coincident. Accordingly, the 
 time of descent through the arc Amcm'^, would be shorter 
 than that of the descent through Ambm'B, contrary to the 
 supposition. If, then, the latter arc is that of swiftest de- 
 scent between its extreme points, the same property must 
 helong to each of its parts as mhm'. As this inference is in- 
 dependent of the magnitude of that part, mm' may be sup- 
 posed indefinitely small : and the body arriving at jn, with a 
 velocity due to the vertical height aw, must describe the arc 
 7nhm' in a less time than any other arc between the limits 
 m and m'. 
 
OF THE SIMPLE PENDULUM. 381 
 
 To express this condition, let nn be divided at the point 
 e into two equal parts dz\ then putting mi^z, mnzzy, 
 hm — s, the velocity at m shall be ^^2^^, and the time taken 
 
 ds 
 to describe mh or ds shall be . In like manner, put- 
 
 \\x\g ke — z-[-dzzzz , he—y, and pJi — s-\-ds—s\ the time of 
 describing the arc hm or (//, shall be ■ , " — . And the 
 
 time taken to describe the whole arc mm shall be 
 
 ds ds' 
 
 + 
 
 The condition requires that this should be a minimum : i. e, 
 less than for any other curve that can be drawn betw^een the 
 points m and m, i. e. 
 
 ^f ds ds' \ 
 
 0. 
 
 The character S denotes the variation of the position of a 
 point from one curve to another ; differing in this respect 
 from the character d, which denotes the change of position 
 in the same curve. In the present instance, z, z' are inde- 
 pendent of those variations ; being the same whether the 
 cuvxe is mbm or 7nc?n'. Therefore, §2=: 0, dz' — Q. Accord- 
 
 , S.ds ^.ds' r\ rt 1 ^ , o o - 
 
 i"gly' —1 h -y-r= 0. But ds—{dz- + dirY, and because dz 
 
 \/z V~ 
 
 has no variation, ^.ds = ^dz'-{-dyr =~l^^-'^^- 
 
 Vdz'-^dif ds 
 
 And in like manner, S.ds'zz- ' '' . Therefore, 
 
 ds 
 
 dyXdy dj/Ml_ 
 
 dsy/z '^ ds'V^' ~ ^^ 
 
 Now, whatever be the curve described between the 
 
 points m, m, the difference between the ordinates at those 
 
3S2 STATICS. SECT. VII. 
 
 points is constant, i. e. dy-\-dy is a constant quantity: 
 wherefore, ^[dij-\-cli/)—0. i.e. S.c/y=—S.c///', which changes 
 equation (2) to 
 
 dji dy' _^ 
 
 ds^ z ds V z 
 
 The second term of this equation being what the first be- 
 comes, when the point m is changed for the point h, the 
 
 equation expresses the same thing as d ( ):^0. Where- 
 
 \dss/z/ 
 
 fore, -— ^- — = c. But -^ is the sine of the ansle which the 
 </sV~ ds ^ 
 
 arc ds makes with the axis of z. And where the arc is 
 
 horizontal, this angle is right ; and, therefore, its sine=l. 
 
 Wherefore, putting a for the unknown abscissa of the point 
 
 where the tangent is horizontal, there will be c =: — ; — 
 
 Whence 
 
 dy 
 
 1 % A /^ 
 
 = 77:;' °^:7;=V; 
 
 dsy/z \J a ds ^ a 
 
 Eliminating dy between this and the equation ds^-::^dy'^-\-dz^, 
 
 there is ds — dz'x/ . 'Whence, s-=i —2y/ a[a—z)-\-c. 
 
 For *z=0, there is z — Q. Wherefore, c'=:2«, and 
 
 s:=i2a — 2^a{a — z). 
 
 This is the equation of a cycloid, the arc s being 
 measured from the horizontal base, and a being the 
 axis. 
 
 To find a, the axis of the cycloid, or the diameter of its 
 generating circle, let a cycloid be described on the base ad, 
 of any length, measured from the point a, on the horizontal 
 line AY : and let this curve intersect the right line ab at k. 
 
OF THE SIMPLE PENDULUM. 383 
 
 Then drawing kd, and from b the parallel be, ae shall be 
 the base equal to ira. 
 
 This construction is founded on the principle, that all 
 cycloids are similar curves. Such are all curves defined by 
 the same equation, provided that equation involves but one 
 constant quantity. 
 
JS4 
 
 DYNAMICS. — SECT. VIII. 
 
 SECTION VIII. 
 
 OF MOTION CONSTRAINED BY A SURFACE. 
 
 1. The case of motion restrained by a surface, is to be 
 treated much in the same way as that of motion restrained 
 by a hne. The difference of conditions will, however, create 
 some difference in the theory: for which reason, it seemed 
 expedient to reserve this part of the subject for a distinct 
 consideration. 
 
 The fundamental equations are those already given in 
 Sect. 6. Art. 1. And the expressions for the time and velocity 
 are derived from them by the method employed in that arti- 
 cle. But if it is desired to express the relation between the 
 time or the velocity, and one of the coordinates, there being 
 but one equation of the surface, it will be necessary to use a 
 second equation to be obtained by eliminating n between two 
 of the equations (1). By this proceeding, the cosines are 
 not made to reappear, inasmuch as they may be expressed 
 by the co-efficients of dx, dy, dz, in the differential equation 
 of the surface. That equation being represented generally 
 by ldx-\-mdy-\-ndz = 0. gives 
 
 COfi,ff— . — . cos.W = 
 
 cos.t/ = , - * 
 
 V/2 + wi-^ + «-^ 
 
 Whereby the three equations, (1), become 
 
 * See Lardner's DifTeiential and Integral Calculus. 
 
OF MOTION CONSTRAINED BY A SURFACE. 385 
 
 -— :=Y-f N. — - ■ 
 
 dt' V /2 + m- 4- n"- 
 
 --^^z+N. 
 
 V l^ + iri^ + w' 
 
 Multiplying the first of these hy dx, the second by dy, and 
 the third by dz, we have, as before, 
 
 dx.d^x-\-dii.d^il-\-ds.(^z , , , . , / n 
 — j.y^^ —i^.dx-\-x.dij-\-L.dz. (a) 
 
 From which we get the expressions for the time and velo- 
 city, as in Sect. 6. Articles 3 and 4, with the other conse- 
 quences there derived, relative to those quantities. 
 
 A second independent equation is obtained by eliminating 
 N between any two of the fundamental equations. Thus, 
 multiplying the first by m, and the second by /, and then 
 subtracting one from the other, we have 
 
 m.d^x — l.d^y , ,,. 
 
 — — -, =mx — Iy. (b) 
 
 di- ^ ' 
 
 By means of this and of the equation of the surface, two 
 -of the coordinates may be eliminated from equation (a), and 
 t or V thereby expressed as a function of one of the co- 
 ordinates only. 
 
 The trajectory itself is not given; but by eliminating dt 
 between the equations {a) and (i), we get an equation, which, 
 with that of the surface, exhibiting the relations of the co- 
 ordinates, furnish the two equations of the trajectory. 
 
 % The expression for the pressure is found by adding 
 
 the three fundamental equations multiplied by I, m, n, re- 
 
 ds^ 
 spectively; and substituting for dt- its value, viz. -^. By 
 
 v 
 
 this process we get 
 
 2c 
 
386 DYNAMICS. — SECT. VIII. 
 
 _ 1)^ LdP^x + m.<Py + n.d^z Ix 4- »«y + n'Z 
 
 This is the reaction ; to which the pressure on the surface is 
 equal and opposite. 
 
 Instead of applying to this formula, the pressure may be 
 
 conveniently derived from the forces r and — . In the use of 
 
 which method, it is to be observed, that when the motion is 
 restrained by a surface, the reaction at any point of the tra- 
 jectory is in the perpendicular to the surface : and as there 
 is but one normal at each point of the surface, the forces 
 
 R and — , are to be reduced to this direction. Wherefore, 
 
 P 
 putting a, a' for the angles made by the normal with the di- 
 rection of R, and with that of the radius of the circle of cur- 
 
 vatureatthe same point, the pressure is R.cos.aH .cos. a'. 
 
 P 
 3. Some important conclusions are to be derived by con- 
 sidering the other components of the same forces. Where- 
 fore, let the three axes be the normal to the surface, the 
 tangent to the curve, and a perpendicular to both at the 
 same point, which will be in the plane tangent to the surface. 
 
 t;2 
 
 The forces r and — , resolved according to the normal, con- 
 stitute the pressure already considered. In the tangent to 
 
 v 
 the curve the force — has no effect ; and the force R, re- 
 
 P 
 solved in that direction, exerts no pressure, the effect of this 
 
 resolved force being altogether expended in changing the 
 velocity. The sum of the two forces reduced to the third 
 axis is cipher : otherwise, the pressure would not be perpen- 
 dicular to the surface. Accordingly, putting y, y for the 
 
 angles made by the directions of the forces r and — , with the 
 
 P 
 third axis, this condition is expressed by the equation 
 
OF MOTION CONSTRAINED BY A SURFACE. 387 
 
 R.COS.'y ^ .COS.7 =0. (c) 
 
 This equation will serve to determine the inclination of 
 the plane of the osculating circle to the constraining surface 
 at each point. For the tangent to the trajectory is the in- 
 tersection of the plane of the osculating circle and the tan- 
 gent plane : and the angle -y' being contained by two lines 
 in those planes, and perpendicular to their intersection, is 
 that by which the inclination of those planes is measured. 
 
 When R=0, the last equation gives cos.7'zi:0, oryziOO". 
 Which shows, that when the body is set in motion by an im- 
 pulse, and not afterwards solicited by any moving force, the 
 plane of the osculating circle is, at every point, perpendicular 
 to the constraining surface. 
 
 The same thing would appear by supposing the con- 
 straining surface to be polygonal. In passing from one face 
 to another, the body is deflected by a force perpendicular to 
 that on which it enters : and as its motion on this face is 
 compounded of that in the preceding, and of that due to the 
 reaction, it follows, that the two contiguous elements of its 
 path are in a plane perpendicular to the face containing the 
 second of those elements. The same thing is to be under- 
 stood of every two contiguous elements of the path de- 
 scribed on the polygonal surface. And the observation being 
 extended to the limiting curved surface, it is inferred that 
 the radius of the circle of curvature of the trajectory is every 
 where perpendicular to the surface. 
 
 This being the exclusive property of the shortest Hue 
 that can be drawn on the surface, between two given points, 
 it follows, that if a body moves on a curved surface, un- 
 solicited by any accelerating or retarding force, the course 
 it takes between any two points of its path, is that of the 
 shortest line that can be drawn on the surface between those 
 points. But this inference is not limitetl to the case of 
 
388 DYNAMICS. — SECT. VIII. 
 
 iirrO. For if this were a tangential force, such as the force 
 of friction^ or the resistance of a fluid medium, the angle y 
 would be right, and cos.y^rO. In which case, equation (c) 
 equally gives cos.y' = 0. or, ^' = 90". 
 
 It is scarcely requisite to observe, that when RzrO. in 
 which case the velocity is constant, the pressure is every 
 where reciprocally proportional to the radius of curvature. 
 
 4. To exemplify this theory, let a body influenced by 
 the force of gravity receive an impvdse communicating to it 
 the velocity v, and let the motion be restrained by a surface 
 of revolution whose axis is vertical. 
 
 Putting r for the radius of the circle described by any 
 point of the generating curve, and ^ for the abscissa; the 
 equation of that curve will be of the form jxh + qdrz^O. 
 The normal at any point of the surface being in the plane of 
 the generating curve passing through that point, the re- 
 action of the surface may be resolved into two, of which one 
 is in the vertical, and the other in the direction of r, or per- 
 pendicular to the axis of revolution. Putting / for the arc 
 of the generating curve, measured from the vertex, these 
 
 resolved forces are N.-7-,. and — n.-t-,. Of which the latter 
 as as 
 
 is again resolved into — n.-^,. -. — n.-^,. -. in the directions 
 
 as r as r 
 
 of X and y respectively. Accordingly, the fundamental 
 
 equations are 
 
 df ~ ' ds" r 
 
 dht dz 7/ y f » 
 
 rPz dr 
 
 Eliminating N from the two first of these equations, we 
 
OF MOTION CONSTRAINED BY A SURFACE. 389 
 
 2/ dbOij ' ' oc d li 
 get ^-^ — '- — ^= 0. and by integration, y.dx — x.dy= 
 
 celt, or putting w for the angle contained between r and the 
 axis of a:, 
 
 r'^dn) zz cdt. {e) 
 
 This equation shows that the areas projected on a horizontal 
 plane are proportional to the times. 
 Equation («) becomes in this case, 
 
 dx.d^x-\-dy,d'^y-\-d%.d^z 
 
 ^^-T^— = —mh. 
 
 dt^ ^ 
 
 or, 
 
 d^M±cll±df)_ 
 
 dt- ^ 
 
 which, by integration, gives 
 
 dx I (/?/ — |— d^ 
 
 yr, = — 2gz-{-c. or, v^= — 2gz-{-c. 
 
 Putting z' for the height of the point of projection, the 
 value of c will be expressed by the equation c = v"^-{-2gz'. 
 Wherefore, v^ = v"^-\-2g{as'—z). Or putting // for the height 
 due to the velocity v, it is 
 
 v'=2g{h'+z'-z). if) 
 
 The path of the body shall be ascertained by find- 
 ing an equation between w and ;;-. And for this, we have 
 
 dx^-{-dif-\-dz- r. /T ' \ 1 / o 7 , 
 j-:^ zr 2g\li -\-z — z). ana cat = v. du). between 
 
 which, dt being eliminated, there is 
 
 c\dx'-^dy'' + dz'') = 2r^g{/i'+x'- z)doj\ 
 
 Moreover, dx'^-\-dy- is the square of the element of the arc 
 described by the moving point, when projected on the plane 
 of xy. Its value, therefore, is dr'-^-r-dw^. Also, r=fz, 
 given by the equation of the generating curve ; and d^—pdr : 
 wherein, p denotes the tangent of the inclination of the ele- 
 
390 DYNAMICS. SECT. VIII. 
 
 merit of the generating curve to the horizon. Making these 
 substitutions, the equation is 
 
 Whence, 
 
 d..= ^VT+P'-d^^_ . ^g^ 
 
 and 
 
 c p^ 2g{fa)\h'+z'~z)~c'' 
 
 Expanding the second member of this equation, and inte- 
 grating any number of the terms, there is found an approxi- 
 mate value of t, as a function of z. 
 
 If the apsides are sought, we have only to make 
 
 dz 
 
 — =0. or 2g{fzf{h'^z'-.'z)-c'-Q, and to solve this 
 
 equation for z. 
 
 To derive the expression of the pressure from the equa- 
 tions [d), the first of those equations is to be multiplied by 
 
 — x.dz . , , — y.dz , . 1 . T , dr 
 
 -T— , the second by — '-rr-, and the third by -r-,, and 
 
 rds r.ds "^ ds 
 
 then adding, we have 
 
 _ r.dr. d^z — dz{x.d^.v -{-y. d^y) dr 
 ~ r.ds .df" ^' ds' 
 
 ds 
 And as dt= — , the equation is 
 
 _ v"^ [r.dr.d^z—dz{x.d^a: + y.dh/)] dr 
 
 To perceive the agreement of this formula, with the 
 rule already given in Art. 2., it is to be observed, that 
 x^-\-y^ — '>''> twice differenced, gives x.d^x-\-7/.dr^y—r.d'r-{- 
 dr'^—dx'—di/'f which is reduced by the equations dz'-^dr"^ 
 
OF MOTION CONSTRAINED BY A SURFACE. 391 
 
 -ds'\ dx^-\-dt/+ds''=d^ to x.d'x + i/d''i/-r.(Pr-\-ds"'-ds\ 
 Wherefore, 
 
 v"" {r{dr.d:'z~dz.d'r)-[-dz(ds'~dP)'\ dr 
 
 ds^ r.ds' ^ ^ ds' 
 
 Further, p being put for the radius of curvature of the 
 generating curve at any point, its vahie is given by the 
 
 ds^ 
 "^^^"*^"" ^ - drA~d.,dV Wherefore, 
 
 _^ cJf_, o {ds--ds"') dz dr 
 
 ^^ p" ds'' '^''' ds' r.ds'^^'ds" 
 
 In this expression, we find the velocity resolved into 
 
 two, viz. v.-r- J according to the tangent of the generating 
 
 ds '^ ^ too 
 
 y/ds^—ds'^ 
 curve ; and v. , according to the tangent of the 
 
 (ZS 
 
 circle described by a point in that curve. The square of the 
 
 former divided by p' is the centrifugal force arising from 
 
 that velocity : and this force is directed in the normal. 
 
 ^2 ( els-— dp) 
 
 — P5 — - is the centrifugal force, arising from the velocity 
 
 r ds~ o o J 
 
 reduced to the tangent of the circle. The direction of this 
 is that of r, the radius of that circle ; and being multiplied 
 
 dz 
 by -7-,, it is reduced to the normal. Wherefore, the sum of 
 ds 
 
 the two first terms in the value of n is the expression of the 
 
 centrifugal force reduced to the normal. The last term is 
 
 the force of gravity reduced to the same direction. Whence 
 
 it appears, that the whole expression is equivalent to R.cos.a-f- 
 
 o 
 
 ^ f 
 
 — . COS. aJ. 
 P 
 
 5. If it is required to find the velocity of projection, in 
 
 order that the motion of the body should be horizontal, it is 
 
392 DYNAMICS. — SECT. VIIL 
 
 to be observed that r and ^ are then to be treated as con- 
 stant' quantities. This, however, will not affect the differen- 
 tials in the second members of the equations {d) which are 
 there employed only hypothetically to denote the angles. 
 This being observed, the third of those equations becomes 
 
 = — ^-f-N.y-/. Eliminating n between this and the first of 
 
 those equations, we have — -tr- =:—£(•.-.— =z —«,£f.-. Buta:=: 
 ^ dt- ^ r dr '■ ^ r 
 
 r. cos.w. And this twice differenced for r constant, gives 
 
 (V'x r. COS. w.d(t)^ T-< ■ 1 1 ,^ d''^' i 
 
 -7--= r^ . xLiquatmff these values 01 —r^, we have 
 
 dt^ dt- ^ ^ dt' 
 
 dti)^ A 1 ^'d(ii , . . ., >^ 
 
 ^''-jji^Jg- And as —-zzv, the equation is v'=2).g.r. Or 
 
 putting for n^ its value, viz. '2gh, it is 
 
 Now JO is the tangent of the angle made by the element 
 of the generating curve with the horizontal ordinate r ; 
 wherefore, pr is the subtangent, i. e. the portion of the axis 
 between the ordinate r and the tangent : and therefore, //, 
 the height due to the velocity of projection, is half the sub- 
 tangent. If the body is projected with this velocity in the 
 tangent of a horizontal section of a surface of revolution, 
 whose axis is vertical, it shall continue to move in that sec- 
 tion. 
 
 The time of the revolution, or the periodic time of a body 
 
 . , . . . ^TT.r 27rr Stt . / r „ ^ 
 moving in this section, is Tr= ~ , - n:— --.y -. l>ut 
 
 V yf pgr V g P 
 
 -, is the subnormal, i. e. the portion of the axis contained 
 P 
 
 between the normal and the horizontal ordinate r. Where- 
 fore putting n for this line, we have 
 
OF MOTION CONSTRAINED BY A SURFACE. 393 
 
 T-=z—--.vn. 
 
 Vg 
 
 If the surface is that of a paraboloid, the subtangent is 
 bisected at the vertex ; wherefore, the velocity for circular 
 motion is that due to the abscissa, or the portion of the axis 
 cut ofFby the plane of the circle. 
 
 If the surface is that of a sphere, the velocity of projec- 
 tion for circular motion near the vertex or lowest point of 
 the segment, as in the case of the paraboloid, is that due to 
 the abscissa or height above that point. But this height 
 increasing, the subtangent increases much more rapidly, be- 
 coming infinite for a height equal to the radius of the sphere. 
 Wherefore for motion in a horizontal plane passing through 
 the centre of the sphere, the velocity of projection should be 
 infinite. 
 
 The time of the revolution given by the equation 
 
 Tzz— — -.-i/w, varies as the square root of the portion of the 
 
 Vg 
 
 vertical diameter between the centre of the sphere, and the 
 plane of the circle described by the movement. 
 
 The condition of a body moving on the surface of a 
 sphere, is the same as if it were suspended from the centre 
 by a cord equal to the radius : the tension of the cord 
 then taking the place of the pressure, and the reaction of the 
 fixed point that of the constraining surface. Such is called 
 a conical pendulum. And as the time of oscillation in a 
 cycloid or small circular arc, was given by the equation 
 
 T—ttA/-. it is evident that a conical pendulum shall 
 
 ^ g 
 perform its circular revolution in a time equal to that 
 
 of the double oscillation of a pendulum whose length is 
 equal to n, or the distance of the plane of the circle, de- 
 scribed by the conical pendulum, from the point of sus- 
 pension. 
 
 2d 
 
394' DYNAMICS. SECT. VIII. 
 
 In the case of horizontal movement on a surface of 
 revolution, the pressure on the surface is at once given 
 
 by the third of the equations (d) ; which for -T-2- = 0.is 
 
 ds 
 N=rg.— . The coefficient of g, in this equation, is the se- 
 cant of the angle made by the element of the generating 
 curve with the horizon ; and this angle is equal to that 
 between the normal and vertical axis. Wherefore, in the 
 case of the horizontal movement of a body influenced by 
 gravity, the pressure on the surface is the weight mul- 
 tiplied by the secant of the angle contained between the 
 normal and axis. 
 
 It seemed proper to show how these conclusions respect- 
 ing the horizontal motion of a heavy body on a surface of 
 revolution described round a vertical axis, were to be ob- 
 tained from the general theory of constrained motion. But 
 the problems respecting such motions are readily solved, 
 by equating the force of gravity and the centrifugal force, 
 both reduced to the direction of the tangent to the 
 generating curve : or which is the same thing, by resolv- 
 ing the force of gravity into two forces, of which, one is 
 in the direction of the normal, and the other in that of the 
 radius of the circle ; and equating this last to the centri- 
 fugal force. 
 
 Putting fi for the normal, and n, as before, for the sub- 
 normal, the force in the direction of the former is -.£«•. This 
 
 n ® 
 
 which is the pressure on the surface, is equilibrated by the 
 
 J- 
 reaction. The central force is -.g. And this is equal and 
 
 opposite to the centrifugal force. The expression for this 
 
 force, already given, is — . or —-. Wherefore, -^zz-.s. 
 
OF MOTION CONSTRAINED BY A SURFACE. 395 
 
 or li-=.—, i. e. the height due to the velocity of projection is 
 
 equal to half the subtangent ; which is the conclusion before 
 obtained, respecting the velocity of projection required for 
 the horizontal motion of a heavy body, on a surface of revo- 
 lution about a vertical axis. 
 
 END OF VOLUME THE FIRST. 
 
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