- 
 
 
 REESE LIBRARY 
 
 -1 Itll: 
 
 NIVERSiTY OF CALIFORNIA. 
 
 rS8 
 
 Accessions No. ^ ^ ^2 3 Sheh A ' 
 
KINEMATICS. 
 
 A TREATISE ON THE MODIFICATION OF MOTION, AS AFFECTED BY 
 
 THE FORMS AND MODES OF CONNECTION OF THE 
 
 MOVING PARTS OF MACHINES. 
 
 FULLY ILLUSTRATED 
 
 BY ACCURATE DIAGRAMS OF MECHANICAL MOVEMENTS, 
 AS PRACTICALLY CONSTRUCTED ; 
 
 FOR THE USE OF DRAUGHTSMEN, MACHINISTS, AND STUDENTS OF 
 MECHANICAL ENGINEERING. 
 
 BY 
 
 CHARLES WILLIAM MAC CORD, A.M., Sc.D., 
 
 PROFESSOR OF MECHANICAL DRAWING IN THE STEVKN8 INSTITUTE OF TECHNOLOGY, HOBOKEN, II. J. 
 
 AUTHOR OF "LESSONS IN MECHANICAL DRAWING," "A PRACTICAL TREATISE ON THE 
 
 BLIDE VALVE AND ECCENTRIC," AND VARIOUS MONOGRAPHS ON MECHANISM. 
 
 UNIVERSITY 
 
 NEW YORK : 
 
 JOHN WILEY & SONS. 
 
 1883. 
 
COPYRIGHT, 1883, 
 BY CHAS. WILLIAM MAC CORD. 
 
Snscribeb 
 
 TO THE REV. JOHN T. DUFFIELD, D.D., LL.D., 
 
 OF PRINCETON^ COLLEGE, 
 BY HIS FRIEND AND FORMER PUPIL, 
 
 
PREFACE. 
 
 A word of explanation is due to the reader, in view of the fact that 
 the following pages relate to but a small number of the vast array of 
 devices included in the broad term Mechanism. 
 
 Having in the opening chapters considered the methods by which 
 motion, and the modification of motion, may be represented and an- 
 alyzed, and the basis upon which a proper classification of elementary 
 mechanical combinations may be made, the next question was, what 
 classes of such combinations should be first examined. So large a 
 proportion of these consist of pieces rotating in contact about fixed 
 axes, that they seemed to have a natural claim to precedence, which 
 was duly recognized. 
 
 Attention was accordingly next directed to the discussion of the 
 pitch surfaces, and in natural sequence to the forms of the teeth, of 
 gear wheels of all kinds. Which having been done, it appeared 
 proper to publish so much as was completed, because notwithstanding 
 that many treatises upon these special topics have been written, there 
 would seem to be room for another ; the more particularly since even 
 in sweeping out this part of the shop, scraps of new material and cut- 
 tings of old have been found, in quantity and of quality to be worth 
 using. 
 
 The endeavor has been made to treat the theory of the subject in a 
 practical manner, for the benefit of the practical man. That is to 
 say, the demonstrations are made as far as possible directly dependent 
 upon the diagrams ; and the latter, in most cases reduced from work 
 actually executed upon a large scale, are accompanied by explanations 
 which it is hoped will enable any ordinarily expert draughtsman to 
 "lay out" the movements with ease and accuracy. 
 
 In order to avoid interrupting the argument by subordinate discus- 
 sions, as well as for more ready reference, an Appendix has been 
 added, containing the methods of construction, and other graphic 
 
Till CONTEXTS. 
 
 CHAPTER VI. 
 
 TAGE 
 
 Tooth Gearing 1. Classification of Toothed Gearing. 2. Spur Wheels. 
 Epicycloidal Teeth, Outside Gear, Generation of the Outline. Pitch, 
 Angles of Action, Backlash, Clearance, Approaching and Receding 
 Action. Interchangeable Wheels. Size of Describing Circle. Rack and 
 Pinion. 3. Inside Spur Gearing. Intermediate Describing Circle. Limit- 
 ing Diameters of Generating Circles 88 
 
 CHAPTER VII. 
 
 Low-Numbered Pinions Practical Limit of Obliquity Pinions with 3, 4, and 
 5 Leaves Least Number that can be Used with a Given Driver The Two- 
 Leaved Pinion Low Numbers in Inside Gear Various Examples Infe- 
 rior and Superior Limits Two-Leaved Pinions in Inside Gear Least Num- 
 ber that can be Used with a Given Follower 110 
 
 CHAPTER VIII. 
 
 Spur Gearing, Continued Epicycloidal System. Limiting Numbers of Teeth 
 for Various Arcs of Action. Details of Trigonometrical Process of Deter- 
 mination. The Nomodont or Curve of Limiting Values 128 
 
 CHAPTER IX. 
 
 Spur Gearing, Continued Involute Teeth. Involute Generated by Rolling of 
 Right Line on Base Circles. Peculiar Properties. Original Pitch Circle. 
 Rack and Pinion. Annular Wheels. Low-numbered Pinions. Involute 
 Tooth with Epicycloidal Extensions. Limiting Numbers for Given Arcs 
 of Recess. Comparison of the Involute and Epicycloidal Systems. Invo- 
 lute Generated by Rolling of Logarithmic Spiral on Pitch Circles 143 
 
 CHAPTER X. 
 
 Spur Gearing, Continued. Conjugate Teeth. Sang's Theory. Path of Con- 
 tact. Unsymmetrical Teeth. Approximate Forms. The Odontographs. 
 Diametral Pitch. Manufacture of Gear Cutters. Determination of Series 
 of Equidistant Cutters 167 
 
 CHAPTER XI. 
 
 1. Twisted Spur Gearing. Hooke's Stepped Wheels. Twisted Wheels. Rota- 
 tion not due to Screw-like Action. Elimination of Sliding. Neutraliza- 
 tion of End Pressure. 2. Pin Gearing. Generation of Elementary Tooth. 
 Derivation of Working Tooth. Peculiarities of the Action. Racks and 
 Pinions. Annular Wheels. Determination of Angle of Action. Deter- 
 mination of Limiting Numbers of Teeth. Wheels with Radial Planes. 3. 
 Non-Circular Spur Gearing. Construction of Teeth for Elliptical and 
 Lobed Wheels. . . 195 
 
CONTENTS. i x 
 
 CHAPTER XII. 
 
 PACK 
 
 1. The Teeth of Bevel Wheels. General Principles of their Correct Forma- 
 tion by means of a Describing Cone. Tredgold's Approximate Method. 
 Details of Correct Process. Results of the two Methods Compared, for 
 Epicycloidal Teeth. Inside Bevel Gearing. The Involute System. Action 
 of Bevel and Spur Wheels Compared. Teeth of Conical Lobed Wheels. 
 Methods of Cutting the Teeth. Corliss's Bevel Gear Cutter. Twisted 
 
 Bevel Wheels. 2. The Teeth of Skew Wheels. Theory of Willis and 
 
 Rankine. Describing Hyperboloid Approximate Methods. Direct Con- 
 struction of Teeth thus Generated. Fallacy of this Theory. Such Teeth 
 Impracticable. A New Theory. Oliviers's Involute in Different Planes. 
 The Fronts of the Teeth Single Curved. They Vanish at the Gorge. The 
 Length of the Tooth Limited. Backs of Teeth Warped. The Conjugate 
 Teeth Unlike. Action Reversed on Crossing the Gorge. Twisted Skew 
 Wheels. 3. The Teeth of Screw Wheels. Common Worm and Wheel. 
 Construction Referred to that of Rack and Pinion. Distinctive Features 
 cf the Action. Close-fitting Tangent Screw. Practical Proportions. 
 ' Sang's Theory Embodied. Multiple- threaded Screw Wheels. Screw and 
 Rack. Oblique Screw Gearing. Construction of Teeth from Oblique Rack 
 and Pinion. Peculiar Modification of the Action. Close-fitting Oblique 
 Worm. Oblique Screw and Rack. Construction for Least Amount of 
 Sliding. Resemblance to Skew Gearing. Hour-glass Worm Gear. 
 General Arrangement. Form of the Pitch Surface of the Worm. Forms 
 of the Threads. Action Confined to one Plane. Forms of the Wheel-teeth. 
 Multiple-threaded Hour-glass Worm with Face-gear Wheel. Rollers Sub- 
 stituted for Teeth upon the Wheel. 4. The Teeth of Face Wheels. Equal 
 Wheels with Cylindrical Pins. Equal Wheels with Axes perpendicular to 
 each other. Unequal Wheels Similarly Situated. Unequal Wheels with 
 Axes in Different Planes. Miscellaneous Arrangements of Face Wheels. 
 Combination of Face and Screw Gearing. Spherical Screw and Wheel. . . 232 
 
 Appendix 295 
 
 Index.., 329 
 
PRACTICAL KINEMATICS. 
 
 CHAPTER I. 
 
 1. Mechanism is the science which treats of the designing and con- 
 struction of machinery. Its objects are, to investigate those abstract 
 principles which are involved in planning correctly, and to describe 
 the practical operations involved in successful execution. 
 
 2. A Machine is very properly said, in a general way, to be an arti- 
 ficial work which serves to apply or to regulate moving power. 
 
 This definition will not answer our purpose, for which it is not suf- 
 ficiently minute ; but it is correct so far as it goes, and close and clear 
 enough for the ordinary employment of the word. 
 
 From its terms we infer that a machine applies or regulates ex- 
 traneous power for some useful purpose. That is to say, the existence 
 of the machine presupposes the existence of something to be done 
 and of power to do it ; and it also implies the necessity of modifying, 
 in some way or other, both the force and the motion caused by the 
 force. No machine can move itself, nor can it create motive power ; 
 this must be derived from external sources, such as the falling of a 
 weight, the uncoiling of a spring, or the expansion of steam. 
 
 3. Motive power has different characteristics, according to the 
 nature of the source. It may be constant, as in the case of a head of 
 water kept at the same level by an unfailing stream ; it may vary ac- 
 cording to regular laws, as when derived from expanding steam ; it 
 may vary irregularly, like the strength of animals : or it may be wholly 
 fitful and uncertain, like the wind. 
 
 But these characteristics, as well as the supply of power itself, are 
 beyond our control. We cannot create power as we want it, but must 
 take it whence we can get it. We cannot stipulate conditions, but 
 
2 PURE AND CONSTRUCTIVE MECHANISM. 
 
 must take the power as we find it, be thankful for it at that, and 
 adapt it to our needs and purposes as best we can. 
 
 4. This is done by the use of machinery : and it is clear that in the 
 construction of every machine, reference must be had to the charac- 
 teristics of the motive power as well as to the nature of the work to be 
 done. We may therefore amplify the definition above given, and say 
 that A Machine is an assemblage of moving parts, interposed between 
 the power and the work, for the purpose of adapting the one to the 
 other. 
 
 It is not, however, always necessary to trace back the source of 
 power to its origin. For example, a line of shafting, whether itself 
 driven by wind, water, or steam, may properly be considered as the 
 "source of power" in reference to the various drills, lathes, planers, 
 etc., driven by it. 
 
 Pure and Constructive Mechanism. 
 
 5. The operation of any machine depends upon two things, viz. : 
 definite force and determinate motion. And in the process of design- 
 ing, due consideration must be given to both these, so that each part 
 may be adapted to bear the strains put upon it, as well as to move 
 properly in relation to other parts. 
 
 But the nature of the movements does not depend upon the strength 
 nor upon the absolute dimensions of the moving pieces, and may often 
 be clearly illustrated by a model whose proportions are very unlike 
 those of the actual working machine. Consequently the force and the 
 motion may be considered separately ; and thus the science of Mechan- 
 ism is divided into two branches, called respectively Pure and Con- 
 structive. 
 
 6. The selection of materials, and the proportioning of details with 
 reference to strength and durability, are governed directly by consid- 
 erations relating to the forces involved. Closely connected with these 
 are other considerations relating to facility in manufacturing, con- 
 venience in repairing, and kindred features essential to practical ex- 
 cellence ; and the whole fall properly within the scope of Constructive 
 Mechanism. 
 
 But we may examine the action of a machine by merely putting 
 it in motion, without actually setting it at work ; and we can plan 
 its movements without regard to the requisite strength of the 
 parts. 
 
 The laws of motion may be discussed quite independently of any 
 consideration of the force involved, and without reference to either 
 
MOTION AND BEST. 3 
 
 the power or the work ; and this constitutes the branch called Pure 
 Mechanism. 
 
 7. Purposing now to confine our attention to this latter branch of 
 the science, it is necessary, before entering upon its study, to define 
 and explain the sense in which certain terms and phrases of frequent 
 occurrence shall be used. And first of all, we perceive that we have 
 not, as yet, assigned to the word machine a meaning which is precise 
 and in accordance with the above limitation. 
 
 We must therefore modify the definition once more ; and consider- 
 ing it with reference to its motion only, 
 
 A Machine is a combination of parts so connected that when one 
 moves according to a given law, the others must move according to cer- 
 tain other laws. 
 
 8. Motion and Rest are essentially relative terms, within the limits 
 of our knowledge. We can conceive a body to remain in a fixed posi- 
 tion in space, but we cannot know that there is one which does. If 
 there be any such body, it is in a state of absolute rest. 
 
 If two bodies, although both are moving in space, retain the same 
 relative positions, each is said to be at rest with respect to the other : 
 if they do not, either may be said to be in motion with respect to the 
 other. 
 
 Path. A point moving in space describes a line, called its path; 
 which may be rectilinear or curvilinear. The motion of a body, or 
 geometrical magnitude, may be defined by the paths of one or more of 
 its points, selected at pleasure. 
 
 Direction. In a given path, a point can move in either of two direc- 
 tions only, which may be defined in various ways, as up or down, to 
 the right or left, with the clock or the reverse ; direction, as well as 
 motion, being entirely relative. 
 
 9. Velocity, however, is not essentially relative. Whatever the 
 form of the path, the speed of the motion is estimated by comparing 
 the distance, or space, through which a point or body moves, with the 
 tune occupied in doing it. And since both space and time are abso- 
 lute magnitudes, the velocity itself is absolute. 
 
 Velocity is either uniform, equal spaces being traversed in equal 
 times ; accelerated, the spaces increasing, or retarded, the spaces de- 
 creasing, while the times remain equal. And the rate of acceleration 
 or of retardation may itself be either uniform or fluctuating. But it 
 is not necessary to consider all the complications which may arise in 
 this way ; for our purposes it will suffice to make one general distinc- 
 tion, viz. : that between motions with uniform velocities and motions 
 with variable velocities. 
 
4 ANGULAK VELOCITY. 
 
 10. In the case of uniform motion, the s^pace varies directly with 
 both the time and the velocity. Thus if one body move twice as long 
 and twice as fast as another, it will clearly travel four times as far. 
 This is otherwise expressed by saying that the space increases in the 
 compound ratio of the time and the velocity ; or, still better, in the 
 form of an equation, thus, 
 
 Space = Time x Velocity. 
 
 The space and the time are measured by comparing them with fixed 
 standards, or units, and may therefore be expressed by abstract num- 
 bers. And so in consequence may the velocity ; for from the above 
 equation we have 
 
 = 3ity ' 
 
 in which the first member being composed of abstract terms, the 
 second member will also be an abstract number, showing how many 
 units of space are traversed in a unit of time. 
 
 And this is the measure of absolute velocity when the motion is 
 uniform. 
 
 11. Angular Velocity. This expression relates to rotatory motion, 
 like that of a wheel turning in its bearings ; the speed of which may 
 be measured by the linear velocity of any point in the rotating body 
 whose radial distance from the axis is equal to the unit of space. 
 This is called the angular velocity, and may be either uniform or 
 variable. 
 
 If the angular velocity be uniform, the linear velocity of any point 
 varies directly as its distance from the axis : for the angles are propor- 
 tional to the times, and the arcs to the radii. Thus if one point be 
 two feet, and another four feet, from the axis, the outer will move 
 twice as rapidly as the inner, since in the same time it describes twice 
 as large a circle. 
 
 The speed of a wheel may also be conveniently expressed by stating 
 the number of turns it makes in a given time ; which evidently varies 
 as the angular velocity, if the latter be uniform. 
 
 The most concise and useful value, however, is the equation 
 
 , TT , . Linear Velocity 
 Angular Velocity = Radiug 
 
 12. Revolution and Rotation. A point is said to revolve about a 
 right line as an axis, when it describes a circle of which the centre is 
 
VARIOUS KINDS OF MOTION. 5 
 
 in, and the plane perpendicular to, that line. ^ When all the points 
 of a body thus revolve, with the same angular velocity and therefore 
 without changing their relative positions, the body itself is said to re- 
 volve about the .axis. 
 
 If the axis passes through the body, as in the case of a wheel, the 
 word rotation may be properly used synonymously with revolution. 
 
 But it frequently occurs in mechanical combinations, as for instance 
 in Watt's Sun-and-Planet Wheels, that a body not only rotates about 
 an axis which passes through it, but at the same time moves in an 
 orbit about another axis. In order to make a distinction between the 
 two motions, we shall in such cases speak of the first as a Rotation, and 
 of the second as a Revolution, just as we say that the earth rotates on 
 its axis, and revolves around the sun. 
 
 13. Continuous Motion. Motion is in its nature continuous, in the 
 sense that a point cannot move from one position in space to another, 
 without passing through all the intermediate positions, whether its 
 path be rectilinear or otherwise. But in the nature of things it is im- 
 possible for a point to go on moving indefinitely in the same direc- 
 tion, unless its path be one that returns into itself, like a circle, ellipse, 
 or other closed curve. 
 
 And the possibility of such indefinite continuance is what is im- 
 plied in the expression continuous motion, as technically employed. 
 A wheel turning freely in its bearings affords an example of motion 
 continuous in this sense, which naturally occurs oftener in circular 
 paths than in any others. 
 
 14. Reciprocating Motion. If a point traverses the same path alter- 
 nately in opposite directions, its motion is called reciprocating, whether 
 the path be rectilinear or not. But if the point travel in a circular or 
 other arc, the use of this term will be confined to those cases in which 
 the arc traversed is less than a circumference. For if a wheel make 
 a number of complete turns, first in one direction and then in the 
 other, it is manifestly improper to style such motion reciprocating, 
 notwithstanding the recurring reversals in direction. 
 
 Reciprocating circular motion, like that of a pendulum, or of a lever 
 swinging on a fixed centre, is also called vibration. 
 
 15. Intermittent Motion. When a reciprocating piece has reached 
 the end of its excursion in one direction, there must evidently be an 
 instant of rest, before it begins to return. But it frequently is re- 
 quired that a piece shall remain still for a definite time, after which 
 it again moves, either in the same direction as before or in the oppo- 
 site. When a piece in its action thus alternates motion with definite 
 periods of rest, it is said to have an intermittent motion. If the mo- 
 
6 CYCLE AND PHASES OF MOTION. 
 
 tions occur alternately in opposite directions, the action may be called 
 .an intermittent reciprocating motion. 
 
 16. Ifechanical Movements. The different kinds of motion above 
 specified are to a certain extent interchangeable. That is to say, one 
 kind may be converted into another, by means of various devices, 
 which are called mechanical movements. 
 
 It is ordinarily the case, that a machine is composed of a number 
 of such movements, or subsidiary combinations of parts, each fulfilling 
 a distinct function in the general operation. They are to the machine 
 what the members are to the body ; but each one, serving a definite 
 purpose in respect to its motion, may be regarded as a little machine, 
 whose action may be studied by itself. For example, the valve gear 
 of a steam engine may be entirely disconnected from the other parts, 
 and its operation investigated without reference to them. 
 
 17. Cycle of Motions. When a mechanical combination is set in 
 action, its parts go through a certain series of motions, involving 
 various changes in direction, velocity, or kind of motion, in a regular 
 order. It is usually the case that the parts finally return to their 
 original positions, after which the same motions will recur in the same 
 order, and so on perpetually. The whole series is called the Cycle. 
 Under these circumstances the combination is also said to have a 
 Uniform Periodic Motion. These terms, however, are used only for 
 want of better ones. Neither the word " Cycle " nor the word " Peri- 
 odic " have any reference to the time required to go through the series 
 of motions, nor does the word "Uniform" imply that .the time occu- 
 pied is always the same. 
 
 18. The terms are intended to convey the idea of regularity of 
 succession and constancy of relation, as obtaining among the motions 
 which make up the scries. 
 
 To illustrate : One revolution of the crank of a steam engine pro- 
 duces a reciprocation of the piston and a series of different angular 
 positions of the connecting rod, which itself vibrates on a moving 
 axis. And if the speed of the crank be uniform, the velocities of 
 both piston and connecting rod will vary according to a definite law. 
 Now it is clear that the parts will go through the same series of 
 motions, in the same relative order, and with the same variations in 
 velocity as compared with the speed of the crank, at every turn of the 
 latter, whether it go quickly or slowly, uniformly or variably, in one 
 direction or the other. 
 
 19. Phases of Motion. This term is used to designate the successive 
 phenomena of varied motion. Thus, one phase in the movement of a 
 steam engine would be represented in a, diagram showing the relative 
 
ELEMENTARY COMBINATIONS. 7 
 
 positions of the piston, connecting rod, and crank, at the beginning 
 of a stroke ; another if the middle of the stroke had heen selected, 
 and so on. Such diagrams, representing different phases of the mo- 
 tion, are often of the greatest utility in conveying clear ideas of the 
 action of complicated mechanical movements. Since the order of 
 recurrence is the same, we may select at pleasure any phase as the 
 beginning of a cycle. 
 
 20. Recapitulating a little, we see from the foregoing that every 
 machine, regarded broadly as a contrivance for utilizing power, con- 
 sists essentially of three classes of parts ; the function of the first 
 being to receive the power, that of the second to transmit and modify 
 the force and the motion, and that of the third to do the work. Evi- 
 dently, the nature and form of the first class depend directly and 
 largely on the character of the motive power, those of the third class 
 upon the nature of the work to be done, which also to a great extent 
 determines the proper actual velocity of the machine while in practical 
 operation. It is also apparent that these three classes of parts are in- 
 dependent of each other, in so far that any kind of work may be done 
 by any kind of power, and by means of different combinations of 
 interposed mechanism. 
 
 21. Now in what follows, we shall have to do only with the second 
 of these classes. Our object is to investigate the laws which govern 
 the determinate motions involved in the action of the machine : and 
 the motions as well as the form of the first class are determined by the 
 manner of action of the motive power, and those of the third by the 
 nature of the work and the manner in which it is to be done. The 
 motion of the one class has then to be transmitted to the other ; and 
 as the given motion of the former may be and usually is different from 
 the required motion of the latter, it follows that, during transmission, 
 the motion must be modified according to specific conditions. These 
 objects are accomplished by the second of the three classes of parts 
 above enumerated ; and it is the province of Pure Mechanism to dis- 
 cuss the methods by which motion may be transmitted, and to inves- 
 tigate the laws which govern its modification during the process. 
 
 22. Elementary Combinations. If two pieces be so connected and 
 arranged that a given motion imparted to one compels the other to 
 move in a determinate manner, these two constitute an elementary 
 combination. Practically, the motions are made determinate by means 
 of a rigid frame, in relation to which, as well as in relation to each 
 other, the two pieces move. But, obviously, the modification of the 
 motion is best seen by comparing the movements of the two pieces with 
 each other, so that for our purposes it is sufficient to take note of the 
 
8 TRANSMISSION OF MOTION. 
 
 motion of which each one is capable, without regard to the means 
 by which it is restrained or limited. 
 
 23. Driver and Follower. That piece of an elementary combina- 
 tion to which motion is supposed to be imparted, is called the Driver : 
 and the one whose motion is made compulsory by the action of the 
 other upon it, is called the Follower. 
 
 A Train of Mechanism consists of a series of parts, composing the 
 whole or a portion of a machine, each of which receives motion 
 from the preceding one, and transmits it to the next in order. The 
 train is, therefore, made up of elementary combinations, and each 
 piece is a, follower to the one which comes before, and a driver to the 
 one which comes after it. As the motion may be modified at each 
 successive step, it is necessary to begin by considering the modifica- 
 tions which may be effected by means of elementary combinations 
 only. 
 
 24. Modes of Transmission. Strictly speaking, if we leave out of the 
 question the agency of attractive or repulsive forces, such as magnet- 
 ism, one piece cannot compel another to move unless the two are in 
 actual contact. 
 
 But in many cases the motion of one piece is communicated to 
 another by the intervention of a third one, under such circumstances 
 that the movements of the latter are of no possible consequence, the 
 proper action of the whole depending entirely on the relative motions 
 of the first and second : and these two may then be properly regarded 
 as forming an elementary combination. We have, then, that motion 
 may be transmitted from a driver to a follower, 
 
 1. By Direct Contact. 
 
 2. By Intermediate Connectors. 
 
 25. Links and Bands. Such an intermediate connector must be either 
 rigid or flexible. If it be rigid, it is called a Link, and can either 
 push or pull, like the connecting rod of a steam engine : being neces- 
 sarily pivoted or otherwise jointed to both the driver and the follower. 
 
 If the intermediate connector be flexible, it is called a Band : for 
 our purposes it is supposed to be inextensible, and it can transmit 
 motion only by pulling. 
 
 26. Modification of Motion. In the action of an "elementary combi- 
 nation, the motion of the follower may differ from that of the driver 
 in kind, in velocity, in direction, or in all three. For example, a con- 
 tinuous rotation with uniform velocity may transmit continuous rota- 
 tion whose velocity is greater or less, uniform or variable, in the same 
 direction or the reverse ; it may transmit rotation intermittently : or 
 
MODIFICATION OF MOTION. 9 
 
 the follower may receive a reciprocating motion with a varying velocity 
 in a rectilinear or a curvilinear path. In an elementary combination, 
 as has been pointed out, the path of each piece is determined by its 
 connection with the frame- work of the machine, and it remains for 
 us to ascertain, at each instant of the action, its direction and velocity. 
 If this be done, the question is already settled as to whether the mo- 
 tion has been changed in kind. Hence we may properly say that the 
 most important function of an elementary combination is to modify 
 motion in velocity and direction. 
 
 27. The laws which govern this modification are determined by the 
 comparative movements of the two pieces. It is apparent on reflection 
 that, at every instant of the action, or, in other words, for every possi- 
 ble position of the driver with respect to the follower, there will exist 
 a certain definite proportion between the velocities, and an equally 
 definite relation between the directions, of their motions ; which will 
 depend entirely upon the pieces themselves and the manner in which 
 they act upon each other, and cannot be affected by the absolute 
 directions or velocities. 
 
 Consequently, whatever the nature of the combination, the analysis 
 of its action will be complete, if throughout its range we are able to 
 determine, as between the driver and the follower, 
 
 1. The Velocity Ratio. 
 
 2. The Directional Relation. 
 
 28. Now, the velocity ratio of the two motions may remain the same 
 during the entire action, or it may vary ; and this is also true of the 
 directional relation. To illustrate : If two circular wheels gear with 
 each other, turning about fixed axes, it is clear that the velocity ratio 
 is constant. If one wheel is twice as large as the other, it will at any 
 instant be turning half as fast, whether the motions be uniform or not. 
 And so of the relative directions of the rotations ; if the wheels are in 
 external, gear, they will turn in opposite directions, if they are in in- 
 ternal gear they will turn in the same direction : but the directional 
 relation, whichever it may be, does not change. If two elliptical 
 wheels engage, the directional relation, as before, is constant : but the 
 velocity ratio will change as the radii of contact vary. In the case of 
 the piston" and crank of a steam engine, neither the velocity ratio nor 
 the directional relation is constant ; supposing the crank to turn at a 
 uniform rate in one direction, the piston travels to and fro with a 
 varying speed. 
 
 It is this feature of constancy or the reverse, in these two partic- 
 ulars, which distinguishes the actions of elementary combinations 
 
10 BEPKESENTATIOK OF MOTION. 
 
 from each other, and forms the true basis for their proper classifi- 
 cation. 
 
 29. Graphic Representation of Motion. It has already been suggested, 
 that the action of a combination may often be most clearly illustrated 
 by drawings which show the parts in their proper relative positions at 
 convenient phases of the motion. 
 
 But the motions themselves of any points in either piece also admit 
 of perfect graphic representation at any given phase. For though 
 the path of a moving point may be a curve of any kind, yet, as the 
 direction of a curve at any point is that of its tangent at that point, 
 the direction of the motion at any instant may be indicated by that 
 of a right line^ And the velocity being an abstract number, may be 
 properly represented by the length of that line. 
 
 30, Geometrical Method of Investigation. Between the lines thus 
 representing the' motions of properly selected points, and other lines 
 closely connected with the moving pieces, definite relations may usu- 
 ally be established, in such a manner that, by means of diagrams thus 
 constructed, the velocity ratio and the directional relation may be as- 
 certained in the particular phase represented, and the law governing 
 the modification of motion throughout the action deduced, by simple 
 geometrical reasoning. 
 
 The method here outlined is peculiarly appropriate to this subject, 
 as directly leading to, if not directly involving, the accurate construc- 
 tions of the movements considered, which are essential in practical 
 operations : and to it we shall adhere throughout. But there are 
 some general principles relating to motion and contact, by a previous 
 study of which the analysis of motion as modified by mechanical de- 
 vices will be much facilitated : and these will accordingly receive our 
 attention in the following chapter. 
 
HE a r 
 
 If \ 
 
 -{UNIVERSITY^ 
 
 CHAPTER II. 
 
 COMPOSITION AND RESOLUTION" OF MOTION INSTANTANEOUS AXIS OF 
 ROTATION MOTIONS OF TRANSLATION COMPOSITION OF ROTATION 
 AND TRANSLATION SLIDING, ROLLING, AND MIXED CONTRACT. 
 
 Composition and Resolution of Motion. 
 
 31, Resultant. If a material point receive a single impulse in any 
 direction, it will move in that direction with a certain velocity. If it 
 receive at the same instant two impulses in different directions, it will 
 obey both, moving in an intermediate direction, and with a velocity 
 different from that due to either impulse alone. Now such a point 
 may receive at the same instant any number of impulses, differing in 
 magnitude and direction. But the point can move only in one direc- 
 tion and with one velocity; this actual motion is called the Resultant; 
 and the separate motions, which the different impulses taken singly 
 tended to give it, are called the Components. These components being 
 represented by* right lines, which may lie in the same or in different 
 planes, the resultant may be found graphically by the following con- 
 structions : 
 
 32. Parallelogram of Motions. In Fig. 1, let the point A have two 
 component motions, represented by AB 9 AC. 
 
 These two lines determine a plane, in which the 
 resultant must lie. In this plane draw through 
 B a parallel to AC, and through C a parallel to A < 
 AB. These parallels intersect at D ; and AD FIO. 
 
 is the resultant sought. 
 
 This fundamental proposition may be thus stated : If two compo- 
 nent motions be represented in magnitude and direction by the adjacent 
 sides of a parallelogram, the resultant will be similarly represented by 
 tJie diagonal passing through the point of intersection. 
 
12 COMPOSITION AND KESOLUTION OF MOTION. 
 
 83. Composition of Motions. This process of finding the resultant of 
 simultaneous independent motions is called Composition : and any 
 number of them may be compounded by repeating it. Thus in Fig. 
 
 2, let there be three components, AB, 
 AC, AE, all lying in the plane of the 
 paper. We first compound any two of 
 them, say AB and A C, as in the figure, 
 giving a resultant AD, which is next 
 compounded with AE, by which finally 
 we find AF, the required resultant. The 
 process is the same for any number of 
 components, and it makes no difference in what order they are taken. 
 
 34. Parallelopipedon of Motions. The above holds true whether the 
 components all lie in the same plane or not. In Fig. 3, let AB, AC, 
 
 AE, be three components, the first two 
 lying in the plane of the paper, while 
 the last does not. Proceeding as before, 
 AD, the resultant of AB and AC, will 
 lie in the plane of the paper also ; but 
 
 AF, the final resultant, found by com- FIG. 3. 
 pounding AD with AE, will lie in a different plane, determined *by 
 those two lines. Had there been a greater number of components, we 
 should have continued in a similar manner, compounding AF \f\i\\. 
 one of them, their resultant with another component, and so on. But 
 the figure sufficiently illustrates not only the process, but also another 
 fundamental proposition, relating to three components not in one 
 plane, which may be thus expressed : 
 
 If three component motions be represented by tlie three adjacent edges 
 of a parallelopipedon, the resultant will be represented in magnitude 
 and direction by the body diagonal which passes through the point of 
 intersection. 
 
 35. Resolution of Motion. This is the inverse process to that above 
 explained. It is obvious that if two or more independent motions can 
 be compounded into a single equivalent motion, that resultant can be 
 again separated, or resolved, into its components. This would be 
 
 #0 ^.. 9 just as true if the resultant .thus found had 
 
 been originally given as an independent sin- 
 gle motion. Consequently any motion may 
 be resolved into two components, either of 
 these into two others, and so on ad libitum. 
 
 And these components may be given any direction at pleasure. In 
 Fig. 4 let AB represent the given motion ; through A draw A C in any 
 
COMPOSITION AND KESOLUTION OF MOTION. 13 
 
 direction, and through B draw BC, cutting A C in any point C. Then 
 completing the parallelogram, it is evident that AB is the resultant 
 of the two motions represented by A C, AD. Of these the direction 
 of the first was assigned ; that of the other is not wholly arbitrary, 
 since it must lie in the same plane with AB and A C. But AD may 
 be again resolved, the direction of one component being assumed as 
 before : and this may be repeated until components of the original 
 motion have been found, in as many different directions in space as 
 may be desired. 
 
 36. From the above it will be perceived that if the given motion 
 and the required components all lie in one plane, whatever the num- 
 ber of the latter, the resolution can be exactly effected. If they do 
 not, there may be one component in addition to those whose directions 
 are assigned : should these all lie in one plane which does not contain 
 the given motion, this must obviously be the case. 
 
 But a motion can be exactly resolved into three components not in 
 one plane, provided that it does not lie in the plane determined by 
 either two. 
 
 Thus in Fig. 3, let AF be the given motion, and Ax, Ay, Az, the 
 directions of the three components. Now these three lines last men- 
 tioned determine three planes ; and by passing through F three 
 other planes respectively parallel to these, a parallelopipedon is con- 
 structed, of which AF is the body diagonal, and the edges pass- 
 ing through A are therefore the required components. 
 
 37. Normal and Tangential Components. It is often required to resolve 
 a motion into two components, of which one shall be perpendicular, the 
 other parallel to a given plane : usually a plane tangent to the surface 
 of some moving piece under consideration. In this case the former is 
 called the normal, and the latter the tan- 
 gential, component. They are easily found 
 
 as in Fig. 5. Let A be the moving point, 
 AB representing its motion : draw through 
 A the plane MN parallel to the given 
 plane ; upon this let fall the perpendicular 
 BC, and draw A C. Completing the paral- 
 lelogram CD, it will be perceived that the 
 
 tangential component is AC, the orthographic projection of AB upon 
 the plane MN, and that AD, the normal component, is equal and 
 parallel to BC, the projecting perpendicular of the point B. 
 
 38. Motions of Connected Points. If two points be so connected that 
 the distance between them is invariable, they may be supposed to be 
 connected by a right line. The motions of the points at any instant 
 
14 
 
 INSTANTANEOUS AXIS OF ROTATION. 
 
 may be represented by right lines, which may be in the same or in 
 different planes. Each of these may be resolved into two components, 
 
 one of which .s in the direction of 
 the connecting line, the other per- 
 pendicular to it. And the simulta- 
 neous motions are subject to this con- 
 dition, that the components along the 
 line, thus found, must have the same 
 magnitude and direction. Thus in 
 FlG - 6 - Fig. 6, the components A C, BF, along 
 
 the line AB, must be equal and in the same direction ; otherwise 
 the distance AB would change, which is contrary to the hypothesis. 
 These components are at once found by drawing through E and D 9 
 planes perpendicular to AB, cutting it in C and F. 
 
 The Instantaneous Axis of Rotation. 
 
 39. A right line, moving in any manner in space, may at any given 
 instant be regarded as revolving about some other right line more or 
 less remote. The latter may, from instant to instant, change its posi- 
 tion not only in space, but in relation to the line whose motion is 
 under consideration, with reference to which it is therefore called the 
 Instantaneous Axis. 
 
 40. If the motions of two given points in a right line are known, 
 the motion of the whole line is fully determined. Consequently, if it 
 can be shown that the actual motions of these two points, at any one 
 instant, are the same as though they were revolving with the same 
 angular velocity and in the same direction about any axis, the motion 
 of the line is at that instant equivalent to one of revolution about the 
 same axis. 
 
 First, considering the motion of a single point in space : whatever 
 the path, the motion is, at any given instant, equivalent to one of 
 revolution about any right line in the plane normal to the path at the 
 position occupied at that instant by the moving point. Thus in Fig. 
 7, let P be the moving point, PT the tan- 
 gent to its path, and MN the normal plane. 
 In this plane draw any right line AB, and 
 PC perpendicular to it. Then if P be sup- 
 posed to revolve about AB as an axis, it 
 will describe a circle whose plane, being FlG . 7 . 
 
 perpendicular to MN, will contain the line PT: and this line being 
 perpendicular to the radius CP at its extremity, will be tangent to 
 
INSTANTANEOUS AXIS OF ROTATION. 
 
 15 
 
 the circle thus described. Similarly the actual motion of P may be 
 proved equivalent to one of revolution about HL, or any other right 
 line lying in the normal plane MN. 
 
 The angular velocity will depend upon the distance of the moving 
 point from the assumed axis, and is conveniently measured by divid- 
 ing the given linear velocity by that distance, since in general we 
 have (11) 
 
 ang. vel. 
 
 lin. vel. 
 rad. 
 
 Thus if in the figure, PT represent the linear velocity of P at the 
 given instant, we shall have for the equivalent rotations, 
 
 and 
 
 ang. vel. about A B z 
 
 ang. vel. about HL = 
 
 PT 
 PD 
 
 41. Now, if the motions of two given points on a right line be 
 known, the above reasoning applies to each. Consequently, if we draw 
 two planes, respectively normal to the paths at the positions simulta- 
 neously occupied by the moving points, the intersection of these planes 
 will be the only right line about which both points can at the given 
 instant be regarded as revolving. 
 
 But we have already seen that the 
 motions of two connected points 
 are subject to another condition 
 (38), and it remains to be shown 
 that these equivalent revolutions 
 will have the same direction and 
 the same angular velocity when 
 that condition is satisfied. 
 
 42, We will first consider the case 
 in which all the motions of the line 
 upon which the given points are 
 situated, are confined to one plane. 
 In Fig. 8, let AB be the moving 
 line, A and B the given points. Let 
 AD represent the motion of A in 
 
 magnitude and direation, and sup- FIO. 9. 
 
 pose B to move in the direction BE\AD may be resolved into the 
 
16 INSTANTANEOUS AXIS OF ROTATION. 
 
 two components, A C in the direction of AB, and AF perpendicular 
 to it. The motion of B must have a component BG along AB, 
 equal to AC and in the same direction. Having set this off, draw 
 at G a perpendicular to AB, cutting BE at E. This determines the 
 linear velocity, BE, of the point B, and completing the parallelo- 
 gram, this is seen to be the resultant of the two components BG, BH. 
 Suppose all these lines to lie in a horizontal plane parallel to the paper ; 
 then AO perpendicular to AD, and BO perpendicular to BE, will be 
 the traces of the two normal planes, and .their intersection will 
 represent a vertical line. Now draw OL perpendicular to AB, then 
 the triangles ACD, ALO, are similar, and also the triangles EGB } 
 BLO ; whence 
 
 and 
 
 .AC _ 
 OL ~ OA 
 
 OL ~- OB 
 
 AD BE 
 
 ~ OB 
 
 But since AD, BE, are the linear velocities of A and B, the frac- 
 
 tions - and are the angular velocities of the revolutions of those 
 C/_/l (J Jj 
 
 points about ; which are thus proved to be equal, and their direc- 
 tions are the same by construction. 
 
 43. The same pairs of similar triangles will therefore give 
 
 A~L " UA ~ OB ~ IL 
 
 but CD = AF, and GE = BH ; 
 whence 
 
 AF BH 
 
 Therefore the right linS FH must pass through L, the foot of the 
 perpendicular from upon AB. This fact is important, for the 
 angle between AO and BO may be so acute as to render it difficult 
 to determine the exact point of intersection. In that case, if the rel- 
 ative velocities are known, we have only to find the components per- 
 pendicular to AB, which may be laid off upon any scale, so that FH 
 need not cut AB acutely ; and the point L can thus be found v/ith 
 precision. 
 
INSTANTANEOUS AXIS OF KOTATION. 17 
 
 44. The general case is that in which the motions of the two given 
 points are in different planes ; in illustration of which, Fig. 8, still 
 regarded as a horizontal projection, is to be studied in connection with 
 the vertical projection given in Fig. 9. AD and BR are parallel 
 to the horizontal plane, to which AB is inclined, though par- 
 allel to the vertical plane. In Fig. 9, therefore, AB appears 
 as A'B', and AD is foreshortened into A'D' : and the actual 
 component A'M along the line is found as in Fig. 6, by passing 
 through D' a plane perpendicular to A'B'. Then making B'N, 
 the corresponding component of the motion of B, equal to A'M, 
 and drawing a plane through ^perpendicular to A'B', we have B'E' 
 the vertical projection of the resultant ; from this the horizontal pro- 
 jection BE, which is seen in its true length, is determined, and is 
 evidently of precisely the same length as in the previous case. Now 
 AD and BE may as before be resolved into the components AF, AC, 
 and BG, BH, lying in horizontal planes. The preceding argument 
 depends upon these only, and accordingly applies without change. 
 
 45. In general, then, the instanta- 
 neous axis of a moving right line may 
 be found if the directions only of the 
 simultaneous motions of two points 
 upon it are given : but this may not be 
 possible if those directions are parallel. 
 Thus in Figs. 10 and 11, let A C and BE, 
 the motions of the points A and B at the 
 given instant, be parallel to each other 
 and to the paper, and perpendicular to 
 the moving line AB. In this case the 
 two normal planes coincide ; yet there 
 is an instantaneous axis, in order to 
 
 locate which we must know also the FlG> 
 
 relative velocities at least of the two points. Setting these off as AC, 
 BE, of their proper proportionate lengths by any scale, CE will cut 
 AB or its prolongation in D. If we conceive a right line perpendic- 
 ular to the paper at D, it will lie in the normal planes, and the motion 
 of AB is evidently equivalent to one of rotation about it. And this will 
 as obviously be true, whatever may be the inclination of the moving 
 line to the plane of the paper. Any line in the normal plane, there- 
 fore, which passes through D, maybe taken as the instantaneous axis. 
 
 46. Now, in Figs. 10 and 11, let AB, AC, be regarded as parallel 
 to the plane of the paper, while BE is the projection of a line inclined 
 to that plane, though perpendicular to AB. Then the planes normal 
 
 2 
 
18 
 
 MOTION OP TRANSLATION. 
 
 FIG. 12. 
 
 to A and BE will intersect in the line A B itself ; which is then its 
 own instantaneous axis, about which we may for the sake of consist- 
 ency say that it is revolving with infinite velocity. It is, certainly, 
 the only line about which it can be said to have a motion of rotation 
 only ; it does not follow, however, that it will remain stationary in 
 space, which it will not do : but, as will be shown subsequently, its 
 actual motion will be helical. 
 
 47. Motion of Translation. From consideration of Fig. 11 it will be 
 seen that the more nearly equal A C and BE are, the more remote will 
 be the intersection D : and that if they be exactly equal, the instanta- 
 neous axis will be at an infinite distance. Since in that case all the 
 points of AB are at the instant traveling in the same direction with 
 the same velocity, the consecutive position of the moving line will be 
 
 _ parallel to the present one. This 
 
 will occur whenever the simulta- 
 neous motions of the two given 
 points are in parallel lines and in 
 the same direction, whether per- 
 pendicular to the moving line or 
 not. Thus in Fig. 12, the normal 
 planes AM, BN, are parallel, al- 
 though the motions of A and B are inclined to the line AB. It is 
 evident that in this case A C and BE, the actual linear velocities of 
 the two points, must be equal, since (38) the components A H and 
 BG, along the line, must be equal and lie in the same direction. 
 The motion of AB in Fig. 12 may then be regarded as one of rota- 
 tion about an infinitely remote line parallel to BN, and lying in a 
 plane parallel to AB : if AC, BE, are made perpendicular to AB, 
 the instantaneous axis will be infinitely remote and parallel to the 
 moving line. 
 
 48. The motion of the line, at any instant when it is thus revolving 
 about an axis at an infinite distance, is called Translation. It may be 
 continuous ; and not only a right line, but any geometrical magnitude, 
 is said to have a motion of translation, when all its points move with 
 the same velocity and in the same direction, in equal and similar 
 paths. Under these circumstances, evidently, all the tangents to 
 these paths, at the positions simultaneously occupied by the moving 
 points, will be parallel : and the right line joining any two of these 
 points will remain parallel to itself during the whole movement. If 
 the paths are rectilinear, the motion is called straight translation, or, 
 more commonly, sliding : if they are curvilinear, the motion is called 
 circular, elliptical, or helical, translation as the case may be, the name 
 
COMPOSITION OF ROTATION AND TRANSLA 
 
 depending on the form of the curve. A familiar instam 
 translation is afforded by the coupling-rod of a locomotive, shown in 
 Fig. 13 : the points A and B move as indicated by the arrows, in the 
 circles whose centres are D and E, 
 with the same velocity. And any 
 other point C, in a rigid exten- 
 sion of the bar AB, will move 
 similarly in the equal circle whose 
 centre is F. 
 
 49, In relation to a single right 
 line, it is to be noted that a mo- 
 tion of circular translation, if the 
 
 planes of the circles be perpendicular to the moving line, is identical 
 with one of revolution about the axis of the cylinder thus generated. 
 Thus, in Fig. 13, the point C may be considered as a line perpendicu- 
 lar to the paper ; and its motion is the same as though it revolved 
 about a parallel axis passing through the point F. 
 
 It is also to be observed that the motion of a single right line of a 
 rigid body does not determine the motion of the whole body, which if 
 not otherwise controlled would be free to rotate about that line. But 
 if the motions of two right lines of the body be given, those of all its 
 other points are thereby fully determined. 
 
 Composition of Rotation and Translation. 
 
 50. Now, just as a simple motion in a right line can be resolved into 
 two others, so a simple rotation about a fixed axis may be regarded as 
 a resultant of two other motions, viz., a rotation about another axis 
 parallel to and revolving around the first, and a translation in which 
 the paths are circles whose radii are equal to the distance between 
 these axes ; the angular velocities and the di- 
 rections of both components being the same 
 as those of the rotation which was to be re- 
 solved. 
 
 Thus in Fig. 14, let C represent a fixed axis 
 perpendicular to the paper, and let MAN re- 
 volve about this axis into the position 8BT. 
 We may also let A represent a right line per- 
 pendicular to the paper, about which MN may rotate into the posi- 
 tion UV, afterward moving by circular translation to the position 
 ST. Or, MN may move by circular translation into the position PR, 
 afterward rotating about B into the position ST ': the path of A being 
 
20 RESOLUTION OF CIRCULAR TRANSLATION. 
 
 in all cases the arc AB. On either supposition the result is the same 
 as if both motions progress simultaneously and uniformly, which pro- 
 duces the original revolution about C : and the angles J/^T*y RBT, 
 
 and A CB 9 are equal by construc- 
 tion. In Fig. 15, the traveling 
 axis passes through N; now let 
 the points M and A, during trans- 
 lation from position MN to position 
 PT, describe circles whose radii are 
 equal to CN. Then if in the same 
 time MN rotate about N, in the 
 same direction, through an angle 
 PTS = NCT, it will come into the same position, ST, as in Fig. 14. 
 In short, any axis parallel to the fixed one may be assumed as that of 
 the component rotation, and thus the revolution around C may be re- 
 solved in an infinite number of ways. 
 
 51. On the other hand, a motion of circular translation may be con- 
 sidered as a resultant. For (49) any 
 
 line perpendicular to the planes of the 
 translation, and partaking of that mo- 
 tion, may be regarded as revolving 
 about a fixed axis. Assuming such M 
 line as a traveling axis of rotation, 
 the component motions arc, first: a 
 revolution about that fixed axis, the c 
 
 direction and angular velocity being 
 
 the same as those of the translation; and, second, a rotation in the 
 opposite direction, but with the same angular velocity, about the as- 
 sumed traveling axis. Thus in Fig. 16, if MAN revolve about the 
 fixed axis C, until A reaches B, and this be its only motion, it will 
 have the position SBT. If in the same time it also rotates in the 
 opposite direction about A, as shown by the arrows, through an angle 
 TBR, equal to A CB, its new position will be PBR, parallel to the 
 original position MAN; and the resultant motion will be one of cir- 
 cular translation, in which M, N, and all the moving points, describe 
 circles whose radii are equal to A C. 
 
 52. If any geometrical magnitude have a motion of rectilinear 
 translation parallel to an axis about which it is revolving, the result- 
 ant is a helical motion about that axis. But a motion may be helical 
 with respect to one axis, and yet simply rotatory with respect to an- 
 other. For instance, the rectilinear generatrix of an oblique helicoid 
 has a continuous helical motion about the fixed axis of the surface ; 
 
KESULTAKT MOTION OF A KIGID BODY. 
 
 but in any given position of the moving line, its motion is equivalent 
 to one of rotation only about an instantaneous axis, determined (41) 
 by the intersection of planes normal to the paths of any two of its 
 points : and either may be regarded as the actual motion of the gene- 
 ratrix at that instant, the latter being the less complex. In the case 
 of the right helicoid, the intersection of these normal planes is the 
 generatrix itself ; and the motion of that line in space cannot be re- 
 duced to any simpler form than that of a helical one about an axis, 
 which, in this special case, is at every instant the same, being the 
 fixed axis of the surface. 
 
 53. But, in general, if the motions of two connected points are 
 perpendicular to the right line joining them, 
 and are not parallel to each other, the mo- 
 tion of that line will be a helical one about 
 an instantaneous axis : which may be de- 
 termined, if the relative velocities are given, 
 in the following manner : 
 
 In Figs. 17 and 18, the moving line AB 
 is perpendicular to the vertical plane, and 
 B' is its vertical projection. The motions 
 of the points A and B are seen in their true 
 lengths and directions in the vertical pro- 
 jection as B'C', B'E' (for convenience so 
 drawn that C'E' is horizontal), and their 
 horizontal projections are A C, BE. Through 
 AB pass a plane parallel to C'E', and resolve these motions normally 
 and tangentially with reference to this plane. The tangential com- 
 ponents are B G, B'F, horizontally projected in AC, BE. Draw EC, 
 
 and produce it if necessary to cut BA or 
 its prolongation in D, whose vertical pro- 
 jection is B'. Then the motion due to 
 these components, which are horizontal 
 and parallel, will be (45) a rotation about 
 any axis passing through D, and lying in 
 the vertical pla*ne which contains AB. 
 The normal components are vertical and 
 equal, the true length, B'H, being seen in 
 the vertical projection. The motion of 
 AB due to these components, therefore 
 is one of vertical translation ; and the 
 actual motion in space is a helical one about a vertical axis LL pass- 
 ing through D. 
 
 FIG. 17. 
 
 B' 
 
 r. 
 
 FIG. 18. 
 
22 RESULTANT MOTION OF A RIGID BODY. 
 
 54, Let the points of a material right line be so connected with 
 those of a rigid body that the distances between them cannot change. 
 We may then suppose that line to move, and the body primarily to 
 move in the same manner. If now the body also rotate about the 
 line as an axis, then these two motions may be compounded : and (49) 
 if it can be shown that any two lines of the rigid body are at the same 
 instant moving in a rotatory or helical manner about any axis, the whole 
 body may at that instant be regarded as doing the same. Evidently 
 the moving line itself may be taken as one of these two, and any line 
 intersecting it as the other. 
 
 The problem of determining the resultant presents several different 
 cases, depending upon the nature of the motions primarily assigned 
 to the rigid body and the line with which it is connected ; and these 
 will be considered separately. 
 
 55. I. Let the primary motion be one of revolution about an axis 
 parallel to the line. 
 
 The resultant will then be a rotation of the body about an instan- 
 taneous axis, which will be parallel to the other two axes, and lie in 
 the same plane. 
 
 In Fig. 19, let C represent the moving line, P the axis about which 
 
 it is revolving, both being perpen- 
 dicular to the paper ; and let CD, 
 perpendicular to CP, and parallel to 
 the paper, represent the linear veloc- 
 ity of any point of the moving line 
 in its revolution around P. Join 
 any point of this line with any point 
 A of the body : then CA represents 
 a line which may or may not be in- 
 clined to the paper. Let AE, per- 
 pendicular to CA, be the linear velocity of A in its rotation about Cj 
 the motion of A due to the revolution about P will be AB, perpen- 
 dicular to AP, the magnitude being determined by making the angle 
 APB equal to the angle CPD. Both AE and AB, and therefore 
 their resultant AF, the actual motion of A, are parallel to the paper. 
 Consequently the normal planes NN, MM, of which the latter con- 
 tains the axes C and P, intersect at in a line perpendicular to the 
 paper, which is the instantaneous axis. 
 
 The location of this axis may be found otherwise, thus : regarding 
 CP as a line of the rigid body (which may be inclined or parallel to 
 the paper), the rotation of P around tf will be represented by PG, 
 the angle PCG being made equal to the angle ACE : then (45) GD 
 will cut CP in 0. 
 
COMPOSITION AND KESOLUTION OF MOTION. 
 
 56. II. Let the primary motion be one of translation, in which the 
 paths are in planes perpendicular to the line. 
 
 In this case also the resultant will be a rotation about an instanta- 
 neous axis parallel to the line. 
 
 This will be apparent from inspection of Fig. 20, the construction 
 differing from the preceding only in 
 this ; that AB is parallel and equal to 
 CD, the motion of C in its path RS, 
 whose radius of curvature at C is for 
 the purpose of comparison made equal 
 to CPof Fig. 19 : also CD and AE are 
 of the same lengths in both diagrams, 
 in which therefore themo tions of C at FIG. 20- 
 
 the given instant are identical, and the rotations about the moving 
 line are alike. 
 
 57. III. Let the primary motion be one of revolution about an 
 axis which intersects the line. 
 
 The resultant will then be a rotation about an instantaneous axis 
 lying in the plane of the other two, and passing through their point 
 of intersection. 
 
 In Fig. 21, let CO be the line, OA the axis, both in the vertical 
 plane. Draw CA at pleasure, and suppose 
 it a line of the rigid body. Then the mo- 
 tion of A is due only to the rotation about 
 CO, that of C is due only to the primary 
 revolution about OA, and both are perpen- 
 dicular to the vertical plane. Let these mo- 
 tions be represented by A 'E, C'D, in the hor- 
 izontal projection ; and draw DE, cutting 
 C'A' in P', whose vertical projection is P. 
 Then (45) CA may be regarded as rotating 
 about any axis lying in the vertical plane and 
 passing through P, and CO as rotating 
 
 'A 
 
 about any 
 
 axis in the 
 
 same plane, 
 
 and passing through 0. Consequent- 
 ly OP is the instantaneous axis of 
 both CO and CA, and therefore of 
 the whole body. 
 
 58. Draw IPK perpendicular to 
 OP and consider it another line of FIG. 22. 
 
 FIG. 21. 
 
24 
 
 COMPOSITION AND RESOLUTION OF MOTION. 
 
 the rigid body : then -in Fig. 22, draw KG perpendicular and PL 
 parallel to 0(7, also IH perpendicular and PM parallel to OA, and 
 join PG. 
 Now let 
 
 V ang. vel. of K around 00, 
 
 y> = i OA) i Then 
 
 V = 
 
 /or K 
 
 OP. 
 
 v 
 V 
 
 Iff 
 
 By construction, OPK, OGK are right angles, therefore a circle 
 will go round OGPK, whose diameter = OK ' ; 
 in which KOP = KGP, standing on same arc 'PK, 
 and GKP = GOP, " " " " GP ; 
 
 but OLP = GOP, by reason of parallels PL, OG: 
 
 . . triangles GPK, OPL, are similar. 
 
 whence 
 
 V PK _PL OM 
 
 v ~ KG - OP ~ OP : 
 similarly 
 
 . - 
 v ~ IH 
 
 OP* 
 
 That is to say : that if we lay off upon 
 the axes the distances OM, OL, propor- 
 tional to the angular velocities of the rota- 
 tion and the revolution about those axes 
 respectively, and complete the parallelo- 
 gram ML, its diagonal OP will lie in the 
 instantaneous axis, and be proportional to 
 the angular velocity of the resultant rota- 
 
 tion about it. 
 In Figs. 21 
 and 22, the 
 direction i n 
 which the rigid 
 body rotates 
 about the 
 traveling axis, 
 and that in 
 which the 
 traveling axis 
 revolves around 
 
 FIG. 23. 
 
KESULTAKT MOTIOH OF A EIGID BODY. 
 
 the fixed one, are the same. But the above reasoning applies equally 
 well if they are opposite ; and Figs. 23 and 24 show the modifications 
 in the diagrams due to this change in the conditions. In regard to 
 which it will be noted that in both cases OH is set off in the same 
 direction from ; but OL is measured in one direction if the rota- 
 tion and the revolution are alike, and in the other if they are unlike. 
 
 59. IV. Let the body and the line primarily revolve about another 
 line in a different plane. 
 
 In this case the resultant will be a helical motion of the whole body 
 about an instantaneous axis lying in a 
 plane parallel to the other two lines, 
 and intersecting their common perpen- 
 dicular. 
 
 In Fig. 25, let OA be the fixed axis, 
 OC the moving one, both parallel to the 
 vertical plane ; being the vertical 
 and DG the horizontal projection of 
 their common perpendicular. Consider- 
 ing DG as a line of the rigid body, let 
 the motion of D be represented by OE 
 perpendicular to OA, that of G by OS 
 perpendicular to 00; the axes being 
 for convenience so placed that BE is horizontal. 
 
 Eesolve these motions normally and tangentially with reference to 
 a horizontal plane through : the normal components will be verti- 
 cally projected in OR, the tangential ones in ON, OM, of which the 
 horizontal projections are DF, GU: therefore UP is the horizontal 
 projection of a line joining the points M and N. 
 
 Produce MB to cut OA in A, also NE to cut OC in C; then 
 A C will be parallel to MN. For the triangles OMA, ONE, are 
 similar : so are also the pair OMB, ONC: and BM = EN. Hence 
 we have 
 
 AM. OM ..CNOM A ^ 
 ON ~ EN* and OJf= ' * AM = 
 
 and UF is also the horizontal projection of A C. 
 
 60. Moreover, UF is the trace on a horizontal plane through A, of 
 a plane passing through the point C and the axis A . The actual 
 motion of C about that axis, is perpendicular to this plane ; therefore 
 its horizontal projection F8 must be perpendicular to UF. Its verti- 
 cal projection CH must (see Fig. 9) be parallel and equal to OE ; 
 whence the magnitude of F8 is determined. With reference to the 
 
26 RESULTANT MOTION OF A RIGID BODY. 
 
 vertical plane, this motion may be resolved into a normal component 
 FQ, vertically projected in C, and a tangential one of which the pro- 
 jections are FL, CH. The latter component being parallel and equal 
 to OE, may be resolved into the vertical component CI equal to 
 OR, and the horizontal one CK parallel and equal to ON. The hori- 
 zontal projection of CK is FL, equal to DF, and compounding it with 
 FQ, the resultant FS is a horizontal line, vertically projected in CK. 
 
 61. The motions of the two lines DG, OC, then, have each a com- 
 ponent of vertical translation equal to OR. The motion of OC has 
 for its remaining components, DF for the point 0, and FS for the 
 point C, both horizontal. The resultant of these is a rotation about 
 the instantaneous axis OP, whose horizontal projection is T, the inter- 
 section of the two vertical planes normal to DF and FS. The re- 
 maining component of the motion of G, is the horizontal line G U, and 
 the resultant of GU and. DF is a rotation of DG about an instanta- 
 neous axis determined as in Fig. 10 ; which axis is also the vertical 
 line through T. And the rotations of the two lines about this axis 
 have the same angular velocity ; for, drawing TS, we shall have from 
 the similar triangles FLS, FDT, in which DF '= FL, the ratios 
 
 TD TD TF 
 FL ~DF = FS> 
 
 but TFS, TDF, are both right angles, hence those triangles are sim- 
 ilar, and the angles STF, FTD, GTUzTQ equal. 
 
 Combining this rotation with the translation, the final resultant is 
 a helical motion of both lines and therefore of the whole body about 
 OP. 
 
 62. In the vertical projection, draw PY parallel to OA, and PZ 
 parallel to OC, making PY equal to OZ, and the triangles PCY, 
 AGO, similar. 
 
 Let V = ang. vel. of D around OA, 
 
 V " " " G " OC, 
 
 v = " " " D or G " OP. 
 
 We shall then have, since ang. vel. = m * J e ', 
 
 rad. 
 
 V _OE ON V OE DT 
 
 V - DG ' V ~~ ~~ Dl ' * "' v ~ ON X DG ' 
 
 But from aim. triang. ONE, OP A, 
 
RESULTANT MOTIOH OF A RIGID BODY. 27 
 
 and from sim. triang. TDF, TGU. . . = - 
 
 PC 
 
 OP' 
 
 -.. .-. 
 
 UF~ OP X ~AG~- 
 
 Now from sim. triang. ACO, PCY, 
 
 ^ 
 
 AC~ P0 
 
 . - . - - 
 
 '* v ~ PC X OP~ OP ~~ OP 
 
 V OY 
 And similarly, = 
 
 Draw A X, CW 9 respectively perpendicular to 00 and OA ; then 
 by reasoning similar to that used in connection with Fig. 22, it may 
 be shown that 
 
 V _ CP V'_ AP 
 
 " a ~ 
 
 63. From which it appears, that in the parallelogram ZY, of which 
 the adjacent sides OZ, OY, are parallel to the axes and proportional 
 to the angular velocities of the revolution and the rotation about those 
 axes respectively, the diagonal OP is parallel to the instantaneous axis 
 and proportional to the angular velocity of 
 
 the rotation about it. And that the seg- 
 ments AP, PC, into which A C perpendicu- 
 lar to OP and limited by the prolongations 
 of OZ and Y, is divided at P, are propor- 
 tional to those into which DG, the com- 
 mon perpendicular of the two axes, is cut 
 by OP. 
 
 In Fig. 25, as in Figs. 21 and 22, the ro- 
 tation about OC and the primary revolution 
 about OA are in the same direction. Should 
 they have contrary directions, the argu- 
 ment would still apply without change; 
 and the modifications in the diagram caused 
 by that change are shown in Fig. 26, which 
 being lettered throughout to correspond 
 with Fig. 25, requires no explanation. 
 
 64. V. Let the primary motion be one of 
 
 translation, in which the paths are not in planes perpendicular to the line. 
 
MOTION OF A KIGID BODY 
 
 SPACE. 
 
 The resultant is then a helical motion of the body about an in- 
 stantaneous axis parallel to the line. 
 
 In Fig. 27, let the line OC be vertical, and D its horizontal pro- 
 jection ; the motions OE, CH, of the points and C, being parallel 
 to the vertical plane, and horizontally projected in DF. These 
 motions being equal and parallel, may be resolved with reference to 
 the horizontal plane into the equal vertical components OR, 01, and 
 the equal and parallel horizontal ones ON, CK, both of the latter 
 being horizontally projected in DF. 
 
 Let be the vertical and DG the horizontal projection of a line of 
 the rigid body, and let OB, GU, be the corresponding projections of 
 the rotation of G about OC. Now G has also a motion of translation, 
 of which OE is the vertical and GL the horizontal projection, which 
 l H may be resolved into a vertical component 
 
 OR, and a horizontal one, of which the 
 projections are ON, GL. Compounding 
 this latter with GU, the resultant is 
 GS = GU LG ; the vertical projec- 
 tion is OM. Hence the resultant of these 
 horizontal components is a rotation of 
 OC and DG about the instantaneous axis 
 whose horizontal projection is T, and 
 vertical projection OC; with which is to 
 be finally compounded the vertical mo- 
 tion of translation, producing the helical 
 resultant motion of the whole body. 
 FIG - 27 - If OE and CH coincide with OC, that 
 
 is to say, if the line primarily move endlong, the resultant is, obvi- 
 ously, a helical motion about an axis coinciding with the line it- 
 self. 
 
 65. VI. Let the primary motion be a helical one about an axis per- 
 pendicular to the line. 
 
 In this case also the resultant will be a helical motion of the whole 
 body about an instantaneous axis. For the original helical motion is 
 compounded of a rotation about the fixed axis and a translation par- 
 allel to it ; this rotation being compounded with that about the trav- 
 eling axis, the resultant is (57) a rotation about a third axis, in the 
 plane of the other two, and passing through their common point. 
 The translation may again be resolved into components respectively 
 perpendicular and parallel to the third axis; compounding the former 
 with the rotation atiput that axis, the resultant is (56) a rotation about 
 a fourth axis parallef to the third, which finally compounded with the 
 
MOTION OF A KIGID BODY 
 
 SPACE. 
 
 component of translation parallel to that axis, determines the instan- 
 taneous helical motion of the body about the fourth axis. 
 
 66. Thus in Fig. 28, let the vertical fixed axis OA, and the hori- 
 zontal traveling one OC, both lie in the vertical plane, and let OG, 
 CK, be the vertical compo- 
 nents of translation of the 
 
 helical motion of OC about 
 OA. Draw in the vertical 
 plane any line CA, and con- 
 sider it as a line of the rigid 
 bocly ; its horizontal projec- 
 tion is AC'. Let C'D be the 
 rotatory component of the mo- 
 tion of C, in its helical path, 
 and let A'B be the motion of 
 A in its rotation about OC ': 
 then as in Fig. 21 we find OP 
 
 the instantaneous axis of the FIG. 28. 
 
 resultant rotation. Since P lies in this axis, its only motion is one 
 of vertical translation, PY, equal to OG and CK. These three mo- 
 tions may be resolved into PW, OH, and CE, perpendicular to OP, 
 and PZl OX, CF, parallel to OP. The projections of 06 Y and PC, 
 on a plane perpendicular to OP, will coincide in RL, and those of 
 the perpendicular components OH, PW, will coincide in RS. In 
 this projection, the motion of C will be represented by L U, the re- 
 sultant of L M, perpendicular to LR and equal to C 'D, compounded 
 with LN, the projection of CE. The planes normal to L U and RS 
 intersect in T, which represents a line perpendicular to this plane of 
 projection and therefore parallel to OP j this is the instantaneous axis 
 about which OC and PC are rotating, and the helical motion results 
 from the final addition of the parallel components OX, CF and PZ. 
 
 67. It will be seen that the preceding cases include every possible 
 motion of a rigid body in space. For*if the motion of one of its right 
 lines be determined, the body can have no other motion except one 
 of rotation about that line as an axis. And whatever the law by 
 which the movement of this axis may be governed, its motion at any 
 instant is either one of translation, or one of simple rotation about 
 another axis in the same or a different plane, or a helical one about an 
 axis perpendicular to it. Hence it appears that the most complicated 
 motion of which any rigid body is capable at any instant, is a helical 
 one about some right line : that is to say, all its points have at the 
 instant a uniform rotation about the axis, combined with a uni- 
 
30 VAKIOUS KINDS OF CONTACT MOTIONS. 
 
 form motion parallel to it, their paths being helices of the same 
 pitch. 
 
 Sliding, Rolling, and Mixed Contact. 
 
 68. Sliding Contact. The motions of a piston in a cylinder, of a 
 journal in its bearing, and of a screw in a nut, afford instances in 
 which the contact during the action is purely sliding. 
 
 Kegarding closely the action of a single point in the surface of the 
 piston, the journal, or the screw, we see that it moves over a certain 
 path in the surface of the cylinder, the bearing, or the nut, and that 
 it comes into contact, one after another, with all the points in th^t 
 path. 
 
 In these instances the contact of the two pieces is superficial, the 
 moving surface being identical with the fixed one. But one piece 
 may touch another at discontinuous points, or even in only one point, 
 and yet the same peculiarity may characterize the action. 
 
 We may therefore define the condition of Pure Sliding Contact to be, 
 Such relative motion of two pieces that every point of contact in the one, 
 is brought into coincidence with all the successive points in their order, 
 of a line in the other. 
 
 69. Sliding contact in the nature of things as they are, is attended 
 with friction, because there is not any known substance whose surface 
 is absolutely smooth. We might however admit that friction would 
 exist even if all surfaces were perfectly polished and free from asperi- 
 ties. But its existence even then would imply relative motion of two 
 pieces in contact, of such nature that a given point of one should, be- 
 fore quitting contact, be brought into coincidence with more than 
 one point of the other. 
 
 Could this be entirely avoided, and the action be made such that 
 no point of either should touch two consecutive points of the other, 
 there would be no friction, which cannot be conceived of as being pro- 
 duced by contact alone. 
 
 Practically, this is impossible*, owing to imperfections of materials 
 and workmanship : but the abstract condition, compliance with which, 
 if it were attainable, would effect the result, is an exceedingly simple 
 one. 
 
 70. In Fig. 29, let the tangent 
 AB remain fixed ; if the circle now 
 rotate about the fixed centre C, the 
 points c, d, e., etc., of its circumfer- 
 ence will successively be brought 
 into coincidence with the point A 
 
KOLLING CONTACT. 31 
 
 of the tangent. Or if the circle be translated without rotating, as 
 shown by the horizontal arrow, the point A of the circumference 
 will come into coincidence successively with the points c', d', e', 
 etc., of the tangent : in either case the action is one of pure sliding 
 contact. 
 
 But if we suppose the circle to roll, like a hoop, along the tangent, 
 from C to D, it is readily seen that the point A will instantly quit 
 contact, and the points c, d, e of the circumference will come succes- 
 sively into contact with the tangent at c', d', e'. As it is commonly 
 expressed, the circle measures itself off upon the tangent ; thus if AK 
 be equal to the arc A G, then the point G will come into contact at 
 K, and if AB be equal to the semicircumference AGE, then E will 
 come into coincidence with B, and so on. 
 
 No point of the circumference can however be said to move in con- 
 tact with the tangent. The circle has in this case both the motions 
 of rotation and translation ; and the movement of the point A to the 
 left, due to the former, is neutralized by an equal movement toward 
 the right, due to the latter. That point is therefore for the instant 
 at rest ; and the same will evidently be true of the other points c, d, e, 
 etc., as they successively become the points of contact. 
 
 71. In Fig. 30, let ML be the common tangent, NN the common 
 normal, of the two plane curves AB, EG, 
 in contact at P. Let 0, o', be points on these 
 curves respectively, consecutive to P ; and 
 let EG be fixed. 
 
 Then if AB move so that P goes in the 
 direction PL, the effect will be to bring this 
 point P of the upper curve, into coincidence 
 with 0' on EG ; but if P goes in the direc- 
 tion PM, the point o of. AB will be brought 
 into coincidence with the point P of the 
 fixed curve EG. Now the motion PL may 
 be considered as either a translation, or a 
 
 rotation about any centre D in JVJV. If it * [if 
 
 be the former, evidently the curve AB can- FlG - 30 - 
 
 not be at the same time translated in the opposite direction. But in 
 either case the motion of P can be neutralized by an equal and op- 
 posite motion PM, which may be considered as a rotation about any 
 centre (other than D) in NN, as for instance C. The resultant of 
 these two simultaneous motions of AB will be a rotation about P as 
 an instantaneous centre, the effect of which will be to bring the point 
 into coincidence with o', the points of the two curves which now 
 
32 ROLLING CONTACT. 
 
 fall together at P, instantly separating : and if these conditions be 
 continuously maintained, the action will consist of a rolling contact 
 between AB and EG, precisely similar to that in the preceding case. 
 The point of contact of the moving curve is always at rest, and con- 
 sequently the action is unattended by friction. 
 
 72. It is apparent that the same kind of contact will ensue if in 
 Fig. 29 the circle merely rotates about C as a fixed centre, while AB 
 moves in contact with it, if the linear velocities and directions at the 
 point of tangency are the same : so also in Fig. 30, if the points of 
 the two curves which fall together at P, move in the same direc- 
 tion at the same rate, as represented by PM, the consecutive points 
 o, o' will come into coincidence, and the action will be that of roll- 
 ing contact. 
 
 It is also obvious that these two figures perfectly represent the roll- 
 ing of a cylinder upon a plane, and of one cylindrical surface upon 
 another, the rectilinear elements being perpendicular to the paper. 
 These illustrations sufficiently explain the nature of the action under 
 consideration, which may be defined as follows : 
 
 Pure Rolling Contact consists in such a relative motion of two lines 
 or surfaces, that the consecutive points of one come successively into 
 contact with those of the other in their order. 
 
 73. It will be readily seen that a double curved surface, like that of 
 a ball or an egg, can roll without slipping upon a surface of any kind, 
 plane, single curved, warped, or double curved, convex or concave ; 
 provided that the curvature of the rolling surface at the point of con- 
 tact be the greater. Tangency existing in this case at a single point 
 only, the action in fact consists merely in the rolling of a line of one 
 surface in contact with another line lying on the other surface, and 
 one or both of these lines may be of double curvature. 
 
 We are thus led to consider this kind of moving contact between 
 
 lines, in its most general aspect : and in 
 illustration, let NN 9 Fig. 31, represent 
 the common normal plane of two curves 
 of any kind, AB and EG: ML being 
 their common tangent at P, and o, o' 
 points on those curves, consecutive to P, 
 as in Fig. 30. Considering EG as sta- 
 tionary then, it appears, from the forego- 
 ing, that the rolling of AB upon it is effected by communicating to 
 that curve two simultaneous motions, one of which imparts to P the 
 motion PL, and the other imparts to the same point the equal and 
 opposite motion PM. Of these two motions, one may, and the other 
 
 / 
 
 /v 
 
KOLLING CONTACT. 
 
 33 
 
 must, be regarded as a rotation about a line at a finite distance, lying 
 in the plane NN: their resultant is a rotation about an instantaneous 
 axis also lying in NN, and passing through P. But the directions 
 of the axes of the component rotations, and also their distances from 
 P, are arbitrary : hence any line RPS, drawn through P in the 
 plane NN 9 may be assumed as the instantaneous axis ; a rotation 
 about which will bring the points o, o', into coincidence as required. 
 
 74. But though the instantaneous axis must pass through P, it is 
 not essential that it lie in the plane NN. For if any line be drawn 
 through P, then both PL and PM may be resolved into components, 
 perpendicular to, and lying in, that line. The latter may be regarded 
 as motions of translation, and, being equal and opposite, they will 
 neutralize each other. The former will also be equal and opposite, 
 and regarding one or both as rotatory, the irresultant will be a rota- 
 tion about the assumed line through 
 P, as an instantaneous axis. 
 
 Thus in Fig. 32, let PR, PS, be 
 the vertical projections of two he- 
 lices, lying on the surfaces of two 
 cylinders tangent along PB ; and let 
 MPL be the horizontal projection of 
 their common tangent, and NN the 
 horizontal trace of their common 
 normal plane. It will be apparent 
 that PL, PM, may be considered as 
 rotations about an axis in NN. But 
 also, with reference to OPO, we may 
 resolve PL into the components PC, 
 PD, and also PM into the com- 
 ponents PE, PG. Then regarding PD, PG, as motions of transla- 
 tion, they neutralize each other : and PC, PE, may be regarded as 
 rotations about axes lying in the plane of which 00 is the trace. These 
 also neutralize each other, leaving P for the instant at rest : and it 
 will readily be seen that as a matter of fact, the instantaneous axis 
 of the upper cylinder, in rolling around the lower, must in the present 
 position be the element of contact, of which PB is the vertical, and 
 OPO the horizontal projection ; as also that if one cylinder rolls upon 
 the other, the two helices move in rolling contact. 
 
 75. In so far then as the rolling of any one line upon another is con- 
 cerned, the moving line may at any instant be regarded as rotating 
 about any right line passing through the point of tangency. This, 
 as above pointed out, includes the case of rolling contact between sur- 
 3 
 
 FIG. 32. 
 
34 ^ ROLLING CONTACT. 
 
 faces which, touch each other in only one point. We have now to 
 consider the conditions under which surfaces having more than one 
 point of tangency can roll together without sliding. It has been 
 stated by Prof. Reuleaux,* that helicoids and certain other warped 
 surfaces are capable of moving in perfect rolling contact. From a 
 practical point of view it is of comparatively little consequence whether 
 they are or are not : but the fact that such a statement is made by 
 such an authority, warrants the presentation of the following consid- 
 erations : 
 
 76. 1. The rolling of one surface upon another is a motion of rota- 
 tion about an instantaneous axis, which must pass through every 
 point of tangency. 
 
 2. An axis is a right line ; hence if there be more than one point of 
 tangency, all of them must lie upon a rectilinear element of contact. 
 
 3. This rotation about the instantaneous axis may be resolved into 
 two component motions, of which one must be a rotation about 
 another axis. 
 
 4. In this component rotation, each point in the moving surface de- 
 scribes a circle with plane perpendicular to and centre in the second axis; 
 of which the radius is a perpendicular from the point to that axis. 
 
 5. The other component motion must be such that each point of 
 tangency shall by it be made to move with a velocity equal to that due 
 to the first component rotation, and in the opposite direction. 
 
 6. If this second component be a motion of revolution, the axis 
 must lie in a plane normal to the path of every point of tangency, de- 
 termined as above. Hence those paths must be parallel ; which re- 
 quires the axis of the first component rotation to lie in the same plane 
 with the common element. 
 
 7. If the second component be a motion of translation, all the points 
 of tangency are in consequence moving in parallel directions. There- 
 fore the axis of the first component rotation must lie in a plane contain- 
 ing the common element, and perpendicular to those parallel motions. 
 
 8. Consequently the axis of the first component rotation must in 
 all cases and at all times lie in the same plane with the common ele- 
 ment : and the motions of the points of contact, due to this rotation, 
 must at any instant be in parallel directions. 
 
 77. 9. Now the effect of the first component rotation must be, to 
 bring the consecutive element of the moving surface into coincidence 
 with the present line of contact on the fixed one. But as all the points 
 of the line of contact are moving in parallel directions, by virtue of 
 
 -"Kinematics of Machinery, 1876, pp. 81, 82. 
 
MIXED CONTACT. 35 
 
 this rotation, the consecutive position (that is to say the consecutive 
 element of the moving surface) must lie in the same plane with the 
 original one : thus determining a plane tangent all along an element. 
 
 10. Again the effect of the second component must be, to bring the 
 present line of contact on the moving surface, into coincidence with 
 the consecutive element of the fixed one. But whether this motion 
 be one of translation or of revolution, it imparts parallel motions to 
 all points of the common element, which consequently will lie in a 
 plane containing the consecutive position, and tangent to both sur- 
 faces all along an element. 
 
 11. Pure rolling contact, then, is not possible between surfaces 
 touching each other in more than one point, unless they arc plane or 
 single curved. 
 
 12. Nor is it possible between all such surfaces. No argument is 
 needed to show that a cone cannot roll upon a cylinder, nor upon an- 
 other cone unless the two have a common vertex. The rolling of one 
 single curved surface upon another involves the simultaneous rolling 
 of both upon the common tangent plane. Hence it is an essential 
 condition, that through any point of the common element, it shall 
 be possible to draw two lines, one upon each surface, of which the 
 developments upon the common tangent plane shall coincide. 
 
 78. Mixed Contact. By this is meant a combination of rolling and 
 sliding contact ; the nature of such action will be readily understood 
 by reference to Fig. 20, if we suppose the circle to rotate about the 
 centre, and at the same time to be translated, but in such manner 
 that the points c, d, e, do not come into coincidence with the points 
 c' d' e'. If, for instance, the circle rotate through an arc A G, and in 
 the same time be translated through a space AB or Ad', greater or 
 less than that arc, it is clear that though the action partakes to some 
 extent of the nature of rolling, yet there must be a sliding between 
 the circumference and the tangent, to an amount depending upon 
 the difference between the arc AG, and the distance AB or Ad'. 
 The resultant of the two motions imparted to the circle will be a ro- 
 tation about an instantaneous axis, which will however not pass 
 through the point of tangency A : and this affords a means of ascer- 
 taining whether in any given case the contact motion is of this de- 
 scription or not. 
 
 But we may for all the practical purposes of mechanism, disregard 
 this particular combination ; and we shall, in what follows, take note 
 only of the distinction between those contact motions in which the 
 action is purely rolling, and those in which it is not ; all the latter 
 being called cases of sliding contact. 
 
CHAPTER III. 
 
 ELEMENTARY COMBINATIONS DRIVER AND FOLLOWER ROTATING 
 ABOUT FIXED PARALLEL AXES DETERMINATION OF VELOCITY 
 RATIO AND DIRECTIONAL RELATION, 1, IN LINK MOTIONS ; 2, 
 IN BAND MOTIONS ; 3, IN CONTACT MOTIONS CONDITIONS OF 
 ROLLING CONTACT AND OF CONSTANT VELOCITY RATIO SIMI- 
 LARITY OF ACTION IN ALL THE MODES OF TRANSMISSION. 
 
 Cr 
 
 FIG. 33. 
 
 79. It has been shown, that even the simplest motion, a rectilinear 
 
 one, may be resolved into compo- 
 nents. But if there be any motion 
 employed in mechanical structures, 
 entitled to be called a simple one by 
 reason of the ease with which it can 
 be produced and continuously main- 
 tained, that motion is certainly the 
 one of rotation about a fixed axis. 
 Naturally enough, therefore, it is more frequently met with than 
 
 any other, and the very mention of machinery suggests ideas of levers 
 
 and Avheel-work. Nothing can be 
 
 simpler than the means by which 
 
 rotation is determined, nor can the 
 
 axes have a simpler relation than 
 
 that of parallelism : and, accord- 
 ingly, by far the largest and most 
 
 important class of elementary com- 
 binations is that in which both 
 
 driver and follower rotate about 
 
 fixed axes which are parallel to 
 
 each other. In the analysis of their 
 
 action it will be most convenient to 
 
 consider first those in which the 
 
 motion is transmitted by means of a link. 
 
 FIG. 34. 
 
TELOCITY RATIO 
 
 LINK MOTIONS. 
 
 37 
 
 Velocity Ratio in Link Motions. 
 
 80. In Figs. 33 and 34, let C and D be fixed axes, about which turn 
 the levers A C, BD, whose free ends are connected by the link A B ; 
 the axes being perpendicular, and the planes of motion parallel, to 
 the paper. 
 
 The motions of A and B are necessarily perpendicular to A C and 
 BD respectively ; whence the instantaneous axis, E, is found by pro- 
 longing those radii till they intersect. 
 
 From C, D, E, let fall the perpendiculars CG, DH, EF, on the 
 line of the link AB or its prolongation. Also draw the line of centres 
 CD, cutting the line of the link, prolonged if necessary, in the point /. 
 Now let 
 
 v ang. vel. of A around C, 
 v' = " " " B " D, 
 
 w = " " " A or B " E. 
 
 Then by similar triangles, 
 
 v^_AE_ EF 
 w~~~AC~~ ~CG 9 
 
 ~CG ~~ ci 
 
 ^ 
 v'~ BE~~ EF 
 
 81. This result may be deduced 
 otherwise, thus : In Fig. 35, let the 
 linear velocity of A be represented 
 by AL, perpendicular to AC: then 
 AM is its component in the line of 
 the link, to which BN must be 
 equal. The direction of BO, the 
 motion of B, is perpendicular to BD, 
 and its length is ascertained by draw- 
 ing NO perpendicular to AB. 
 
 Then by similar triangles we have 
 
 AL _AM 
 AG~ CG' 
 
 V = 
 
 __ 
 
 = BD ~ DH' 
 
 v__DH_DI 
 v'-~CG~ CI 
 
 We have, then, two simple and convenient values for the velocity 
 ratio, which may be thus expressed : 
 
38 
 
 DIRECTIONAL RELATION. 
 
 1. The angular velocities of the arms are to each other inversely as the 
 perpendiculars from their centres of motion upon the line of the link. 
 
 2. The angular velocities of the arms are to each other inversely as 
 the segments into which the line of the link cuts the line of centres. 
 
 82. Directional Relation. The perpendiculars CG and DH, from 
 the centres of motion to the line of the link, are called the Effective 
 Lever Arms. When they lie on the same side of AB, as in Fig. 33, 
 the arms will turn in the same direction. When they lie on opposite 
 sides of the line of the link, as in Figs. 34 and 35, the rotations will 
 be in opposite directions. 
 
 Dead Points. It is evident that in the combination shown in Fig. 
 33, the shorter lever BD is capable of turning completely round. 
 When a lever thus makes entire revolutions, as is very often the case, 
 it is usually called a Crank ; but whether it does or not, it is obviously 
 possible for the system to come into either of the positions shown in 
 Figs. 36 and 37, in which AB and BD coincide. 
 
 Fia. 36. 
 
 Fio. 37. 
 
 When this happens the system is said to be at a dead point, because, 
 the effective lever arm DH having disappeared, and the motion of B 
 having no component in the line of the link, the arm A C is moment- 
 arily at rest. That arm has a reciprocating circular motion ; and 
 the dead points obviously occur at the extreme limits of its travel. 
 By way of distinction, the crank BD is said to be at an outward dead 
 point in Fig. 36, and at an inward dead point in Fig. 37. 
 
 83. Momentary Constancy of Ve- 
 locity Ratio. In general, the veloc- 
 ity ratio in such combinations is 
 varying, as the ratio between the 
 effective lever arms changes from 
 instant to instant : though if, as in 
 Fig. 13, the levers are equal and 
 parallel, the velocity ratio is con- 
 stant throughout the action. But 
 it may be momentarily constant 
 under other circumstances. ^ Thus in Fig. 38, the arms A C, BD, are 
 
 FIG. 38. 
 
BAND MOTIONS. 
 
 39 
 
 parallel, but not equal. The motions of the points A and B are parallel 
 to each other, and since they are not perpendicular to the line of the 
 link, their linear velocities must be equal, whence we have directly 
 
 ^ 
 
 AC~ CG 
 
 Now, the positions of the link consecutive to the present one will 
 be parallel to AB, and cut the line of centres in points consecutive 
 
 FIG. 39. FIG. 40. 
 
 to and practically coincident with /. Hence during such elementary 
 motion the variation in the velocity ratio will be inappreciable. 
 
 84. Again, the line of the link in Figs. 33 and 34 is tangent at F to 
 the circle described by that point about the instantaneous axis E. 
 The consecutive positions of AB therefore in- 
 tersect each other in F : as may be seen very 
 
 clearly in Figs. 36 and 37, where F coincides 
 with A, which is momentarily at rest, while H r 
 BD moves in either direction to its consecu- 
 tive position. 
 
 Now in Figs. 39 and 40, the foot F of the 
 perpendicular upon AB from the instanta- 
 neous axis, coincides with /, the intersection 
 of the link with CD the line of centres. In 
 its consecutive positions, therefore, the line of 
 the link will cut the line of centres into prac- 
 tically the same segments as in its present 
 one, and the velocity ratio is momentarily 
 constant in this case also. 
 
 Band Motions. 
 
 85. Let the two curved pieces whose fixed 
 
 centres of motion arc C and Z>, Fig. 41, be FIG. 41. 
 
 connected by a flexible and mextensible band or cord, wrapped upon 
 
40 
 
 CONTACT MOTIONS. 
 
 their convex edges, and fastened to them at the points K and F. Let 
 the lower curve turn to the right, as shown by the arrow : the band 
 will then be wound upon AE, and unwound from BF, thus com- 
 pelling the upper curve to rotate. It is evident that the parts which 
 are wrapped upon the curves being always idle, the effective length 
 of the band is the distance between the points of tangency ; which 
 may or may not change during the action. 
 
 In the position shown, the points of tangency are A and B, whose 
 motions at the instant must be perpendicular to the contact radii, A C 9 
 BD. Their linear velocities, AL, BO, must be such that the com- 
 ponents AM, BN, along the line AB, shall be equal, because the 
 band is inextensible ; the components AR, BS, perpendicular to AB, 
 are ineffective, because the band is flexible. Now drawing the per- 
 pendiculars CG, DH, from the centres of motion to the line of the 
 band, and also the line of centres CD cutting A B in /, it is evident 
 on comparison that this combination at the instant differs from that 
 of Fig. 35 in no particular except the flexibility of the connector AB. 
 And we shall have, precisely as in that case, 
 
 CG ~ CI 
 
 And also as in that case, the rotations will be in the same direction 
 
 if CG and DHlie on the same side 
 of AB, and in contrary directions if 
 they lie on opposite sides of that line. 
 
 Contact Motions. 
 
 86. In Fig. 42, let C and D be the . 
 fixed centres of motion of the two 
 curved pieces in contact at P, their 
 common tangent being TT, and 
 their common normal NN : it is 
 evident that if the upper curve turn 
 as shown by the arrow, the lower 
 one will be compelled to rotate. 
 Draw the radii of contact, P C and 
 PD ; then the motion of P, con- 
 sidered as a point of the upper piece, 
 will be in the direction PA, perpen- 
 dicular to PC, while the direction of 
 the motion of P on the lower curve must be PB, perpendicular to PD. 
 
^ 
 
 CONTACT MOTIONS. mi " 
 
 Kesolve P^4 into the components PM, PO; of these the former 
 obviously does not tend to move the lower curve, and it is also evident 
 that the magnitude of PB must be such that its normal component 
 shall be PO; for were it greater than PO, the lower" curve would 
 quit contact with the upper, and if it were less the two curves would 
 intersect, both of which are contrary to the hypothesis. Draw CG, 
 DH, perpendicular to NN; then the triangles PCG, APO, are simi- 
 lar, as also are PDH, BPO. Draw CD cutting JOT at /; then ICG, 
 IDH are similar ; and letting 
 
 v ang. vel. around C ; 
 v' " " " D. 
 we have 
 
 V'=PJB_PO 
 
 PI) ~DH 
 
 Which may be thus expressed : 
 
 1. The angular velocities of two pieces in contact are to each other 
 inversely as the perpendiculars let fall upon the common normal from 
 the centres of motion. 
 
 2. These angular velocities are to each other inversely as the seg- 
 ments into which the common normal cuts the line of centres. 
 
 87. Condition of Constant Velocity Ratio. This is at once deduced 
 from the second of the above values, viz ; 
 
 v^_DI 
 v'~~ Cl'' 
 
 for DI + CI = CD, which is constant, therefore / must be fixed, and 
 the two segments at all times the same. 
 
 In other words, the curves, in order to maintain a constant velocity 
 ratio, must be such that their common normal shall always cut the 
 line of centres in the same point. 
 
 88. Condition of Compulsory Rotation. A rigid link can both push 
 and pull, so that the direction of the driver's motion is arbitrary. A 
 band is capable of pulling only : while on the other hand one piece 
 can drive another in direct contact with it only by pushing. In Fig. 
 43, the lower piece is the driver, turning as shown by the arrow. Let 
 x, z, be points consecutive to and on opposite sides of P, the former 
 being in advance. Then CP is greater than Cx, but less than Cz ; 
 and it is clear that in order to produce compulsory rotation, the 
 
SLIDING COMPOHEHTS. 
 
 curves and their motions must be such that the contact radius of the 
 
 driver shall he on the increase. 
 
 89. Rate of Sliding. The tangential components PM, PL, repre- 
 sent the linear velocities with 
 which the curves are at the in- 
 stant sliding on the common tan- 
 gent. In Figs. 42 and 43, these 
 components lie in the same di- 
 
 O N. 
 
 FIG. 43. 
 
 FIG. 44. 
 
 rection from P, and consequently their difference LM represents the 
 rate at which the two curves are sliding upon each other. In Figs. 
 44 and 45 these components fall in opposite directions, and the rate 
 of sliding is represented by their sum. 
 
 Directional Relation. From these various diagrams, which are 
 lettered to correspond throughout, we also perceive that the direc- 
 tional relation depends upon the positions of the perpendiculars CG, 
 DH, with reference to the common normal ; if they lie upon the same 
 side of NN, the rotations will be in the same direction ; if on oppo- 
 site sides, the rotations will be in contrary directions. 
 
 90. Condition of Rolling Contact. In order that there may be no 
 
 FIG. 46. 
 
 sliding between the two moving pieces, the tangential components 
 PM, PL, must have the same magnitude and the same direction, as 
 in Fig. 46. The normal components PO being the same in both 
 
CONDITION OF ROLLING CONTACT. 43 
 
 motions, the resultants PA and PB must also be alike in magnitude 
 and direction ; and these being perpendicular to the contact radii PC, 
 PD, the latter must also coincide in one right line, which can be no 
 other than CD. 
 
 That is to say : the condition of rolling contact is, that the point of 
 contact shall lie upon the line of centres. 
 
 The curves may be of such a nature that this condition is continu- 
 ously satisfied, the point of contact traveling along the line of centres, 
 and the velocity ratio varying accordingly. 
 
 On the other hand, that point may travel across the line of centres, 
 the action taking place partly on one side and partly on the other ; in 
 which case the velocity ratio may or may not vary ; but whether it 
 does or not, there will be more or less of sliding between the curves, 
 except at the instant when the point of contact crosses that line. 
 
 91. Similarity of Action in all the Modes of Transmission, It will be 
 observed that the common normal in contact motions has a very 
 striking resemblance to the lines of the link and the band, previously 
 discussed. In fact, we may select any two points in this normal, 
 and, drawing right lines from them to the centres of motion, form 
 thus a pair of arms and a link, which combination will at the instant 
 have precisely the same action as the original contact pieces. 
 
 AVo perceive, in short, that the motion of the driver is transmitted 
 to the follower in a right line, whether they be in contact or not ; and 
 that the action of all elementary combinations in which the two pieces 
 rotate about fixed parallel axes, is governed by these laws : 
 
 I. The angular velocities are to each other inversely as the perpen- 
 diculars from the centres of motion upon the Line of Action ; or, in- 
 versely as the segments into which the line of action divides the line 
 of centres. 
 
 II. The rotations have the same direction if the centres of motion 
 lie on the same side of the line of action ; and contrary directions if 
 they lie on opposite sides. 
 
 III. The rate of sliding, in contact motions, is the difference of those 
 components of the actual motions of the common point about the two 
 centres respectively, which are perpendicular to the line of action, 
 when they lie on the same side of that line ; and their sum if they lie 
 on opposite sides. 
 
 92. We may now proceed either to examine separately the action of 
 given combinations, or to ascertain, by the aid of the principles thus 
 far explained, the forms and proportions which may or must be given 
 to the elements in order to satisfy assigned conditions. The latter is 
 plainly the more systematic and fruitful course ; and wheels whose 
 
44 COMBINATIONS TO BE DISCUSSED. 
 
 axes are parallel, enter so largely into the composition of machinery 
 of all descriptions, that the fact is a sufficient reason for at present 
 confining our attention to the transmission of motion by direct con- 
 tact ; considering first those cases in which the action is purely roll- 
 ing, and taking up afterward those in which the contact is of the 
 mixed variety. 
 
CHAPTER IV. 
 
 ROTATION BY ROLLING CONTACT, FIXED PARALLEL AXES. 
 
 1. VELOCITY RATIO CONSTANT. Toothless Circular Wheels. Friction Gearing. 
 
 2. VELOCITY RATIO VARYING. The Logarithmic Spiral, and Toothless Lobed 
 
 Wheels formed from it. The Rolling Ellipses, and Lobed Wheels derived 
 from them by Contraction of Angles. Interchangeable Multilobes, derived 
 from Ellipse and from Logarithmic Spiral. Irregular Lobed Wheels. 
 
 1. Velocity Ratio Constant. 
 
 93. If the teeth of a pair of ordinary circular spur wheels be in- 
 definitely reduced in size and in- 
 creased in number, they will ulti- 
 mately become mere lines, or rec- 
 tilinear elements of cylinders of 
 
 revolution, as shown in Fig. 47. 
 
 These are technically called the 
 Pitch Surfaces of the wheels. Evi- 
 dently they are capable of moving 
 in perfect rolling contact about 
 their axes, the linear velocities of FlG - 47 - 
 
 their circumferences being the same, and the angular velocities in- 
 versely proportional to the radii, or to the original numbers of the 
 teeth. Which agrees with the deductions of the preceding chapter ; 
 the condition of rolling contact is (90), that the point of tangency 
 shall always be upon the line of centres : and the condition of a con- 
 stant velocity ratio is, (87), that the common normal, which passes 
 through that paint, shall always divide the line of centres into the 
 same segments. Hence in order to satisfy both conditions at once, 
 the contact radii must be constant ; or in other words the contact 
 curves must be circles whose centres are in the axes. 
 
 94. But it is important to note that although circles are the only 
 curves which can move in rolling contact about fixed axes, yet one has 
 
46 FRICTION GEARING. 
 
 no tendency to drive the other. The condition of compulsory rota- 
 tion is, (88), that the contact radius of the driver shall be on the in- 
 crease ; which is here impossible. 
 
 And again, in Fig. 47, let the upper circle turn as shown by the ar- 
 row, the motion of the contact point being represented by PB ; then 
 this motion, being tangent to both circles, has no normal component 
 at all. Consequently the driver tends merely to slip upon the fol- 
 lower, not to cause the latter to move. And it is a well-known fact, 
 that two polished rollers cannot be used to transmit rotation with a 
 constant velocity ratio : the more nearly they approach theoretical 
 perfection as cylinders, the farther they depart from practical perfec- 
 tion as wheels, and if motion be transmitted at all, it is by reason of 
 adhesion between them. 
 
 95. Friction Gearing. Nevertheless such toothless wheels arc quite 
 extensively used in practice, constituting what is known as Friction 
 Gearing. The axes arc pressed together with considerable force, and 
 in order to secure the greater adhesion, the peripheries in contact are 
 usually made of different materials : thus if one wheel be of iron, 
 the other may be made of wood, or have its rim covered with leather 
 or india-rubber. There is, of course, always a possibility that such 
 wheels may slip upon each other ; in point of fact they usually do, 
 and the amount of slip, moreover, varies under different conditions of 
 pressure and speed. There is, then, no certainty that a constant ve- 
 locity ratio will be preserved, and they cannot be employed where 
 that is imperative ; but there arc many cases in which it is not. 
 
 96. Grooved Friction Gearing. Another arrangement of friction 
 
 gearing, very generally used in hoisting 
 machines, especially for mining pur- 
 poses, is shown in Fig. 48 ; the two 
 wheels having a series of angular 
 grooves turned in their peripheries, so 
 M that by pressing them together an effec- 
 tive degree of adhesion is secured : both 
 wheels are usually made of cast iron. 
 The action is evidently attended with 
 considerable sliding in any case, and the 
 amount may vary between wide limits : 
 FIG. 48. but for such work a constant velocity 
 
 ratio is not requisite, and the possibility of slipping is a practical ad- 
 vantage, as it reduces the shock if the machine be suddenly started 
 or stopped, thus diminishing the risk of breakage. The velocity ra- 
 tio when in full work is substantially the same as that between two 
 
VELOCITY KATIO VARYING. 
 
 pitch cylinders whose line of tangency is midway between the tops 
 and bottoms of the grooves, as L M in the figure. 
 
 2. Velocity Ratio Varying. 
 
 97. If the contact curves are not required to maintain a constant 
 velocity ratio, then, since the point of tangency must still be always 
 upon the line of centres, the contact radii will vary. But their sum, 
 evidently, remains constant, if the point of contact lies between the 
 centres of motion ; or if those centres lie on the same side of that 
 point, the difference of the contact radii must be constant. 
 
 And by reference to Fig. 46, it will be seen that if PR, PS, be two 
 equal arcs of the curves, the latter must be of such form that 
 
 CR + DS = CP + DP = CD. 
 
 We proceed now to show how some curves may be constructed, which 
 will satisfy the above condition, and the manner in which various 
 changes in the velocity ratio may be effected by means of them. 
 
 98. The Logarithmic Spiral. In Fig. 49, let the parallels YU, TW, 
 be cut at any angle by TB, and , 
 
 perpendicularly by the other I i 
 
 parallels CD, IN, BN, etc. 
 About C as a centre, describe 
 an arc with radius CE A M ; 
 and about P as centre, describe 
 an arc with radius PA, cutting 
 the first in E, thus constructing 
 the triangle CPE. With cen- 
 tres C and E, and radii CG, 
 EG, respectively equal to BN 
 and AB, describe arcs intersecting in G. Continuing in a similar 
 manner, the result is the formation of the broken line RP V. 
 
 Now let us imagine C to turn first about P until P^coincides with 
 PA, and then about A until EC coincides with AM; then RPV 
 will have the position shown in dotted lines, the triangle EP C ap- 
 pearing as A OM. After which in a similar manner G may be brought 
 to coincide with B, the point C meantime advancing to N, and soon. 
 
 99. The polygon 7?PFthus travels with a sort of imperfect rolling 
 action, measuring off its perimeter on TB, since PE, EG, are by con- 
 struction equal to PA, AB, respectively; also PV = PH, etc. 
 
 Now if the subdivisions on TB be made indefinitely minute, so that 
 A and H are points consecutive to P, then RP Fwill become a curve 
 
 FIG. 49. 
 
48" 
 
 LOGARITHMIC SPIRAL. 
 
 to which TB is tangent at P. This curve will not, however, pass 
 through the angles of the polygon here shown ; since the arc EP, for 
 instance, must be equal to AP, the angle ECP will in the curve be 
 less than it now is. 
 
 By reason of the parallels LH, CD, etc., the angles LHT, CPH, 
 MAP, etc., are equal ; hence the curve must be such that its tangent 
 at any point shall make a constant angle with the radiant from C to 
 that point. It is apparent from the mode of derivation that the radi- 
 ants from C will increase as we go from CP to the left : and these 
 conditions determine the curve to be the logarithmic or equiangular 
 spiral, of which C is the pole and TB the tangent at P. 
 
 Supposing this curve to replace RP V, the rolling action will be 
 perfect, C traveling in a right line to M and N 9 while E and G go to 
 A and B. 
 
 100. The same process being repeated below TB, with D as the 
 pole, we shall have a polygon of the same description as RPV, but in 
 a reversed position. "We have in this second one, PF equal to PA, 
 and DF equal to IA. In the upper one, PE = PA, and CE ' AM-, 
 hence if the two rotate in opposite directions about C and D as fixed 
 centres, E and F will come together at S on the line CD. So in like 
 manner will G and Z, or F and H, meet upon CD. 
 
 It is apparent that these two broken lines will at the limit become 
 parts of equal, similar, and opposite equiangular spirals. The curves 
 cannot be constructed in this manner, except approximately : but this 
 mode of derivation exhibits in the clearest light the fact that they will 
 move in perfect rolling contact about fixed axes passing through their 
 poles. Referring the reader to the Appendix for detailed instructions 
 relating to the accurate delineation of these and other curves, we 
 proceed to consider the conditions which may be assigned in some ap- 
 plications of this spiral in mechanism. 
 
 101. Since the curve does not return into 
 itself, continuous rotation cannot be trans- 
 mitted from one shaft to another by means of 
 a single pair of the spiral arcs. But for 
 reciprocating circular motions they are well 
 adapted ; of which two cases are illustrated in 
 Figs. 50 and 51. 
 
 In the former, let the centres C and D be 
 given ; and suppose it to be required, that while 
 the upper piece rotates through a given angle, 
 the velocity ratio shall vary between two as- 
 FIG. 50. signed limits. 
 
LOGAKITHMIC SPIEAL CAMS. 49 
 
 Divide CD at A into two segments, whose ratio is one limit, and 
 at G into two others whose ratio equals the other limit. Set off 
 the given angle A CB, and make CB equal to CO. We have then two 
 given radiants CA, CB, and the pole C, to construct the spiral AB. 
 The other sector is a portion of the same curve ; of which one radiant 
 is AD, and the other ED, is equal to GD : whence the angle EDA is 
 determined by simply cutting the spiral by arcs described about the 
 pole D, with those given radii. 
 
 Letting v = ang. vel. of Upper Sector, 
 
 v' = " " Lower " 
 we have in the position shown, 
 
 t = AD - 
 
 v' AC' 
 
 and when J& and E meet at G, 
 
 _ ^ 
 
 v 1 " ~GC ' 
 
 102. It may be required that the two shafts shall turn through 
 equal angles. In that case one limit 
 
 may be assigned to the velocity ratio, 
 but the other will of necessity be the 
 reciprocal of the first. 
 
 For, as shown in Fig. 51, we must 
 have CB = AD ; also the angles 
 BCA, EDG, are to be equal, whence, FIG. 51. 
 
 from the nature of the curve, ED = AC, and the velocity ratio will 
 vary between the limits, 
 
 v_ = AD v __ GD __ AC . 
 v' ~ AC 9 v' ' GC AD' 
 
 of which the one is determined when the other is assigned. 
 
 103. Lobed Wheels. In the transmission of continuous rotation, it 
 may be required that a certain number of determinate changes in the 
 velocity ratio shall take place during the revolution of one of the 
 shafts. If the same number of changes is to occur during the revolu- 
 tion of each shaft, the equiangular spiral may be employed as follows : 
 
 In Fig. 52, let C, D be the given centres, and let the angles ACB, 
 ADE, each be equal to 60. Make DE - AC, and CB = AD : con- 
 struct the equal spiral arcs AE, AB ; draw AH, BF, similar curves 
 4 
 
50 
 
 LOBED WHEELS. 
 
 symmetrically placed with, reference to AD and BC. The velocity 
 ratio in the position shown is 
 
 AC 
 
 when B and E meet at G, the ratio will be 
 
 -' -in Fig. 51. 
 
 The angles EDH, A CF, each including 
 one-third of a circumference, the pair 
 of trilobes may be completed as shown. 
 They will roll together, the velocity ratio 
 varying from maximum to minimum and 
 back again, three times in each revolution 
 of each shaft. 
 
 In this construction the limits are neces- 
 sarily reciprocals of each other, so that only 
 one of them is arbitrarily assignable. 
 
 The angles A CB 9 ADE, may include any 
 aliquot part of the circle whose denominator 
 is even ; and in this way pairs of wheels 
 may be made with any desired number of 
 lobes, which will roll together. In Fig. 53 
 those angles are each equal to 180, the re- 
 suit being a pair of unilobes ; in Fig. 54, 
 they are made equal to 90, for constructing bilobed wheels. 
 It is to be observed that the wheels laid out as above will work to- 
 
 52. 
 
 FIG. 53. 
 
 FIG. 54. 
 
 gether only in equal and similar pairs, trilobe with trilobe, one uni- 
 lobe with another one, and so on. 
 
THE KOLLItfG ELLIPSES. 
 
 51 
 
 FIG. 55. 
 
 104. The Rolling Ellipses. Two equal and similar ellipses, of what- 
 ever eccentricity, will also work in per- 
 fect rolling contact, each revolving 
 
 about one of its foci as a fixed centre ; 
 the distance between the centres being 
 equal to the major axis. 
 
 In Fig. 55, let C and D be the fixed 
 centres, A and B the free foci, and E 
 a point of contact. It is to be proved 
 that E will always lie upon the line of 
 centres CD. 
 
 From E draw lines to both foci of 
 each ellipse. Then since the common 
 tangent FG makes equal angles with 
 all these lines, we have AEG = BEF, 
 and DEG = CEF, therefore AEB, 
 CED, are both right lines. 
 
 Also, EG + EA = EC + ED = EB 
 + ED. Consequently AB = CD, and EG EB. 
 
 The latter equality shows that the elliptical arcs EL, EM, meas- 
 ured from any point of tangency to those extremities of the major 
 axes which will come together, are equal. The rolling action is there- 
 fore perfect. 
 
 105. From the fact that AB is constant, -we are enabled, when the 
 rolling ellipses are fixed on the ends of shafts which overhang their 
 bearings, to connect the free foci by means of a link, as shown in the 
 figure, which is sometimes advantageous in practice, as will subse- 
 quently appear, even when the ellipses are provided with teeth. These 
 free foci, obviously, describe circles of which the radius is A C. 
 
 106. During each revolution of either ellipse, the velocity ratio va- 
 ries once from maximum to minimum and back again. The limits 
 are, evidently, mutually reciprocal, and determined by the ratio of 
 the segments into which either focus cuts the major axis. Hence if 
 one limit be assigned, and the distance, between the axes given, it is 
 only necessary to divide that distance into two parts whose ratio is the 
 assigned limit : we have then the major axis and the foci, whence the 
 ellipses are determined. 
 
 107. Transformation of Rolling Curves Contraction of Angles. 
 If any two curves are capable of rotating in rolling contact about 
 fixed axes, producing a given variation in the velocity ratio during 
 definite angular movements, they may be transformed into two others, 
 also rolling in contact about the same axes, and producing the same 
 
CONTKACTIOtf OF ANGLES. 
 
 FIG. 56. 
 
 FIG. 57. 
 
 variation in the velocity ratio, during angular movements which are 
 greater or less than the original ones in any assigned ratio. This is 
 effected by a process called the Contraction of Angles, since usually 
 the angular movement of the derived curves is less than that of the 
 original ones during the same variation in the velocity ratio. 
 In Fig. 56, let ^4. be the point of contact of the rolling curves AH, 
 AI, which turn with given angular velocities 
 about C and D : then at the end of a definite 
 time the points / and H will meet at G. In 
 Fig. 57, let the lengths of the radiants be the 
 same as in Fig. 56, but the angles included be- 
 tween them only half as great. Let these new 
 curves turn with the same angular velocity as 
 the original ones : then in half the time, H and / will also meet at G. 
 All the angles being reduced in the same proportion, while the radi- 
 ants remain the same, it follows that intermediate points, as E, B, 
 which meet on CD in the first pair, will do the same in the second ; 
 which, therefore, will move in pure rolling contact. 
 
 108. Lobed Wheels Derived from the Ellipse. -By applying the 
 above process to the rolling ellipses, pairs of wheels may be con- 
 structed which will roll together, the velocity ratio varying between 
 the same limits and according to the same law as in the ellipses them- 
 selves, but the maxima and minima recurring two, three, or any de- 
 sired number of times in each revolution. 
 
 The operation is illustrated in Fig. 58, in which C is a focus of the 
 
 ellipse whose major axis is AB. 
 For convenience of construction, 
 a circle is described around C, of 
 which the upper half is divided 
 into any number of parts at /, 
 77, ///, etc., and the lower left- 
 hand quadrant is divided in a 
 similar manner, at 1, 2, 3, etc. 
 Radii are drawn to the points of 
 division, the upper ones cutting 
 the ellipse at D, E, F, etc. : and 
 the segments intercepted between 
 the focus and the curve are set off 
 from C on the corresponding radii in the quadrant below, as Cd = CD, 
 on (71, Ce CE, on 02, and so on, CB going to CH. 
 
 109. Regarding the ellipse as a unilobe, is is evident that the curve 
 AeH, thus determined, is one fourth of the outline of abilobed wheel, 
 
SIMILAR LOBED WHEELS. 
 
 which when completed will roll in contact with 
 similar one, as shown in Fig. 59. 
 By contracting the angles to one 
 third instead of to one half, as 
 shown in Fig. GO, the contour of 
 a trilobe is determined, the com- 
 plete wheel with its rolling mate 
 being shown in Fig. 61 ; and in 
 like manner wheels with any 
 number of lobes may be con- 
 structed. 
 
 110. But these wheels, like 
 those previously constructed from 
 the equiangular spiral, will Work 
 only in equal and similar pairs, 
 bilobe with bilobe, trilobe with 
 trilobe, and so on : consequently 
 they can only be used when the 
 number of changes in the 
 velocity ratio is to be the same 
 during each revolution of each 
 shaft. Now, the above pro- 
 cess may be so modified that 
 from a pair of rolling ellipses 
 a dissimilar pair of wheels may 
 be derived, as for instance, a 
 bilobe rolling with a trilobe, or 
 
 53 
 
 FIG. 59. 
 
 FIG. 60. 
 
 FIG. 81. 
 
 a unilobe with a wheel of two or more lobes ; so that the numbers of 
 changes in the velocity ratio will be different in the two revolutions. 
 
54 
 
 DISSIMILAR MULTILOBES. 
 
 The eccentricity of the original ellipses, however, is not entirely arbi- 
 trary, but will vary between limits determined by the conditions of 
 each particular combination. 
 
 The modification just mentioned consists merely in this, that a part 
 only, instead of the whole, of the elliptical outline, is transformed into 
 a new curve. It will be readily apprehended that equal arcs of two 
 rolling ellipses may subtend at the fixed foci unequal angles, the ratio 
 of whose magnitudes shall be thafc of two integers : and that when 
 this is the case, those arcs may be transformed as above into portions 
 of the contours of lobed wheels. 
 
 111. For example, in Fig. 62, let C be the fixed and A the free 
 
 focus of the ellipse whose major 
 axis is HP ; let D be the fixed focus 
 of its rolling mate, and B and E, 
 two points which will meet at G, 
 the elliptical arcs PB, PE, being 
 equal. Now if the angles BCP, 
 EDP, are to each other as 3 to 2, 
 then by contracting them in the 
 same proportion, the former may 
 be reduced to 90 and the latter to 
 60. Thus the elliptical arc PB 
 
 ^ N . .y, f .. will be transformed into PI, one 
 
 flu ^^^^}-^^^^ fourth of the outline of a bilobe, 
 
 and PE into PM, one sixth of the 
 contour of a trilobe, and these new 
 curves will roll together. 
 
 Let be the free focus of the 
 second ellipse, then, drawing OE 
 and AB, it is evident that the 
 angles EDP, BA C, are equal. 
 
 If, therefore, on the indefinite 
 right line HD, we construct any two 
 angles BAG, BCD, whose magnitudes have the ratio of 2 to 3, and 
 produce the lines which limit them, till they intersect at B; then 
 from the ellipse whose foci are A and C, and major axis is equal to 
 AB + BC, a bilobe and a trilobe may be derived, which will roll 
 together. 
 
 The velocity ratio in the position shown, is 
 
 PD 
 
 PC 
 
 CH 
 TC 
 
LOBED WHEELS DERIVED PROM ELLIPSES. 
 
 and when B and E meet at G, it will be 
 
 55 
 
 ^ 
 JO BC~ BC 
 
 112. In Fig. 62, the angles B CP, BA C, arc both obtuse ; but the con- 
 struction holds true if they be made acute, as in Fig. 63. But whereas 
 
 they were contracted in the first case, 
 they have in the second case to be ex- 
 panded in order to produce the re- 
 quired contours of the lobed wheels ; 
 
 a circumstance which points directly to the conclusion that there 
 must be one ellipse so proportioned that it is not necessary to do 
 either. This is shown in Fig. 64, where the angle BCP being 90, 
 and EDP, or its equal BAG', being 60, the equal elliptical arcs PB 
 and PE form at once portions of the outlines of the bilobe and the 
 trilobe respectively. 
 
 From these same ellipses another pair of dissimilar multilobes can 
 be constructed. Erecting the perpendiculars A I, DK, and drawing 
 CI, OK, the elliptical arcs PBI, PEK, are equal. But the angles 
 ICP, KDO, measuring 120 and 90 respectively, are to each other 
 in the ratio of 4 to 3 : therefore, contracting them to one half, as in 
 
56 
 
 GENERAL PROCESS OF CONSTRUCTION. 
 
 Fig. 65, we have portions of the contours of a trilobe and a quad- 
 
 rilobe which will work in rolling contact. 
 
 113. In general, then, we may pro- 
 ceed as follows : If it be required to 
 construct a wheel with m lobes, to roll 
 with another having n lobes, supposing 
 m to be the greater number : Assuming 
 any two points A, C, on the indefinite 
 right line HD, Fig. 66, construct the 
 angles BCD, BAD, with magnitudes 
 whose ratio is that of m to n, the lines 
 AB, CB 9 intersecting at B. Then the 
 sum of those two lines will be the major 
 axis, and the points A and C the 
 foci, of an ellipse from which the 
 wheels can be constructed as above ex- 
 plained. 
 
 Other pairs of angles having the 
 same ratio may be constructed, deter- 
 mining the points I, G, etc. : these 
 points will lie in a curve, AID, and it 
 is evident that if from any point of this 
 curve, lines be drawn to A and C, they 
 
 will form with HD, angles in the required ratio, and their sum may 
 
 be taken as the major axis of the proposecT ellipse : as for instance, 
 
 UP = EA + EC, if E be the point selected. 
 114. The curve itself may be constructed either by determining a 
 
 sufficient number of points 
 
 as above described, or by 
 
 aid of the following prop- 
 erty. The centre of the 
 
 circle circumscribing any 
 
 one of the triangles whose 
 
 base is AC, and vertex in 
 
 the curve, as for example H 
 
 ABC, must lie on the 
 
 line LM perpendicular to and bisecting A C. 
 Now let the angle BCD = m, 
 then " " BAG =11, 
 and " " ABC=m-n-, 
 
 and in the circumscribing circle 
 
 we shall have 
 
 FIG. 66. 
 
LIMIT OF ECCENTRICITY. . 57 
 
 m n measured by -J arc A C y 
 n " " 4- " CB, 
 
 m " "I " ACB: 
 
 that is to say, 
 
 arc CB : arc A C : : n : m n. 
 
 If then we had taken the centre at .pleasure on LM, and, de- 
 scribing the circle through A and G, had set off the arc CB in the 
 above ratio to the arc A C, the point B would have been thus de- 
 termined. 
 
 115. Now, this curve springs from the point A ; for when m 
 becomes 180, as shown in the two small diagrams marked x and z, 
 A C will divide the circle whose centre is 0, into segments such that 
 CRA 9 above A C, is to ASC below that line, as n is to m n : and the 
 curve will be tangent at A to that circle. 
 
 The limit D of the curve will be reached when m becomes zero : 
 that is to say when the radius of the circle A CB becomes infinite, in 
 which case the circle itself will coincide with the right line A CD. 
 We shall then have 
 
 But 
 
 AD + CD = 2 CD + AC 
 
 m n 
 
 = AC 
 
 \m n 
 
 or, 
 
 + AC 
 
 m n 
 whence 
 
 AC m n 
 
 AD + CD ~~ m + n * 
 
 We have, therefore, this limit to the eccentricity of the ellipses 
 from which dissimilar wheels with assigned numbers of lobes can be 
 derived, viz : that the distance between the foci, divided by the 
 major axis, must be greater than the difference of the numbers divided 
 by their sum. 
 
 116. Each wheel derived from the ellipse has thus far been sup- 
 posed to consist of two or more lobes. But the construction last dis- 
 
58 
 
 UNILOBES WITH MULTILOBES. 
 
 cussed will hold good when n is to m in the ratio of unity to any 
 integer ; therefore by expanding the elliptical arc subtending the 
 angle n, until the latter equals 180, it will be seen that a unilobe 
 may be formed, which will roll with a wheel having any given num- 
 ber of lobes. This is sufficiently illustrated by Figs. 67 and 68, 
 which represent respectively a unilobe rolling with a bilobe and one 
 rolling with a trilobe. 
 
 In the former case, it will be observed that since the angles ABC, 
 
 BAC, are each equal to BCP, the curve, AID of Fig. 66, in Fig. 
 67 becomes the semicircle ABG, of which the centre is C and radius 
 AC, the distance between the foci of the primitive ellipse. The 
 elliptical arc PB, becomes by expansion the curve PLA, forming half 
 the outline of the unilobe ; while the equal arc PE is expanded into 
 PM, bounding one quadrant of the bilobe. 
 
 117. Interchangeable Multilobes, It appears, then, that from a 
 given ellipse there can be constructed pairs of similar wheels having 
 any number of lobes ; and if the eccentricity be great enough, several 
 different pairs of dissimilar wheels may be derived from the same 
 ellipse. But these wheels will work only in pairs as constructed, and 
 are not interchangeable ; thus, the bilobe which rolls with a trilobe, 
 
HOLDITCH'S LNTEECHANGEABLE MULTILOBES. 
 
 59 
 
 is not the same as the one which works with a unilobe, though derived 
 from the same ellipse, and if a unilobe and a trilobe be derived from 
 the same primitive ellipse, they will be different from either of the 
 others. 
 
 There are, however, two methods by which a series of lobed wheels 
 may be constructed, which are interchangeable, any one working in 
 perfect rolling contact with any other one of the series. 
 
 118. In the first of these methods, which was discovered by the 
 Rev. ITamnett Holditch, the multilpbes are derived, each from one of 
 a series of different ellipses. Their rolling properties do not appear 
 to admit of geometrical demonstration, but the process of construct- 
 ing them is simple and practical. 
 
 A series of ellipses is first made, having the same foci, but with 
 minor axes which are to each other as 1, 2, 3, 4, etc. Then the 
 primitive ellipse, whose minor axis is 1, will roll with a bilobe con- 
 structed as in Fig. 58 from the ellipse whose minor axis is 2, with a 
 trilobe derived as in Fig. GO from the one whose minor axis is 3, etc., 
 and these with each other indifferently. 
 
 119. In Fig. 69, let be the centre, 
 A and C the foci, and HP the major 
 axis of the primitive ellipse. Draw 
 CG perpendicular to HP, and cut it 
 at B by an arc with centre and 
 radius Off, then setting off BD, DE, 
 etc., each equal to CB the semi-minor 
 axis, describe arcs about through 
 D, E, F, etc., thus determining OK, 
 
 FIG. 70. 
 
 OM, OR, the semi-major axes of tho ellipses from which the bilobe, 
 trilobe, and quadrilobe are to be derived. In Fig. 70, the first three 
 of a series are shown in action, the unilobe, or primitive ellipse, 
 being the same as in Fig. 69. Since the difference between the 
 greatest and the least radii is the same in all the wheels, being equal 
 
60 
 
 MAC CORD'S INTERCHANGEABLE MULTILOBES. 
 
 to the distance between the foci, it will be seen that if one nmltilobe 
 be given, the original ellipse may be found by reversing the process, 
 and wheels with any desired numbers of lobes constructed, which will 
 roll with the one given. 
 
 120. The other method of constructing a set of lobed wheels capa- 
 ble of rolling with each other 
 indifferently, was devised by 
 the author, and the funda- 
 mental curve is the equiangular 
 spiral. The process is illus- 
 trated in Fig. 71, where C is 
 the pole of the spiral SMN. 
 Drawing through C any right 
 line, cutting the curve at A 
 and B y the portion A SB is one 
 half the outline of a unilobe 
 which will roll with a similar 
 and equal one, as already 
 shown. In order to construct 
 a bilobe which shall roll with 
 this unilobe, it is necessary 
 to find two radiants at right 
 angles to each other, whose 
 difference is the same as that 
 between CB and CA. Pro- 
 duce BA, and intersect its 
 prolongation at G by a circle 
 through B, with centre at the 
 pole ; and draw CF perpendic- 
 ular to BA, cutting this circle 
 
 FIG. 71. 
 
 at F and the spiral at H. 
 Then 
 
 and 
 
 CBCA = AG, 
 CBCH = HP. 
 
 If, now, two radiants be found, which are to CB and CH respect- 
 ively, in the same proportion that AG bears to HF, they will be 
 perpendicular to each other, and their difference will be equal to A G. 
 
 121. This may be done graphically, as it involves merely the con- 
 struction of a fourth proportional to three given lines, thus, 
 
 A G : HF : : CB : CM, 
 AG'.HF::CH: OS. 
 
UNSYMMETRICAL WILOBES. 
 
 61 
 
 The arc SBM of the spiral thus found is one quadrant of the con- 
 tour of the required bilobe ; the triangle ADK is therefore made sim- 
 ilar and equal to MCS, and the completion of the wheel requires no 
 explanation. 
 
 For the trilobe, draw any two radiants including an angle of GO , 
 ascertain their difference, and, comparing it with A G, proceed as above 
 to increase or diminish the assumed radiants, as the case may be, in 
 the same proportion, until the difference is equal to AG; the lengths 
 thus determined will be those of the greatest and least radii of the 
 trilobe ; and in a similar manner wheels of any number of lobes may 
 be constructed. 
 
 122. Since in any system of interchangeable multilobes, the differ- 
 ence between the greatest and least radii must be constant, while their 
 actual lengths vary with the number of lobes, the limits of the vari- 
 ation in the velocity ratio will not be the same for different pairs ; and 
 obviously the law which governs the rates of variation as well as the 
 differences between those limits, depends upon the peculiarities of the 
 fundamental curve upon the eccentricity of the primitive ellipse or 
 the obliquity of the original spiral. 
 
 123. Irregular Lobed Wheels. The non-circular wheels thus far de- 
 scribed, whether consisting of one lobe or many, are symmetrical ; and 
 moreover, the whole contour of each is made up of curves of the same 
 kind. Neither of these things is so of necessity ;. and as illustrating 
 the possibilities of varied motion in rolling contact, we give a few ex- 
 amples of irregular wheels composed of different curves. 
 
 In Tig. 72, let O and D be the poles of the equal and similar loga- 
 rithmic spiral arcs ALB, A ME ; 
 then these arcs will roll together, B 
 and E meeting at G. Now, AB may 
 be taken as the major axis, and C as 
 one focus, of the semi-ellipse ANB, 
 which will roll with the equal and 
 similar one A OE. 
 
 In each revolution, then, the veloc- FIG. 72. 
 
 ity ratio will vary between the reciprocal limits 
 
 CB 
 CA> 
 
 CA 
 
 but by reason of the different natures of the curves, the rates of vari- 
 ation during the decrease, will not be the converse of those during 
 the increase. 
 
62 DISSIMILAE UNILOBES. 
 
 124. In Fig. 73, let C be the pole of the logarithmic spiral arc 
 AFB. With centre C, and radius equal to %AB, describe an arc cut- 
 ting the spiral at F '; draw FA, 
 FB, FC, also FP perpendicular 
 to AB, and on BA produced set 
 off PL, PM, each equal to FC. 
 Then because CF- CA -CB- 
 CF, the spiral arcs AF, FB are 
 
 pToTra"" equal. Consequently the chord 
 
 FB is greater than the chord FA, whence PJS is greater than PA. 
 That is to say, P is not the middle point of AB ; but it is the middle 
 point of ML by construction, therefore PL is greater than PA, and 
 PM is less than PB. 
 
 The spiral arc AFB, turning round 6 Y as a fixed centre, will roll 
 with an equal and similar one, the distance between their centres of 
 motion being equal to A B. Set off PR = PC', then R and C are 
 the foci of the ellipse whose major axis is ML, and semi-minor axis 
 PF: which also will roll with an equal and similar one, turning 
 round the same centre C, the distance between centres being equal to 
 ML, which by construction is equal to AB. 
 
 125. From these data we can construct, as in Fig. 74, two dissimilar 
 unilobes, whose contours are com- o s 
 
 posed partly of the spiral and 
 
 partly of the elliptical arcs. We 
 
 have first the semi-ellipse FLO, of 
 
 which C is the focus and P the 
 
 centre, PL being the semi-major 
 
 axis ; this rolls with the equal FIG. 74. 
 
 semi-ellipse GL8: then the spiral arcs OB, FB, corresponding to FB 
 
 of the preceding diagram, and these roll with the equal arcs SA, GA, 
 
 which correspond to FA of Fig. 73. 
 
 During the revolution, the velocity ratio varies between the 
 limits 
 
 v DL v_ AD^ 
 v' =: ~CL> v' '~~~ ~BC 9 
 
 which, it will be observed, are not reciprocals, as in the unilobes 
 previously described. 
 
 126. In Fig. 72, the wheels, though similar, are not symmetrical ; 
 in Fig. 74 they are symmetrical but not. similar : and finally, those 
 represented in Fig. 75 are neither the one nor the other. Above the 
 
IRREGULAR MULTILOBES. 
 
 63 
 
 line of centres, each wheel is formed of a single arc of a logarithmic 
 
 spiral, A MB rolling with 
 
 ANE. Below that line, we 
 
 have BH, which rolls with EF, 
 
 each being a quadrant of the 
 
 bilobe derived, as in Fig. 58, 
 
 from the ellipse whose major 
 
 axis is AB, O being one focus ; 
 
 AL, which rolls with AT, these 
 
 curves being parts of quadri- FIG. 75. 
 
 lobes derived in the same way from the same ellipse : and LH, rolling 
 
 with IF, these two being equal arcs of another logarithmic spiral. 
 
 127. It may be added, that there are a number of other curves 
 capable of rolling in contact about fixed axes. But it is the province 
 of the mathematician rather than of the mechanician to investigate 
 their properties, which do not admit of simple geometrical demon- 
 stration ; nor aro they so easily constructed and adapted to use in me- 
 chanical devices. It -may be safely said that in general the number of 
 changes in the velocity ratio, and the limits of its variation, are of 
 greater importance than the precise law according to which it varies 
 between those limits : and it is believed that the principles involved 
 in the examples already given will be found to meet most if not all 
 the requirements of praclical mechanism. 
 
 128, These various curves, whose action has been discussed as 
 though they were mere lines rolling with each other in their own 
 plane, are of course practically to be considered as the bases, or trans- 
 verse sections, of cylindrical surfaces, tangent to each other along an 
 element. Strictly, therefore, the expression that <e the point of con- 
 tact lies always on the line of centres," means that "the element of 
 tangency lies always in the plane of the axes," which is one condition, 
 as shown in Chap. II., of pure rolling contact between two surfaces. 
 
 And equally of course, the consideration of these surfaces is only 
 the step preliminary to the investigation of the surfaces of teeth to 
 be formed upon them : for, the motion of the followers being com- 
 pulsory only so long as the contact radii of the driver are increasing, 
 continuous rotation can not be transmitted by means of these rolling 
 surfaces themselves, with any certainty that the desired velocity ratio, 
 whether constant or varying, will be exactly maintained. 
 
CHAPTER Y. 
 
 ROTATION BY ROLLING CONTACT, AXES NOT PARALLEL. 
 
 1. AXES INTERSECTING. Velocity Ratio Constant or Varying. The Rolling Cones, 
 
 or Pitch Surfaces of Circular and Elliptical Bevel Wheels. Conical Lobed 
 Wheels. 
 
 2. AXES IN DIFFERENT PLANES. Velocity Ratio Constant. The Rolling Hyper- 
 
 boloids, or Pitch Surfaces of Skew-Bevel Wheels. 
 
 1. Axes Intersecting. 
 
 129. Velocity Ratio Constant. It follows directly from the deduc- 
 tions at the conclusion of Chap. II, that the only surfaces capable of 
 rolling in line contact about two fixed axes which intersect each other, 
 
 are cones, the common vertex being 
 the point of intersection, and the com- 
 mon element lying in the plane 
 of those axes. And in order to 
 maintain a constant velocity ratio, 
 the transverse sections of these 
 cones must be circles. Thus, in Pig. 
 76, let AB, AC, be the axes, and AP 
 the common element, all lying in the 
 plane of the paper : then in the rota- 
 tion indicated by the arrows, the 
 motion of the common point P will 
 Draw PD perpendicular to AB, PE 
 
 FIG. 76. 
 
 be perpendicular to the paper, 
 perpendicular to A C, and let 
 
 v = 
 
 V' 
 
 ang. 
 
 vel. about AB , 
 " " AC; 
 
 then at the instant we shall have 
 
 PJ2 
 PD 
 
THE ROLLING CONES. 65 
 
 and in order that this ratio may be constant as required, the contact 
 radii PD, PE, must be constant also. That is to say, the plane 
 bases of the cones, PDF, PEG, perpendicular to the axes, must 
 be circles ; and the cones themselves are generated by the revolution 
 of the common element AP about each axis in succession. 
 
 130. These then are the pitch surfaces of the ordinary conical or 
 bevel wheels. In practice it is of course sufficient to use compara- 
 tively thin frusta of the cones, as indicated in the figure : in the great 
 majority of cases they are provided with teeth, but are to some extent 
 used without, constituting what is called Bevel Friction Gearing. 
 
 As above deduced, the angular velocities are always inversely pro- 
 portional to the perpendiculars let fall from any point of the common 
 element upon the axes, whatever the relative positions of those three 
 lines. But it is easy to see that those positions may be very different 
 from what they are in Fig. 76 ; and it is of interest to consider what 
 results may follow from various assigned or assumed conditions. 
 
 131. First, however, it may be remarked that in practice two cases 
 may present themselves for solution. Thus, the axes AB, AC, being 
 given, as in Fig. 76, suppose it to be required to construct a pair of 
 wheels so proportioned that m revolutions about AB shall produce 
 n revolutions about AC. Draw xx parallel to AB, at a distance from 
 it measuring n parts on any convenient scale of equal divisions, and 
 zz parallel to AC, at a distance from it equal to m parts on the same 
 scale. These two lines intersect in o ; and APo is evidently the com- 
 mon element of a pair of cones which will roll together with the re- 
 quired velocity ratio. 
 
 Or, the cone APF being given, it may be required to find another 
 which will roll with it, the velocity ratio being also assigned as above. 
 In that case draw xx as just ex- 
 plained, cutting AP in oj then 
 about o as centre, describe an arc 
 yy with radius equal to m parts on 
 the scale, and the axis A C of the 
 required cone will be tangent to 
 this arc as shown. 
 
 132. Now in Fig. 77, let the axes 
 AB, CD, cut each other obliquely 
 at V, the angle of inclination being 
 given and the velocity ratio as- 
 
 . -, TV- 1 FlG - 77 - 
 
 signed. Drawing xx and zz as de- 
 scribed in the preceding article, Vo is .determined and the cones 
 constructed. 
 
66 
 
 BEVEL WHEELS IK DOUBLE PAIRS. 
 
 It is to be noted in this case, that these surfaces are tangent along 
 the single line oVl ; and that if one shaft be carried past the vertex, 
 two pairs of frusta may be used at once, cut from opposite nappes of 
 the cones, as E, F, and G, H. These two pairs may be equidistant from 
 the vertex and therefore exactly alike, as shown ; or either pair may 
 be placed nearer the vertex if desired, as they roll together without 
 any interference, with the same velocity ratio, and as indicated by the 
 arrows, with the same directional relation. 
 
 133. In this instance the lines xx, zz, are drawn within the acute 
 angle formed by the intersecting axes. But in Fig. 78, they are 
 
 , z drawn within the obtuse an- 
 
 gle, the inclination of the 
 axes and the velocity ratio 
 being the same as before. 
 The result is a different pair 
 of cones ; the velocity ratio 
 is the same as in Fig. 77, by 
 construction ; but the direc- 
 tional relation is changed. 
 
 Also, the opposite nappes 
 of these cones intersect each 
 other in such a way, that 
 though under the condi- 
 tions here assumed one shaft can be carried past the vertex, and two 
 pairs of frusta employed at once if desired, they cannot be placed at 
 equal distances from the vertex : the wheels, therefore, will not be 
 alike, but one pair must be larger than the other in order to avoid 
 interference. 
 
 134. But it may not be possible to use two pairs at once, whatever 
 
 FIG. r ,8. 
 
 Fia. 79. FIG. 80. 
 
 their distances from the vertex. Thus in Fig. 79, the cones being 
 constructed so as to produce an assigned velocity ratio and also an 
 assigned directional relation, it is clear that neither shaft can be car- 
 ried past the other wheel. 
 
DOUBLE TAKGEITCY OF COtfES. 
 
 67 
 
 Again, it is to be observed that in all these cases the tangency is 
 external : and it obviously always will be when the common element 
 lies within the acute angle formed by obliquely intersecting axes. 
 But when it lies within the obtuse angle, it may happen, as in Fig. 
 80, that one cone shall touch the other internally. Or, the common 
 element may be perpendicular to one of the axes, as in Fig. 81, the 
 pitch cone degenerating into a plane. 
 
 135. Thus far the axes have been supposed to cut each other 
 obliquely ; and whatever the forms of the pitch surfaces, it will be 
 noted that in each they are tangent along one line only. We have 
 
 IQ 
 
 FIG. 81. 
 
 yet to consider the case in which the axes are perpendicular to each 
 other, as in practice they are, more frequently than otherwise. And, 
 as shown in Fig. 82, we are confronted by the singular circumstance 
 that the pitch cones are tangent along two elements, or, mn. The 
 frustum ^Tcan then roll simultaneously with the frusta F and G, 
 
 FIG. 83. 
 
 cut from opposite nappes of the other cone, but these two are tangent 
 to H along different lines, and consequently they will turn in oppo- 
 site directions. Equal and opposite frusta, like those in Fig. 77, 
 
68 
 
 HOLLOW CONE WITH DOUBLE PAIR. 
 
 cannot, of course, be simultaneously employed : but either of the 
 shafts can be carried past the vertex, and the unequal pairs of wheels 
 cut from the opposite nappes, being tangent along the same line, will 
 have the same directional relation. And, as shown also in Figs. 83 
 and 84, this directional relation is optional, since either of the two 
 common elements of the cones may be selected as the line of prag- 
 matic contact. 
 
 136. In the case of internal tangency, equal frusta of the opposite 
 nappes of the hollo w cone may be combined in a single wheel, as 
 
 shown in Fig. 85. Here the frusta 
 F and H are formed in opposite sides 
 of a solid ring, which is carried by 
 arms LL, so curved as to avoid 
 interference with the wheel G, which 
 engages with H, and is fixed on the 
 same shaft with E, engaging with 
 F. By this arrangement, 7 " the forces 
 transmitted act upon the large wheel 
 in the manner of a couple, in oppo. 
 site directions, on opposite sides of 
 and equidistant from the axis -AB 
 thus relieving the bearings from side 
 pressure. 
 
 137. We find, then, that whatever the angle included between the 
 axes, it is always possible to construct two pairs of cones, rolling to- 
 gether with the same velocity ratio, but having different directional 
 relations, and we are at liberty to employ whichever may best serve 
 the purpose to be accomplished. 
 
 The cones can always be determined when the axes, velocity ratio 
 and directional relation are given, by the process described in (131). 
 
 Another method, differing slightly in detail, is shown in Figs. 86 
 and 87, which also illustrate the following property, viz : that if dis- 
 tances be set off on the axes from the point of intersection, directly 
 proportional to the angular velocities, and the parallelogram' com- 
 pleted of which these are the adjacent sides, the common element 
 will lie in the direction of the diagonal. 
 
 Thus let v' the angular velocity about AB' be represented by VL, 
 and v', that about CG, by VM, then VP will be the common element. 
 
 For, draw PD perpendicular ioAB, PE perpendicular to CG, and 
 join DE. Now VP divides the parallelogram ML into two triangles 
 
 FIG. 85. 
 
 Which was first suggested by Mr. 0. A. Benton. 
 
PARALLELOGRAM OF ANGULAR YELOC 
 
 F I V E^R SIT 
 
 x/r o:Er *-tV 
 which are similar to each other ; and also to DPE, because a circle 
 
 will go round VEPD, in which the angles DEP, D VP, stand on th( 
 
 FIG. 86. FIG. 87. 
 
 same arc, and are therefore equal. For a like reason the angle EDP 
 is equal to EVP 9 which is equal to VPL. Then if VP be the com- 
 mon element, we have 
 
 PE 
 PD 
 
 VL 
 LP 
 
 VL 
 
 ~VM 
 
 If the angle between the axes be acute, it may be better to set off 
 VL representing v, and then to draw LP parallel to CG, and of a 
 length representing v' on the same scale ; thus avoiding the uncer- 
 tainty arising from attempting to locate P by an acute intersection. 
 
 In these two diagrams the inclination of the axes, and the Telocity 
 ratio, are the same, but the directional relations are unlike. Conse- 
 quently VM has the same ratio to VL in both cases, but is set off so that 
 in one diagram the angle MVL is acute, while in the ot*her it is obtuse. 
 
 138. Velocity Ratio Varying, In Fig. 88 let E, F, be two pins 
 fixed in the sphere whose centre is 0; 
 let EFP be a loop of fine inextensi- 
 ble thread, passing around the pins 
 and a marking point at P. Then as 
 P is moved along, the thread being 
 kept always taut and in contact with 
 the sphere, it will trace the curve 
 APBHj which \sthesphericalettipse. 
 
 Evidently, EF, EP and FP will 
 always be arcs of great circles ; and 
 since the part EF of the thread is 
 always the same, and always idle, 
 
70 THE SPHERICAL ELLIPSE. 
 
 while the total length is invariable, this curve may be defined as the 
 locus of the vertices of a series of spherical triangles, of which the base 
 is common and the sum of the other two sides constant. This sum 
 will be equal to the major axis AB ; now let AD be the major axis of 
 an equal and similar curve tangent to the first at A, of which G and 
 H are the foci : then BD will be a continuous arc of a great circle, 
 and EHw\\\ be equal to AB or AD. 
 
 Next, let these two curves be taken as bases of cones whose com- 
 mon vertex is 0, the centre of the sphere : then they will be tangent 
 to each other along the element OA, and they will roll in contact 
 about OE and. OH as fixed axes. For, let AP, AR be any equal arcs 
 of the two spherical ellipses ; then we shall have 
 
 
 PE=RG, PF=RH, 
 
 whence, 
 
 PE + RH = PE + PF = EH-, 
 
 consequently the points P and R will meet on the arc DAB, or in 
 other words, the common element will always lie in the plane of the 
 axes. 
 
 139. AVe have, then, a pair of non-circular cones, capable of rolling 
 in contact about fixed intersecting axes, with a varying velocity ratio, 
 and forming the pitch surfaces of what may be properly called Ellip- 
 tical Bevel Wheels, since their bases, as thus constructed, are spheri- 
 cal ellipses. But though these curves are easily traced upon the sur- 
 face of a sphere in the mechanical manner above described, it will be 
 found practically more convenient to employ others, which may be 
 constructed as follows. Referring to Fig. 88, it is seen that the sides 
 of the spherical triangles are the measures of the faces of a series of 
 trihedral angles whose common vertex is 0. Join with the 
 centre C of the spherical ellipse ; then a plane perpendicular to 00 
 will cut the edges of these pyramids in points readily determined, and 
 the problem of finding the plane base of the cone resolves itself into 
 the construction of a number of trihedral angles, of which one face is 
 common and the sum of the other two constant. The base thus found 
 will not be a true ellipse, but it will closely resemble one, and evi- 
 dently will be symmetrical about two lines at right angles to each 
 other. 
 
 140. The processes of constructing one of the trihedral angles in ac- 
 cordance with the above conditions, and of determining by means of 
 it a point in the required curve, are illustrated in Figs. 89 and 90. 
 
CONSTRUCTION OF TRIHEDRAL ANGLES. 
 
 71 
 
 FIG. 90. 
 
 The lines OE, OC, OF, of Fig. 89 correspond to those similarly lettered 
 in Fig. 88 ; they all lie in the hori- 
 zontal plane, and OE is perpendicu- 
 lar to GL the ground line. EOF, 
 then, is the common face of the tri- 
 hedral angles, bisected by OC. As- 
 sume ono of the two remaining faces, 
 as EOA, producing OA to cut GL 
 in A. Subtract this assumed face 
 from the given constant sura, thus 
 determining the third, which lay off 
 in the horizontal plane as FOB, mak- 
 ing OB equal to OA. 
 
 Draw BH perpendicular to OF, 
 produce it to cut GL in p ; erect 
 pp' perpendicular to GL, and of 
 a length equal to the altitude of a right-angled triangle of which HP 
 is the base and HB the hypothenuse : then p is the horizontal and p' 
 is the vertical projection of the point where B will fall in the vertical 
 plane if OB is revolved about OF. 
 
 Also, if OA be revolved about OE, A will describe a circle in the 
 vertical plane; which will pass through p', because OA = OB; 
 hence OP is the horizontal and Ep' is the vertical projection of the 
 third edge of the trihedral angle required. 
 
 141. Now let J/jVbe a plane perpendicular to OC '; it will cut this 
 third edge in a point whose horizontal projection is r, and r's' will 
 be the vertical projection of a perpendicular let fall from this point 
 to the horizontal plane. 
 
 In Fig. 90, which is a projection upon this new vertical plane MN, 
 the horizontal plane is seen as G'L, and the point r appears at R, 
 the distance RS being equal to r's' of Fig. 89 : the line 0(7 appears 
 as the point C, and i and &, the points in which OE and. OF are cut 
 by MN, are seen as / and K. 
 
 In this projection the required plane base of the cone is seen in its 
 true form ; as many points' as may be necessary to determine its con- 
 tour with precision, being found exactly as R was, by the construc- 
 tion of other trihedral angles. 
 
 142. In Fig. 91, O.E'and 05" are the fixed axes of a pair of frusta 
 cut from cones of this description ; the extremities of the major 
 axes of the bases are shown in contact at A, the plane of the paper 
 in this figure corresponding to the plane ODAB of Fig. 88. 
 
 The action . is obviously akin to that of two ellipses rolling 
 
72 
 
 ELLIPTICAL BEVEL WHEELS. 
 
 together in their own plane ; letting fall upon the fixed axes 
 
 the perpendiculars AM, AN, 
 the velocity ratio in the present 
 
 AM 
 
 position is ^ > and at the end 
 
 of a half revolution it will be 
 
 AN 
 
 AM' 
 
 This combination also resem- 
 bles the elliptical spur-wheels 
 in the respect that if these con- 
 ical frusta overhang their bear- 
 ings, as here shown, a link, L, 
 may be used to connect what 
 it is proper to call their free foci : 
 the axes of the pins by which 
 this link is pivoted to the wheels, 
 of course converging in the 
 FIO. 91. common vertex. 
 
 143. Any reciprocal limits may be assigned to the velocity ratio ; 
 
 and if the angle between the axes be also given, the pitch cones may 
 
 be constructed as follows. 
 
 Let the assigned values of the velocity ratio be - 
 
 m n 
 
 - 5 and let OE, 
 m 
 
 OH, Fig. 92, be the given axes. 
 Draw xx, zz, parallel to the 
 axes, and at distances from 
 them which are to each other 
 in the ratio of m to n ; these 
 lines intersect at P. Draw PO, 
 and lay off the angle POB 
 equal to EOH-, draw OCbisect- 
 ing POB, and from any point 
 A of PO, draw AB perpendic- 
 ular to 00, cutting OB in B. 
 Then A B may be taken as the 
 major axis of the plane base of 
 the cone, which from these data can now be constructed in the 
 manner previously explained ; for making the angle BOF equal to 
 A OE, we have EOF the common face, and A OB the constant sum 
 of the other two faces, to be used in determining the trihedral angles, 
 whose edges are the elements of the cone. 
 
ELLIPTICAL BEVEL WHEELS. 
 
 73 
 
 FIG. 93. 
 
 144. In Fig. 93, the inclination of the axes is the same as in Fig. 
 92, but the lines xx, zz, are so drawn that their intersection P falls 
 
 within the obtuse instead of in the 
 acute angle : the result being that 
 the directional relation is different. 
 
 From the very nature of the case, 
 an engaging pair of elliptical bevel 
 wheels must be equal and similar, 
 and always in external contact. 
 And it will be seen that, since the 
 angle subtended at the vertex by the 
 major axis of the base, is equal to 
 that included between the fixed axes 
 of rotation, neither cone can degen- 
 erate into a plane, until that angle 
 becomes 180, and the axes coincide : 
 much less can either be hollow. 
 
 We have then always the choice between two pairs of wheels, hav- 
 ing different directional relations but the same action in respect to 
 the velocity ratio. 
 
 If the axes be perpendicular to each other, as in Fig. 94, the pitch 
 cones will, like the circular 
 ones under the same condi- 
 tion, have two common ele- 
 ments, either of which may 
 be selected as the line of prag- 
 matic contact. 
 
 145. Conical Lobed Wheels. 
 From these elliptical bevel 
 wheels, it is possible to con- 
 struct conical wheels with 
 various numbers of lobes, by 
 a process of contraction or ex- 
 pansion of angles analogous to that applied to the plane ellipses. 
 
 Let the elements cut from one of a pair of these pitch cones by a 
 series of radiating planes through the fixed axis, bo revolved about that 
 axis until the dihedral angles between the planes are expanded or con- 
 tracted in any desired proportion. Any part of the elliptical cone 
 may be thus transformed into a new conical surface, which will roll 
 in contact with the one formed by treating in like manner the corre- 
 sponding portion of its rolling mate. 
 
 The bases of these lobed cones will be similar to the outlines of the 
 
74 COKICAL LOBED WHEELS. 
 
 multilobes derived from the plane ellipses. By applying the process 
 of contraction to one half the perimeters of the spherical ellipses, pairs 
 of similar wheels, with any desired number of lobes, may be con- 
 structed analogous in form and action to those shown in Figs. 59 
 and 61. 
 
 146. It is also practicable to construct conical lobed wheels which 
 will roll together in dissimilar pairs, unilobe with bilobe, trilobe with 
 quadrilobe, and so on. Eef erring to Fig. 88, it will be seen that the 
 dihedral angles PFE, 1?HG, are equal. Hence if the spherical ellipses 
 be such, for example, that the dihedral angles RGA, RHG, are to 
 each other as 3 is to 2, the former may be contracted to 90, and the 
 latter will by contraction in the same proportion be reduced to 60 ; 
 the result being the formation of a bilobe rolling with a trilobe, the 
 bases resembling those of Fig. 62. 
 
 It is sufficient merely to state that by constructing a series of spher- 
 ical triangles, as was done with the plane triangles in Fig. GG, a spher- 
 ical curve, analogous to the one there shown, may be drawn, by the 
 aid of which we can determine the limits of the eccentricity of the 
 spherical ellipses from which it is possible to derive such dissimilar 
 pairs with assigned numbers of lobes. It may be said, and no doubt 
 with truth, that the difficulty of making such wheels would prevent 
 their being used under any ordinary circumstances. But it is equally 
 true that only extraordinary conditions would require them to be used : 
 and should cases occur in which it would be desirable, it is worthy of 
 note that very many cf the combinations of non-circular cylindrical 
 wheels described in the preceding chapter, may be replaced by combi- 
 nations of conical wheels nearly identical in their action. 
 
 147. Of these, it is probable that the ones derived as above suggested 
 from the spherical ellipse, would be found practically preferable, as 
 being the ones most easily constructed, and also as allowing a wider 
 range in the selection of the limits of variation in the velocity ratio. 
 But the equiangular spiral has also its spherical analogue, which may 
 be made the base of a conical surface, capable of rolling in contact 
 with an equal and similar one, the fixed axes of rotation passing 
 through the poles of the bases and the centre of the sphere upon which 
 they lie. 
 
 If from any point on the sphere arcs whose lengths arc in geometri- 
 cal progression be set off successively on equidistant meridians passing 
 through the point, the curve drawn through the extremities of these 
 arcs will be the one required. 
 
 148, This will be readily seen by the aid of Fig. 95. In the side 
 view let 0(7 be a vertical radius of the sphere, and AB a plane tan- 
 
NON-CIRCULAR CONES. 
 
 75 
 
 gent to it at C. Let C in this plane be the pole of the logarithmic 
 spiral unilohe shown in dotted outline, as AMNm the horizontal pro- 
 jection. 
 
 The radiants of this spiral which include equal angles, are in geomet- 
 rical progression ; and planes passing through them and also through 
 OC, cut equidistant meridians 
 from the sphere. On each me- 
 ridian set off from C an arc 
 equal in length to the corre- 
 sponding radiant of the spiral : 
 the result will be the spherical 
 curve shown in full lines ; the 
 vertical projection being A'P'B'. 
 
 Now let A' RE be the vertical 
 projection of an equal and simi- 
 lar curve, of which D is the 
 pole : the vertical plane being 
 that of the great circle contain- 
 ing OC, OD, and also the arcs 
 A'C'B', ADE. It is then obvi- 
 ous that these curves will be tan- 
 gent to each other at A; also, 
 that if A'P, AR be equal arcs 
 of the two, then P and R will 
 meet in the plane of the axes, if 
 the curves revolve about OC and 
 OD respectively, as shown by 
 the arrows. For the angular 
 distances of these points from 
 that plane are the same as in the original spirals, which are known 
 to roll together, and by construction the sum of the arcs CP, DR, is 
 equal to the arc CD. 
 
 149, It will be seen, then, that if these curves be made the direc- 
 trices of cones whose common vertex is the centre of the sphere, 
 those surfaces will work in rolling contact about OC and OD as fixe*d 
 axes ; and that the velocity ratio will vary from maximum to mini- 
 mum and back again once in each revolution, this combination being 
 analogous to that of the two logarithmic spiral unilobes shown in Fig. 
 53. 
 
 In fact the plane original ANN is identical with one of those 
 unilobes. But the actions of the two combinations, though very sim- 
 ilar, are not identical. Draw AS, AT, perpendicular to 0(7 and 
 
 Fia. 95. 
 
76 NOX-CIRCULAR COXES. 
 
 OD, the limiting values of the velocity ratio in Fig. 95 are 
 
 A_8 AT , 
 A'T 9 A'S' 
 while in Fig. 53 they are 
 
 AC _ arc A'C 
 ~BC 9 arcB'C' 
 and 
 
 BC_ _ arc B'C 
 AC 9 arc A'C' 
 
 By comparing in a similar manner the values of the velocity ratio 
 for intermediate positions, it will be found that neither the limits nor 
 the laws of variation are precisely alike in the two cases. 
 
 150. If the angle COD between the axes be given, and the values of 
 the reciprocal limits of the velocity ratio be assigned, the cones may 
 be constructed, by processes closely resembling those employed in pre- 
 ceding cases. 
 
 Thus, it is evident that OA' must pass through the intersection of 
 two lines xx, zz, drawn parallel to OC and OD respectively, and at 
 distances from them which are to each other in the proportion of A'S 
 to A'T-, then CB' = A'D, and the arcs A'C and CB' being rectified 
 give AC and CB, whence the original spiral A MB may be recon- 
 structed, and the spherical curves derived from it as just described. 
 
 Like preceding combinations, too, this might be discussed, with 
 reference to the effects of dividing the obtuse instead of the acute 
 angles, of placing the axes of rotation at right angles, and in short, of 
 assuming any special conditions. It will also readily be seen that the 
 bilobes, trilobes, etc., derived from the equiangular spiral and rolling 
 in contact about parallel axes, have their analogues in conical multi- 
 lobes whose axes intersect. All these matters, however, we shall leave 
 the reader, if so. disposed, to pursue farther at his leisure : the exam- 
 ples already given illustrating sufficiently the principles involved in 
 the construction of this class of combinations, which, it is proper to 
 add in conclusion, we believe to be a new one. 
 
 2. Axes in Different Planes. 
 
 151. Velocity Eatio Constant. If a right line revolve about an axis 
 in a different plane, the surface generated is the hyperboloid of revo- 
 lution. Any normal to this surface will intersect the axis ; it will 
 also be perpendicular to both generatrices through the point of nor- 
 malcy, since these two elements determine the tangent plane. 
 
 If then a series of normals be drawn through different points of the 
 
AXES IK DIFFERENT PLANES. 
 
 77 
 
 revolving line, they will lie in planes perpendicular to that line, and 
 therefore parallel to each other. They are, consequently, elements 
 of one generation of a hyperbolic paraboloid, of which the directrices 
 are the axis and the revolving line, and the plane directer is perpen- 
 dicular to the latter. 
 
 Any plane parallel to those directrices will therefore cut the series 
 of normals in points which will lie in one right line, an element of 
 the second generation of the hyperbolic paraboloid. 
 
 Now the line thus determined may be taken as a new axis ; and by 
 revolving around it, the same line which generated the first hyper- 
 boloid will generate another. These two surfaces of revolution, hav- 
 ing, at every point of a common rectilinear element, a common nor- 
 mal and a common tangent plane, will be tangent to each other all 
 along that element. 
 
 152. Since these hyperboloids are warped surfaces, perfect rolling 
 contact between them, as has been shown, is not possible under any 
 circumstances whatever. 
 
 But they are capable of rotating 
 in contact about fixed axes, with a 
 constant velocity ratio; and the 
 sliding between them is quite dif- 
 ferent from that between two tan- 
 gent cones or cylinders whose peri- 
 metral velocities are not the same, 
 in the respect that it is wholly in 
 the direction of the common ele- 
 ment. And these hyperboloids, like 
 the cylinders and the cones, are 
 practically used as the pitch sur- 
 faces of toothed wheels. In view of 
 these facts, it is proper to consider 
 their action in this place, notwith- 
 standing the imperfection in their 
 rolling. 
 
 153. The form of the surface, and 
 the manner of constructing it, are 
 shown in Fig. 96. As the inclined 
 line AB revolves about the vertical 
 axis, every point in it describes a 
 
 circle in a horizontal plane, whose radius is seen in its true length in 
 the horizontal projection. In that projection, the axis appears as the 
 point C, and CD is the common perpendicular of the axis and the 
 
78 
 
 THE ROLLING HYPERBOLOIDS. 
 
 revolving line. Then the point D describes the circle of the gorge, 
 EDE; A describes the circle of the upper base, FAF; B, that of 
 the lower base, GBG, in this case equal to FAF, because A and B 
 are equidistant from D : and any intermediate point L describes a 
 circle 00, whose radius is OL. In this way any number of points in 
 the meridian outline GOJEJFmay be determined. 
 
 The same surface may be generated by the revolution about the 
 same axis, of another right line MDN\ MNan&AB having the 
 same horizontal projection, and being equally inclined to the plane of 
 rotation, but in opposite directions. 
 
 For the paths of A and M, also those of B and N, coincide, and D 
 is common to both lines : consequently any two points, one on each 
 line, equidistant from D, as for example L and P, will describe the 
 same circle. 
 
 154. Through any point of the surface, then, two rectilinear ele- 
 ments, or companion generatrices, may be drawn ; whose projections 
 on a plane perpendicular to the axis will be tangent to that of the 
 gorge circle on the same plane. 
 
 Thus AH, tangent at K in the horizontal projection . to the circle 
 EDE, is the second generatrix through A. In this figure, the verti- 
 cal plane contains the axis ; AH 
 pierces this plane at /, as deter- 
 mined from the horizontal projec- 
 tion : and in the vertical projec- 
 tion AH is tangent at / to the 
 hyperbolic outline, of which E is 
 the vertex, and AB and MN are 
 the asymptotes. 
 
 These two lines, AB and AH, 
 determine the plane tangent to the 
 hyperboloid at A ; the horizontal 
 trace of this plane is therefore 
 parallel to BH ; and its vertical 
 trace is parallel to AB, since that 
 line is parallel to the vertical plane. 
 Consequently AC, perpendicular 
 to BH, is the horizontal, and AR 
 perpendicular to AB, is the verti- 
 cal, projection of the normal to 
 the surface at A. 
 
 155, Now in Fig. 97, two hy- 
 perboloids are shown in contact, the axes A C and FE, and the com- 
 
DETERMINATION OF VELOCITY KATIO. 79 
 
 mon element AB, being parallel to the vertical plane. In the hori- 
 zontal projection the vertical axis appears as the point C', the in- 
 clined one as F'E', and the common element as AB 1 , which at A' 
 cuts the common perpendicular of the axes, here seen in its true 
 length as (7T, into segments A 'C', C'l', which are the radii of the 
 two gorge circles. Also C'B'E' is the horizontal, and CBE is the 
 vertical, projection of the common normal to the surfaces at B. In 
 the vertical projection, the upper bases and the gorge circles will 
 appear as lines perpendicular to the axes : the latter intersecting at A, 
 and the former, which in the figure passes through B, cutting the axes 
 at M and N. 
 
 156. Let these hyperboloids revolve about their axes in the direc- 
 tions indicated by the arrows. The point A of the inclined surface 
 will at the instant move in the direction of the tangent to the gorge 
 circle at that point : which being parallel to the vertical plane, this 
 motion may be represented in magnitude and direction by AHm the 
 vertical projection : and it may be resolved into the components, AS 
 in the direction of AB, and AO perpendicular to that line. 
 
 Considering the inclined hyperboloid as the driver, the rotation of 
 the vertical one will not be compulsory, because the contact radii are 
 constant. 
 
 But the common element may here, as in the cases of the cylinders 
 and the cones, be regarded as the line of contact of teeth formed upon 
 the two surfaces, so that AO would be the normal, and AS the tan- 
 gential component of AH ; because AB would necessarily be a line 
 of the plane tangent to both teeth at A. Therefore the resultant 
 motion of that point of the vertical hyperboloid (whose direction is 
 tangent to the larger gorge circle), must be of such magnitude, AL, 
 as to have the same normal component A 0, the other component 
 being A Tin the line AB. 
 
 157. Now let v = ang. vel. about FE ' , 
 
 v' = " " " AC; 
 then 
 
 a). 
 
 A'F 
 
 But from similar triangles AHL, ACE, 
 
 AH _ AE_ 
 ~AL ~~ AC ' 
 
80 DETERMINATION OF VELOCITY RATIO. 
 
 and by the principles of projection, 
 Substituting in (I), 
 
 A^_(7 _ I?Cr_ _ BC 
 AT' ^ B'E 1 ~ BE 
 
 v' BE AC 
 From similar triangles ABC, ABM, 
 
 Afi _ AB m 
 BE~~ B~M' 
 
 and from similar triangles ABC, ABN, 
 
 _ BN 
 AC ~~ ~AB ' 
 
 Substituting in (2), we have, finally, for the velocity ratio, 
 
 v_ BN 
 
 v' ~ BM ' 
 
 That is to say, the angular velocities are to each other in the in- 
 verse ratio, not of the perpendiculars let fall from any point of the 
 common element upon the axes, but of the projections of those per- 
 pendiculars upon a plane parallel to both axes and the common 
 element. 
 
 158. Supposing the axes to be given in position, then, and the 
 velocity ratio assigned, the inclination of the common element is 
 found exactly as in the case of intersecting axes : two lines, xx, zz, 
 are drawn parallel to ^C'and AE, at distances from them which are 
 to each other in the inverse ratio of the angular velocities about those 
 axes respectively ; and AB must pass through their intersection P. 
 Then drawing CE perpendicular to AB, it is the vertical projection 
 of the normal, therefore the horizontal projection of E must be at E' 
 on the horizontal projection of FE ; and projecting B to B' on C'E', 
 we draw B'A' parallel to F'E', thus determining the horizontal pro- 
 jection of the common element and also the radii of the gorge circles, 
 AC' and A I' : whence the surfaces may be constructed as before 
 described. Or, if for example the vertical surface be given, and it 
 be required to find the other, the velocity ratio being also assigned : 
 then, knowing j?^and its ratio to BM, the length of the latter may 
 be found, and with that as radius, an arc is described -about B as 
 
HYPERBOLOIDS IXTEEtfALLY TAKGENT. 
 
 81 
 
 centre. The vertical projection FE of the second axis is drawn 
 through A, tangent to this arc. The vertical projection of the nor- 
 mal at B y is CB perpendicular to AB, and it cuts FE in E, whose 
 horizontal projection must therefore be at E' on the prolongation of 
 C'B'y and F'E' parallel to A ' B' is the horizontal projection ol the 
 axis of the required surface. 
 
 159. In Fig. 97, the common element lies between the axes, and 
 the two surfaces touch other exter- 
 nally. But if one be larger than 
 
 the other, the smaller one may be 
 placed within it, and that in such 
 a position as to touch the larger 
 along a line of the concave surface. 
 Thus, in Fig. 98, the same pair of 
 hyperboloids as in the preceding 
 figures, are shown in this new rela- 
 tion : the demonstration of the ve- 
 locity ratio applies, it will be seen, 
 without any change whatever, but 
 the directional relation in the two 
 cases is different. The surfaces are 
 tangent along an element of the 
 same generation as before, but it 
 now lies on the same side of both 
 axes. 
 
 It will be observed that in the 
 case in which the vertical hyper- 
 boloid and the velocity ratio are 
 given, FAE is drawn tangent to 
 the same circle, described about B 
 with radius BM, but on the side opposite to that selected in Fig. 97. 
 
 Also, that in the other case, where the two axes and the velocity 
 ratio are given, the lines xx, zz, are so 'drawn that their intersection 
 P lies in the obtuse angle formed by the axes, instead of in the acute 
 one as in the preceding figure. 
 
 160, In the above determination of the velocity ratio, the motions 
 of the coincident points of the two gorge circles were selected merely 
 because they were the most convenient ones for the purpose. It is 
 evident that the angular velocity of every point in either surface about 
 its axis must be the same. The motion of the line AB of the vertical 
 surface is one of revolution about AC '; and AL being the linear ve- 
 locity of A y every point in AB must have a component motion along 
 
 FIG. 98. 
 
82 BATE OF SLIDIKG. TRANSVERSE OBLIQUITY. 
 
 that line equal to A T, and in the same direction, as shown in Figs. 8 
 and 9. And in revolving about FE, the motion of every point of 
 AB must have a component in the direction of and equal to AS, 
 since AH is the linear velocity of the point A in the inclined hyper- 
 boloid. 
 
 The action of the combination, therefore, consists in rolling, com- 
 bined with a sliding in the line of the common element, which is rep- 
 resented by ST, the sum or the difference, as the case may be, of 
 AS and. AT. 
 
 The rate of sliding, then, is at any instant the same at every point 
 of contact ; but as the linear velocity of each point depends upon its 
 distance from the axis of the surface upon which it lies, it is evident 
 that the proportion of the sliding component to the perimetral veloc- 
 ity will be greatest at the gorge, and diminish as the point under con- 
 sideration recedes therefrom. 
 
 161. Although in Figs. 97 and 98, the hyperboloids, in order to save 
 space, are limited by the gorge planes, it is obvious that they may be 
 extended to any distance on both sides of those planes, and will be 
 tangent from end to end. But practically, as in bevel gearing, com- 
 paratively thin frusta only of the pitch surfaces are used ; and their 
 location is optional within certain limits in some special cases here- 
 after to be noted. Thus in Fig. 99 the surfaces are in contact all 
 along the line mn ; and we may use either of the three pairs of frusta 
 A and B, C and Z>, E and F, or any two or all .three pairs at once : 
 and the steadiness of the motion will practically be greater when two 
 pairs equidistant from the gorge planes are employed. 
 
 162. Transverse Obliquity. The plane tangent to the hypefboloid at 
 .any point is determined (154) by the companion generatrices through 
 the point. If a meridian plane be passed through the same point, it 
 will cut the tangent plane in a line tangent to the hyperbolic outline 
 ;at that point : and the angle included between this line and either 
 generatrix is the measure of what is called the transverse obliquity, 
 or slceiv, of the teeth which mifst be used in practice. 
 
 As before mentioned, teeth which work in line contact ultimately 
 reduce to rectilinear elements of their pitch surfaces. But when they 
 are of sensible magnitude, there is one instant in the action of each 
 engaging pair, when their line of tangency coincides with the common 
 element of the pitch surfaces. When those are cylinders or cones, the 
 common element lies in the plane of the axes : this is impossible in 
 the case of the rolling hyperboloids, but the less the departure from 
 that condition, the more advantageous will be the action. And this 
 transverse obliquity, which measures the inclination of the generatrix 
 
SIMILARITY OP HYPEKBOLOIDS TO CO^ES. 83 
 
 to the meridian plane at any point, is obviously greatest at the gorge, 
 and diminishes as the element is extended, which is another reason 
 for employing, when the circumstances of the case admit of it, frusta 
 at some distance from the gorge planes. 
 
 Keferring to Fig. 96, it will be seen that the transverse obliquity at 
 the gorge is measured by the angle TDA in the vertical projection. 
 That at A is half the angle between AB and AH\ to construct it, de- 
 scribe with radius the true length of AB (seen in the vertical projec- 
 tion), arcs about B and H in the horizontal projection, where BH ap- 
 pears in its true length. These arcs intersect at 8 on the prolonga- 
 tion of CA ; and C8B or CSHis the required angle. 
 
 163. Now since, when the axes are given in position, the angle 
 between their projections on a plane parallel to both is divided, in 
 order to produce any assigned velocity ratio, precisely as though the 
 axes intersected and the pitch surfaces were cones, it will be seen that 
 in every case the problem admits of two solutions. That is to say ; 
 with any given pair of axes, it is possible to construct two pairs of 
 hyperboloids, having the same velocity ratio, but different directional 
 relations. 
 
 The cone is, in fact, but the limiting form of the hyperboloid, in 
 which the radius of the gorge circle becomes zero : and all the pecu- 
 liarities which have been pointed out as resulting from special condi- 
 tions with intersecting axes, will be found to have almost their exact 
 counterparts under analogous circumstances when the axes lie in 
 different planes. 
 
 164. Thus, when the projections of the axes cut each other oh- 
 liquely, we may divide the acute angle, as in Fig. 97, and the resulting 
 hyperboloids are externally tangent : a comparison of Fig. 99, which 
 represents the same pair of surfaces, with Fig. 77, will clearly show 
 the close resemblance between the two combinations. 
 
 If the obtuse angle be divided instead, one hyperboloid may be 
 internally tangent to the other, as in Fig. 98 ; which condition of 
 things is at once seen to correspond to the case of a cone rolling 
 within a hollow one, as shown in Fig. 79. 
 
 But whether the axes intersect or not, it does not follow that the 
 phenomenon of internal tangency will always occur when the obtuse 
 angle between their projections is selected. This, in relation to coni- 
 cal surfaces, was illustrated in Figs. 78 and 79 : and the truth of it 
 in regard to the surfaces now under consideration will be evident on 
 examination of Fig.. 100, in which the relative positions of the axes, 
 and the velocity ratio, are precisely the same as in Figs. 97 and 99, 
 the only difference being in the directional relation. This, as above 
 
84 
 
 RESULTS OF DIVIDING THE OBTUSE ANGLE. 
 
 stated, is due to the division of the obtuse instead of the acute angle ; 
 but the tangency of the pitch surface is still external, as it always will 
 be if the acute angle be divided. 
 
 165. And this figure calls attention to another point of similarity 
 between the cone and the hyperboloid. The latter, although a sur- 
 
 FIG. 99. 
 
 FIG. 100. 
 
 face of one nappe, is divided by the gorge plane into two parts, whose 
 relations are very like those of the opposite nappes of the cone. In 
 Fig. 100, it will be observed that the shaded frusta, f and D, might 
 be extended to the gorge planes. Not beyond, however, because the 
 extension G of the inclined surface will intersect the frustum D of 
 the vertical one, and F, the extension of the latter surface, will inter- 
 sect H : a condition of things very similar to that in Fig. 78. 
 
 Equal and opposite pairs of frusta, then, cannot be used in this 
 case : nor yet can we employ a pair like the central one of Fig. 99, 
 of which the mid-planes are the gorge circles. But if H and D be 
 very near the gorges, and very thin, it will be possible to carry the 
 inclined shaft past D, and to use at the same time another pair, 
 G and F\ if they be sufficiently far from the gorge planes to clear 
 the first pair. 
 
 166. Again, ifc may happen that when the projection of the com- 
 mon element lies in the obtuse angle, it shall be perpendicular to the 
 projection of one of the axes. This case, shown in Fig. 101, presents 
 the remarkable feature that the pitch hyperboloids retain their limit- 
 ing forms, the one remaining a cone, the other a plane. 
 

 DEGENERATED HYPERBOLOIDS. (i 83 
 
 V OJET 
 
 It will be seen that the common element AB 
 inclined axis FE, by revolving around which it generates the cone, 
 ABL : while in revolving 
 about the vertical axis, the 
 points A and B describe the 
 circles whose radii are C'A' 9 
 C'B', in the horizontal pro- 
 jection. 
 
 The determination of the 
 velocity ratio will be most 
 conveniently made by con- 
 sidering the motions about 
 the axes of the two points 
 which fall together at B. 
 The motion of the point 
 which belongs to the plane 
 hyperboloid, may be repre- 
 sented by B'H, perpendicu- FIG. 101. 
 lar to C'B' in the horizontal projection ; and it may bo resolved into 
 the components, B'Tm the line of the common element, and B' 
 perpendicular to it. 
 
 Now B'O is tangent to the circle described about the inclined axis 
 by the point which belongs to the cone, and therefore represents its 
 resultant motion, which has no component along A'H'. 
 
 167. Then as before, let 
 
 
 and we have 
 
 v = ang. vei. about inclined axis, 
 v' = " " " vertical " , 
 
 B'C' 
 
 v' B'H BN 
 
 0)- 
 
 B'C' ) 
 But from similar triangles B' OH, B'A'C', we have, 
 
 B'O A'B' AB 
 
 B'H B'C' 
 
 since A'B = AB : 
 
 Substituting in (1), 
 
 v' " BN 
 
86 
 
 DOUBLE TANGENCY OF HYPERBOLOIDS. 
 
 As in the other cases, then, the velocity ratio depends entirely upon 
 the projections of the axes and the common element on a plane par- 
 allel to all three, and the singular circumstance results that the angu- 
 lar velocity of the cone is not affected by the lateral separation of the- 
 axes or the resulting variation in the diameter of the perforated disc 
 , with which it works in contact. 
 
 The sliding component B'T de- 
 pends wholly upon the revolution of 
 AB about the vertical axis, and 
 since it must be the same at every 
 point of the moving line, its ratio 
 to the linear velocity is greatest at 
 the vertex of the cone, diminishing 
 as we recede from that point. 
 
 168. Finally, in the case in which 
 the projections of the axes intersect 
 each other perpendicularly, as in 
 Fig. 102, it is seen that the pitch 
 hyperboloids are tangent to each 
 other along two elements, as were 
 the cones under the same circum- 
 stances. 
 
 FIG. 102. These elements, mn, op, are neces- 
 
 sarily the comparion generatrices, and evidently tend to establish 
 the same velocity ratio, but different directional relations. Either of 
 
 them may be made the line of prag- 
 matic contact, and at the limit, as 
 shown in Fig. 103, a frustum B of 
 one hyperboloid may work in con- 
 tact with two frusta, A and C, on 
 opposite sides of the gorge plane of 
 the other. But this combination 
 could practically be used only as an 
 arrangement of friction gearing, the 
 wheels being parts of the pitch sur- 
 faces only, for teeth of sensible magnitude must be disposed in the 
 direction of a generatrix, common to the engaging frusta ; now B 
 and C are tangent along one line, B and A along another, and the 
 teeth of B cannot slope both ways at once. 
 
 169, Evidently this double tangency prevents in this case also the 
 use of frusta whose mid-planes are the gorge circles, as well as of equal 
 and opposite pairs. But pairs on opposite sides of the gorge planes 
 
 FIG. 103. 
 
SKEW FRICTION GEARING. 87 
 
 may be used, having the same common element, and therefore the 
 same directional relation, if they be placed at different distances from 
 those planes. Either line of contact maybe selected, so that as in the 
 case of the cones the directional relation is optional. The resulting 
 combinations are shown in Figs. 104 and 105 ; comparing these with 
 
 FIG. 104. t FIG. 105. 
 
 Figs. 83 and 84, the analogy between the cone and the hyperboloid, 
 considered as the pitch surfaces of wheels, will be most clearly seen. 
 
 170. It may be remarked that frusta of these hyperboloids can be 
 employed in the manner of friction gearing. When this is to be done 
 the frusta are preferably placed at some distance from the gorge 
 planes ; for the curvature of the hyperbola diminishes so rapidly as it 
 recedes from the vertex, that it very soon becomes almost inappreciable. 
 Consequently, if at the mid-planes of the frusta thus located cones be 
 drawn tangent to the hyperboloids, frusta of these cones may practi- 
 cally be employed. It is hardly necessary to repeat that since the 
 transmission of the motion depends wholly upon the adhesion be- 
 tween the surfaces in contact, no absolute dependence can be placed 
 upon the constancy of the velocity ratio, and it may be noted that, 
 since in order to secure such adhesion, it is necessary to press the sur- 
 faces together with considerable force, an amount of friction is thus 
 caused in the bearings^ so great that it may be questioned whether 
 such gearing has in many cases any advantage other than that of 
 original simplicity and possibly in freedom from noise at high veloc- 
 ities, over toothed wheel- work ; which Jatter we will now proceed to 
 discuss. 
 
CHAPTEE VI. 
 
 1. CLASSIFICATION OF TOOTHED GEARING. 
 
 2. SPUR WHEELS. Epicycloidal Teeth, Outside Gear, Generation of the Outline. 
 
 Pitch, Angles of action, Backlash, Clearance, Approaching and Receding Ac- 
 tion. Interchangeable Wheels. Size of Describing Circle. Rack and Pinion. 
 
 3. INSIDE SPUR GEARING. Intermediate Describing Circle. Limiting Diameters 
 
 of Generating Circles. 
 
 1. Tlie Different Varieties of Gearing Classified. 
 
 171. It has been pointed out, that when a definite Telocity ratio is to 
 be maintained, the pitch surfaces previously discussed are unsuitable 
 for the transmission of rotation, on account of their liability to slip 
 upon each other ; and we propose now to consider the forms of the 
 teeth which in practice must be employed in order to prevent this 
 slipping. 
 
 But it is desirable first to gain clear ideas of the general nature of 
 the various kinds of toothed wheels in use, and of the peculiarities 
 upon which their classification is based. 
 
 172. Not only may the axes of a pair of engaging wheels have dif- 
 ferent relative positions, but the teeth themselves may be of different 
 kinds, and act upon each other in different ways ; for example, the 
 mode of action of a pair of screw-wheels is quite dissimilar in its in- 
 trinsic nature from that of a pair of skew-bevel wheels, although the 
 relative positions of the axes may be the same in each case. There 
 are, in consequence, six varieties or classes of toothed gearing to be 
 met with in practice, viz : 
 
 1. Spur Gearing. 4. Twisted Gearing. 
 
 2. Bevel Gearing. 5. Screw Gearing. 
 
 3. Skew Gearing. G. Face -Gearing. 
 
 173. Eegarding this matter from a new point of view, it is seen that 
 the teeth of engaging wheels act upon each other by contact what- 
 ever their number. If then that number be indefinitely increased, the 
 size being correspondingly diminished, the teeth will ultimately be- 
 
CLASSIFICATION OF GEARING. 
 
 89 
 
 come, in general, mere lines, or elements of surfaces in contact. The 
 relative motions of these surfaces will be the same as those of the 
 wheels from which they are thus derived, their forms and disposition 
 depending on the nature of the class of gearing to which those wheels 
 originally belonged : these are the pitch surfaces, whose action in 
 some combinations has already been investigated. For the purpose 
 of comparison, and to illustrate the distinctive features, a pair 
 of wheels of each class, and also their pitch surfaces, are shown in 
 Figs. 106 to 111, inclusive. 
 
 174. In the first three classes, the engaging teeth, which are bounded 
 
 FlG. 106, SPUll GEARING. 
 
 FlG. 107, BEVEL GEARING. 
 
 by ruled surfaces, touch each other along right lines, and by the process 
 above indicated they are reduced to rectilinear elements of the pitch 
 surfaces, which by the mode of derivation must be tangent all along a 
 right line. As already stated, the axes of spur wheels are parallel, 
 and their pitch surfaces are cylinders ; the axes of bevel wheels inter- 
 sect, and their pitch surfaces are cones whose common vertex is the 
 point of intersection : the axes of skew wheels lie in different planes, 
 and their pitch surfaces are hyperboloids. 
 
 175. Let us now suppose one of a pair of engaging circular wheels, 
 
90 
 
 CLASSIFICATION OF GEAKING. 
 
 belonging to either of these three classes, to be uniformly twisted on 
 its axis, each successive transverse plane being rotated through a 
 greater angle than the preceding one ; then the other wheel of the 
 pair will receive a corresponding twist, as will be readily understood 
 by the aid of Fig. 109. It will hereafter be shown that the twisted 
 
 FlG. 109, TWISTED GEARING. 
 
 | 
 
 wheels thus formed will gear together as well as before, and in sub- 
 stantially the same manner. The teeth are now distorted into sur- 
 faces of a helicoidal nature, and by the above process of indefinite sub- 
 division, they reduce to helical lines. 
 
 But it is to be noted that these lines lie upon surfaces which are 
 tangent along a right line, whether the axes are parallel, intersecting, 
 or neither. And it will also be seen that whatever of screw-like 
 action may be involved in their motions, tends only to cause pressure 
 in the direction of the common element of the pitch surfaces, and 
 has nothing to do with the transmission of rotation. 
 
 176. In these respects there is a marked distinction between these 
 twisted wheels, and those belonging to the next class, of Screw Gear- 
 ing properly so called ; although they are frequently confounded with 
 each other. In all the latter, the teeth, it is true, are also of helicoidal 
 
TWISTED AND SCREW WHEELS CONTRASTED. 
 
 91 
 
 form, and reduce to helical lines : but these helices lie upon cylinders 
 whose axes are in different planes, and the pitch surfaces therefore 
 touch each other in a single point only. Moreover, as illustrated by 
 the familiar combination of the "worm and wheel," it is the screw- 
 like action alone of one wheel upon the other, by which the rotation is 
 transmitted in that class of gearing. 
 
 177. Face Gearing is not much used in modern machinery ; the 
 name is derived from 
 
 the fact that the 
 wheels were usually 
 formed with teeth 
 consisting of turned 
 pins projecting from 
 the faces of cir- 
 cular' disks, as shown 
 in Fig. Ill : a mode 
 of construction well 
 adapted to wooden 
 mill work, and to 
 that only. In the 
 case illustrated here, 
 the axes are perpen- 
 dicular to each other ; 
 but turned pins may 
 be inserted in other surfaces than planes, and in this way such wheels 
 can be made to work* together when the axes have different relative 
 positions. All these may be properly said to belong to the same 
 class ; of which the distinguishing features are, that whatever the 
 relation of the axes or the general forms of the wheels, the teeth are 
 circular in their transverse sections, touch each other in a single 
 point, and ultimately become points in the circumferences of circles 
 which are in contact. The reason of this last peculiarity is, thafc an 
 increase of the number involves a diminution of the length as well as 
 of the diameters of the teeth, so that at the limit they vanish alto- 
 gether : whereas, in the other classes of gearing, the length of the 
 teeth is not affected by any variation in the height or thickness, 
 and they reduce to lines. Face wheels, then, have no pitch 
 surfaces properly so called, although in constructing them, surfaces 
 of some kind must be provided in which to secure the teeth or pins. 
 
 178. The following table exhibits in a convenient manner the pecu- 
 liar features of the different kinds of gearing above mentioned : the 
 teeth, of whose linear elements the forms are given in the last column, 
 
 FlO. Ill, FACE GEARING. 
 
TABLE OF DISTINCTIVE FEATUEES. 
 
 being supposed to be of sensible magnitude, in order that the circular 
 sections of those in the sixth class may be kept in view. 
 
 CLASS OP GEARING. 
 
 RELATIVE POSITIONS 
 OF AXES. 
 
 PITCH SURFACES. 
 
 ELEMENTS OF TEETH. 
 
 1. SPUR. 
 
 Parallel. 
 
 Cylinders. 
 
 Rectilinear. 
 
 2. BEVEL. 
 
 Intersecting. 
 
 Cones. 
 
 Rectilinear. 
 
 3. SKEW. 
 
 In Different Planes. 
 
 Hyperbole-ids. 
 
 Rectilinear. 
 
 4. TWISTED. 
 
 Any. 
 
 Either. 
 
 Helical. 
 
 5. SCREW. 
 
 In Different Planes. 
 
 Cylinders. 
 
 Helical. 
 
 C. FACE. 
 
 Any. 
 
 None. 
 
 Circular. 
 
 2. The Teeth of Spur Wheels. 
 
 Epicycloidal System. 
 179. Generation of the Tooth Outline, In Fig. 112, let C, D be the 
 
 T FIG. 112. 
 
 centres of the pitch circles LM 9 EN. Tangent to these at A, is a 
 smaller circle whose centre is 0. Suppose all the centres to be fixed, 
 
GENERATION OF TOOTH-OUTLINE. 93 
 
 then the three circles can move in rolling contact, with equal linear 
 velocities. Set off from A the three equal arcs AB, AE, A P. Suppose 
 a marking point fixed originally at A, in the circumference of the 
 smaller circle ; then while this travels to P, it will trace, with refer- 
 ence to RX, the curve BP, and with reference to LM, the curve EP. 
 Now the relative motions of the circles are precisely the same as 
 though the smaller one, which carries the marking point, had rolled 
 upon the outside of RN, and upon the inside of LM, regarding these 
 two as fixed base lines : the curves tire, therefore, an epicycloid and 
 a hypocycloid respectively ; and AP is their common normal at P, 
 because on whichever of the two fixed lines we regard the small circle 
 to be rolling at the instant, the point of contact A is the instantaneous 
 axis (73), so that the motion of P in cither curve is perpendicular to 
 A P. If the tracing point go on to P, the arcs AP', AE', AB' 9 
 being equal, the resulting curves B'P', E'P, are, clearly, but exten- 
 sions of the first pair, and AP' is their common normal. 
 
 180. We perceive, then, that the curves thus simultaneously gener- 
 ated are tangent to each other at some point, throughout the gener- 
 ation ; that the point of tangency is always in the describing circle : 
 and that the common normal always passes through the fixed point 
 A 9 upon the line of centres. 
 
 Consequently these curves are correct outlines for parts, at least, of 
 teeth ; if the curved lever CE turn, as shown by the dotted arrow, it will 
 drive the other before it, the point of contact following the arc P' PA, 
 until E' and B' meet at A, and as the common normal always cuts the 
 line of centres at the same point, the velocity ratio will be constant. 
 
 181. It is to be noted, that this condition would have been satisfied, 
 had the tracing point been fixed, not in the circumference of the de- 
 scribing circle, but at a greater or less distance from its centre. And 
 it will also be readily perceived, that the tracing point need not be 
 carried by a circle at all ; any other describing curve might have been 
 used, provided that it were capable of rolling in contact with the 
 pitch circles; the point of contact would not travel in a path coin- 
 ciding with the describing curve, but the generated curves would 
 always have a point of tangency, and the common normal would 
 always have passed through A. And conversely, any two curves 
 of which these two things are true, can be generated in the manner 
 above described. In general, then : The tooth-outlines which act in 
 contact, must be such as can be simultaneously traced upon the planes 
 of rotation of the two wheels while in action, by a marking point 
 which is carried by a describing curve moving in rolling contact with 
 both pitch circles. 
 
94 LAYIKG OUT THE TEETH. 
 
 By using various describing curves, then, an infinite number of 
 tooth-outlines may be generated, all of which geometrically satisfy 
 the conditions. But many of them are of impracticable forms ; of 
 those which are not, none have been more extensively employed than 
 the Epicycloid and the Involute, to which therefore we shall at pres- 
 ent confine our attention ; and we now proceed to the practical oper- 
 ations of "laying out the teeth " of a pair of wheels in outside gear. 
 
 182. Circular Pitch, Supposing the distance between the centres of 
 a pair of wheels to be given, and the velocity ratio to be assigned ; the 
 first step is to divide the line of centres into segments having the 
 given ratio, thus determining the radii of the pitch circles. The cir- 
 cumference of each circle is next to be divided into as many equal 
 parts as its wheel is to have teeth. 
 
 The pitch of the teeth is the length of the circular arc measuring 
 one of these subdivisions ; or, in other words, it is the distance meas- 
 ured on the pitch circle, occupied by a tooth and a space. This pitch 
 arc, it is obvious, must be the same on each wheel ; although the 
 teeth may be smaller, and the spaces larger, on one wheel than upon 
 the other. The numbers of the subdivisions, then, are proportional 
 to the diameters of the pitch circles; and a fractional tooth being 
 impossible, the pitch must be an aliquot part of each circumference. 
 
 The pitch as above defined is sometimes called the Circular pitch, 
 in distinction from what is called the Diametral pitch, which will be 
 explained hereafter. 
 
 183. Face and Flank. The part of a tooth-outline which lies out- 
 side its pitch-circle, as B'P' in Fig. 112, is technically called ihefaoe 
 of the tooth ; and the part which lies within the pitch circle, as 
 E'P' in the same figure, is called \\\Q flank. Usually, each tooth has 
 both ; but wheels can be made, and sometimes used to great advan- 
 tage, in which one of a pair has faces only, the other one having only 
 flanks ; we will consider this case first. 
 
 184. Arc and Angle of Action. The angle through which a wheel 
 turns, while one of its teeth is in contact with the engaging tooth of 
 another wheel, is called the angle of action : and the arc of the pitch 
 circle by which it is measured, is called the arc of action. The latter 
 must, evidently, be at least equal to the pitch arc, in order that each 
 tooth may continue in gear until the next one begins to act; and it 
 ought in practice to be considerably greater. 
 
 185. Backlash. The size of the tooth depends, partially at least, 
 upon the pitch, since, as above stated, the pitch arc is to be divided 
 between a tooth and a space. Practically, of course, the teeth of both 
 wheels are made of the same thickness ; hence, were perfect work- 
 
A LIMITING CASE. 
 
 95 
 
 man ship attainable, the tooth and the space might be made exactly 
 equal. But since it is not, the space must be made a little wider than 
 the tooth ; the difference is called backlash, and should be as small as 
 it is practicable to make it. For our present purposes we may neg- 
 lect it altogether, and make the thickness of the tooth just one half 
 the pitch. 
 
 A Pair of Wheels. Limiting Case. 
 186, In Fig. 113 the pitch and describing circles being drawn as in 
 
 II 
 
 the preceding figure, let AB, AE, be the pitch arcs, and AP an equal 
 arc on the describing circle. Then iheface for the tooth of RN can- 
 not be less than BP, since if made of just that height, as shown, con- 
 tact is ending at P at the very instant when the next tooth begins to 
 act at A. Bisect AB in H, which gives the thickness of the tooth ; 
 and draw through Ha, reversed face similar to BP. 
 
 The conditions are purposely so chosen that this reversed face passes 
 through P : the case is, therefore, a barely possible one, the tooth 
 being pointed, and just high enough to make the angle of action equal 
 to the pitch angle. 
 
 We founjl that the face must be of the height BP, in order to se- 
 cure this angle of action : drawing PD, which cuts the pitch circle 
 in 0, we see that in this case BG is just half the thickness of the 
 
96 APPROACHING AND RECEDING ACTION. 
 
 tooth. Had BG been greater, GHmust have been less, so that the 
 reversed face through H would not have passed through P, but be- 
 tween P and G, and the case would have been impracticable, without 
 reducing the pitch and giving both wheels more teeth. 
 
 But if BG had been less than half the thickness of the tooth, the 
 tooth itself could have been made higher by extending the face above 
 P, or it might be left of the height BP, which would have given it 
 some thickness at the top, as in the next figure. 
 
 187. Clearance. The acting flank is EP ; but in order to let the 
 teeth of the other wheel pass, the hypocycloid is continued to 7, mak- 
 ing the depth of the space a little greater than PG ; the difference is 
 called clearance, and a similar provision is made in the other wheel 
 by cutting in radially, as shown at A, H, B, a little below the pitch 
 circle. The tooth of LM is completed by bisecting the pitch-arc AE 
 at T, and drawing the curves AK, FJ, similar to EL 
 
 188. A Practical Case. Limiting cases like the preceding are to be 
 avoided in practice. A pointed tooth is bad, as being weak and liable 
 to wear at the top. And even if it be not pointed, the angle of action 
 should be greater, as otherwise the least wear at the top reduces the 
 face below the requisite height, and causes one tooth to quit correct 
 driving contact before the next one properly begins to act. We say 
 correct driving contact, for it will be seen that if in Fig. 113 we sup- 
 pose all the teeth to be removed except the pair in contact at P 9 the 
 face EB would push the flank IE out of its way, even if the height 
 were slightly reduced, but the two acting curves would not be tangent 
 to each other, and the velocity ratio would not remain constant, but 
 the speed of the wheel C would diminish. 
 
 A reasonable case is shown in Fig. 114 ; the arc of action in this 
 instance is 1-J- times the pitch, and drawing the radial line PS, we 
 find BG to be much less than ^ BH, thus enabling us to give the 
 tooth a substantial thickness, PK, at the top. 
 
 189. Approaching and Receding Action. In Figs. 113 and 114, the 
 action takes place wholly on one side of the line of centres. If RN 
 be the driver (the directions being as shown by the arrows), the action 
 begins at A and ends at P, the point of contact continually receding 
 from the line of centres, in which case AB, AE, are called arcs of re- 
 cess, or of receding action. If LM drive (in the opposite direction), 
 the action begins at P, ending at A : the point of contact is always 
 approaching the line of centres, and AB, AE, are then called arcs of 
 approach, or of approaching action. 
 
 It has been 'found by experience that the friction is greater and 
 more injurious in the latter case than in the former : when such 
 
A PRACTICAL CASE. 
 
 97 
 
 wheels as those under consideration are used, therefore, the one whose 
 teeth have faces only should always drive. 
 
 190. But even then there is one drawback, which will be understood 
 by reference to Fig. 112. The longer the arc of action, the longer the 
 face of the tooth, and the greater the obliquity of the line of action, 
 that is, its inclination to TT, the common tangent of the pitch cir- 
 cles. Now the pressure, as well as the motion, is transmitted in the 
 line of action, and the greater its obliquity, the greater will be the 
 component of pressure in the line of centres, tending to cause friction 
 and wear in the bearings. 
 
 The amount of sliding also increases more rapidly as the point of 
 
 C 
 
 contact recedes from the line of centres, so that upon the whole such 
 wheels are better suited for use in light mechanism where the teeth 
 can be made small and numerous, and smoothness of action is impor- 
 tant, than for the transmission of heavy pressures. 
 
 Teeth with loth Faces and Flanks. 
 
 191. It will now readily be seen that by using another describing 
 circle on the other side of the point of contact of the pitch circles, 
 thus giving both faces and flanks to the teeth of each wheel, two 
 things will be accomplished : a given angle of action may be secured 
 with shorter faces, and therefore with less sliding, and this angle will 
 be divided into an angle of approach and an angle of recess, thus en- 
 abling us to use either wheel as the driver. 
 
98 
 
 TEETH WITH BOTH FACES AND FLANKS. 
 
 If a wheel has both to drive and to follow, the arcs of approach and 
 of recess may be made equal or nearly so, but if one wheel of a pair is 
 always to be the driver, it may be desirable to make the arc of recess 
 the greater, in order to reduce the amount of the more detrimental 
 friction. 
 
 192. The construction is shown in Fig. 115 ; all that relates to the 
 face BP for RN, and to the flank EP for LM, is precisely the same 
 as in Fig. 113, and the lettering being made so far to correspond in 
 
 N 
 
 FIG. 115. 
 
 the two diagrams, no further explanation is needed in reference to 
 those curves. To complete the teeth, another describing circle is 
 used on the opposite side of the pitch circumferences, which generates 
 the face OF for LM and the flank OK for RN. 
 
 If we assume the arc of action on that side of CD, as for instance 
 A F or AK, the possibility of securing it with a given number of 
 teeth is at once ascertained by making the arc AO equal to AF, and 
 drawing OC to cut LM in J. If FI be less than half the thickness of 
 the tooth which is required by the given pitch, or equal to it, the 
 construction is possible, the tooth in the latter event being pointed ; 
 if greater, it is impracticable. 
 
 Should it prove to be feasible, we have only to draw the epicycloid 
 OF, which joined to EP completes the outline of the tooth for LM, 
 and the hypocycloid OK, joining the latter to BP, which finishes the 
 outline of the tooth of RN. 
 
COMPLETE WHEELS A PRACTICAL CASE. 
 
 99 
 
 That is to say, these are the whole of the acting outlines ; the 
 flanks are, as already explained, extended to a greater depth in order 
 to give clearance. 
 
 193. The operation will be readily traced, as in the diagram the 
 acting side of a tooth of each wheel is drawn in two positions, show- 
 ing the state of affairs at the beginning and at the termination of the 
 action respectively. Supposing RN to drive, the action begins at 0, 
 the driver's flank pushing the face of the follower, and the point of 
 contact moving in the arc OA, until the points K and F meet at A. 
 The face of the driver then urges the follower's flank, the point of 
 contact now traveling in the arc AP, and at P the action ends. 
 
 We see, then, that the angle of approach depends upon the length 
 of the follower's face, and the angle of recess upon that of the driver's 
 face : and if these lengths be assumed or given, the angles are easily 
 found. For instance, had the length of the follower's face been as- 
 signed, as ES, then a circle described about C through S cuts the de- 
 scribing circle of that face in 0, determining A the length of the 
 arc of approach, to which AF and A K are to be made equal. 
 
 194, A Practicable Example. The diagram, Fig. 115, is drawn 
 without regard to practical proportions, the only object being to illus- 
 trate the eonstruction clearly ; but in Fig. 116, we have shown a 
 feasible case. The cut is. half size, and the conditions assigned are as 
 follows : 
 
100 
 
 INTERCHANGEABLE WHEELS. 
 
 Distance between centres,, 27 inches. 
 Wheels to have 63 and 45 teeth respectively. 
 The smaller wheel to be the driver. 
 Whole arc of action to be 2s times the pitch. 
 Angle of recess to be ^ greater than angle of approach. 
 We have, then, 
 
 63 : 45 :: 7 : 5, 7 + 5 = 12, f| = 2}, 
 
 also 
 
 Again, 
 
 2} x 7 = 15|, = radius of larger pitch circle. 
 2} x 5 = 11}, = " " smaller " " 
 
 which is to be divided into parts in the ratio of 3 to 4 ; 
 whence, 
 
 3 + 4=7, V -v- 7 = f ; 
 and 
 
 t x 3 = li times the pitch = angle of approach, 
 
 1x4=1* 
 
 recess. 
 
 Interchangeable Wheels. 
 
 195. Inasmuch as the face and the flank which act upon each other 
 are generated by the same describing circle, it makes no difference 
 
 FIG. 117. FIG. 118. FIG. 119. 
 
 whether the diameter of the one which traces the other face and flank 
 be the same or not, in laying out a single pair of wheels, and in Fig. 
 115, the describing circles are of different diameters. But for the very 
 reason just stated, it is clear that if we wish to make a number of 
 wheels, any one of which will gear with any other one, we must use 
 the same describing circle for all the faces and all the flanks. 
 
 196. Size of the Describing Circle. In making such a set of inter- 
 changeable wheels, the question at once arises, how large should the 
 describing circle be ? 
 
LIMITING DIAMETEE OF DESCEIBING CIKCLE. 101 
 
 The answer to this depends upon properties of the hypocycloid, 
 which are illustrated in the next three diagrams. In Fig. 117, the 
 describing circle is half as large as the pitch circle ; and the generated 
 curve degenerates into a right line, so that the tooth, having radial 
 flanks, is comparatively weak at the root;. In Fig. 118, the describ- 
 ing circle is smaller, and the flank curves away from the radius, out- 
 wardly with respect to the body of the tooth, as it recedes from the 
 pitch circle, giving a stronger form. In Fig. 119, on the other hand, 
 the describing circle is larger, and the flank curves in the opposite 
 direction, rendering the tooth both weak and difficult to make. 
 
 The safe practical deduction would seem to be, that the dia- 
 meter of the describing circle should not be more than half that 
 of the pitch circle of the smallest wheel of a set. Still, it 
 will be found that if it be made five-eighths instead of one- 
 half that diameter, the curvature of the flanks will not be so great, 
 with the customary proportions of height to thickness of the teeth, 
 as to make the spaces any wider at the bottom than at the pitch cir- 
 cle ; the teeth can therefore be made as usual by means of a milling 
 cutter, and a describing circle of the size last mentioned has been 
 employed with excellent results. 
 
 197. In special constructions, as, for instance, in laying out a single 
 pair of wheels for which cutters are to be made expressly, good results 
 for general purposes may be attained by the use of two describing cir- 
 cles, the diameter of each being three-eighths that of the pitch circle 
 within which it rolls. 
 
 However, with a given arc of action, the face is shorter, and the 
 obliquity of the line of action less, the larger the describing circle. 
 Consequently in very delicate mechanism, as, for example, in watch- 
 work or clock-work of the finest grades, the advantages thus gained 
 may make it advisable to use teeth of the form shown in Fig. 119, 
 notwithstanding the difficulty of making them, and their inherent 
 weakness ; the latter may be to some extent obviated by using large 
 fillets at the junction of the sides and bottom of the spaces, which is 
 quite admissible, because, as already shown, the acting flank is com- 
 paratively short, and the exact outline of the clearing space is of no 
 consequence, so long as the space is great enough. 
 
 198. Rack and Wheel. A rack is simply an infinitely large wheel. 
 The curvature of a circle diminishes as the radius increases, and dis- 
 appears when the radius becomes infinite. Thus the pitch line of a 
 rack is only a straight tangent to the pitch circle of the wheel with 
 which it works, and the line of centres becomes a perpendicular to 
 this pitch line, passing through the centre of the wheel. 
 
102 
 
 BACK AND WHEEL. 
 
 The rack will travel through a distance equal to the circumference 
 of the pitch circle of the wheel during one revolution of the latter, 
 
 whatever the num- 
 ber of teeth, and 
 in the same propor- 
 tion for any frac- 
 tion of a revolu- 
 tion. The pitch of 
 the rack teeth, 
 therefore, is found 
 by rectifying the 
 pitch arc of the 
 wheel, whatever 
 that may be, and 
 setting off that length upon the pitch line. The construction is 
 shown in Fig. 120 ; the two describing circles are here made of the 
 same diameter, and it is clear that if the same circle be used to gen- 
 enite the faces and flanks of a set of wheels, any one of them will gear 
 with the rack if the pitch be also the same. 
 
 199. Evidently, both faces and flanks of the rack teeth are cycloids, 
 being generated by the rolling of a circle upon the pitch line. If the 
 length of the face be assumed, as FFFfor instance, a line parallel to 
 RN, through Fthc highest point, cuts the pitch circle in P, thus de- 
 termining AP, to which AB must be made equal, and fixing the part 
 of the action which will take place on the right of CD. Or if AB 
 be assigned, we make AP equal to it, thus ascertaining the necessary 
 length of face. In either case, PS is now to be drawn perpendicular 
 to the pitch line, which it cuts at G ; and as in the preceding con- 
 structions, BG cannot be greater, and should be less, than half the 
 thickness of the tooth as determined by the pitch. The part of the 
 action which will take place on the left of CD, depends upon the 
 length of the face of the wheel tooth, and is ascertained as in the 
 cases previously explained. 
 
 200. If in constructing the teeth for a pair of wheels, we employ 
 two describing circles, the diameter of each being one-half that of the 
 pifcch circle within which it rolls, the teeth of each wheel will have 
 radial flanks. A similar course may be pursued in laying out a rack 
 and wheel, as shown in Fig. 121. The describing circle whose diam- 
 eter is A C, generates the radial flanks of the wheel-teeth and the 
 cycloidal faces of the rack-teeth. 
 
 The diameter of the pitch-circle of the rack being infinite, one-half 
 of it is also infinite, and EN is therefore the describing line for the 
 
ARBITRARY PRACTICAL PROPORTIONS. 
 
 103 
 
 faces of the wheel-teeth, which consequently aro involutes of the pitch- 
 circle LM, and for the flanks of the rack-teeth. The latter being 
 
 M 
 
 FIG. 121. 
 
 radii of the infinite circle RN, would, strictly, be simply straight 
 lines perpendicular to it, as shown in the teeth on the right of CD ; 
 and they were formerly made so. But when these parts of the teeth 
 are in action, the point of contact moves in the describing line, that 
 is to say, in the pitch line RN itself. 
 
 The fact is, that the acting flank of the rack-tooth degenerates into 
 a point ; thus a marking point carried by RN, in moving from A to 
 0, traces on the plane of the wheel, which turns as indicated by the 
 arrow, the involute F0 9 the arc A F being equal to AO : while, hav- 
 ing no motion relatively to the plane of the rack, it marks on that 
 plane simply the point 0. The action is consequently bad, this point 
 being subjected to excessive wear : but the flank of 
 the rack tooth may be made, as shown on the left 
 of OD, an arc of a circle whose centre is on RN, 
 and radius equal to the radius of curvature of the 
 face of the wheel-tooth at its highest point. This 
 radius is found thus : let V be the highest point of 
 the involute tooth ; draw through V a tangent to 
 the pitch-circle LM, lying, as shown, on the concave 
 side of the involute, and find the point of tangency 
 T \ then VTis the required radius of curvature. 
 
 201, Arbitrary Proportions. It is not necessary 
 in all cases, to pay particular attention to the rela- 
 tive amounts of the approaching and the receding action. And it is 
 a very common practice to make the whole radial height of the tooth 
 a certain fraction of the pitch ; the part without the pitch circle 
 being a little less than that within, by which the clearance is pro- 
 vided for. In Fig. 122, these parts are marked h and d respectively. 
 I being the whole height. Three of these arbitrary proportions, as 
 
 FIG. 122. 
 
104 BACKLASH. 
 
 they may be called, which have been extensively adopted, are as fol- 
 lows : 
 
 1. I = -f pitch ; h : d : : 11 : 13. (b = T V pitch.) 
 
 2. 1= f pitch; hid:: 4: 5. (b = & pitch.) 
 
 3. I = j\ pitch ; h : d : : 3 : 4. (b = T 1 T pitch.) 
 
 The whole angle of action, as well as the ratio of the approach to 
 the recess, will of course vary according to the numbers of the teeth, 
 in the use of any such system : but either of these rules will give sat- 
 isfactory results for most purposes, the wheels acting as drivers or 
 followers indifferently, provided that there are at least twelve teeth 
 upon the smallest wheel. Less than that should not be used unless 
 it is necessary : sometimes, however, the use of lower numbers cannot 
 be avoided, in which event it will often be requisite to extend the 
 faces of the pinion's teeth beyond these limits ; and the proper length 
 should be determined as before explained. 
 
 202. It was also formerly the custom to make the backlash a defi- 
 nite fraction of the pitch, which we have added above in parenthesis 
 as usually given in connection with each of the preceding rules. But 
 although these values may have been proper in many cases, as allow- 
 ing for imperfections of workmanship in wooden mill-work or when 
 the wheels were simply to be cast, it is certain that they are in many 
 cases too large, if the teeth are to be cut with the slightest pretension 
 to accuracy. Nor does there appear to be any reason why the back- 
 lash should vary directly with the pitch ; on the contrary, it seems 
 almost self-evident that the coarser the pitch, the smaller will be the 
 proportion borne to it by any unavoidable error. From this point of 
 view, it appears more reasonable to say that the backlash should vary 
 inversely as the pitch ; and perfectly safe to insist that it ought in 
 every case to be as small as the skill of the workman will enable him 
 to make it with the facilities at command. 
 
 Since theoretically the teeth may be in contact on both sides at once, 
 we have in the diagrams entirely disregarded the backlash. If it were 
 introduced, the constructions, evidently, would be modified only in 
 this respect, that the " thickness of the tooth as required by the pitch," 
 instead of being exactly half the pitch arc, would be less than that by 
 just the amount of backlash allowed. 
 
 3. Annular Wheels. 
 
 203. In Fig. 123, the smaller pitch circle lies within the greater, 
 which it touches internally ; the teeth of the outer wheel being there- 
 fore formed on the concave circumference of an annular rim. But 
 neither the mode of generating the tooth-outlines, nor the nature of 
 
ANNULAR WHEELS. 
 
 their action, are in any way changed; two describing circles are 
 shown, each of which generates a face for one pitch circle and a flank 
 for the other, and in short, a comparison of this diagram with Fig. 
 115, which is lettered similarly throughout, will show that the two 
 are identical in all particulars relating to the construction. 
 
 It will also be ob- 
 served that the contour 
 of the annular wheel, 
 in respect to the forms 
 of the acting curves, is 
 identical with that of 
 an ordinary spur wheel 
 having the same pitch 
 and describing circles, 
 the tooth of the one cor- 
 responding precisely to 
 the space of the other. FIG. 123. 
 
 The describing circles in the figure are of different diameters : but 
 they might have been equal, and it will readily be seen that the spur 
 wheels which are thus made interchangeable with each other (195), may 
 also be made interchangeable with annular ones. But in the construc- 
 tion of inside gearing the diameters of the describing circles are limited 
 by considerations which relate entirely to the peculiar condition of 
 internal tangency between the pitch circles. The manner in which 
 these limits are determined, will be most clearly seen by first regard- 
 ing the tooth-outlines as generated 
 in another way. 
 
 204. Intermediate Describing Cir- 
 cle. In Fig. 124, let C be the 
 centre of the inner pitch circle, D 
 that of the outer, and A OL a curve 
 lying between the two circumfer- 
 ences, tangent to both at A, and 
 capable of rolling in contact with 
 them. Let the arcs AK, AF 9 of 
 the pitch circles, be equal to each 
 other, and during the rotation indi- 
 cated by the arrow, let a marking 
 point be carried from A to 0, by the 
 describing curve : it will in its 
 progress, evidently, trace upon 
 the planes of rotation, the face OK for the pinion, and the face OF for 
 
 FIG. 124. 
 
106 INTERMEDIATE DESCRIBING CIRCLE. 
 
 the annular wheel. If these be used as the outlines of teeth, it 
 follows from the mode of their generation that they will transmit ro- 
 tation with a constant velocity ratio, whatever the form of the de- 
 scribing curve or of the locus of contact : if the pinion drive, the 
 action begins at A, and ends at 0. 
 
 Now, if it be required that these faces shall be epicycloidal, the 
 curve AOL must be, as in the figure, a circle whose centre-./? lies 
 between C and D. It is true that OK and OF may then be generated 
 in a different manner ; which, however, though an important coinci- 
 dence, does not alter the fact that the driving contact between these 
 two curves is due solely to the mode of generation here explained. 
 And it will subsequently be seen that annular wheels may be required 
 to work under conditions which can be satisfied only by this method 
 of construction : upon which also depends the determination of the 
 limiting diameters of the describing circles in either method. 
 
 205. Limiting Diameter of Intermediate Describing Circle. In making 
 these determinations, we avail ourselves of the peculiar property 
 above alluded to, viz : that every epicycloid, internal or external, as 
 well as every hypocycloid, is capable of two generations. (See Ap- 
 pendix,) 
 
 The face OF being now a true hypocycloid, may be generated not 
 only by the intermediate circle whose centre is E, but by another 
 whose radius is ED. In Fig. 124, AE is equal to CD, whence ED = 
 A C the radius of the smaller pitch circle. Consequently on rolling 
 the pinion within the wheel, the hypocycloidal path traced by the 
 point A of the former, coincides with the face AB of the adjacent 
 tooth of the wheel. The radius AE may be increased, in which case 
 this path AM will lie to the right of AB ; but ;f it be decreased, 
 AM will lie to the left of AB, that is, within the body of the wheel's 
 tooth, which is clearly impracticable. In this construction then we 
 have the limit, that 
 
 The radius of the intermediate describing circle can not he less than 
 the line of centres. 
 
 206. Drawing through a circle about centre D, it cuts the inner 
 pitch circle in R ; at which point the action begins, the face KO 
 then having the position RN. 
 
 A marking point at R, carried by the inner pitch circle as a de- 
 scribing curve, will, in going to A, obviously generate the face R8 
 similar to OF; against Avhich this point R of the pinion will act 
 during the approach, the locus of contact being the arc RA. 
 
 The face of the pinion having now reached the position AP, the 
 receding action begins ; and here a phenomenon peculiar to inside 
 
TEETH WITH DOUBLE CONTACT. 
 
 107 
 
 gearing presents itself. For the face AP is not only an internal 
 epicycloid whose generating circle is A OL, but an external one which 
 may be generated by a circle whose radius is AH, equal to EC. It 
 will, therefore, work correctly with a flank A Q traced by tho same 
 describing circle. This action ter- 
 minates at T, the point in which 
 a circle* through about C cuts the 
 circumference of the exterior describ- 
 ing circle, and during its continuance 
 the locus of contact is the arc 
 AT. 
 
 This flank may or may not be 
 used ; but if it be, since the action 
 between the faces AP, AB, pre- 
 viously explained, also begins at A, 
 the singular fact appears, that while 
 the pinion is turning through the 
 angle PCT, its face has two points 
 of driving contact. This circum- 
 stance is of some practical impor- 
 tance, not only on account of tho FIG. 125. 
 division of the pressure, but also as affecting tho resultant obliquity 
 of the line of action. 
 
 207. Limiting Diameters of Exterior and Interior Describing Circles. 
 In Fig. 125, the radius AE of the intermediate describing circle is 
 greater than CD. The face OF may also be generated by an interior 
 describing circle whose radius A G is equal to ED, and OJTby an 
 exterior one of which the radius is AH, equal to EC. In all that 
 relates to the action the only new feature is that the pinion now has 
 a flank of sensible magnitude, traced by the circle whose centre is G ; 
 as will readily be seen, the lettering throughout being similar to that 
 of Fig. 124. Now, either radius, AG or AH, may be diminished 
 without changing the other ; the only result being that the faces OF, 
 OK, as shown in dotted lines, will not be in contact during the recess, 
 and will consequently act only against the corresponding flanks, as in 
 Fig. 123. But if either of these radii be increased without diminish- 
 ing the other to the same extent, it is apparent that the faces OF, 
 OK, will intersect each other, rendering the construction impractica- 
 ble : and we have as a limit, that 
 
 TJie sum of the radii of the two describing circles cannot ~be greater 
 than the line of centres. 
 
 208. This holds true when either radius vanishes, the other then 
 
108 
 
 LIMITING DIAMETERS OF DESCRIBING CIECLES. 
 
 becoming equal to the line of centres. Thus, if in Fig. 125, we make 
 AE = AD, we shall have 
 
 ED = = AH, whence AG = EG = CD. 
 If, on the other hand, we make AE = AC, we shall have 
 
 EG = = AG, whence A H = ED CD. 
 These two limiting cases are illustrated in Figs. 126 and 127 respect- 
 ively. In the former, it is apparent that when the pinion drives the 
 arc of recess is greater than the arc of approach, but since the action 
 
 FIG. 126. 
 
 FIG. 127. 
 
 during the recess is confined to the single point of the pinion's teeth, 
 the advantage thus gained is more than neutralized. The fact that the 
 pinion can have no face, will be seen from the consideration that if there 
 were one, it must be tangent at to the radius OC, and would, there- 
 fore lie within the body of the adjacent tooth of the annular wheel. 
 In Fig. 127, on the other hand, the wheel can have no face, for a 
 
 similar reason. The pinion 
 consequently having no 
 flank, there is no approach- 
 ing action, but its face has 
 two points of driving con- 
 tact during a part of the 
 recess. 
 
 209. It is evidently more 
 frequently practicable in 
 inside than in outside gear- 
 ing, to secure an angle of 
 recess greater than the 
 pitch, and thus to avoid 
 altogether the more injurious friction of approaching action. An in- 
 stance of this is shown in Fig. 128, where, the wheel driving, an in- 
 
 FIG. 128. 
 
SPECIAL CASES. 
 
 109 
 
 terior describing circle only is employed, whose radius is in this case 
 half that of the inner pitch circle, thus giving the pinion radial 
 flanks. Attention is called to the fact that when the tooth thus has 
 no face, it should, nevertheless, be allowed to project beyond the pitch 
 circle for the sake of 
 strength, the corner being 
 finished by a circular arc 
 tangent to the radius at its 
 extremity. 
 
 Fig. 129 shows the ap- 
 pearance of an annular 
 wheel and pinion which 
 differ but little in size ; the 
 teeth are necessarily very 
 short in order that they 
 may escape from engage- 
 ments and pass each other. 
 Whether they will do so or 
 not, if the height be as- FIG. 129. 
 
 sumed, is readily determined by constructing the epitrochoid traced 
 by the highest point of either tooth when its pitch circle is rolled 
 upon the other one ; this path, obviously, must not intersect the tooth 
 outline of the engaging wheel : and the clearing spaces of both the 
 wheel and the pinion must also be such as not to touch the epitro- 
 choids thus described. 
 
CHAPTER VII. 
 
 LOW-NUMBERED PINIONS PRACTICAL LIMIT OF OBLIQUITY PIN- 
 IONS WITH 3, 4, and 5 LEAVES LEAST NUMBER THAT CAN BE 
 
 USED WITH A GIVEN DRIVER THE TWO-LEAVED' PINION LOW 
 NUMBERS IN INSIDE GEAR VARIOUS EXAMPLES INFERIOR 
 AND SUPERIOR LIMITS TWO-LEAVED PINIONS IN INSIDE GEAR 
 LEAST NUMBER THAT CAN BE USED WITH A GIVEN FOLLOWER. 
 
 On the Use of Low-numbered Pinions. 
 
 210. In the operation of epicycloidal teeth, the obliquity of the line 
 of action is continually varying ; diminishing during the approach, it 
 becomes zero when the point of contact reaches the line of centres, and 
 again increases during the recess. 
 
 If wheels are to do heavy work, it has been found by experience that 
 the mean obliquity should not in general exceed about 15, nor the 
 maximum about 30. A high maximum is less objectionable when 
 several pairs of teeth are engaged at once, since the greatest portion of 
 the pressure will be acting less obliquely. Such distribution of the 
 pressure, it will readily be seen, is more often to be attained in inside 
 gearing, and the obliquity is one serious disadvantage when low-num- 
 bered pinions act in outside gear. Another is the excessive amount 
 of sliding, due to the necessarily great length of the faces of the teeth. 
 And from both combined is deduced the practical rule that, in mill- 
 work and machinery in general, no pinion of less than twelve teeth 
 should be used if it be possible to avoid it. 
 
 But it is not always possible ; and in lighter mechanism, such as 
 clock-work, it is often necessary to use much lower numbers. In work 
 of this description a greater obliquity is often admissible ; and when 
 it is not so considered, the convexity of the flanks caused by using a 
 large interior describing circle, is not so objectionable when the work 
 to be done is light, and strength of form not an imperative necessity. 
 
 For the present, then, we will confine ourselves to the following 
 limits, viz : that the maximum obliquity shall not exceed 36, and 
 
LOW-NUMBERED PINIONS. 
 
 Ill 
 
 that the diameter of the describing circle for the flanks shall not 
 
 exceed -f- that of the pitch 
 
 circle within which it rolls. 
 211. In Fig. 130 are shown 
 
 two equal pitch circles, and 
 
 two equal describing circles 
 
 of half the diameter, TAS 
 
 being the common tangent. 
 
 Draw the line of action with 
 
 an obliquity of 30 ; it cuts 
 
 the describing circles in 
 
 and P, making the arcs A 0, 
 
 AP, each equal to 60. Draw- 
 ing <7PJ?and DOK, the arcs 
 
 AE, AK, are each equal to 
 
 30, and we thus have two 
 
 similar wheels which will just 
 
 work, the angles of approach 
 
 and of recess being each equal FIG. 130. 
 
 to half the pitch, and the mean obliquity exactly 15. It is seen to 
 
 be possible, with only a slight increase in the obliquity, to lengthen 
 
 the faces and secure an arc of 
 action greater than the pitch, 
 as in practice it should be. 
 
 In this case the flanks are 
 radial. But if without chang- 
 ing the describing circles, we 
 reduce the pitch circles to f 
 their present diameter, the arc 
 AP of 60 will be equal to an 
 arc of 36 on the new pitch cir- 
 cumference. It is, then, prac- 
 ticable to make two pinions of 
 five teeth each, which, like those 
 in the figure, will just work, with 
 the same obliquity. The flanks 
 
 - 
 
 will now be convex; still, the di- 
 ameter of the describing circle is 
 within the assumed limit, being 
 FIG. LSI. but -jfcj- that of the pitch circle. 
 
 212. In Fig. 131 the diameter of the larger pitch circle is 1} times 
 that of the smaller one ; also the diameter of the upper describing 
 
LOW-ls TJMBEKED PINIONS. 
 
 circle is equal to CA, so tjiat as in the preceding figure the arc AP 
 of 60 is equal to the arc AE of 30, and PAS, the maximum obli- 
 quity on the right of CD, is also 30. The diameter of the lower de- 
 scribing circle is -| that of its pitch circle, therefore the arc A of 73 
 is equal to the arc AK of 45, and OAT, the maximum obliquity on 
 the left of CD, is 36. Completing the construction, we perceive that 
 the pinion of four teeth will just work with the other of six. The 
 arcs of approach and recess are equal, the mean obliquity being 18 
 during the one and 15 during the other, giving an average of 16|- 
 for the whole action if the Tatter be made just equal to the pitch ; and 
 the faces may also be made a little longer than here shown, as in the 
 previous case. 
 
 The leaves of the larger pinion still have radial flanks, but proceed- 
 ing as before, we may reduce the upper pitch circle to f of its present 
 diameter, without changing the circle which rolls within it. TVe shall 
 then have the arc AE equal to 36, and the five-leaved pinion thus 
 constructed will work with the four- leaved one, the obliquity remain- 
 ing unchanged, since the points, 0, A, P, retain the same positions 
 as in the figure. 
 
 213, Now produce OA to I, making AI = AO. Then, were 
 another describing circle drawn above T8, equal in diameter to the 
 one below it, the point /would lie upon its circumference ; and since 
 the arc AI would be equal to the arc AB, the epicycloid traced by roll- 
 ing it upon the lower pitch circle would extend to the point B. 
 
 Draw ID ; it will then be apparent that were the angle IDB equal 
 to or less than the angle IDA, two four-leaved pinions would be capa- 
 ble of working together, all the acting curves being traced by a de- 
 scribing circle whose diameter is that of the pitch circles. Now in 
 the triangle ADI, the sides AI and AD, also their included angle, 
 are known ; whence it will be found that the angle IDA is 22 27' 
 42", whereas it should be 22 30' in order that one pinion should 
 drive the other through a total arc of action just equal to the pitch, 
 even if the teeth were pointed. 
 
 Five, then, is the least number of teeth that can be used if the pin- 
 ions are to be alike, and four will work with five or any higher number. 
 
 214. The force of the objections urged against the use of small 
 pinions for heavy work, is clearly shown in Fig. 130. Letting OA 
 represent the pressure when is the point of contact, this may be 
 resolved into the two components Om, On ; of which the latter tends 
 simply to force the journals apart, thus increasing the friction in the 
 bearings. Since only one pair of teeth is in action, and the angle 
 OA T is 30, this objectionable component is equal to half the total 
 
THREE-LEAVE D PIKIOH OUTSIDE GEAK. 
 
 113 
 
 pressure at 0, which again is greater than that at the point K on the 
 pitch circumference, in the ratio of DKto DO. 
 
 215. Thus far, the angles of approach and recess have been made 
 equal. Should it be desired to make them otherwise, the question 
 whether any assumed conditions can be satisfied or not, is readily set- 
 tled by the construction of a diagram, as explained in connection 
 with Fig. 115. 
 
 If, as usual in such cases, the angle of recess is to be the greater, 
 it will be apparent that with a wheel of a given number of teeth a 
 smaller pinion may in general be used to drive than to follow. For 
 the angle of recess depends upon the length of the face of the driver's 
 tooth ; and the smaller the pinion the smaller will be the describing 
 circle which can be used within it, and, consequently, the less will be 
 the greatest possible length of the face of the wheel-tooth, ^ince the 
 pitch is already assigned by the conditions. 
 
 216, An illustration of this is incidentally afforded by Fig. 132, in 
 the construction of the three-leaved pinion. Taking for its flanks a 
 describing circle whose diame- 
 ter is | that of its pitch circle, 
 
 and making OAT, the maxi- 
 mum obliquity on the left of 
 CD, 36 as before, the angle of 
 action, KDA, on that side of the 
 line of centres is 45. The 
 pitch arc KB is 120, there- 
 fore the angle of action ADB, 
 on the right of CD, must be 
 75. It will be found that this 
 can be secured, with a very lit- 
 tle to spare, by using for its 
 faces a describing circle five 
 times as large as the one used 
 for its flanks. 
 
 And keeping to the limits 
 heretofore observed, it follows 
 that the second pitch circle can- 
 not be less than five times as large 
 as that of the pinion ; which, 
 therefore, will just work with a 
 wheel of fifteen teeth 1 , and no 
 
 le$s, under the conditions as- FIG. 132. 
 
 signed. In other words, a three-leaved pinion will drive a wheel of 
 8 
 
114 
 
 LEAST NUMBER OF TEETH. 
 
 fifteen teeth, the angle of approach being |, and that of recess f, of 
 the pitch angle. Seasonably well, too, in respect to the obliquity of 
 the line of action ; during the approach the maximum is 36 and the 
 mean is 18, during the recess the maximum is 13 and the mean 6 : 
 but the latter angle is larger than the former in the ratio of 5 to 3, 
 whence we have, as the average obliquity during the whole action, 
 
 18 x 3 + G x 5 84 
 
 3 + 5 8 
 
 217. It will be perceived that the above construction involves the 
 solution of the following problem, viz. : 
 
 Given the pitch circle, number of teeth, and arc of recess, of the 
 driver, to find the least number of teeth which can be assigned to the 
 follower.. 
 
 This may be determined graphically, as shown in Fig. 133, where 
 CD is the line of centres, the centre of the driver, and AF the 
 
 given arc of recess. Make FL equal 
 to J the pitch, that is, half the thick- 
 ness of the tooth, and draw through 
 L a radial line of indefinite length ; 
 then if the tooth be pointed, its point 
 will lie upon the prolongation of CL. 
 Make A G, perpendicular to CD, equal 
 to the arc AF (see Appendix (1)), and 
 set off A M = i A G. With centre M 
 and radius MG j AG, describe an 
 arc cutting CL produced in 0. Draw 
 OA, bisect it in N by a perpendicu- 
 lar cutting CD in E\ then (see Ap- 
 pendix (1)), the arc OA, whose centre 
 is E, will be equal to A G and therefore 
 to AF. 
 
 Evidently, then, E is the centre of a 
 describing circle which by rolling on 
 the given pitch circle will trace an 
 epicycloid from to F, and this will 
 FIG. 133. be the face of the driver's tooth. Ob- 
 
 serving the limits before assigned, the radius of the required pitch 
 circle cannot be less than f of AE ; but it must be such that the given 
 pitch, viz., four times the arc FL, shall be an aliquot part of the cir- 
 cumference. 
 
 It may be noted, that the conditions assigned as above furnish 
 
FIVE-LEAVED DRIVEN 
 
 115 
 
 sufficient data to enable us to verify the results by trigonometrical 
 computation, should it be considered necessary. 
 
 218. By the above process we ascertain at once the size of the de- 
 scribing circle which will give the driver a pointed tooth, and from 
 this derive the diameter of the follower's pitch circle. Should the 
 latter correspond to a fractional number of teeth, the next higher 
 integer must be used, the pitch circle being increased accordingly. 
 In this case the diameter of the describing circle may be increased or 
 not, at pleasure ; though it is better that it should be, since then the 
 driver's tooth may be topped off. 
 
 And it may be necessary to increase it for another reason ; because 
 the very process of determining the minimum radius AE, Fig. 131, 
 fixes also the maximum obliquity OA G corresponding thereto. At- 
 tention must, therefore, be given to this, for it is possible that the 
 obliquity thus fixed may be too great, although the numbers of the 
 teeth be practicable ; and in that event the describing circle, and, if 
 necessary, the pitch circle as well, must be increased until the ob- 
 liquity is reduced to the desired limit. 
 
 219. Fig. 134 illustrates the converse case to that of Fig. 132, the 
 conditions being that the wheel of 15 teeth is to drive, and that the 
 angle of recess shall be f of the pitch 
 
 angle. Making the construction as 
 explained in (217), the least pitch 
 circle of the follower corresponds, 
 as found by computation, to 4.45 
 teeth, with a maximum obliquity of 
 40 30' 2". Five, therefore, is the 
 least practicable number of leaves 
 that the pinion can have, thus mak- 
 ing the pitch 72, and the arc of 
 recess 45, and by using a describing 
 circle for their flanks, whose diame- 
 ter is f that of the pitch circle, the 
 maximum obliquity during the re- 
 ceding action is reduced to 36, 
 while at the same time the driver's 
 tooth is made of reasonable breadth FlG - 134; 
 
 at the top. In this diagram, as in Fig. 132, the angle of approach is 
 | of the pitch angle, but it is obvious that although the driver's flanks 
 are radial, the angle of approach might be very considerably increased 
 in this case, although in that of the 3-leaved driver it could not be. 
 
 220. The two-leaved Pinion. When a spur-wheel is made in the or- 
 
116 
 
 TWO-LEAVED DKIVING 
 
 dinary way, that is, with the teeth in the same plane, a pinion of three 
 leaves is the smallest that can be used either to drive or to follow. 
 But if the alternate teeth are placed in different planes, a two-leaved 
 pinion can be made to drive in a very satisfactory manner ; the ar- 
 rangement is clearly shown in Fig. 135, the pinion being composed of 
 two heart-shaped cams or teeth, U and W, fixed side by side on the 
 same shaft, and the wheel of the two star-shaped plates similarly fixed 
 upon another shaft, of which the one, M, works with the cam U, and 
 the other, N, with W. The diameter of the describing circle is equal 
 to the radius of the pitch circle of the follower, whose flank is there- 
 fore radial. The diameters of the pitch circles arc in this case in the 
 
 ratio of 6 to 1 ; the arc AP of 60, 
 is therefore equal to the arc AE of 
 30, and to the semi-circumference 
 AB. In the position here shown 
 P is a point of contact, but the epi- 
 cycloid BP is continued to R, so 
 that the action is not yet ended, 
 although the tooth A IS is just be- 
 ginning to drive a flank of the other 
 plate JVof the wheel. 
 
 It is obvious that a half revolution 
 of the pinion will bring the point 8 
 to the position R, turning the wheel 
 through the angle ACE of 30 ; G 
 and B then meeting at A, the ac- 
 tion will be continuous. 
 
 The obliquity, it will be observed, 
 is not great, but the amount of sliding is excessive, owing to the great 
 length of the face PB as compared with the flank PE ; still the ac- 
 tion is smooth and noiseless, since the driving contact is wholly re- 
 ceding, there being no arc of approach. 
 
 The number of teeth in the wheel may be reduced, for the flank need 
 not be radial : by,.using a describing circle of | the diameter of the 
 pitch circle, and making AP 72, the angle ACE will become 45, 
 the velocity ratio thus obtained being that of 4 to 1, and the maxi- 
 mum obliquity very little over 36. 
 
 221, Low-numbered Pinions in Inside Gear. In making investiga- 
 tions similar to the preceding in relation to annular wheels, different 
 cases will be found to arise under varying conditions, the treatment of 
 which will perhaps be best developed by beginning with a specific 
 example. 
 
 FIG. 135. 
 
THREE-LEAVED PINION INSIDE GEAR. 
 
 11? 
 
 Lei us take in illustration the three-leaved pinion, Fig. 13G, the 
 conditions being as follows : 
 
 Diam. Inner DCS. Circle = f Diam. Pitch Circle. 
 
 Arc of approach. 45 c 
 
 recess, 
 
 75 C 
 
 Total = 120 = Pitch. 
 
 Eequired to find the least annular wheel which can be driven. 
 
 Evidently a describing circle externally tangent to both pitch cir- 
 cles, as in Fig. 123, 
 must be used to gener- 
 ate the faces of the pin- 
 ion and the flanks for 
 the wheel; and its least 
 radius may be found as 
 in Fig. 133. 
 
 Knowing, then, this 
 radius and that of the 
 pinion's pitch circle, 
 for the reason given in 
 (208) the radius of the 
 larger pitch circle must 
 be greater than their 
 sum. Also, it must be 
 such that the pitch 
 which is given shall be 
 an aliquot part of the circumference. 
 
 Taking, for convenience, 5 as the radius of the inner describing 
 circle, then A C 8, whence by computation we find the least radius 
 of the exterior describing circle to be 24.93. 
 
 AD, then, must be greater than 8 + 24.93 = 32.93 ; let it equal 33 
 for instance. 
 
 Now the numbers of the teeth are proportional to the radii of the 
 pitch circles ; and letting 
 
 FIG. 136. 
 
 V = No. teeth of Annular Wheel, 
 3 : 8 :: N : 33, 
 
 we must have 
 
 which gives 
 
 N = 12| 
 
 But since JV^must be an integer, the least number that can be used 
 is 13 ; and the corresponding radius of the pitch circle is 34f. 
 
118 INFERIOR LIMIT IK INSIDE GEAR. 
 
 222. With this radius accordingly, the wheel shown in Fig. 136 ia 
 constructed ; the radius of the outer describing circle is taken as 25, 
 so that the teeth of the pinion are not exactly pointed, though their 
 breadth at the top is so small as to be practically inappreciable. 
 
 And a glance at the figure is sufficient to show that no matter how 
 many more teeth are given to the wheel, the pinion will work with it 
 equally well. If the outer pitch circle be increased, the chord of the 
 arc AE will also increase, and the flank PE lying always outside of 
 the face PB, will continue to be, as now, a practicable curve ; which 
 is fcelf-evidently true in regard to the face OF oi the wheel-tooth. 
 
 This pinion, it will be observed, is precisely the same as the one 
 shown in Fig. 132 ; and is, therefore, capable of working with a 
 rack, or with a wheel -of any number of teeth whatever, between the 
 limits of 
 
 13 and oo in Inside Gear, 
 oo and 15 in Outside Gear. 
 
 223. From this example, then, we have thus far ascertained that 
 with a given pinion, under certain conditions at least, there is an 
 inferior limit, below which the number of teeth in the annular wheel 
 cannot be reduced, but no superior limit beyond which the number 
 may not be increased. 
 
 To these results we call attention, the more particularly because so 
 high an authority as Professor Willis makes, without further com- 
 ment, this sweeping statement, viz. : 
 
 " The case of annular wheels differs from that of spur-wheels in 
 this respect, that, with a given pinion a small-numbered wheel works 
 with a greater angle of action than a large-numbered one, and there- 
 fore we have to assign the greatest number that will work with each 
 given pinion." It is true that he tacitly recognizes the absence under 
 some conditions of a limit in that direction, by assigning "any num- 
 ber" as the greatest, in his tables of limiting cases; but neither in 
 this fact, nor in the remarks just quoted, is there any recognition of 
 an inferior limit under any conditions. It is still more singular that 
 he gives nowhere any explanation of the circumstances under which 
 there is a maximum number, nor yet of the only mode in which the 
 teeth can possibly be generated when there is one ; neither does any 
 other writer that we know of. 
 
 224. Let us now suppose that a pinion of three leaves is to drive, 
 under the conditions that the tooth shall be equal to the space, and 
 the arc of recess just equal to the pitch. 
 
SUPERIOR LIMIT IIT INSIDE GEAR. 
 
 119 
 
 Then in Fig. 137, A being the diameter of the pitch circle, we 
 shall have on its circumference the space A L = the tooth LB = 60 ; 
 and if the tooth be pointed, its point must lie upon M C, perpendicu- 
 lar to CD, the line of centres. Evidently, if we draw AL and OB, 
 these lines produced will intersect in P upon MC, making OP OA, 
 and the arc AP 9 upon the 
 circle whose centre is and 
 radius OA, will be equal to 
 the arc ALB. That circle 
 may therefore be used as a 
 describing circle, and by roll- 
 ing upon the pinion's pitch 
 circle it will generate the 
 cardioidai faces PB and PL 
 for its tooth. This describ- 
 ing circle cuts CD in N ; 
 drawing PN and LO, they 
 will be parallel to each other 
 and perpendicular to AP. 
 Erect a perpendicular to PL 
 at its middle point S ; this 
 will pass through B, and bi- 
 sect ON in D, whence FIG. 137. 
 
 DA : OA :: 3 : 2, . . DA : CA :: 3 : 1 ; 
 
 and taking D as the centre of the outer pitch circle, an arc, AE, of 40 
 upon its circumference will be equal to the arc AP of 60 and the arc 
 AB of 120. 
 
 Bisect AE in 0, and draw ED, GD ; then the angle included be- 
 tween these radii is bisected by SD, since SDA = 30 ; consequently, 
 L is the highest point on the face LG of one tooth of the annular 
 wheel, just as P is in the face PE of the next one. 
 
 225. From this example we draw these conclusions, viz. : 
 
 1. When the conditions are such that the highest point in the face 
 of the pinion's tooth, when quitting contact, lies above the common 
 tangent YY, the describing circle must be an intermediate one, as in 
 Fig. 125. 
 
 2. The intermediate describing circle is largest when the tooth is 
 pointed. 
 
 3. The larger that describing circle, the larger may be the annular 
 
120 LIMIT AFFECTED BY OBLIQUITY. 
 
 wheel, for -(205) AD may be equal to AO + AC, but cannot be 
 greater. 
 
 Now by the construction above explained we have ascertained the 
 maximum diameters of the describing circle and of the outer pitch 
 circle, the latter being three times that of the inner ; therefore 9 is 
 the greatest number of teeth that can be given to the annular wheel. 
 The number assigned by Prof. Willis under the same conditions 
 is 12. 
 
 226. It will be perceived that when an intermediate describing cir- 
 cle is used, the obliquity varies inversely as its diameter. If, then, a 
 maximum value of the obliquity be assigned, this will determine a 
 minimum diameter of the describing circle, and consequently of the 
 outer pitch circle, which fixes in this case also an inferior limit to the 
 number of teeth in the annular wheel. Thus, in Fig. 137 it is clear 
 that although if we reduce the outer pitch circle we may also reduce 
 the describing circle, we cannot doit without increasing the obliquity, 
 of which the maximum value in the figure is 30. Now, if it be stip- 
 ulated that this value shall not be exceeded, we must retain the pres- 
 ent describing circle, but we may still diminish the outer pitch circle. 
 If this be done, the face PE will become shorter, and the limit, 
 obviously, will be reached when AD becomes equal to AO = 2 AC. 
 The hypocycloid PE will then have degenerated into the point P, to 
 which the whole action will be confined ; all the points of BP, which 
 remains unchanged, coming successively into coincidence with P. 
 We find, then, that under this additional restriction the least number 
 of teeth that can be employed is 6. If, however, it should under any 
 circumstances be desirable for the modification of motion to use such 
 a combination, regardless of excessive obliquity, it is proper to note 
 that by making the describing circle smaller the number may be re- 
 duced to 4 ; and in general, a pinion of any given number of teeth 
 may thus be made to work, more or less satisfactorily, with an annu- 
 lar wheel having one more, a fact of considerable importance in the 
 construction of differential trains of wheels. 
 
 227. It is also to be observed that in Fig. 137, the point L upon 
 the inner pitch circle is at once the highest point of the face GL and 
 the lowest one of the face PL ; whence, as will readily appear if we 
 suppose the motion to be reversed, the arc of approach is equal to 
 half the pitch. 
 
 Again, the face A Q will work with a flank generated by the circle 
 whose radius is AH = OC AC. And the point Q lies upon the 
 circumference of this circle ; for it lies on PA produced, and AQ 
 AL = AC = AH, whence, if HQ be drawn, the triangles ACL, 
 
GENERAL SOLUTION. 
 
 AHQ, will be equilateral and equal. But ACQ ACT 30 ; so 
 that the angle QCT, while turning through which the pinion's face 
 has two points of driving contact, is also equal to half the pitch. 
 
 In rolling the outer pitch circle upon the inner, the point L traces 
 an epifcrochoid, to which AL is normal and LO is tangent at that 
 point (these lines corresponding to AR and R Fin Fig. 124). The 
 outline of the clearing space in the pinion must be such that L can 
 move freely within it, but may also be tangent to LO at L. But the 
 face PL is tangent at the same point to the radius LC r ; consequently 
 the tooth must be formed with a positive, although an obtuse, inter- 
 section at that point. Still, were such a wheel and pinion made of 
 the proper material and finish, as for instance of hardened steel finely 
 polished, the combination might be used in light mechanism. 
 
 228. The above is a special case, in which the argument is based 
 upon peculiarities of the assigned conditions ; and was selected because 
 it seemed most simply and clearly to illustrate the principles involved. 
 
 In general, however, a direct geometrical solution like this is not 
 possible, and the required results are reached, as in outside gearing, 
 by the process explained in connection with Fig. 133. In adapting 
 it to the case of inside gearing, the diagram, as shown in Fig. 138, is 
 modified only in this respect, that OA, 
 and consequently the centre E of the 
 intermediate describing circle, will lie 
 above instead of below the tangent line 
 A G. In this case it is necessary to pro- 
 duce CO and AG in order that they 
 may intersect in H ; which being done, 
 we have before us all the data for trigo- 
 nometrical verification. 
 
 If the obliquity be assigned, the con- 
 struction is made in the following order : 
 Draw the indefinite line AQ, making 
 with AG an angle equal to the given 
 obliquity ; then the arc GO cuts AQ in 
 0, the highest point of the tooth. Then 
 as before bisect OA by a perpendicular 
 cutting AD in E '; this determines AE, FlG - 138 - 
 
 the radius of the least intermediate describing circle. 
 
 229. The conditions might be such that the point should fall 
 exactly upon the common tangent. In this event OA, equal to OF, 
 is itself the arc of the describing circle, whose radius AE, whether 
 we choose to regard it as a maximum or a minimum, is infinite. The 
 
122 
 
 TWO-LEAVED INTERNAL PINION. 
 
 pinion, then, can just work with a rack, but if the tooth be pointed 
 it can drive nothing less ; and the face will be an involute of the 
 pitch circle, precisely as in Fig. 121. Since, however, the tooth need 
 not be pointed, the pinion, with the same pitch and arc of recess, can 
 be made to drive any annular wheel which has at least one more tooth 
 than itself, by the use of an intermediate describing circle (see 226). 
 230. Two-leaved Pinions in Inside Gear. An annular wheel can be 
 driven by a pinion of only two leaves, the teeth working in the same 
 plane, as shown in Fig. 139. Let AC, AO, AD, be to each other in 
 the proportion of 1, 2, 3 ; then the intermediate describing circle is 
 of maximum diameter. Let the arc of recess, AB, = 135, then the 
 angle AOB = 67J, and prolonging OB to P, the arcs AB, AP, will 
 
 FIG. 139. 
 
 be equal ; also P will be the highest point of the cardioidal face PB, 
 and of the hypocycloidal face PE. Draw AP, cutting the inner 
 pitch circle in R ; then RO and PU will be perpendicular to A P. 
 Draw D8 also perpendicular to AP ; then, since A U is bisected at 
 0, and OUatD, PR is also bisected at S. 
 
 Therefore, DP = DR, that is to say, the path of P will cut the 
 inner pitch circle in R. If then we suppose the motion to be reversed, 
 the root L of the pinion's face LM must have met the point N of the 
 wheel's face NG, at R. In other words, the arc of approach is equal to 
 AP, which again is equal to RB, because R O bisects the angle A OP. 
 
 231. The total arc of action, then, is ample, that of ap- 
 proach being f, and that of recess f , of the pitch ; and the 
 maximum obliquity being 37J, the combination might be used 
 
TWO-LEAVED IKTEEKAL PI^TIOK. 
 
 123 
 
 in light mechanism. But attention is called to a peculiarity 
 of the action which would require a modification in the finish 
 of the teeth for practical purposes. When contact is just end- 
 ing at P between the fronts, or acting faces, of one pair of teeth, the 
 backs of the next pair are just coming into contact at /. Now, were 
 the teeth of the pinion merely "topped off'*' in the usual way in the 
 lathe, there will bo danger that the points would catch upon each 
 other near the point I, in case of any inaccuracy of workmanship or 
 wear in the bearings. This risk might be obviated by allowing some 
 backlash, but this is highly objectionable in mechanism of the only 
 description for which, if for any, this combination is suitable. A 
 much better expedient in such cases is shown in the figure, the top 
 of the tooth being bounded by a circular arc whose centre is the in- 
 tersection of the normals PA, MO. The action still ends at P, but 
 the other tooth of the pinion will have been guided into its space with- 
 out risk of jamming. And it may be added that a similar finish of 
 the wheel's tooth, by lengthening it a little and rounding off the 
 corner, is advisable in order to prevent its catching upon the root of 
 the pinion's tooth, since, as explained in (227), there will be a blunt 
 angle at L. 
 
 The tooth of the pinion here shown has a considerable breadth, 
 PM, at the top, and it will be seen that the action might have been 
 extended ; also, that a larger pitch circle might have been used for 
 the wheel ; but as will sub- 
 sequently appear, the max- 
 imum number of teeth 
 which can be used with this 
 pinion is seven. 
 
 232. A pinion with two 
 leaves in different planes may 
 also be made to drive an an- 
 nular wheel, as shown in Fig. 
 140. The velocity ratio be- 
 ing as four to one, let A C 
 1, AD 4 ; then the 
 limiting value of AH = 3, and the pinion is constructed exactly as 
 in Fig. 135, being in fact identical with the one there shown, which 
 is, therefore, capable of driving wheels of any number of teeth be- 
 tween the limits of 
 
 FIG. 140. 
 
 6 and oo in Outside Gear. 
 4 and oo in Inside Gear. 
 
124: 
 
 ANGULAR DRIVER GIVEX. 
 
 The tooth of the wheel should be finished as shown in clotted line at 
 EW, by which the injurious action of a sharp corner maybe avoided ; 
 and the movement is very smooth and noiseless, while the amount of 
 sliding is much less than in outside gear. 
 
 233. In the cases thus far considered the pinion has been the driver ; 
 let it now be required to determine the least number of teeth that can 
 be given to a pinion which is to be driven by an annular wheel whose 
 diameter, pitch and arc of recess are assigned. In Fig. 141 let D be 
 the centre of this wheel, ^jPthe arc of recess, FL the half thickness 
 of the tooth, whose face, evidently, must be generated by an interior 
 describing circle. On the tangent at A, set off A G = arc AF, also 
 
 FIG. 142. 
 
 AM \ A G ; with centre M and radius M G describe an arc cut- 
 ting DL in ; draw OA, and bisect it by a perpendicular cutting AD 
 in E. Then E is the centre and EA the radius of the least interior 
 describing circle, and if the pinion's flanks are to be radial the mini- 
 mum radius of its pitch circle will be AC 2 AE\ but if we adopt 
 the limit mentioned in (210) it will be A C =. -f AE. Since, 
 however, AE cannot exceed CD, it follows that if its value as above 
 determined prove to be greater than 1 AD, in the one case, or T 5 ? AD 
 in the other, the assigned conditions cannot be satisfied. 
 
 If a maximum obliquity be assigned, the teeth may or may not be 
 pointed, and the diagram is constructed by first drawing AQ. making 
 with A G an angle equal to the given obliquity, and then describing 
 
FOLLOWER GIVEN OUTSIDE GEAR. 
 
 125 
 
 the arc about M, to cut A Q in 0, which will be the highest point of 
 the wheel's tooth -face, whether the tooth itself be blunted or not. 
 
 234. Low- numbered Pinions. Follower Given. A new phase of the 
 question presents itself when the assigned conditions relate to the 
 follower, and the least number of teeth for the driver is to be de- 
 termined. This requires a different mode of operation, which, as 
 before, will be best explained by first considering a case in outside 
 gear. 
 
 In Fig. 142, let C be the centre of the follower; then A K, the 
 assigned arc of recess, is equal to the arc A of a describing circle 
 whose radius AE is known, and will be the highest point of the 
 
 UNIVERSITY 
 
 FIG. 143. FIG. 144. 
 
 driver's tooth. Let D be the centre of the driver ; then AD will be 
 a minimum when the tooth is pointed as in the figure, in which case 
 OD bisects and is perpendicular to the chord FG subtending the tooth. 
 Draw a parallel to FG through A, cutting the driver's pitch circle in 
 P\ then the chord A P is also bisected by OD, and AO, PO, are 
 equal. We have also AF = PG, and 
 
 AP = AF + PG - FG, = 
 
 - FG, 
 
 AP = 2 (arc of recess) (thickness of tooth). 
 Hence the construction, Fig. 143, in which A is, as before, the 
 
 as- 
 
126 FOLLOWER GIVEN INSIDE GEAR. 
 
 signed arc of recess laid off on the describing circle of the follower's 
 flank. Draw a tangent to this arc at A, on which set off 
 
 AO = 2 (arc A 0) (thickness of tooth), 
 also 
 
 AM= 
 
 About centre M with radius M G describe the indefinite arc GK ; 
 about centre with radius OA describe another arc cutting GK in 
 P. Draw AP, and bisect it by a perpendicular which will pass 
 through and cut CA produced in /), then AD will be the minimum 
 radius of the driver. 
 
 235. Should the point P fall upon the tangent line, the assigned 
 conditions can just be satisfied by a rack with pointed teeth, but by 
 nothing less in outside gear. 
 
 It may, however, fall upon the same side of the tangent with the 
 centres (7 and E, as in Fig. 144. In that case the driver will be an- 
 nular, and its radius AD will be a maximum when the tooth is 
 pointed. But A E cannot in any event exceed CD ; consequently, if 
 AD prove to be less than *-- AC (or f AC if the follower have radial 
 flanks) the assigned conditions cannot be satisfied. 
 
 If it prove to be greater, the maximum value may be used or not, 
 at pleasure, for the tooth need not be pointed, and by topping it off, 
 as shown in Fig. 145, the radius of the outer 
 pitch circle may be reduced, but not below the 
 limit just named. 
 
 It appears, then, that with a given pinion there 
 may or may not bo a superior limit to the num- 
 ber of teeth for the annular driver, but in either 
 case there is an inferior one if there be any re- 
 ceding action ; the wheel cannot have less than 
 one and a half times as many as the pinion, when 
 the latter has radial flanks. 
 
 236. One case remains to be considered, viz., when the given fol- 
 lower is annular ; in regard to which it has already been pointed out 
 (226) that in general the wheel may be driven by a pinion having one 
 tooth less than itself, which may be called a natural maximum. 
 
 By reference to Fig. 129, it will be seen that the larger the pinion 
 the broader will the teeth be at the top, and that its diameter will be 
 a minimum when its teeth are pointed. Now in all the previous cases 
 the assigned conditions have enabled us to fix the position of the 
 highest point of the tooth at the instant of quitting contact. This, 
 
ANNULAR FOLLOWER GIVEN. 127 
 
 / 
 
 however, is not so in the present case, and it is therefore impossible 
 to determine that minimum by direct means. 
 
 But it may be found indirectly ; for if we ascertain the limiting 
 numbers of teeth for the annular wheels driven by various given pin- 
 ions, we shall know from mere inspection of these results, the least 
 number for the pinion which can be used to drive, under like condi- 
 tions, in inside gear, a wheel of any assigned number of teeth. 
 
CHAPTER VIII. 
 
 SPUB GEAKING, CONTINUED EPICYCLOIDAL SYSTEM. 
 
 Limiting Numbers of Teeth for Various Arcs of Action. Details of Trigonomet- 
 rical Process of Determination. The Nomodont, or Curve of Limiting 
 Values. 
 
 Computation of Tables. 
 
 237. It can always be determined by construction, whether a proposed 
 pair of wheels will work under given conditions as to pitch, arc of 
 action, etc. But in order to avoid wasting time by attempting impos- 
 sible cases, it is well that the limiting numbers of teeth, within a 
 reasonable range of varying conditions, should be ascertained and 
 tabulated for reference. 
 
 Prof* Willis, in his "Principles of Mechanism,"* gives a diagram 
 and deduces from it an expression, confessedly "so involved as to 
 make the direct solution of the equation impossible, although approx- 
 imations may be obtained." He adds : " However, on account of the 
 practical importance of the question, I have arranged in the following 
 Tables the exact required results, which I derived organically from 
 the diagram by constructing it on a large scale with movable rulers." 
 
 This rather obscure expression evidently indicates the use of an ad- 
 justable mechanical device of some kind ; but since this, whatever it 
 was, must have been capable of being drawn in any phase of its move- 
 ments, the inference is a fair one that his process was equivalent to 
 that of deducing results graphically by trial and error. 
 
 238. Prof. Willis may perhaps have overestimated the practical im- 
 portance of the question, since limiting cases are in general to be 
 avoided when possible. Still, the discussion has developed some 
 hitherto overlooked points of considerable abstract interest; and 
 again, the practical value, whatever it may be, of such tables, depends 
 
 *2d. Ed. 1870, pp. 106,107. 
 
COMPUTATION OF TABLES. 129 
 
 entirely upon their correctness. And by the application of Prof. 
 Rankine's singularly elegant processes relating to circular ares, as ex- 
 plained in the preceding paragraphs, we have been enabled to deter- 
 mine these limiting numbers accurately, we believe for the first time. 
 For the numbers in the tables above mentioned are those of teeth in 
 complete circular wheels, capable of transmitting -rotation continuously 
 in the same direction ; and though the processes of Prof. Rankine are 
 only approximations, they are such close ones as to preclude the pos- 
 sibility of any error in the computations arising from this fact, of 
 such magnitude as to affect the integers in the final results. 
 
 Prof. Willis calls particular attention to the statement that his 
 tables are "geometrically exact." * It has already been demonstrated 
 (224, 225) that in at least one instance this is very far from being the 
 case. And the results obtained by the above methods differ from his 
 in so many other instances, and so widely, that in view of his high 
 standing as an authority, we feel called upon, before presenting our 
 own Tables of Limiting Numbers, to illustrate in detail the manner 
 in which the values there assigned were computed. 
 
 239. As the first example we select the following case. 
 
 p . j Driving Pinion of 5 leaves, Radial Flanks. 
 '' \ Arc of Recess = Pitch. Tooth = Space. 
 
 The numbers of teeth being directly proportional to the radii, let 
 rad. pinion = 5 ; constructing the diagram, we find the point to 
 lie in relation to AG as in Fig. 138, showing that the pinion cannot 
 under these conditions drive a wheel in outside gear, and in that figure 
 we have 
 
 ACF = 72, arc AF = } x 5 x 2 x 3.1416 = 6.2832 
 ACH= 54, AG = arc AF = 6.2832 
 
 AHC= 36, AM = i AG = 1.5708 
 
 AC = 5, OM = | AG = 4.7124 
 
 Rad. 10. 
 
 tan ACff= 54 10.138739 
 
 A C = 5. 0.698970 
 
 All = 6.8819 0.837709 
 AM 1.5708 
 HM = 5.3111 
 
 * Principles of Mechanism, p. 110. 
 
130 COMPUTATION OF TABLES. 
 
 (OM = 4.7124 ar. com. 9.326758 
 
 : HM= 5.3111 0.725183 
 
 :: sin OHM = 36 9.769219 
 
 : sin HOM = 138 30' 44" 9.821160 
 OMA = 174 30' 44" 
 
 180 
 
 0AM + AOM = 5 29' 16"; -f- 2 = 2 44' 38" 
 
 OM + AM = 6.2832 ar. com. 9.201819 
 
 Triangle j : QM - AM = 3.1416 0.497151 
 
 :: tan. (A + 0) = 2 44' 38" 8.680580 
 
 : tan $ (A 0) = 1 22' 22" 8.379550 
 
 0AM = ~1' = Max. Obliq. = AEN. 
 
 ( siuOAM= 4 7' ar.com. 1.143951 
 
 Triangle I : s i n QMA = 174 30' 44" 8.980609 
 
 OM= 4.7124 0.673242 
 
 OA 0.797802 
 
 2, 0.301030 
 
 = \ OA 0.496772 
 
 sin AEN '= 4 7' ar. com. 1.143951 
 
 Triangle I : AN 0.496772 
 
 AEN. I :: Rad. 10. 
 
 : AE 43.7343 = Rad. Interm. Des. Circle. 1.640723 
 
 AC = 5. (85 = Max. No. given by Prof. Willis.) 
 
 48.7343 .-. 48- Maximum No. Teeth. 
 
 240. We will next take a case in outside gear, thus : 
 
 Given -S Follower of 10 Teeth. Eadial Flanks. 
 
 ' 1 Arc of Recess = Pitch. Tooth = Space. 
 
 Then in Fig. 143 we shall have 
 
 AG = | x 10 x T V x 2 x 3.1416 = 9.4248. 
 
 180 C 
 
 0AM = 36 ( 
 
 AOM + AMO = Iff 
 i (M + 0) = 72< 
 
 AM = AG = 2.3562 
 
 PM = IAG = 7.0686 
 AC = 10 = Rad. Follower. 
 
 ACM = OAM = 3Q= Max. Obliquity. 
 
COMPUTATION OF TABLES. 
 
 131 
 
 Triangle 
 CAM. 
 
 Triangle 
 0AM. 
 
 Bad. 
 
 
 sin 0AM = 
 
 36 
 
 AC = 10 
 
 
 OA = OP 
 
 = 5.8778 
 
 AM 
 
 = 2.3562 
 
 OA + AM = 8.2340 
 : OA - AM = 3.5216 
 
 :: tan J (M + 0) = 72 
 
 : tan J- (M - 0) = 52 46' 32" 
 AOM = 19 13' 28" 
 AOM= 9 36' 44" 
 
 10. 
 
 9.769219 
 JL 
 
 0.769219 
 
 ar. com. 9.084389 
 
 0.546740 
 
 10.488224 
 
 10.119353 
 
 Triangle 
 0AM. 
 
 sin AOM = 19 13' 28" 
 sin OA M = 36 
 AM = 2.3562 
 OM = 4.2061 
 OP = 5.8778 
 PM^ 7.0686 
 
 2) 17.1525 = OM + OP + PJf. 
 8.5763 = 8. 
 1.5077 = 8 - PM. 
 
 ar. com. 0.482449 
 
 Triangle 
 POM. 
 
 cos 
 
 , /fiad 2 . S 
 
 . (S PM} 
 
 ~Y OP 
 
 x 0Jf 
 
 'Bad. 
 
 9 
 
 
 8 
 
 = 8.5763 
 
 
 s 
 
 PJf = 1.5077 
 
 
 OP 
 
 = 5.8778 
 
 0.769219) 
 
 OM 
 
 = 4.2061 
 
 0. 623880 f 
 
 . cos -J 
 
 P03f = 43 40' 51" 
 
 
 J 
 
 AOM 9 36' 44" 
 
 
 AON= 53 17' 35" 
 
 OAD = 90 + 0AM = 126 
 
 9.769219 
 0.372212 
 
 0.623880 
 
 20. 
 
 0.933300 
 
 0.178315 
 21.111615 
 
 1.393099 
 2) 19.718516 
 
 9. 859258 
 
 ADN = 180 - (179 17' 35") = 42' 25" 
 
 f sin ADN= 42' 25" ar. com. 1.908777 
 
 Triangle j : s i n AON = 53 17' 35" 9.904014 
 
 ADN. ::OA= 5.8778 0.769219 
 
 [ : AD= 381.9531 2.582010 
 
 . . 382 = Minimum No. for Driver. 
 
132 
 
 COMPUTATION OF TABLES, 
 
 241. Prof. Willis correctly observes that when the action begins at 
 the line of centres, no pinion of less than ten leaves can be driven, but 
 in reference to the above case he says (p. 210) "nothing less than a 
 rack can drive a pinion of ten ; " whereas it appears from these figures 
 that it can be driven by any wheel of 382 teeth or more. In corrob- 
 oration of which we will now assume a driver of 382 teeth, the arc of 
 recess being equal to the pitch, and the tooth to the space, as before. 
 
 We shall then have in Fig. 133, 
 
 ACF = 56' 32.67" 
 ACH = 42' 24.5" 
 ARC = 89 17' 35.5" 
 OHM = 90 42' 24.5" 
 AC= 382. 
 
 arc AF = ^ x 382 x 2 x 3.1416 = 6.2832 
 AG = arcAF= 6.2832 
 AM = J AG = 1.5708 
 OM = f AG = 4.7124 
 
 Rad. 
 
 Triangle I : t&n ACff = 42' 24.5' 
 ACH. I :: AC = 382 
 
 : AH = 4.7123 
 AM = 1.5708 
 HM= 3.1415 
 
 10. 
 
 8.091169 
 2.582063 
 0.673232 
 
 Triangle 
 OHM. 
 
 I 
 
 OM- 4.7124 
 HM= 3.1415 
 sin OHM r-. 90 42' 24.5" 
 smHOM = 41 48' 17.5" 
 OMA = 132 30' 42" 
 180 
 
 ar. com. 9.326758 
 0.497137 
 9.999967 
 9.823862 
 
 0AM + AOM = 47 29' 18" ; + 2 = 23 44' 39" 
 
 OM + AM = 6.2832 
 Triangle I : QM - AM = 3.1416 
 OMA. 1 :: tan $ (A + 0) = 23 44' 39" 
 : tan | ( A - 0} = 12 24' 5.8 
 0AM = 
 
 ar. com. 9.201819 
 0.497151 
 9.643243 
 
 9.342213 
 
 36 8' 45" = Max.Obliq. = AEN. 
 
 sin 0AM = 36 8' 45" 
 Triangle I : s in OMA 132 30' 42 
 OMA. I ''- OM =4.7124 
 : OA 
 2 
 AN = OA 
 
 ar. com. 
 
 0.229236 
 9.867596 
 0.673242 
 0.770074 
 0.301030 
 0.469044 
 
COMPUTATION OF TABLES. 133 
 
 sin AEN = 36 8' 45" ar. com. 0.229236 
 
 Triangle I : AN 0.469044 
 
 AEN. 1 "Rad. 10. 
 
 : AE = 4.9921 Had. Ext. Des. Circle. 0.698280 
 
 2 (Flanks to be Radial). 
 9.9842 . '. 10 = Minimum JSTo. for Follower. 
 
 242. In the case of a driving pinion of 6 leaves, under the same 
 conditions, we find, by a similar computation, 
 
 AE = 71.7083 = Kad. Exterior Des. Circle. 
 Add AC = 6. Ead. Driver, 
 
 then 77.7083 = Rad. Annular Wheel, 
 
 also 2 AE = 143.4166 = Rad. Follower in Outside Gear. 
 
 These being minimum values, the next higher integers must be 
 taken as the limiting numbers of teeth for complete wheels. Thus 
 the 6-leaved pinion can drive an annular wheel of 78 teeth, or an ex- 
 ternally toothed one of 144, but no less numbers can be used when the 
 describing circle thus found is employed. 
 
 Prof. Willis assigns 176 instead of 144, and, as previously stated, he 
 gives a rack, instead of 382 teeth, as the least that can drive a pinion 
 of 10. In the table for outside gear we have merely marked with an 
 asterisk those of our own values which differ from his, because the 
 above-mentioned discrepancies are the most serious ones. But his 
 method, whatever it was, appears to have been singularly defective in 
 its application to inside gear ; he not only ignores entirely the mini- 
 mum values, but assigns maximum values which differ so often and 
 so widely from those computed as above, that we have deemed it 
 proper to present his tables for annular wheels side by side with our 
 own, which we think it safe to assert are numerically exact. Had the 
 object been to determine the limiting radii with the utmost precision, 
 a correction for the error in Prof. Rankine's approximations would have 
 been necessary in every instance. This, however, we have applied 
 only when the arc to be dealt with was so large as to make it requisite, 
 in order to secure in the computation of converse cases, as in (240) 
 and (241), results concordant in respect to the number of teeth for 
 complete wheels. 
 
134 
 
 OUTSIDE G EASING. 
 
 Limiting Numbers of Teeth. 
 
 BECESS = PITCH. 
 
 RECESS J PITCH. 
 
 RECESS = f PITCH. 
 
 D. 
 
 F. 
 
 D. 
 
 F. 
 
 D. 
 
 F. 
 
 5.58 
 
 oo 
 
 3.25 
 
 00 
 
 2.58 
 
 00 
 
 6 
 
 144* 
 
 4 
 
 34* 
 
 3 
 
 37* 
 
 7 
 
 50* 
 
 5 
 
 19 
 
 4 
 
 15 
 
 8 
 
 34* 
 
 6 
 
 14 
 
 5* 
 
 11* 
 
 9 
 
 27 
 
 7 
 
 12 
 
 6 
 
 10 
 
 10 
 
 23 
 
 8 
 
 11* 
 
 7 
 
 9 
 
 11 
 
 21 
 
 9* 
 
 10 
 
 8 
 
 8 
 
 12 
 
 19 
 
 11* 
 
 9* 
 
 11 
 
 7 
 
 13 
 
 18 
 
 16 
 
 8 
 
 21* 
 
 6 
 
 14 
 
 17 
 
 33* 
 
 7 
 
 oo 
 
 5.17 
 
 15 
 
 16 
 
 00 
 
 6,35 
 
 
 
 17 
 
 15 
 
 
 20 
 
 14 
 
 
 24 
 
 13 
 
 
 32* 
 
 12 
 
 OUTSIDE GEARING. 
 
 EPICYCLOIDAL TEETH. RADIAL FLANKS. 
 
 57* 
 
 11 
 
 Tooth = Space. 
 
 382* 
 
 10 
 
 Minimum Values. 
 
 00 
 
 9.83 
 
 
 * Numbers differing from those given by Prof. Willis. 
 
INSIDE EPICYCLOIDAL GEARING. 
 
 Limiting Numbers of Teeth. 
 
 135 
 
 INSIDE EPICYCLOIDAL GEARING. 
 
 FLANKS OF PINIONS RADIAL. TOOTH = SPACE. 
 
 RECESS = PITCH. 
 
 EECESS = J PITCH. 
 
 RECESS = | PITCH. 
 
 D. 
 
 F. 
 
 D. 
 
 F. 
 
 D. 
 
 F. 
 
 3 
 
 9 
 
 2 
 
 7 
 
 2 
 
 11 
 
 4 
 
 16 
 
 3 
 
 41 
 
 3 
 
 CO 
 
 5 
 
 48 
 
 4 
 
 oo 
 
 
 
 6 
 
 00 
 
 Maxim um Followers. 
 
 F. 
 
 D. 
 
 F. 
 
 D. 
 
 F. 
 
 D. 
 
 7 
 
 14 
 
 5 
 
 12 
 
 4 
 
 8 
 
 8 
 
 24 
 
 6 
 
 65 
 
 5 
 
 53 
 
 9 
 
 60 
 
 7 
 
 00 
 
 6 
 
 00 
 
 10 
 
 00 
 
 Maximum Drivers. 
 
 D. 
 
 F. 
 
 D. 
 
 F. 
 
 D. 
 
 F. 
 
 6 
 
 78 
 
 4 
 
 21 
 
 3 
 
 22 
 
 7 
 
 32 
 
 5 
 
 15 
 
 4 
 
 12 
 
 8 
 
 25 
 
 6 
 
 13 
 
 5 
 
 11 
 
 9 
 
 23 
 
 7 
 
 13 
 
 6 
 
 11 
 
 10 
 
 22 
 
 8 
 
 14 
 
 7 
 
 12 
 
 11 
 
 22 
 
 9 
 
 14 
 
 8 
 
 12 
 
 12 
 
 22 
 
 10 
 
 15 
 
 9 
 
 13 
 
 13 
 
 22 
 
 11 
 
 16 
 
 10 
 
 14 
 
 14 
 
 23 
 
 12 
 
 17 
 
 11 
 
 15 
 
 15 
 16 
 
 23 
 24 
 
 Minimum Followers. 
 
136 
 
 INSIDE EPICYCLOIDAL GEARING. 
 
 Limiting Numbers. (Prof. Willis.) 
 
 INSIDE EPICYCLOIDAL GEARING. 
 
 FLANKS OF PINIONS BADIAL. TOOTH = SPACE. 
 
 RECESS = PITCH. 
 
 RECESS = J PITCH. 
 
 RECESS = f PITCH. 
 
 D. 
 
 F. 
 
 D. 
 
 F. 
 
 D. 
 
 F. 
 
 2 
 
 5 
 
 2 
 
 10 
 
 2 
 
 14 
 
 3 
 
 12 
 
 3 
 
 77 
 
 4 
 
 Any No. 
 
 
 
 
 
 
 
 4 
 
 26 
 
 4 
 
 Any No. 
 
 
 
 5 
 
 7 
 
 85 
 Any No. 
 
 Maximum Followers. 
 
 F, 
 
 D. 
 
 F. 
 
 D. 
 
 F. 
 
 D. 
 
 7 
 
 14 
 
 4 
 
 5 
 
 4 
 
 8 
 
 8 
 
 25 
 
 5 
 
 12 
 
 5 
 
 64 
 
 9 
 
 60 
 
 6 
 
 77 
 
 
 
 10 
 
 Rack. 
 
 
 
 
 Maximum Drivers. 
 
 243. The manner of using these tables is best illustrated by an ex- 
 ample or two. In outside gearing, let the given wheel be a driver of 
 12 teeth, the arc of recess to equal the pitch. In this division are 
 two columns, marked D and F, for drivers and followers respectively : 
 and opposite 12 in column D, we find in column F the number 19, 
 which is the least that can be driven. If the same number be assigned 
 for a follower, we find in column D, opposite 12 in column F, the 
 least number, 32, that will drive. 
 
 If the given number be not found in the table, the number for the 
 required wheel will be found opposite the next less number. For in- 
 stance, the arc of recess being three-fourths of the pitch, let the given 
 wheel have 28 teeth. By reference to the table, it is seen that 16 
 
EXPLANATION OF TABLES. 137 
 
 (or over) will drive 8, but 33 is the least that will drive 7 ; hence the 
 given wheel will drive 8, but no less. The given number is not found 
 in column F either ; but 19 can be driven by 5, or any number greater 
 than five, while a pinion of 4 can drive no less than 34 : consequently 
 5 is the least that can be usecj to drive the wheel of 28. Thus this 
 table includes all possible limiting numbers for the three values of the 
 arc of recess, in outside gearing. 
 
 In the first division of the table for inside gearing, marked Maxi- 
 mum Followers, the columns F contain the greatest numbers of teeth 
 for the annular wheels which can be driven by the corresponding 
 numbers in the columns D ; for instance, when recess = pitch, a pinion 
 of 5 can drive a wheel of 48, but not more. Since the least number 
 that can be driven is alwa"ys one more than that on the pinion, it is 
 not given in the table ; it is necessary only to remark that the lowest 
 numbers which can be thus used are 3 for the pinion and 4 for the 
 wheel, as a two-leaved pinion will not drive on account of the exces- 
 sive obliquity when the wheel has but three teeth. In all the com- 
 binations in this division, the receding action can be secured only by 
 the use of an intermediate describing circle. 
 
 In the second division, the columns D contain the greatest numbers 
 for the annular wheels which can drive the corresponding numbers in 
 columns F\ thus when.recess = f pitch, a pinion of 6 can be driven 
 by a wheel of 65 teeth, but not by a larger one. Since the flanks of 
 the pinion are to be radial, the least number for the driver (235) will 
 be one and a half times that of the given pinion, whatever the arc of 
 recess, and therefore is not set clown in the table. 
 
 In the third division, the columns D contain the numbers for given 
 pinions whose teeth are pointed, the faces being, therefore, generated 
 in each case by the least possible describing circle. The columns F 
 contain the least numbers for the annular wheels which can be driven 
 by these pinions, whose faces act against flanks traced by the same 
 describing circles, which are exterior ones ; and when these least num- 
 bers are used, the annular wheels can have no faces. There being, 
 then, no approaching action, it would at first sight appear that the 
 action could only be continuously maintained when the arc of recess 
 is equal to the pitch. Bat the assigned amount of receding action in 
 these cases is secured by the use of the exterior describing circle only : 
 and the actual amount is greater, owing to the peculiarity of the ac- 
 tion explained in (206), there being in fact two points of driving con- 
 tact up to the limit assigned in the table, so that the rotation will be 
 properly maintained in every case. 
 
 244. It remains now to illustrate by an example the use of this 
 
138 EXPLANATION OF TABLES. 
 
 table in finding, as mentioned in (236), the limiting number of teeth 
 for a pinion which is to drive a given annular follower. 
 
 Suppose that the recess is to be equal to the pitch : then, recollect- 
 ing that, as just mentioned, a pinion of 3 can drive a wheel of 4, we 
 deduce at once, in regard to the least number, the following by in- 
 spection of the tablo : 
 
 No. ON GIVEN WHEEL. LEAST No. TOP. PINION. DESCRIBING CIRCLE. 
 
 4 - 9 inclusive 3 j 
 
 nils " :::::::::: J intermediate. 
 
 49-77 " 6 J 
 
 78-oo 6 Exterior. 
 
 The greatest number for the pinion, if an intermediate describing 
 circle be used, is always one less than the given number on the annu- 
 lar wheel. 
 
 If it bo stipulated that the assigned arc of recess shall be secured 
 by the use of an exterior describing circle, the limiting numbers may 
 also be found by simple inspection if within the range covered by the 
 table. But in order to include the superior limit for even moderately 
 large wheels, the table would require to be inconveniently extended. 
 For instance, let the given number be 42 ; this is found by computa- 
 tion to be the least that can be driven by a spur-wheel of 36 teeth, 
 which is therefore the largest possible driver. A pinion of 6 can 
 drive no less number than 78, but one of 7 can drive any number 
 above 32, and is, consequently, the least which can be used under the 
 assigned conditions. 
 
 245. If an interchangeable set of spur wheels be constructed by the 
 use of a constant describing circle, the limiting numbers for the an- 
 nular wheels which will gear with them may be readily found without 
 reference to a table. In the system formerly employed by Brown & 
 Sharpe, of Providence, K. I., the diameter of the describing circle was 
 equal to the radius of a pinion of twelve teeth. If then the teeth of 
 the annular wheels are to have both faces and flanks, the distance be- 
 tween centres of the interior and exterior describing circles may be 
 represented by twelve, since the radii are proportional to the numbers 
 of teeth ; and at the limit this distance is equal to that between the 
 centres of the pitch circles. In these circumstances, therefore, any 
 spur wheel of the set will gear with an annular wheel having twelve 
 more teeth than itself, but no less than that will answer. Either may 
 be used as driver or as follower indifferently, but the relative and 
 the actual amounts of approaching and receding action will depend 
 
THE NOMODOtfT. 139 
 
 not only upon which does drive, but also upon the actual numbers of 
 teeth used in any given case. Since, however, in the use of this sys- 
 tem the smallest pinion ever used was one of twelve teeth, the total 
 angle of action would clearly be always ample. 
 
 By cutting down the pinion to the pitch circle we may use a smaller 
 wheel (as a driver only), and by similarly treating the wheel we may 
 use a smaller pinion for the same purpose ; the least number for the 
 annular wheel being found in either case by adding six, instead of 
 twelve, to the number of teeth upon the given internal wheel. 
 
 In the system first adopted by Pratt & Whitney, of Hartford, Conn., 
 and subsequently by Brown & Sharpe, the diameter of the describ- 
 ing circle is equal to half that of the pinion of fifteen teeth. These 
 limiting numbers are, therefore, found by adding to the number of 
 teeth on the given spur-wheel, fifteen if both wheels are to have faces 
 as well as flanks, and eight, if either is to be cut down to the pitch 
 line ; in the latter case, the exact number to be added is seven and a 
 half, which giving a fractional result, the next higher integer must be 
 taken. 
 
 246. The Nomodont. If the radius of the given wheel be gradually 
 increased, other things remaining unchanged, it is obvious that the 
 radius of the greatest or the least wheel, as the case may be, which will 
 work with it either as a driver or a follower, will vary according to 
 some regular law. As in many other cases, this law can be best illus- 
 trated graphically, by a curve whose abscissas are proportional to the 
 given, and the ordinates to the required, radii of the wheels ; to which 
 we have given the descriptive name of the Nomodont, from the law 
 expressed by it in relation to the numbers of teeth. These limiting 
 numbers, as given in the tables, are integers, as they must be if com- 
 plete wheels are to be used ; this would not be the case were sectors 
 only to be employed, as they often are, and in constructing these 
 curves the fractional values as computed have been given to the 
 ordinates. 
 
 It is hardly necessary to point out that the regularity of the curves 
 affords a quite rigid test of the correctness of the results which they 
 embody. And in order to define them the more perfectly, especially 
 between the origin and the vertices, in the region of the asymptotical 
 ordinates, a great number of intermediate fractional radii were as- 
 sumed for the given wheels, the conditions for all the curves repre- 
 sented being, that the externally toothed wheels shall have radial 
 flanks, that the arc of recess shall be equal to the pitch, and the tooth 
 equal to the space. 
 
 247, In Fig. 146, the abscissas represent the radii of given exter- 
 
140 
 
 THE KOMODOKT. 
 
 nally toothed wheels which are to drive, the ordinates of the curves 
 
 A and B are proportional to the corres- 
 ponding minimum radii of the followers 
 in outside and inside gear respectively, 
 determined as in the case of the 6-leaved 
 pinion, in (243). 
 
 This, as appears from the table, is the 
 least complete pinion which can drive 
 at all in outside gear ; the exact limit- 
 ing radius of one which can just drive a 
 
 THE NOMODONT. 
 
 Epicycloidal System. Recess = Pitch. Tooth = Space. 
 Scale of Ordinatcs j Scale of Abscissas. 
 
 ia 20 28 
 
 RADII OF GIVEN DRIVERS. FIG. 146. 
 
 rack, is 5. 58, at which point the ordinate is therefore infinite, and asymp- 
 totic to the curve A. As the radius of the given driver increases, that 
 of the follower diminishes, at first very rapidly, then more and more 
 slowly, until the driver becomes a rack, when the value of the ordi- 
 nate is 9.83, which agrees with the tabular record that the least whole 
 number which can be driven is 10 ; consequently this curve has also 
 a horizontal asymptote at a distance of 9.83 above the axis of ab- 
 scissas. 
 
 Since the driven rack may be regarded as an infinitely large wheel 
 in inside gear, as well as in outside, the ordinate at 5.58 is also an 
 asymptote to the curves B and C ; the ordinates of the latter being 
 proportional to the maximum radii of annular followers, whose action 
 requires the use of an intermediate describing circle, determined as in 
 (239). 
 
 The ordinates of B also diminish rapidly at first, but reaching a 
 minimum when the radius of the driver is between 10 and 12, subse- 
 quently increase. The rate of increase, however, is not constant ; the 
 value of the ordinate is (239) AC + AE, in which, while AC in- 
 creases uniformly, AE diminishes more and more slowly, reaching the 
 limit of 4.92 when AC = <=*=>. If, then, any two ordinates be drawn, 
 each measuring, by the scale of ordinates, a distance equal to its abscissa 
 
THE NOMODCWT. 
 
 141 
 
 + 4.92, the right line passing through their extremities will be a 
 second asymptote to the curve B. 
 
 Both A and B are, it will be observed, remarkably symmetrical 
 curves, being in fact very nearly true hyperbolas. The ordinates of 
 (7, on the other hand, diminish very rapidly as the radius of the driver 
 is reduced, their values, as shown in (239) and (225), being 48.73 
 when that radius is 5, and exactly 9 when the radius is 3. As ap- 
 pears from the table, only three complete pinions, having respectively 
 3, 4, and 5 teeth, are included in the class here represented. Prof. 
 Willis gives a pinion of two leaves, as capable of driving a wheel of 
 five teeth ; but this will -be found wholly impracticable on account of 
 the excessive obliquity. 
 
 We have, therefore, not considered it worth while to compute an 
 ordinate for a less driving radius than 2.5, at which limit the value 
 is 6.76 ; but the curve is extended in a dotted line to the zero point, 
 on the assumption that driver and follower would vanish simul- 
 taneously. 
 
 248. In Fig. 147 the abscissas represent the radii of given externally 
 
 toothed followers. The ordi- 
 nates of the curve D being pro- 
 portional to the corresponding 
 minimum radii of drivers in 
 outside gear, it is at once appar- 
 ent that this curve will have a 
 vertical asymptote at 9.83, and 
 a horizontal one at a distance of 
 5.58 above the axis of abscissas. 
 
 THE NOMODONT. 
 Epicycloidal System. Recess=Pitch. Teeth=Space. 
 
 O.S3 2O 28 33 
 
 RADII OP GIVEN FOLLOWERS. FIG. 147. 
 
 The curves A and D are very similar, and will become in fact 
 identical if the axes be transposed and the same scale used for both 
 ordinates and abscissas ; for it will readily be seen, by comparing Fig. 
 133 with Figs. 142 and 143, that assuming the processes of construc- 
 tion and computation to be exact instead of approximative, the radii 
 in those converse cases must under similar conditions be precisely the 
 
142 THE NOMODOHT. 
 
 same. As here shown, however, the curve D has been constructed 
 with ordinates computed as in (240). 
 
 When the given radius is less than 9.83, the driver must be annu- 
 lar, and its maximum radius forms an ordinate to the curve E, which 
 is somewhat similar to (7, and like it includes the radii of but three 
 complete pinions ; which have respectively 7, 8 and 9 leaves. 
 
 The maximum radius of the driver for a pinion of six leaves, is 7.45, 
 which is impracticable, as we have seen that it must be at least one 
 and a half times as great as that of the follower. Therefore the com- 
 putations were carried no further, although this curve, like C, and 
 upon the same assumption, is continued in a dotted line to the zero 
 point. 
 

 UNIVERSITY 
 
 CHAPTER IX. 
 
 SPUE GEARING, CONTINUED INVOLUTE TEETH. 
 
 \r 
 
 Involute Generated by Rolling of Right Line on Base Circles. Peculiar Proper- 
 ties. Original Pitch Circle. Rack and Pinion. Annular Wheels. Low- 
 numbered Pinions. Involute Tooth with Epicycloidal Extension. Limiting 
 Numbers for Given Arcs of Recess. Comparison of the Involute and Epicy- 
 cloidal Systems. Involute Generated by Rolling of Logarithmic Spiral on 
 Pitch Circles. 
 
 Involute Teeth. 
 
 249. Next to the epicycloid, the curve most extensively used as the 
 tooth-outline in spur gearing, is the involute of the circle. 
 
 It will subsequently be shown 
 that this curve can be generated 
 in a manner which conforms to 
 the general law enunciated in 
 (181) ; but its fitness for this 
 purpose is proved much more 
 clearly and simply by deriving it 
 in another way. 
 
 In Fig. 148, let LM, RS, be 
 portions of two pitch circles in 
 contact at A ; C and D their cen- 
 tres, and TT their common tan- 
 gent. Draw NN inclined to TT, 
 and upon it let fall the perpen- 
 diculars CE, DF \ with which, as 
 radii, describe the circles EHY, 
 FGQ. Suppose these circles to 
 be disks upon which is wound an 
 inextensible thread EF, carrying 
 a marking point at F. Now let FlG - 148 - 
 
 the upper disk turn as shown by the arrow : it will wind the thread 
 upon itself, unwinding it from the lower one, which will thus be 
 
144 FIRST GENERATION OF INVOLUTE. 
 
 turned in the opposite direction, but with the same perimetral veloc- 
 ity. Let the arcs EH, FG, be equal to each other and to the right 
 line EF\ then while the pencil is drawn from Fto E,it will trace 
 upon the plane of the upper disk the curve KE, and upon the plane 
 of the lower one the curve QE. 
 
 250. These are involutes, not of the pitch circles, but of the base 
 circles EHY, FGQ, the ratio of whose radii CE, DF, by reason of 
 the similar triangles ACE, ADF, is the same as that of AC, AD, the 
 radii of the pitch circles. Being simultaneously generated during the 
 rotation by a point which lies always in the common tangent to the 
 base circles, that is to say, in the common normal to the involutes, 
 these curves will be tangent to each other throughout the generation : 
 and since this common normal always cuts CD at the same point A, 
 they may be used as outlines of teeth for the wheels to which they 
 respectively belong. Thus FI, similar to GE, will drive FH, which 
 is similar to KE, in the direction indicated by the arrow, with a con- 
 stant velocity ratio ; the locus of contact being the line FE. The 
 two curves are shown in the diagram in one intermediate position, so 
 that the action will be readily traced without farther explanation ; 
 and it is evident that this action is precisely equivalent to the rolling 
 together of the pitch circles LM, RS. 
 
 251. Now, since the involute does not extend within its base cir- 
 cle, the curves shown in Fig. 148 are of the greatest possible length ; 
 supposing the lower wheel to drive, the action begins at F and ends 
 at E. Both curves are continuous, and there is no division into face 
 and flank ; but it is evident that the points x and z will meet at A, 
 the involutes then occupying the positions OAU, PAW: also that 
 the ratio of the arc Az to the arc FO, is the same as that between the 
 arcs Ax, HP. 
 
 Seasoning, in like manner, as to the arcs of recess, and recollecting 
 that the arcs FO, HP, are each equal to FA, while OG, PE, are 
 each equal to AE, we have 
 
 Approach _ FA __ CE __ AC 
 Eecess ~ Ztf ~~ ~DF ~ AD ' 
 
 That is to say, if each involute be long enough to extend to the root 
 of the other, the arcs of approach and recess will be to each other in 
 the direct ratio of the radii of the base circles, or pitch circles, of the 
 driver and the follower respectively. 
 
 In these circumstances, the smaller of two wheels must drive in 
 order to secure more receding than approaching action. But as it is 
 not necessary to make the curves so long, we can vary the proportion 
 
APPROACH AND KECESS. 
 
 145 
 
 between the arcs of approach, and recess at pleasure, by properly regu- 
 lating the heights of the teeth, whatever the relative diameters of the 
 wheels. 
 
 252. Thus in Fig. 149, which represents teeth of practical propor- 
 tions, one pair is shown in contact at A, another pair as just quitting 
 
 FIG. 149. 
 
 contact at B, the lower wheel driving as indicated by the arrow. The 
 receding action, then, continues while the point of contact travels 
 from A to B, a distance equal to 01 measured on the base circle. 
 Meantime the linear motion of the driver, measured on the pitch cir- 
 cle, will be A G, which is greater than 01 in the proportion of the 
 radius of the pitch circle to that of the base circle ; but since those 
 two arcs measure the same angle, either is readily found if the other 
 be given. 
 
 Supposing, then, that the angle of recess is assigned ; we first ascer- 
 tain the length of the arc 01 on the base circle by which it is meas- 
 ured, and set off AB on the line of action, equal to it ; then the tops 
 of the driver's teeth are limited by a circle through B, about that 
 wheel's centre. 
 
 A radius through B cuts the pitch circle in H\ if GH be equal to 
 half the thickness of the tooth as determined by the pitch, the con- 
 struction is just possible, and the tooth will be pointed. If GH be 
 greater than that, the conditions are impracticable ; but if it be less, 
 the tooth, as in the figure, will be of sensible breadth at the top. 
 
 By a similar process the height of the follower's tooth for any as- 
 signed angle of approach is determined, and the pitch circles having 
 been subdivided as usual into equal parts, the construction is com- 
 pleted by drawing the reverse involutes for the backs of the teeth, 
 and providing suitable clearing spaces. These may be required to ex- 
 tend some distance within the base circles, and if so they may be 
 10 
 
146 ORIGINAL PITCH CIRCLE. 
 
 bounded as in the figure by radial lines tangent to the involutes at 
 their roots. 
 
 253. Peculiar Properties of Involute Teeth. In the operations above 
 described, the line of action FE was drawn at pleasure, and the ar- 
 gument in no wise depends upon its obliquity. Consequently, for a 
 given pair of pitch circles an infinite number of pairs of base circles 
 may be assigned, and the converse is evidently true, since the common 
 tangent of any two given base circles will always cut the line of cen- 
 tres, be the same greater or less, into segments having the same ratio 
 as their radii. From which follow two important practical deduc- 
 tions, viz.: 
 
 1. Any two wheels with involute teeth, of which the pitch arcs on 
 
 the base circles are equal, will gear correctly with each other. 
 
 2. The velocity ratio will not be affected by any change in the dis- 
 
 tance between their centres. 
 
 "We have seen that wheels with epicycloidal teeth may be made in- 
 terchangeable ; but this second peculiarity gives the involute tooth an 
 advantage over every other, of special importance in mechanism re- 
 quiring the greatest smoothness and uniformity of action. The veloc- 
 ity ratio will be correct, although the wheels be improperly located at 
 the outset, and will remain so in spite of wear in the bearings ; also, 
 the backlash may at any time be reduced to a minimum, by bringing the 
 axes as close together as they can be without causing the teeth to bind. 
 
 On the other hand, the obliquity is constant, and in general greater 
 than the mean obliquity for epicycloidal teeth having the same angle of 
 action. While this may be a serious objection to the involute form for 
 heavy work, it may also be a positive advantage in light mechanism where 
 smoothness of action is all-important. For the side pressure will al- 
 ways keep the axes at the greatest possible distance from each other, 
 which tends to prevent shaking if there be any looseness in the 
 bearings. 
 
 254. Original Pitch Circle. Of the infinite number of pitch circles 
 which may be assigned for any given wheel, there is evidently but one 
 upon which the tooth and space, as in Fig. 149, are measured by equal 
 arcs. This is appropriately called the original pitch circle, being the 
 one given or assumed in laying out a pair of wheels with teeth and 
 spaces equal as above described, and it determines what may be called 
 the proper obliquity for both wheels. 
 
 Suppose now that a wheel has been made, the teeth being involutes 
 of a given base circle, the pitch circle unknown ; and let it be re- 
 quired to construct another which shall gear with it in such wise that 
 the teeth and spaces on the pitch circles shall be equal as in the fig- 
 
RACK AND WHEEL. . 147 
 
 ure. In order to do this, it is necessary first to find the proper obliq- 
 uity of the given wheel. This may be done graphically as follows : 
 draw two radii, the first bisecting a tooth, the second bisecting an ad- 
 jacent space, of the given wheel. Then bisect the angle between these 
 lines by another radius ; this will cut the involute outline of the tooth 
 in a point through which the original pitch circle must pass. This 
 third radius being taken as the line of centres, the tangent to the base 
 circle, through the point last mentioned, will be the line of action 
 having the required obliquity. 
 
 255. Considerations Affecting the Obliquity. The obliquity is, as 
 above stated, abstractly arbitrary. But it will presently appear that 
 under some circumstances there are definite relations between the ob- 
 liquity, the velocity ratio and the arc of recess, such that if either 
 two be assigned, the other, in many cases, can vary only within a quite 
 narrow range. For practical guidance when not trammeled by the 
 considerations alluded to, it may be stated that experience has shown 
 that for ordinary purposes the obliquity should not exceed from 15 
 to 17. When it is no greater than this, it is safe to say that even for 
 heavy work these teeth are in every respect equal to the epicycloidal 
 or any others ; and in addition to the advantages previously men- 
 tioned, it will be noted that their form is essentially a strong one, 
 spreading out rapidly toward the base. 
 
 25G. Rack and Wheel with Involute Teeth. In Fig. 150, LM being 
 the pitch circle, and TT the pitch line of the rack, the base circle is 
 determined as in Fig. 148, by drawing 
 FE at an arbitrary angle through A, 
 the point of tangency, and letting fall 
 upon it the perpendicular CE. Then 
 while a marking point travels from E 
 to F, with a linear velocity equal to 
 that of the circumference EHY, it. 
 will trace upon the plane of the wheel 
 the involute FH. The rack meantime 
 moves to the left side with a linear ve- 
 locity equal to that of the pitch circle, 
 which is greater than that of the base circle in the ratio of A C to CE. 
 
 Consequently, when the pencil reaches F the point E of the rack 
 will have moved to /, and we shall have EF perpendicular to EC, El 
 perpendicular to A C, and also 
 
 iL 
 
 EF ~ : EC ' 
 
148 
 
 INSIDE GEARING. 
 
 M 
 
 pn ._..._.... 
 
 i 
 
 FIG. 151. 
 
 therefore IF, traced by the pencil upon plane of the rack, will be a 
 right line perpendicular to EF. 
 
 Drawing EK similar to HF, and EG similar to IF, and supposing 
 the pinion to drive, the action begins at E, ending at F; and as in 
 Fig. 149, the relative amounts of approaching and receding action 
 
 may be varied at pleasure by alter- 
 ing the heights of the teeth. In 
 this diagram, as in Fig. 148, prac- 
 tical proportions are disregarded 
 for the sake of perspicuity ; the 
 T general appearance of the combi- 
 nation under reasonable conditions 
 is shown in Fig. 151, the obliquity 
 being 20. 
 
 The less the obliquity, the shorter will be the acting face of the rack 
 for a given angle of action, and the less will be the amount of ap- 
 proaching action attainable ; when the obliquity is nil, the rack tooth 
 degenerates into a point, and the wheel-tooth becomes an involute of 
 the pitch circle, as illustrated in Fig. 121. The action is wholly re- 
 ceding, but the sliding is excessive and the wear confined to one point 
 on the rack, thus more than counterbalancing all advantages unless 
 the pressure is very moderate. 
 
 257, Annular Wheels with Involute Teeth. The construction, as 
 shown in Fig. 152, is substantially the same as in the case of outside 
 gearing. And this form of 
 
 tooth would appear to be ex- 
 tremely well adapted for inside 
 gearing requiring unusual 
 smoothness of action, to which, 
 as above suggested, the con- 
 stant obliquity is favorable, 
 while, owing to the small dif- 
 ference in the lengths of the 
 acting curves, there is very 
 little sliding. This diagram is 
 so similar to Fig. 150, that no 
 explanation is necessary, be- 
 yond calling attention to this FIG. 153. 
 fact, viz.: since the action must begin or end at E, the root of the 
 pinion's teeth, the greatest possible height of the wheel-tooth is de- 
 termined by drawing a circle about D through that point. 
 
 258, Low-numbered Pinions with Involute Teeth. If two wheels be 
 
LOW-NUMBERED PINIONS. 
 
 149 
 
 
 of equal size, the least number of teeth which can be given to each 
 depends upon the obliquity. 
 For if this be assigned, the 
 points F and E, Fig. 153, are 
 thereby determined ; and the 
 arcs FG, EH, on the base cir- 
 cles, are each equal to EF. 
 Taking these arcs, then, as the 
 measures of the pitch angles, 
 and making the teeth and 
 spaces on the pitch circles 
 equal, the combination, as rep- 
 resented in the figure, is just a 
 limiting case, and the number 
 of teeth the least possible. 
 
 If the wheels are to be com- 
 plete ones, a fractional tooth 
 being impossible, the arc FG 
 must be contained an exact 
 number of times in the cir- 
 cumference, or, if there be a 
 remainder, the next higher integer must be taken as the least number 
 of teeth. In the latter event the arrangement will be practically ser- 
 viceable, for though the angles of approach and recess will still be 
 equal, their sum will be greater than the pitch. Each of these angles 
 is in point of fact a little greater than the obliquity, but the excess, as 
 can easily be verified by calculation, is so small that the least number 
 of teeth for practical purposes is at once correctly ascertained, by divid- 
 ing the circumference by twice the given obliquity : and thus for equal 
 wheels we have 
 
 FIG. 153. 
 
 OBLIQUITY. 
 
 10 
 
 15 
 
 18 
 20 
 30 
 
 LEAST No. OF TEETH. 
 18 
 
 15 
 
 12 
 
 . 10 
 
 G. 
 
 259. Supposing the lower wheel in Fig. 153 to drive, then the 
 follower may be equal to it, but cannot be less. It may, however, be 
 made as much greater as we please ; a moment's study of the diagram 
 will show that this same wheel can drive even the rack shown in 
 dotted outlines. 
 
150 
 
 LOW-NUMBERED PINIONS. 
 
 A sufficient increase in the diameter of the follower will permit 
 the tooth GE of the driver to be lengthened to an extent limited by 
 its intersection at P with the reverse involute, forming the back of 
 the tooth. About D, describe a circle through P 9 cutting the line of 
 action in B ; draw J5/ parallel to EC, cutting the line of centres in /: 
 then if the driver's tooth be pointed, BI will be the least radius of the 
 follower which will secure the greatest possible amount of receding 
 action, viz. : an arc on the base circle equal in length to AB. 
 
 260. This leads directly to the consideration that the pitch of the 
 driver may be made greater than in the figure, without changing the 
 
 obliquity ; for by increasing the 
 angle FDP, the point of the 
 leading tooth may be made to 
 fall upon the line of action, the 
 tooth and space, as measured on 
 the pitch circle, still remaining 
 equal to each other. 
 
 This is illustrated in Fig. 
 154, FG being the pitch on the 
 base circle, and LM, the corre- 
 sponding arc on the pitch cir- 
 cle P, on the line of action, 
 being the point of the tooth, 
 the involute. PK must bisect 
 LM in N\ then, PD will bi- 
 sect MN in /, and HD, which 
 bisects FK, will also bisect 
 LN in H. The obliquity being 
 given, the angle FDL, which is 
 equal to MDG, may be calcu- 
 lated ; 
 
 then letting FDL = a, 
 FDH = x 
 
 FIG. IM. 
 
 "we shall have 
 
 i Pitch = LDH=x + a, 
 
 FDP = 3(x + a) - a = 3x + 2, 
 FDG = LDM = 4(a? + a). 
 
 But FP, the trigonometrical tangent of the arc FI, is equal to the 
 arc FG ; that is, 
 
 tan (3x + 2a) = arc (4z + 40). 
 
MAXIMUM PITCH FOR GIVEN OBLIQUITY. 151 
 
 This equation cannot, we believe, be solved by any process less 
 laborious than the tentative one of assuming a value for x, and if it 
 fail to satisfy the equation, increasing or diminishing it, as the case 
 may be, until it does. Nor does the situation afford any key to a 
 graphic construction, since the distance to be set off on the line of 
 action, and the angle subtended by an arc equal to it on the circum- 
 ference, are both unknown. 
 
 This, however, is of little consequence, as, if the obliquity be as- 
 signed, there can be no certainty that the pitch arc when found would 
 be an aliquot part of the circumference : and it is of much more prac- 
 tical interest to ascertain the result of assigning a specific number of 
 teeth, or in other words a definite pitch. 
 
 261. When this is done the case becomes more manageable. Consid- 
 ering first the graphic process : The radius FD of the base circle and 
 the pitch angle FDO being given, we have first simply to rectify the 
 arc FG and set off FP equal to it, in a direction perpendicular to 
 DF. Drawing the involutes PG and PK, the tooth is completed as 
 to its acting outlines ; now bisect the angle FDK by the radial line 
 DH, draw PD, and bisect HDP by DN: this latter line will cut the 
 involute PK in the point N, through which is drawn the original 
 pitch circle cutting FP in A, which locates DA the line of centres 
 and determines the proper obliquity. 
 
 The pinion thus formed is barely capable of driving, the whole path 
 of contact FP being just equal to FG, the pitch arc on the base cir- 
 cle : and the minimum radius of the follower will be determined by 
 drawing PC perpendicular to the line of action, cutting DA produced 
 in C the centre. 
 
 262. Otherwise, by computation : Knowing the radius FD and the 
 length of FP, the angle FDPmay be found, whence, the angle FDG 
 being given, the angles GDK, FDK, may also be found, and we shall 
 then have 
 
 FDL = J (GDK - FDK}. 
 
 Now, FA, the trigonometrical tangent of the angle ADF, is equal 
 to the arc of the base circle, which measures the angle ADL ; there- 
 fore, let 
 
 ADF = y, 
 
 FDL = a, 
 
 and we have 
 
 tan y = arc* (y + a). 
 
 This equation may be solved by the tentative process above de- 
 
152 
 
 GREATEST PATH OF COXTACT. 
 
 scribed ; and the angle ADF thus determined is equal to the proper 
 obliquity. Knowing this angle and the side FD, we can now find 
 FA, equal to the arc of approach on the base circle, which taken from 
 FP gives AP equal to the arc of recess, and the triangles ADF, A CP, 
 being similar, the radius PCis readily ascertained. 
 
 263. In Fig. 154 the pitch arc FG is 72, so that the diagram rep- 
 resents the construction of a five-leaved pinion just capable of driving. 
 By comparing this figure with the preceding one, it will be seen that 
 when the point of the tooth thus falls upon the line of action, we have 
 the maximum pitch for a given obliquity, or the minimum obliquity 
 for a given pitch. 
 
 In this case, we find by computation in the manner above explained, 
 the following values, viz. : 
 
 Obliquity 28 8' 
 
 Arc of Recess 4 pitch. 
 
 Least No. for Follower 6.74. 
 
 Practically, then, a pinion of five leaves is capable of working with 
 
 another of seven, but no less 
 without exceeding the obliquity 
 above given. By increasing the 
 obliquity, the number for the 
 follower may be reduced, but it 
 will subsequently be shown that 
 five is not only the least number 
 for equal wheels, but also the 
 least that can either drive or be 
 driven at all., 
 
 264. The greatest possible 
 path of contact is the distance 
 between the feet of the perpen- 
 diculars let fall from the centres 
 of the wheels upon the line of 
 action. It has been pointed out 
 that if the obliquity be assumed, 
 there is no certainty that this 
 distance will be an aliquot part 
 of the circumference of the base 
 circle. But if the numbers of 
 teeth be given for a proposed 
 FIG. 155. pair of wheels, it may be made 
 
 so, and the obliquity thus determined. 
 
INVOLUTE WITH EPICYCLOIDAL EXTENSION. 153 
 
 For example, in Fig. 155, let the numbers be six for the upper 
 wheel and five for the lower. About any centre C, with a radius 
 measuring 6 on any scale of equal parts, describe a circle ; perpendicu- 
 lar to any radius CE, set off EF equal to the arc EH which measures 
 the pitch ; perpendicular to EF, and on the side opposite EC, set off 
 FD measuring 5 parts on the same scale, and draw CD, cutting FE 
 in A. Then taking AC, AD, as the radii of the pitch circles, and 
 CE, DF, as the radii of the base circles, the path of contact, EF, 
 which is equal to one-sixth of the circumference of the upper base 
 circle, will measure one-fifth of that of the lower one. 
 
 And this construction fixes the minimum obliquity for the given 
 numbers of teeth ; it being evident that a decrease of this obliquity 
 will at once shorten the path of contact and enlarge the base circles. 
 
 265. Involute Tooth with Epicycloidal Extension. The path of con- 
 tact, as above determined, is exactly equal to the pitch ; now in prac- 
 tice the arc of action must always be greater than the pitch, and yet 
 it may be desirable to use involute teeth in circumstances like those 
 represented in Fig. 155. The involute does not continue within its 
 base circle, but the face HF of the blunted tooth may be extended as 
 shown in dotted lines, thq curve beyond F being an epicycloid gener- 
 ated by a circle whose diameter is AD, which will work correctly with 
 the radial flank bounding the clearing space in the lower wheel. A 
 similar construction is applicable, of course, to the other wheel, if its 
 tooth be originally so much blunted as to make it desirable ; and in 
 .this way a sufficient angle of action may be secured, and the involute 
 tooth employed in cases where it would otherwise be inadmissible. 
 
 Limiting Numbers of Teeth for Given Arcs of Recess. 
 
 266. In the employment of the involute system, if a definite pitch 
 and arc of recess be assigned for either wheel of an engaging pair, there 
 will be a limit to the number of teeth which can be given to the other, 
 as was shown to be the case with the epicycloidal tooth. 
 
 Practically, a knowledge of these limiting numbers is just as neces- 
 sary or convenient in using one form of tooth as it is in using another. 
 Next to the epicycloid, the involute is the curve most extensively used, 
 and in the mode of generation, as well as in the peculiarities of the 
 action, there is a marked contrast between the two, and for these rea- 
 sons a special interest attaches to a comparison of the limiting num- 
 bers for these forms under like conditions. We shall therefore pro- 
 ceed to investigate a method of determining these limits for the same 
 
154 
 
 LIMITING NUMBERS OF TEETH. 
 
 values of the arc of recess which were assumed in the tables for epi- 
 cycloidal teeth, viz. : 
 
 1. Kecess = Pitch. 2. Recess = |- Pitch. 3. Recess = f Pitch. 
 
 267. In Fig. 156, let C, D be the centres of two wheels, VW and 
 HY arcs of the pitch circles tangent at A, FQ the line of action, 
 CQ and DFihe radii of the base circles, and let the lower wheel drive 
 
 in the direction indicated 
 by the arrow. 
 
 The proportions are such 
 that the tooth of the driver 
 is pointed, and that its 
 point, Q, extends just to 
 the root of the other invo- 
 lute ; also, the breadth MN 
 of the tooth being equal to 
 the space NL, both meas- 
 ured on the pitch circle, 
 there is no backlash. 
 
 This, evidently, is just a 
 limiting case ; the pitch 
 ML and the arc of recess 
 A M being assigned, the in- 
 volutes bounding the tooth 
 GQK must pass through 
 the points M and N, what- 
 ever be the radius of the 
 base circle, and the obliq- 
 uity TAF 9 which causes 
 their intersection to fall on 
 the line of action, is clearly 
 the greatest possible, which 
 
 as clearly gives the least possible value for A C. 
 
 The diagram, then, represents the problem as already solved, and 
 
 it remains to deduce from it the means of effecting the solution in 
 
 any given case. 
 
 268. Draw QD, which cuts the lower base circle in ; then the 
 path of contact FQ is equal to the arc FG, and is also the trigonomet- 
 rical tangent of the arc FO ; that is to say, we have, as an equation 
 of condition, 
 
 tan FO = arc FG. 
 
 FIG. 156. 
 
METHOD OF COMPUTATION. 155 
 
 The line of centres cuts the lower base circle in B ; through L, A, 
 and N, draw LR, A I, NH, involutes of the lower base circle, and 
 through A draw also A J, an involute of the upper base circle. 
 
 Then the arc QJ is equal to QA, the trigonometrical tangent of 
 the angle QCA ; the arc FI is equal to FA, the trigonometrical tan- 
 gent of the angle FDA : and the angles QCA, FDA, are equal to 
 each other and to the obliquity TAF. 
 
 We shall now have 
 
 Fa = FR + 2RH ........................... (1) 
 
 FO = FG - OG 
 
 OG = 
 ( KH= 
 \MN=NL, 
 
 RG = 2I + RH 
 
 B , = 2BI, .' ........ KG = 2BI+ HG 
 
 FO = FR + 2RH - BI-~ 
 
 or FO = FR + - BI 
 
 a 
 
 Also FI = FR + #/= FB + BI 
 
 269. Now when the definite values mentioned in (266) are assigned 
 for the arc of recess, these equations will be modified as follows : 
 
 I. RECESS = PITCH. 
 In this case we have 
 
 RI = 0, ) 
 
 RH = IH, [ whence 
 
 FR = FI,] 
 
 FG = FI + 2IH. ............. 1. 
 
 ......... 
 
 FI= FB + BI ........ 3. 
 
 And for a pinion driving a rack, 
 
 FR = FI = 0, ) 
 
 FB = 0, [ whence 
 BI = 0, ) 
 
156 LIMITING NUMBERS OF TEETH. 
 FG = 2IH 1. 
 
 FO = ^j~- 2. 
 
 II. RECESS = f PITCH. 
 We now have 
 
 r> T HrL r> rr o T> T 
 
 Ml = g- , Or Kli = A lJL) 
 
 which gives 
 
 FG = FR + 47?7 ........ 1. ) vr _ VT , 
 
 FI = FR + RI ......... 3.f FG ~ 
 
 and 
 
 + Mr-BI.... 2.) .. 
 = FB +131.... 3. ) 
 
 FR+ RI 
 When the pinion drives a rack, 
 
 FR = 0, ) j FG = 477, 
 
 ) (FG = 4FI, 
 , J ' 1 FO = FB + 2^7. 
 
 III. RECESS = PITCH. 
 
 With this value 
 
 RI = %RH, or EH = ~. 
 And therefore 
 
 Also 
 
 FR + RI = FB + El. . . . 3. ) 
 And in the case of a pinion driving a rack, 
 
 + SRI. ... 1. ) TTT/^ _. 7:77- 
 
 FI =FR + RI 3. ' ^ " FI 
 
 ~. 
 
 270. How suppose the pitch and the arc of recess to be assigned for 
 the driver. The arcs IH and HO are then known ; and if we assume 
 
BACK DRIVING SPECIAL CONSTRUCTION. 
 
 157 
 
 a value for FI, we can find the corresponding values of BI and FB, 
 which combined with the others will give certain valuer for FO and 
 FG ; and these are finally to be tested to see whether they will satisfy 
 the equation of condition, 
 
 tan FO = arc FG. 
 
 Having at last found in this way the obliquity which under the 
 assigned conditions makes FQ equal in length to the arc FG 9 we 
 know also the length of FA \ whence A C, the minimum radius of 
 the follower, is readily ascertained. 
 
 If, on the other hand, the wheel whose pitch and arc of recess are 
 given be the follower ; we then know the values of QA and the arc 
 QJ, which are equal to each other, and can thence find the angle 
 ACQ, equal to ADF and to the obliquity. We then, knowing the 
 arcs FB, FI, assume a value for RI, and proceed as before to see 
 whether the equation of condition is satisfied. 
 
 271. When the given driver becomes a rack, the least radius for 
 the follower is most readily ascertained by a special construction, 
 which is given in Fig. 157. 
 
 We have here an exact 
 limiting case, the rack 
 tooth being pointed, and 
 the whole path of contact 
 just equal to the pitch arc 
 on the base circle ; thus, 
 the rack driving in the 
 direction of the arrow, a 
 marking point in going 
 from Q to F, generates the 
 right line $7 and the in- 
 volute QG, the latter cut- FIG. 157. 
 ting the pitch circle at M. The action, then, begins at Q, and ends 
 at F, the root of the involute FS 9 which latter curve cuts the pitch 
 circb in L. Since the tooth and space are to be equal, the side XK 
 of the pinion's tooth must bisect the arc ML at N \ and it must also 
 be tangent to the side FV of the rack-tooth, at P its intersection with 
 the other line of action PAZ, to which FVis perpendicular. 
 
 It will now be seen that while the marking point is moving from Q 
 to A, it generates the curve QM, the pinion rotating through the arc 
 MA ; in the reverse motion, a marking point, in going from P to A, 
 traces the curve PN, while the pinion turns through the arc NA. 
 
158 BACK DRIVING SPECIAL CONSTRUCTION. 
 
 But we have 
 
 MA + AN = MN = i ML, 
 whence QA + PA = i QF, 
 
 or PA == i QF - QA ; 
 
 also, QA = QF AF; 
 
 therefore, since QFis equal to the pitch arc, and AFto the arc of re- 
 cess (both measured on the base circle), PA is known when these arcs 
 are given. 
 
 We have, then, this simple graphic process : Draw any line FQ, and 
 let it represent the pitch ; set off upon it a distance FA, making 
 
 FA _ arc of recess . 
 ~FQ pitch arc 
 
 next determine PA as above, and construct the right-angled triangle 
 APF. Then the angle PAF is twice the obliquity, and A T bisecting 
 it is the pitch line of the rack. Draw A C perpendicular to A T, and 
 FC perpendicular to FQ ; these will intersect at C the centre of the 
 pinion, determining CF the radius of the base circle and CA that of 
 the pitch circle. 
 
 272. Now introducing the same definite values as before for the arc 
 of recess, we shall have, 
 
 I. EEC ESS = PITCH. 
 
 II. RECESS = J PITCH. 
 
 III. RECESS = f PITCH. 
 
 In these oases, then, no tentative proceeding is involved in deter- 
 mining the least number of teeth for the pinion, which requires merely 
 the solution of the triangles APF, ACF. 
 
 273. In Fig. 157 the pinion's tooth has a sensible breadth, QX, at 
 the top, and the involutes might therefore be continued as shown in 
 
RACK DRIVING LIMITING CA 
 
 dotted lines, and the approaching action thereby prolonged. 
 
 smaller the assigned arc of recess, the less is AP, and the narrower 
 
 will be the tooth at the top, when as in the figure the total arc of ac- 
 
 
 m \ 
 
 tion is made equal to the ^ , - ; 
 pitch: there will, then, be / w 
 some value of the arc of re- y 
 cess which will cause the 
 points Q and Xto coincide. 
 
 And it is apparent that 
 when the limit is reached, as 
 in Fig. 158, the pinion will 
 have the least possible num-. 
 ber of teeth which can be 
 driven by a rack. 
 
 Now if in this diagram the 
 arc FI be given, we know 
 FA, the trigonometrical tangent of the angle ACF, which is always 
 one half the angle PAF: whence AP maybe found. We have 
 also 
 
 FIG. iss. 
 
 and 
 
 AQ = AF - ZAP, 
 
 AQ + AF= FQ = 
 
 (A) 
 
 Again, in the equations deduced in (268), viz. : 
 
 FG = FR + 2 RH, 
 
 FO = FR + 3 ^ H - BI, 
 
 2 
 
 we have 
 whence 
 
 and 
 or 
 
 FG = 2 RH, 
 
 FO = *J* - BI, 
 
 FO = FG - BI. 
 
 (B) 
 
 274. From these data we may now solve the problem in the tenta- 
 tive manner before described. 
 
 Taking any convenient value for the radius FC, we first assume the 
 arc FI, and find the corresponding values of FA, of the angle A CF, 
 and of the arc BI. Then solving the triangle APF, we determine 
 AP, whence we find FQ, as in Eq. (A). 
 
 The length of FQ being equal to that of the arc FG, we next find 
 the length of the arc FO as in Eq. (B), and knowing the radius, as- 
 certain the value of this arc in degrees. 
 
160 LOW-NUMBERED PINIONS. 
 
 Finally, we apply the test of determining the length of the trigono- 
 metrical tangent of this arc, and comparing it with that of FQ as be- 
 fore found. If these two values are equal, the assumed value of FI 
 is correct ; but if they are unequal a new value must be assumed, and 
 the process repeated until concordant results are obtained. 
 
 In this case we find that the pitch, at the limit which gives a 
 pointed tooth to the pinion, corresponds to 
 
 No. of Teeth = 4.0906 
 
 Obliquity = 41 20' 45" ) , 
 
 Arc of Kecess = -ff Pitch, f Very ' 
 
 275. Now referring to Fig. 153, it will be seen that in constructing 
 a pair of equal wheels, making the pitch arc on the base circle equal 
 to the greatest possible path of contact, the teeth will become nar- 
 rower at the top as the obliquity is increased ; evidently, then, there 
 is a limiting value of the obliquity which will make the teeth of both 
 wheels pointed, as shown in Fig. 159. 
 
 In this case we have [Eq. (B). (273)], 
 
 FO = 1 FQ - BI-, 
 
 and also FQ = 2 FI, . . FQ = 2 FA. 
 
 Whence FO = f FI - BI-, 
 
 but BI = FI - FB, .-. FO = J FI + FB. 
 
 It will now be seen that by assuming a value for FI, all the quan- 
 tities involved in the above expressions may be determined ; and the 
 results, as before, must satisfy the equation of condition, 
 
 tan FO = arc FG. 
 By this process we find 
 
 No. of Teeth = 4.6,256 
 
 Obliquity. = 34 10' 58.5" 
 
 Arc of Eecess (given) = | Pitch. 
 
 If complete wheels are to be used, then, five is the least number that 
 can drive or be driven, since, as shown in the preceding section, four 
 and a fraction are required even when the pinion gears with a pointed 
 rack. By adopting the construction shown in Fig. 155, the obliquity 
 
LIMITING NUMBERS OF TEETH. 
 
 161 
 
 just given may be slightly reduced, and we shall have for the smallest 
 possible pair of complete equal wheels, 
 
 No. of Teeth ........................... 5 
 
 Obliquity .............................. 32 8' 31" 
 
 Arc of Eecess ........... ............... Pitch. 
 
 276. The tentative processes above explained are undeniably tedious, 
 yet when systematically conducted they are less so than might be sup- 
 posed ; and in the preparation of the following tables they have been 
 continued until, with a radius 
 of 10, the linear values of the 
 arc FG and the path of contact 
 FQ agreed up to the sixth deci- 
 mal place, the approximation 
 being therefore to within the 
 one-millionth part of the radius 
 of the base circle. 
 
 It was not to be expected that 
 for a given number of teeth 
 upon either wheel the radius of 
 the other as thus determined 
 would correspond to an exact 
 whole number of teoth upon its 
 periphery. The next higher 
 integer being of course taken 
 as the tabular number, the 
 tooth would not then be abso- 
 lutely a pointed one ; but we 
 have not made any correction 
 for the slight change in the ob- 
 liquity due to this circumstance, considering its value as appearing in 
 the calculation, which is given to the nearest second, sufficiently pre- 
 cise for all practical purposes. 
 
 It is proper to point out that these tables may be of service, not only 
 as indicating exact limiting cases, which are to be avoided when pos- 
 sible, but upon another account, viz. : although in general terms the 
 obliquity is said to be arbitrary, yet in constructing for special pur- 
 poses wheels with a given arc of recess, it may be necessary to employ 
 numbers not far removed from limiting values, and in many instances, 
 it will be seen, the possible variation in the obliquity is confined to 
 quite a narrow range. 
 11 
 
 FIG. 159. 
 
162 
 
 INVOLUTE AND EPICYCLOID COMPARED. 
 
 Limiting Numbers of Teeth. 
 
 EECESS = PITCH. TOOTH = SPACE. 
 
 INVOLUTE. 
 
 EPICYCLOIDAL. 
 
 CONST. OBLIQUITY. 
 
 D. 
 
 F. 
 
 D. 
 
 F. 
 
 MAX. OBLIQUITY. 
 
 0' 
 
 5.58 
 
 CO 
 
 5.58 
 
 00 
 
 
 
 2 28' 42" 
 
 6 
 
 146 
 
 6 
 
 144 
 
 2 30' 
 
 7 0' 47" 
 
 7 
 
 52 
 
 7 
 
 50 
 
 7 12' 
 
 10 17' 4" 
 
 8 
 
 35 
 
 8 
 
 34 
 
 10 35' 18" 
 
 13 45' 5" 
 
 9 
 
 28 
 
 9 
 
 27 
 
 13 20' 
 
 14 40' 33" 
 
 10 
 
 24 
 
 10 
 
 23 
 
 15 89' 8" 
 
 15 54' 56" 
 
 11 
 
 22 
 
 11 
 
 21 
 
 17 8' 34" 
 
 17 26' 26" 
 
 12 
 
 20 
 
 12 
 
 19 
 
 18 56' 50" 
 
 18 17' 57" 
 
 13 
 
 19 
 
 13 
 
 18 
 
 20 
 
 19 2' 37" 
 
 14 
 
 18 
 
 14 
 
 17 
 
 21 10' 35" 
 
 20 17' 4" 
 
 15 
 
 17 
 
 15 
 
 16 
 
 22 80' 
 
 21 26' 24" 
 
 18 
 
 16 
 
 17 
 
 15 
 
 24 
 
 22 43' 40" 
 
 20 
 
 15 
 
 20 
 
 14 
 
 25 42' 51" 
 
 24 10' 12" 
 
 25 
 
 14 
 
 24 
 
 13 
 
 27 41' 32" 
 
 25 47' 45" 
 
 34 
 
 13 
 
 32 
 
 12 
 
 30 
 
 27 38' 12" 
 
 59 
 
 12 
 
 57 
 
 11 
 
 32 43' 38" 
 
 29 44' 6" 
 
 515 
 
 11 
 
 382 
 
 10 
 
 36 
 
 30 
 
 00 
 
 10.88 
 
 00 
 
 9.83 
 
 36 37' 21' 
 
INVOLUTE AKD EPICYCLOID COMPAEED. 
 
 Limiting Numbers of Teeth. 
 
 163 
 
 EECESS = J PITCH. TOOTH = SPACE. 
 
 INVOLUTE. 
 
 EPICYCLOID AL. 
 
 CONST. OBLIQUITY. 
 
 D. 
 
 F. 
 
 D. 
 
 F. 
 
 MAX. OBLIQUITY. 
 
 
 
 
 3.25 
 
 CO 
 
 o o' <r 
 
 16 240" 
 
 546 
 
 CO 
 
 4 
 
 34 
 
 7. 56' 28" 
 
 17 59' 6" 
 
 6 
 
 15 
 
 5 
 
 19 
 
 14 54' 4" 
 
 20 43' 34" 
 
 7 
 
 13 
 
 6 
 
 14 
 
 19 17' 8" 
 
 22 43' 27" 
 
 8 
 
 12 
 
 7 
 
 12 
 
 22 30' 
 
 24 14' 35" 
 
 9 
 
 11 
 
 8 
 
 11 
 
 24 32' 44" 
 
 25 13' 56" 
 
 10 
 
 10 
 
 9 
 
 10 
 
 27 
 
 27 38' 12" 
 
 13 
 
 9 
 
 11 
 
 9 
 
 30 
 
 30 29' 58" 
 
 20 
 
 8 
 
 16 
 
 8 
 
 33 45' 
 
 33 56' 54" 
 
 70 
 
 7 
 
 33 
 
 7 
 
 38 34' 16" 
 
 35 46 57 " 
 
 CO 
 
 6.54 
 
 oo 
 
 6.35 
 
 42 31' 11" 
 
 BECESS f PITCH. TOOTH = SPACE. 
 
 INVOLUTE. 
 
 EPICYCLOIDAL. 
 
 CONST. OBLIQUITY. 
 
 k 
 
 D. 
 
 F. 
 
 D. 
 
 F. 
 
 MAX. OBLIQUITY. 
 
 
 
 
 2,58 
 
 00 
 
 0' 0" 
 
 
 
 
 3 
 
 37 
 
 6 29' 11" 
 
 
 
 
 4 
 
 15 
 
 16 
 
 21 32' 59 " 
 
 5.30 
 
 CO 
 
 5 
 
 11 
 
 21 49' 6" 
 
 23 38' 20" 
 
 6 
 
 10 
 
 6 
 
 10 
 
 24 
 
 24 57' 30" 
 
 7 
 
 9 
 
 7 
 
 9 
 
 26 40' 
 
 27 38' 12" 
 
 9 
 
 8 
 
 8 
 
 8 
 
 30 
 
 30 53' 47" 
 
 12 
 
 7 
 
 11 
 
 7 
 
 34 17' 9" 
 
 34 55' 12" 
 
 28 
 
 6 
 
 21 
 
 6 
 
 .40 
 
 37 58 55" 
 
 oo 
 
 5,36 
 
 00 
 
 5.17 
 
 46 25' 18" 
 
164 
 
 INVOLUTE AtfD EPICYCLOID COMPARED. 
 
 277. But the labor of computing these tables was undertaken 
 mainly with the object of making a fuller comparison than has before 
 been made between the two leading systems of spur gearing : and to 
 facilitate this, the limiting numbers and th3 maximum obliquities for 
 epicycloidal, are here given side by side with those for involute teeth. 
 
 One striking point of difference will at once be noticed. In the 
 epicycloidal system, as the arc of recess diminishes, the least number 
 for the driver also diminishes, and the obliquity is always zero at the 
 limit where the follower becomes a rack. 
 
 In the involute system, on the other hand, this obliquity rapidly 
 increases, while the number for tho least driver remains practically 
 the same, varying but little even at the limit. 
 
 The reason is not far to seek. The generating circle for the driving 
 face of the epicycloidal tooth, which alone has to do with the reced- 
 ing action, is different from and wholly independent of that for the 
 flank, which produces the required amount of approaching action. 
 
 But the outline of the involute tooth is one continuous curve, and 
 the rectilinear generatrix, or line of action, is the same for the part 
 without the pitch circle and the part within. The sum of these two 
 segments must always be at least equal to the length of the pitch arc 
 on the base circle. If then the outer one, upon which the arc of re- 
 cess depends, be decreased, the inner one must be correspondingly 
 increased ; and this obviously involves a greater obliquity. 
 
 278. The difference between the two systems is best illustrated by 
 constructing, as in Fig. 1GO, the nomodont for each under like condi- 
 tions. The arc of 
 recess is here taken 
 at the pitch, and 
 the scale for the ab- 
 scissas is twice that 
 for the ordinates : 
 the distances of the 
 vertical and hori- 
 zontal asymptotes 
 from the axes being 
 respectively 5.30 
 and 5.36 for AA, 
 
 which represents the involute system, and 2.58 and 5.17 for BB, the 
 curve for the epicycloidal system. 
 
 The law of variation in the obliquities is also most clearly exhibited 
 by the same graphic means : thus, the ordinates of the curves CC and 
 DD arc proportional to the tabular values of the obliquity for invo- 
 
SECOND GENERATION OF INVOLUTE. 
 
 165 
 
 lute and epicycloidal teeth respectively, the abscissas, as before, rep- 
 resenting the numbers of teeth of given drivers. 
 
 These ordinates are drawn to the same scale as those of the nomo- 
 donts, the numerical value being in each case the one-hundredth part 
 of the obliquity taken to the nearest minute. Thus, the first ordinate 
 for C is the vertical asymptote to A, and its value is 
 
 As the number for the driver approaches infinity, the obliquity ap- 
 proaches the limit 37 59', as seen by reference to the table, and 
 
 37 59 ' - 22'79 
 
 ~ioo" 23 ' 79 ' 
 
 at which distance above the axis of abscissas this curve will have a 
 horizontal asymptote. 
 
 In like manner, we find that the curve DD has also a horizontal 
 asymptote, at a distance of 27.85 above the axis of abscissas. But 
 this curve, instead of beginning abruptly like the other, and inter- 
 secting the vertical asymptote to BE, is tangent to that line at its foot. 
 
 279. The Involute Generated by Rolling Contact with the Pitch Cir- 
 cle. Before dismissing the invo- 
 lute, it remains to prove that it 
 can be generated in such a man- 
 ner as to accord with the general 
 principle stated in (181). 
 
 In Fig. 161, let A The the com- 
 mon tangent of the pitch circles, 
 and let CQ, DP, the radii of the 
 base circles, be perpendicular to 
 the line of action PQ, drawn at 
 pleasure. Produce DP to inter- 
 sect A T in E; and let POA be a 
 logarithmic spiral, of which P is 
 the pole and A T the tangent at 
 the extremity of the radiant PA. 
 If now the portion AP of this 
 spiral roll upon AE, the pole P 
 will trace the right line PE, 
 which, therefore, may represent 
 the direction and velocity of the FlG - 161 
 
 motion of P due to this rolling. With this motion let there be com- 
 
166 SECOND GENEBATION OF INVOLUTE. 
 
 pounded a motion similarly represented by PF, equal and parallel to 
 EA, then the resultant is PA ; the effect being the same as if, during 
 the rolling of the spiral upon the tangent, the tangent should move 
 to the right as shown by the horizontal arrow. Suppose further that 
 the pitch circles meantime move in rolling contact with the tangent ; 
 they will then have turned about the fixed centres C and J9, through 
 angles measured by the arcs AM, AL, equal to each other and to AE 9 
 and the rectilinear motions PE, PF, being uniform, these rotations 
 will also be uniform. Also, the pitch circles will, during this rota- 
 tion, move in rolling contact with the spiral AP, which being taken 
 as the describing line, will carry the pole P in the right line PA with 
 uniform velocity. But we have already seen that such motion of P 
 will simultaneously trace upon the planes of the two wheels, the in- 
 volutes AH and AI of the base circles. And in like manner it may 
 be shown that the continuations AJ and AK of these involutes may 
 be generated by the similar and equal spiral ANQ, rolling in contact 
 with the pitch circles, its pole Q being the tracing point. 
 
CHAPTER X. 
 
 SPUE GEAEIKG CONTINUED. 
 
 Conjugate Teeth. Sang's Theory. Path of Contact. Unsymmetrical Teeth 
 Approximate Forms. The Odontographs. Diametral Pitch. Manufacture 
 of Gear Cutters. Determination of Series of Equidistant Cutters. 
 
 Conjugate Teeth. 
 
 280. It follows from the general principles stated in (181), that if 
 a tooth-outline of any reasonable form be assumed, it may be gener- 
 ated by some describing curve rolling upon the pitch circle ; and if 
 that describing curve be determined, and used in connection with any 
 other pitch circle, the tooth thus formed will evidently gear correctly 
 with the assumed one. Teeth thus related are said to be conjugate to 
 each other ; since had the second one been given, the application of 
 the above process would have resulted in the formation of a tooth 
 identical with the first. 
 
 Now it may be required to lay out a wheel to gear with one already 
 made ; and supposing the original drawing to be lost, the tooth of 
 this given wheel is to all intents and purposes an assumed one : a 
 ready means of tracing its conjugate is therefore of some practical 
 importance. 
 
 The radius of the original pitch circle, even, may be unknown, but 
 this does not affect the above reasoning. We must assume a radius : 
 and if it should not be the same as the one originally used, the form 
 of the describing curve will be modified accordingly, and the outline 
 of the conjugate tooth will still be correct. 
 
 The form of the describing curve, which, by rolling on the assumed 
 pitch circle will generate the given tooth, can in general be only ap- 
 proximately determined by graphic processes (See Appendix). And 
 even if it were susceptible of exact construction, the method above 
 indicated, of ascertaining by means of it the outline of the conjugate 
 tooth, involves much greater labor, and is withal less satisfactory in 
 
168 
 
 CONJUGATE TEETH. 
 
 FIG. 162. 
 
 all practical respects, than the simple and direct mechanical expedient 
 illustrated in Fig. 162. 
 
 281. Let the form of a tooth, T, of the given wheel, be accurately 
 cut out of a piece of sheet metal, KG, which is properly centred to 
 
 turn upon a fixed pin at D ; and let 
 AD be the radius of the assumed pitch 
 circle. 
 
 Behind this is another piece of sheet 
 metal MN 9 being a portion of a disk 
 turning upon a fixed centre C. Since 
 the radius AD and the number of teeth 
 upon the given wheel are known, the 
 radius A (7 is determined by the velocity 
 ratio required. 
 
 The outline of the tooth, T 9 is now to 
 be traced on the blank disk, MN 9 with 
 a fine marking point : after which let 
 each piece be rotated through a small 
 angle, as though the pitch circles moved in rolling contact. The 
 proper relative amounts of these angular motions are easily regulated 
 by means of subdivisions, on the circular edges, in connection with 
 fixed marks on the board to which this mechanism is secured, as 
 shown at and L. The outline of T is then to be traced again ; and 
 by repeating this process the form of the conjugate tooth may be very 
 accurately mapped out, since each of the marks thus made upon the 
 blank MN 9 by the very nature of the operation, is tangent to its out- 
 line. 
 
 Moreover, if both sides of T be traced upon the disk MN in its 
 various positions, thus determining the boundaries of the space be- 
 tween two adjacent conjugate teeth S 9 S' 9 there will be no backlash ; 
 for by construction the backs as well as fronts of the teeth are in con- 
 tact at all times. It is true that the tooth and the space of the given 
 wheel may not be measured by equal arcs upon the pitch circle, which 
 was drawn at pleasure. Nor is this essential, since the spaces and the 
 teeth of the derived wheel will be correspondingly unequal ; the total 
 pitch arc, or sum of a tooth and a space, being however necessarily 
 the same for both wheels. 
 
 Sang 9 s Theory of the Teeth of WJieels. 
 
 282. By giving different values to AC in Fig. 162, we may construct 
 a whole series of wheels, having various numbers of teeth, all working 
 properly with the given wheel. 
 
169 
 
 Next, let the tooth of any one of these be taken as the assumed 
 tooth in the above process, and used in like manner with various pitch 
 circles, thus forming a second series of wheels. 
 
 From the mode of generation it is evident that any wheel belonging 
 to either series will gear correctly with any one of those constituting 
 the other series. But the wheel of a given number of teeth in the 
 first may not be like that one of the second which has the same num- 
 ber. If the outlines of conjugate teeth upon any two wheels of the 
 same number be made alike by any means, it is clear that the two 
 series will be identical, and all the wheels of both will be inter- 
 changeable. 
 
 Now in the derivation of either series, the assumed* tooth always 
 occupies and maps out the space between two adjacent conjugate 
 teeth ; and this holds true when the latter becomes infinite. From 
 this it follows that the racks of the two series are in any case exactly 
 converse ; their contours are identical, the 
 teeth of the one being precisely like the 
 spaces of the other, and vice versa, as shown 
 in Fig. 163. 
 
 When, therefore, the two series become 
 identical, the two racks will be identical ; 
 each will be its own converse, and have its FlG - 163 - 
 
 teeth and its spaces similar and equal to each other. This requires 
 that the tooth and space shall be equal as measured on the pitch line, 
 and also that the portions , Z>, of the contour, shall be respectively 
 similar and symmetrical to the portions a' l> on the opposite side of 
 the pitch line. 
 
 And if, as all along supposed, the teeth are symmetrical to a radius, 
 that is, have their fronts and backs alike, it is further necessary that 
 a and b shall be similar and equal. 
 
 283. Now it is evident that in the process of Fig. 162, we may 
 assume the tooth of a rack, and from it derive the conjugate teeth of 
 a series of wheels, by a modification in detail which is too obvious to 
 require explanation. 
 
 And from the foregoing, the deduction is that if this original rack- 
 tooth be bounded by four similar and equal lines in alternate rever- 
 sion, the wheels whose teeth are thus determined will form an inter- 
 changeable set. It will readily be seen that if the rack be composed 
 of cycloidal arcs, the result will be the familiar epicycloidal system 
 with a constant describing circle ; while if it be bounded by oblique 
 right lines, the involute system is at once reproduced. 
 
 The contour of the rack being arbitrary, with the limitation just 
 
170 
 
 TO FIND THE PATH OF CONTACT. 
 
 mentioned, any number of systems may be thus derived, each differ- 
 ing from the others in the peculiarities of its action, more particularly 
 in respect to the variation in the obliquity. 
 
 Professor Edward Sang, in a most elaborate treatise, has discussed 
 this method of constructing the teeth of wheels, which consists sub- 
 stantially in reversing the usual order of proceeding. His treatment 
 of the subject is analytical and abstruse, the style obscure and pedan- 
 tic to the last degree ; and although the theory of gearing is fully 
 developed, the equally important practical operations of laying it out 
 are very much neglected. 
 
 284. Determination of the Path of Contact. The theory is, however, 
 more comprehensively seen from this point of view than from any 
 other, since the important part played by the path of contact is 
 brought more prominently to our notice : and Professor Sang's inves- 
 tigations relate more particularly to the form of that path, as depend- 
 ent upon the contour of the assumed tooth. 
 
 It is not necessary to know the describing curve in order to deter- 
 mine the path of contact. In Fig. 164, let A B be the face of a tooth 
 
 for the wheel whose centre is D ; draw 
 a series of normals to this face, cutting 
 the pitch circle at 1, 2, etc. During 
 the action, the common normal of the 
 tooth-outlines must at every instant pass 
 through the common point of the pitch 
 circles ; therefore as the points 1, 2, etc., 
 successively reach the line of centres, the 
 normals 1, 2b, oc, will take the posi- 
 tions Aa' 9 Ah ', Ac ', thus determining 
 the path of contact AB '. 
 
 Let C be the centre of the engaging 
 wheel ; then the form of the conjugate 
 flank may also be found without making 
 use of the describing curve. For con- 
 sidering the given wheel as the driver, 
 the angular motions of the follower cor- 
 responding to the already ascertained an- 
 gular motions aa f , ~bl> , etc., of the driver, 
 may be determined, since the velocity 
 ratio is known. Arcs a'a", W', c'c", 
 FIG. 164. measuring these angular motions of the 
 
 follower, then, are to be set off on circles of which C is the centre, and 
 the curve AE thus determined is the acting conjugate flank ; which 
 
PATH OF CONTACT. 171 
 
 will, of course, require to be extended as shown in dotted lines for the 
 formation of a clearing space, as usual. 
 
 285. But although it is thus possible to determine the path of con- 
 tact and the conjugate tooth by graphic means, this method is not in 
 general well suited for ordinary practical operations, since it involves 
 the drawing of the normals to a curve whose nature may not be known, 
 and this of itself (see Appendix, (7) is a matter of considerable labor ; 
 while the whole process above described is not only tedious, but re- 
 quires most careful manipulation in order to secure satisfactory results. 
 Consequently, if the form of a tooth be given or assumed, the me- 
 chanical method of Fig. 162 is for practical purposes by far the best 
 for determining that of its conjugate. It is true that it gives no indi- 
 cation of the path of contact, and therefore does not enable us to 
 ascertain precisely either where the action begins or where it ends. 
 But if neither wheel has less than twelve teeth, and these of propor- 
 tions in accordance with any of the arbitrary rules given in (201), 
 there can be no question that the total angle of action will be ample. 
 
 For when we consider that in the involute system the path of con- 
 tact is a right line joining the initial and terminal points of action, 
 while in the epicycloidal it is composed of two circular arcs tangent 
 to each other and to both pitch circles at their common point, and 
 also that it must pass through this last-mentioned point in all cases, 
 it will appear more than likely that with any other reasonable form 
 of tooth, this path will be intermediate between these extremes. And 
 if so, the difference, either in respect to the limiting numbers of teeth 
 or the variations in the obliquity of action, is greater between these 
 than between any other two systems of gearing : and either of them, 
 as before stated, will with the above-mentioned numbers and propor- 
 tions give results practically satisfactory for most uses. 
 
 It is beyond the scope of this work to discuss in detail the various 
 forms of teeth which may be constructed by the application of Sang's 
 Theory. In all the processes of " laying out " gearing upon the draw- 
 ing-board, the simplicity of the paths of contact and of the describing 
 lines, peculiar to the involute and the epicycloidal systems, gives 
 them a great and obvious advantage over others ; while years of expe- 
 rience have established the fact that in practical operation these forms 
 have no superiors. 
 
 286. Unsymmetrical Teeth. Thus far, the teeth have been supposed 
 to be symmetrical, that is, to have their fronts and backs alike, as 
 indeed they are made except in rare instances. But this is not at all 
 a matter of necessity, and under some circumstances it may be advan- 
 tageous to make them otherwise. For example, in Fig. 165, the 
 
172 
 
 UNSYMMETRICAL TEETH. 
 
 acting outlines (the lower wheel being supposed to drive to the right) 
 
 are of the epicycloidal form ; the 
 describing circles are large, in order to 
 reduce the obliquity to a minimum, 
 and were the tooth made of the same 
 form on the back, this might render 
 it too weak at the root. But if the 
 motion is never to be reversed, this 
 weakness may be avoided by making 
 the back, as shown, an involute of 
 considerable obliquity. 
 
 The conjugate racks, derived by the 
 
 FIG. 165. FIG. 166. 
 
 process of Fig. 162 from a tooth of this description, would be of the 
 forms shown in Fig. 166; which, it will be observed, are identical, 
 notwithstanding their lack of symmetry. 
 
 Approximate Forms of Teeth. 
 
 287. For the perfect working of mechanism in which wheels are 
 employed, it is unquestionable that the forms of the teeth should be 
 exactly correct. And if they are to be cut, there is no reason why 
 
 the teeth should not be laid out with the 
 utmost precision, since, the cutter once 
 made, it is as easy to cut one shape as 
 another. 
 
 But for ruder machinery, when the 
 wheels are simply to be cast, so great ac- 
 curacy may be sometimes thought unnec- 
 essary ; and, especially if time presses, a 
 more ready means of securing a reasonably 
 close approximation may be desirable. 
 
 In Fig. 167, draw through /, the com- 
 mon point of the pitch circles XY, VW, 
 FIG. 167. a line of action A IP at pleasure, and IE 
 
 perpendicular to it. Draw CA at pleasure, cutting IE in E, and 
 join ED by a right line cutting AP in B. 
 
APPROXIMATE TOOTH-FORMS. 173 
 
 Now if CA, DB, be a pair of levers joined by a link AB, then, 
 letting 
 
 v ang. vel. about D, 
 v = " " " C, 
 we have the value 
 
 v_ _CI f 
 v' ~DV 
 
 and since by the construction E is the instantaneous axis, this velocity 
 ratio will be momentarily constant. (See Figs. 39 and 40.) 
 
 If now through any point P on the line of action we describe two 
 circular arcs, one about B, as a tooth-face for the lower wheel, the 
 other one with centre A being the conjugate flank ; the velocity ratio 
 determined by them will at the instant be exactly the same as that of 
 the combination of link and levers, and will not differ appreciably 
 from it during an elementary motion in either direction. 
 
 288. The above argument, it will be observed, does not depend 
 either upon the obliquity, or upon the position of the point P, both 
 of which are optional. 
 
 But in order to reduce the application of this general principle to 
 a system, let Iff, IK, be each one half the pitch arc, and let IP be 
 an arc of the same length upon a describing circle ; then P/will be 
 the common normal to an epicycloid KP and a hypocycloid HP, tan- 
 gent at P } and these are correct tooth-outlines. Let B and A be the 
 centres of curvature for these two curves respectively, at the common 
 point P ; the circular arcs may then be used as approximating closely 
 enough to the exact forms for many practical purposes, provided that 
 the obliquity be not too great nor the teeth too long. 
 
 And if the same describing circle, and the same arc IP, be used, 
 the obliquity of the line of action, for the mean position when the 
 curves are in contact at P and the velocity ratio is exactly correct, 
 will be the same for all pitch circles. 
 
 289. Willis's Odontograph. The system above explained was origi- 
 nated by Professor Willis, and upon it is based the invention of his 
 well known Odontograph. Making the arc IP 30, the obliquity 
 is 15, and the angle DIA = 75. Then having calculated the dis- 
 tances from / of the centres of curvature at P for the tooth-faces and 
 flanks for various pitches and numbers of teet^h, these are tabulated, 
 and the instrument is constructed as in Fig. 168, which also illus- 
 trates the method of using it. 
 
 It consists merely of two arms of German silver, making an angle 
 of 75 with each other, the edge of one corresponding to the radius 
 
174 
 
 WILLIS'S ODOtfTOGRAPH. 
 
 Fio. 168. 
 
 LI, that of the other to the line of action AP in Fig. 167 ; and this 
 latter edge is divided into a scale of equal parts, numbered both ways 
 from the intersection of the two edges, which corresponds to the 
 point /. We have thus two scales, the one on the right being the 
 " Scale of Centres for Faces," that on* the left for flanks ; and the 
 table above mentioned has two corresponding divisions. 
 
 Suppose it to be required to lay out 
 a wheel of 20 teeth, the circular pitch 
 being 1J inches. These data deter- 
 mine the radius of the pitch circle VW, 
 Fig. 168; upon which set off the pitch 
 arc EF, bisect it at L, and draw the 
 radii DE and DF. Set the odonto- 
 graph so that the slant edge coincides 
 with DE, the point E thus coming 
 just at the zero of the scales. Then in 
 the "Table of Centres for Flanks," in 
 the column for 1^-inch pitch, opposite 
 20 in the column of "Numbers of Teeth," we find the number 37. 
 
 The position indicated by this number on the " Scale of Centres 
 for Flanks," fixes the point A as the centre of the circular arc through 
 L, which gives the approximate form of the required flank. Then 
 setting the odontograph in like manner by the radius DF, by a similar 
 process we find on the other scale the location of the centre B of the 
 circular arc through L, which forms the face of the tooth. 
 
 290, It is evident that wheels of the same pitch, laid out by this 
 odontograph, are interchangeable, their teeth being in effect epicy- 
 cloidal with a constant describing circle. Since half the pitch arc 
 measures 30 upon this circle, the wheel of 12 teeth, the least for 
 which the distinguished inventor designed it to be used, will have 
 radial flanks. 
 
 Of course the approximating circular arcs will deviate more and 
 more from the true curves as the tooth is made longer, and will not 
 answer at all for low-numbered pinions. But if the length of the 
 tooth be limited by the arbitrary rules given in (201), the instrument 
 may be used with confidence for all ordinary purposes when the 
 wheels are to be cast, and none have less than 12 teeth. In the case 
 of a rack, the radii DE, t)F, of Fig. 168 will become perpendicular 
 to the pitch line ; in laying out annular wheels the positions of face 
 and flank are transposed, and due attention must be paid to the limits 
 in regard to the diameters of the pitch circles, which have already 
 been discussed. 
 
TEMPLET ODOKTOGKAPH. 
 
 175 
 
 It is also to be remarked that the tables which accompany the in- 
 strument do not contain radii of curvature for all the consecutive 
 numbers of teeth. The variation in the correct contour, due to the 
 addition of a single tooth, rapidly diminishes as the actual number 
 increases, so that practically the same outline serves for the teeth of 
 several different wheels. If then the number assigned be not found 
 in the table, the nearest number to it is to be taken instead. 
 
 Also, if the given pitch be one not inserted in the tables, the radii 
 required may be found by direct proportion from those of other 
 pitches which are tabulated ; for 4-inch pitch by doubling the radii 
 for 2-inch pitch, halving those given for 1J- inch pitch in order to 
 find the radii for |-inch pitch, and so on, as occasion may require. 
 
 Robinson's OdontograpJi. 
 
 291. This differs entirely from the preceding, both in principle 
 and in the manner of using it. In Fig. 169, let VW be an arc of 
 the pitch circle, to 
 which CTis tangent at 
 A, the middle point of 
 the tooth, A 0, being the 
 half thickness. Then 
 the odontograph is to be 
 so set that CT shall not 
 only be tangent to tho 
 lower curved edge BFH, 
 but also cut the grad- 
 uated edge BE, which 
 must pass through 0, at 
 a certain division C, determined by the aid of tables and instructions 
 which accompany the instrument. The edge CE is then used as a 
 curved ruler for drawing the face of the teeth, after which the odonto- 
 graph, being made of thin metal and graduated on both sides, is 
 turned over, and the opposite face of the tooth is drawn in like man- 
 ner. A different (t setting number" is then determined from the 
 tables, which brings a different portion of the curved edge into posi- 
 tion for describing the flanks of the teeth. By certain modifications 
 in the determination of the setting numbers and in the placing of the 
 instrument, involute teeth also can be drawn, as well as epicycloidal 
 ones with various describing circles, in either outside or inside gear. 
 
 This remarkably ingenious invention, which has a much wider and 
 more flexible range of action than that of Willis, is called the Tern- 
 
 FIG. 
 
176 DIAMETKAL PITCH. 
 
 plet Odontograph, from the fact that the forms of the teeth are drawn 
 by it directly ; and if it be desired to lay out the whole wheel, its use 
 is much facilitated by attaching it temporarily to a wooden rod or 
 radius bar, turning about the centre of the wheel. 
 
 292. Its construction, obviously, depends upon the finding of a 
 curve for the graduated edge, of rapidly changing curvature, and of 
 such a nature that its different portions shall closely approximate to 
 the initial parts, or those next the pitch circles, of teeth of all pitches 
 and sizes, of either the involute or epicycloidal forms. This very 
 peculiar property Prof. Bobinson has shown, by a long and elaborate 
 investigation,* to be possessed by a certain logarithmic spiral, of 
 which form, accordingly, the working edge BCE is made. The other 
 curve BFH, it may be added, is the evolute of BCE, and, therefore, 
 a similar and equal spiral. 
 
 Diametral Pitch. 
 
 293, In designing and laying out wheels upon any of the systems 
 described, it is necessary to find the circular pitch, to which only 
 reference has thus far been made ; not only because it is used in the 
 graphic constructions, but because the strength of the tooth depends 
 upon its thickness. 
 
 Were there nothing to the contrary, it would be most convenient 
 to express the pitch in whole numbers or in manageable fractions, as 
 2-inch pitch, J-inch pitch, and so on. But since for complete wheels 
 the circular pitch must be an aliquot part of the circumference, the 
 diameters of the pitch circles will often contain awkward decimals if 
 this plan be adhered to ; and practically it is much more important 
 to have the diameter a whole number, or a convenient fraction, than 
 that the circular pitch should be either the one or the other. 
 
 This consideration has led to the introduction, and the almost ex- 
 clusive adoption for cut gearing, of what is called the Diametral Pitch ; 
 which is merely the quotient found by dividing the diameter of the 
 pitch circle, instead of its circumference", into as many equal parts as 
 there are to be teeth upon the wheel under consideration : 
 
 whence, 
 
 i T-J. i Diameter Circular Pitch 
 
 Diametral Pitch = =r= ^-=~ r > = - - > 
 
 No. of Teeth 3.1416 
 
 and 
 
 Circular Pitch = Diametral Pitch x 3.1416. 
 
 * See Van Nostrand's Eclectic Engineering Magazine, July, 1876. 
 
DIAMETRAL PITCH. 
 
 294. In the use of this system, convenient values ol 
 pitch are selected, each being a fraction with unity for its numerator 
 and an integer for its denominator, as 1, J, ], -J-, ^ 3^, etc. 
 
 The denominators of these fractions only are commonly used in 
 giving the diametral pitch; thus an " 8-pitch wheel " is one which 
 has eight teeth for each inch of diameter, or whose diametral pitch 
 is J". This is, in fact, merely inverting the fraction, and giving the 
 
 No. of Teeth 
 
 value of ^ ; thus, let a wheel of 16 inches diameter have 
 
 Diameter 
 
 80 teeth ; then, -J$ = -J" diametral pitch, but |$ = 5, and we call 
 it a " 5-pitch " wheel. By this system the calculations as to diameter 
 and number of teeth are made very simple ; as, for example : 
 Required, the diameter of a 4-pitch wheel with 37 teeth : 
 
 ^--c= 9J = diameter. 
 
 How many teeth of 16-pifcch on a wheel of 3J diameter ? 
 3| x 16 = G2 = No. teeth. 
 
 The tooth may be made to project outside the pitch circle a definite 
 fraction of the diametral pitch, as in the case of the circular pitch ; 
 and thus the size of the blank may be readily ascertained. If this 
 projection be made, for instance, 1^ the diametral pitch, the face of 
 the tooth will be nearly as long as that found by the first of the arbi- 
 trary rules given in (201) and the diameter of the blank is deter- 
 mined by simply adding to that of the pitch circle 2J times the 
 diametral pitch. 
 
 On the Manufacture of Accurate Gear Cutters. 
 
 In cutting a spur wheel, it is obviously essential that the contour of 
 the milling cutter conform precisely to that of the space between two 
 adjacent teeth : in order to this the process most extensively, and 
 until recently exclusively, employed, involves : 1st, the drawing of 
 the required curve ; 2d, the filing of a template to that exact form ; 
 and 3d, the turning of the cutter to fit the template. 
 
 The first improvement upon this, we believe, was the construction 
 by Messrs. Brown &. Sharpe, of Providence, R. L, of an ingenious 
 piece of mechanism called an Epicycloidal Engine, by which the 
 curves are traced automatically and with perfect precision upon the 
 template by continuous motion. There still remain the two han el- 
 and-eye operations of filing up the template, and of turning the cutter 
 12 
 
178 
 
 MANUFACTURE OF GEAR CUTTERS. 
 
 to fit it when done ; and these operations are so delicate and difficult, 
 that the exact duplication of templates, cutters or wheels in this 
 manner is well nigh impossible. 
 
 Yet this duplication, it is very easy to see, is quite as desirable as the 
 accurate formation of a single cutter ; and it has been made a matter 
 of perfect certainty and ready execution by means of two machines 
 recently constructed by Messrs. Pratt & Whitney, of Hartford, Conn. 
 Of these, which are remarkable for the ingenuity and beauty of 
 their movements, we will now proceed to describe the principles and 
 mode of action, though not the exact details. The whole process 
 under this system is mechanical and nearly automatic ; a template 
 being not traced, but milled out, by one machine, which is subse- 
 quently used in the other as a guide by which its motions are so con- 
 trolled as by another milling operation to finish the contour of tho 
 gear cutter, whatever the size of the tooth to be cut, to the precise 
 epicycloidal form with a minute and practically unimportant ex- 
 ception, as will presently be explained. 
 
 295, The Epicycloidal Milling Engine. In Fig. 170, A is a portion 
 
 of a flat ring, fixed to the fram- 
 ing; this represents a pitch 
 circle. B, is a disc, represent- 
 ing the describing circle ; this 
 turns freely upon a tubular 
 stud E, fixed in the carrier C, 
 which turns about a pivot />, 
 fixed to the frame at the centre 
 of A ; by means of the clamped 
 socket, capable of sliding upon 
 the rod, the position of D may 
 be adjusted to suit the radius 
 of A. Thus as C moves, the 
 disc can roll upon the edge of 
 A, and is compelled to do so by 
 the flexible steel ribbon shown 
 by the heavy line, which is 
 wrapped round and secured to 
 both pieces, due allowance for 
 its thickness being made in ad- 
 justing their radii. E' is a 
 second tubular stud fixed in the 
 FlG - 17 - carrier, at the same distance 
 
 from the pitch circle as the other, but on the opposite side ; the 
 
EPICYCLOIDAL ENGINE. 
 
 179 
 
 centres of the two studs lying on a right line through D. Upon these 
 two studs turn the two worm wheels F,F', shown in Fig. 171 ; these 
 are in a plane above A and B, so that the axis of the worm, G, is 
 vertically over the common tangent of the pitch and describing circles ; 
 the relative positions of these and other parts will be most clearly seen 
 by a study of the vertical section, I?ig. .175. The worm G, is sup- 
 ported in bearings secured to the carrier (7, and is driven by another 
 small worm turned by the pulley / ; the driving cord, passing through 
 suitable guiding pulleys, is kept at a uniform tension by a weight, not 
 shown, however (7 moves. 
 Upon the same studs, in a plane still higher than the worm-wheels, 
 
 FIG. 171. 
 
 Fro. 172. 
 
 turn the two discs ff, H' 9 Figs. 172, 173, 174. The diameters of 
 these are equal, and precisely the same as those of the describing 
 circles which they represent, with due allowance, again, for the thick- 
 ness of the steel ribbon by which these also are connected. 
 
 It will be understood that each of these discs is secured to the 
 worm wheel below it, and the outer one of these to the disc B ; so 
 that as the worm G turns, H and H' are rotated in opposite direc- 
 tions, the motion of H being identical with that of B ; this last is a 
 rolling upon the edge of A, the carrier C with all its attached mechan- 
 ism moving around D at the same time. Ultimately, then, the motions 
 
180 
 
 EPICYCLOIDAL ENGINE. 
 
 of H, ff', are those of two equal describing circles rolling in external 
 and internal contact with a fixed pitch circle. 
 
 In the edge of each disc a semicircular recess is formed, into which 
 is accurately fitted a cylinder, J 9 provided with flanges, between which 
 the discs fit so as to prevent end play ; this cylinder is perforated for 
 the passage of the steel ribbon, the sides of the opening, as shown in 
 Fig. 172, having the same curvature as the rims of the discs. Thus 
 when these recesses are opposite each other, as in Fig. 173, the cylin- 
 der /fills them both, and the tendency of the steel ribbon is to carry 
 it along Avith H when. C moves to one side of this position, as in Fig. 
 
 FIG. 173. 
 
 FIG. 174. 
 
 174, and along with IT when Amoves to the other side, as in Fig. 
 172. 
 
 This action is made positively certain by means of the hooks 
 K y K ', which catch into recesses formed in the upper flange of 
 J, as seen in Fig. 173. The spindles, with which these hooks turn, 
 extend through the hollow studs, and the coiled springs attached to 
 their lower ends, as seen in Fig. 175, urge the hooks in the directions 
 of their points ; their motions being limited by stops 0, o', fixed not 
 in the discs H, H', but in projecting collars on the upper ends of 
 the tubular studs. The action will be readily traced by comparing 
 
MANUFACTURE OF GEAR CUTTERS. 
 
 181 
 
 Fig. 173 with Fig. 174 ; as C goes to the left, the hook K' is left 
 behind, but the other one, K 9 cannot escape from its engagement with 
 the flange of / ; which accordingly is carried along with H by the 
 combined action of the hook and the steel ribbon. 
 
 On the top of the upper flange of J is secured a bracket, carrying 
 the bearings of a vertical spindle Z, whose centre line is a prolongation 
 of that of J itself. This spindle is driven by the spur wheel N, keyed 
 on its upper end, through a flexible train of gearing, not shown ; at 
 its lower end it carries a small milling cutter, M, which forms the 
 edge of the template, T, firmly clamped to the framing. 
 
 When the machine is in operation, a heavy weight, not shown, acts 
 to move C about the pivot Z), being attached to the carrier by a cord 
 guided by suitably arranged pulleys ; this keeps the cutter M up to 
 
 Fia. 175. 
 
 its work, while the spindle L is independently driven, and the duty 
 left for the worm G to perform is merely that of controlling the 
 motions of the cutter by the means above described, and regulating 
 their speed. 
 
 The centre line of the cutter is thus automatically compelled to 
 travel in the path RS, composed of an epicycloid and a hypocycloid, 
 if A be a segment of a circle as here shown ; or of two cycloids, if A 
 be a straight bar. The radius of the cutter being constant, the edge 
 of the template T is cut to an outline also composed of two curves ; 
 since the radius of J/is small, this outline closely resembles RS; but 
 particular attention is called to the fact that it is not identical with 
 it, nor yet composed of truly epicycloidal curves of any generation 
 ivhatever ; the result of which will be subsequently explained. 
 
 The diameter of the discs which act as describing circles is 7 
 
182 
 
 PANTAGRAPHIC CUTTER ENGINE. 
 
 inches, and that of the milling cutter, which shapes the edge of the 
 template, is J of an inch. 
 
 Now if we make a set of 1-pitch wheels with these describing circles, 
 the one with fifteen teeth will have radial flanks. The curves will be the 
 same whatever the pitch ; but, as shown in Fig. 176, the blank should be 
 adjusted in the epicycloidal engine so that its lower edge shall be T V 
 of an inch (the radius of the cutter M ) above the bottom of the space ; 
 also its relation to the side of the proposed tooth should be as here 
 shown. And, as previously explained, the depth of the space depends 
 upon the pitch. In the system adopted by the Pratt & Whitney Com- 
 pany the whole height of the tooth is 2J times the diametral pitch, the 
 projection outside the pitch circle being just equal to the pitch, so that 
 diameter of blank = diameter of pitch circle + 2 x diametral pitch. 
 
 We have now to show how, from a single set of what may be called 
 1-pitch templates, complete- sets of cutters of the true epicycloidal 
 contour may be made of the same or any less pitch. 
 
 The Pantagraphic Cutter Engine. 
 
 296. In Fig. 176, the edge TT, is shaped by the cutter M, whose cen- 
 tre travels in the path RS, therefore these two lines are at a constant 
 
 FIG. 176. FIG. 177. 
 
 normal distance from each other. Let a roller P, of any reasonable 
 diameter, be run along TT\ its centre will trace the line UV, which 
 is at a constant normal distance from TT, and therefore from US. 
 Let the normal distance between ?7Fand ES be the radius of another 
 milling cutter, N, having the same axis as the roller P, and carried 
 
MANUFACTURE OF GEAR CUTTERS. 183 
 
 by it, but in a different plane, as shown in the side view ; then what- 
 ever .2V cuts will have It/S for its contour if it lie upon the same side 
 of the cutter as the template. 
 
 Now if TT be a 1-pitch template above mentioned, it is clear that 
 N will correctly shape a cutting edge of a gear cutter for a 1-pitch 
 wheel. The same figure, reduced to half size, would correctly repre- 
 sent the formation of a cutter for a 2-piteh wheel of the same number 
 of teeth ; if to quarter size, that of a cutter for a 4-pitch wheel, and 
 so on. 
 
 But since the actual size and curvature of the contour thus deter- 
 mined depend upon the dimensions and motion of the cutter N, it 
 will be seen that the same result will practically be accomplished if 
 these only be reduced ; the size of the template, the diameter and the 
 path of the roller remaining unchanged. 
 
 The nature of the means by which this is effected in the Panta- 
 graphic Cutter Engine is illustrated in Fig. 177. The milling cutter, 
 N 9 is driven by a flexible train acting upon the wheel, ; its spindle 
 is carried by the bracket, B, which can slide from right to left upon 
 the piece, A> and this, again, is free to slide in the frame F. These 
 two motions are in horizontal planes, and perpendicular to each other. 
 
 The upper end of the long lever, PC, is formed into a ball, working 
 in a socket which is fixed to B. Over the cylindrical upper part of 
 this lever slides an accurately fitted sleeve, D, partly spherical exter- 
 nally, and working in a socket which can be clamped at any height 
 on the frame F. The lower end, P, of this lever being accurately 
 turned, corresponds to the roller, P, in Fig. 176, and is moved along 
 the edge of the template, T, which is fastened in the frame in an inva- 
 riable position. 
 
 By clamping D at various heights, the ratio of the lever arms, TD, 
 DC, may be varied at will, and the axis of JVmade to travel in a path 
 similar to that of the axis of P, but as many times smaller as we 
 choose; and the diameter of Nis made less than that of P in the 
 same proportion. 
 
 The template being on the left of the roller, the cutter to be shaped 
 is placed on the right of N, as shown in the plan view at Z, because 
 the lever reverses the movement. 
 
 This arrangement is not mathematically perfect, by reason of the 
 angular motion of the lever. This is, however, very small, owing to 
 the length of the lever ; it might have been compensated for by the 
 introduction of another universal joint, which would practically have 
 introduced an error greater than the one to be obviated, and it has 
 with good judgment been omitted. 
 
184 
 
 MANUFACTURE OF GEAR CUTTERS. 
 
 The gear cutter is turned nearly to the required form, the notches 
 are cut in it, and the duty of the pantagraphic engine is merely to 
 put the finishing touch to each cutting edge and give it the correct 
 outline. It is obvious that this machine is in no way connected with, 
 or dependent upon, the epicycloidal engine ; but by the use of proper 
 templates it will make cutters for any desired form of tooth ; and by 
 its aid exact duplicates may be made in any numbers with the greatest 
 facility. 
 
 TJieoretical Defects of the System. 
 
 297. It forms no part of our plan to represent as perfect that which 
 is not so. And there are one or two facts which at first thought 
 might seem serious objections to the adoption of the epicycloidal sys- 
 tem. These are : 
 
 1. It is physically impossible to mill out a concave cycloid, by any 
 
 FIGS. 178 and 179, 
 
 means whatever, because at the pitch line its radius of curvature is 
 zero, and a milling cutter must have a sensible diameter. 
 
 2. It is impossible to mill out even a convex cycloid or epicycloid, by 
 the means and in the manner above described. 
 
 This is on account of a hitherto unnoticed peculiarity of the curve 
 at a constant normal distance from the cycloid. In order to show 
 this clearly, we have, in Fig. 178, enormously exaggerated the radius, 
 CD, of the milling cutter (M of Figs. 174 and 175). The outer curve, 
 HL, evidently could be milled out by the cutter, whose centre travels 
 
MINUTENESS OF EKROR. 185 
 
 in the cycloid CA ; it resembles the cycloid somewhat in form, and 
 presents no remarkable features. But the inner one is quite different ; 
 it starts at D, and at first goes down, inside the circle whose radius is 
 CD, forms a cusp at E, then begins to rise, crossing this circle at G, 
 and the base line at F. It will be seen then that if the centre of the 
 cutter travel in the cycloid A C, its edge will cut away the part GED, 
 leaving the template of the form OGI. Now if a roller of the same 
 radius CD, he rolled along this edge, its centre will travel in the cy- 
 cloid from A, to the point P, where a normal from G cuts it ; then 
 the roller will turn upon G as a fulcrum, and its centre will travel 
 from P to (7, in a circular arc, whose radius is GP = CD. 
 
 That is to say, even a roller of the same size as the original milling 
 cutter, will not retrace completely the cycloidai path in which the 
 cutter traveled. 
 
 Now in making a rack template, the cutter, after reaching C, 
 travels in the reversed cycloid CR, its left-hand edge, therefore, mill- 
 ing out a curve DK, similar to HL. This curve lies wholly outside 
 the circle/)/, and therefore cuts OG at a point between F and G, 
 very near to G. This point of intersection is marked S in Fig.' 
 179, where the actual form of the template OSK is shown. The 
 roller which is run along this template is larger, as has been explained, 
 than the milling cutter. When the point of contact reaches S (which 
 is so near to G that they practically coincide), this roller cannot now 
 swing about S through an angle so great as PGCof Fig. 12 ; because 
 at the root D, the radius of curvature of /)A^is only equal to that of 
 the cutter, and G and S are so near the root that the curvature of SK, 
 near the latter point, is greater than that of the roller. Consequently 
 there must be some point U in the path of the centre of the roller, 
 such that, when the centre reaches it, the circumference will pass 
 through S, and be also tangent to SK. Let T be the point of tan- 
 gency ; draw $Z7and TU 9 cutting the cycloidai path AR in X and 
 Y. Then, UY being the radius of the new milling cutter (corre- 
 sponding to JVof Fig. 9), it is clear that in the outline of the gear- 
 cutter shaped by it, the circular arc XY will be substituted for the 
 true cycloid. 
 
 The System. Practically Perfect. 
 
 298. The above defects undeniably exist ; now, what do they amount 
 to ? The diagrams, Figs. 178 and 179, are drawn purposely with 
 these sources of error greatly exaggerated in order to make their nature 
 apparent and their existence sensible. The diameters used in prac- 
 tice, as previously stated, are : describing circle, 7 inches ; cutter 
 
186 EQUIDISTANT CUTTEES. 
 
 for shaping template, J of an inch ; roller used against edge of tem- 
 plate, 1J inches ; cutter for shaping a 1-pitch gear-cutter, 1 inch. 
 
 With these data the author has found that the total length of the 
 arc XY of Fig. 179, which appears instead of the cycloid in the out- 
 line of a cutter for a 1-pitch rack, is less than 0.0175 inch ; the real 
 deviation from the true form, obviously, must be much less than that. 
 It need hardly be stated that the effect upon the velocity ratio of an 
 error so minute, and in that part of the contour, is so extremely small 
 as to defy detection. And the best proof of the practical perfection 
 of this system of making epicycloidal teeth is found in the smoothness 
 and precision with which the wheels run, in which respects they have 
 proved, after some years of trial, to be unsurpassed. And we repeat, 
 that objection taken, on whatever grounds, to the epicycloidal form 
 of tooth, has no bearing upon the method above described of produc- 
 ing duplicate cutters for teeth of any form, which the pantagraphic 
 engine will make with the same facility and exactness, if furnished 
 with the proper templates. 
 
 On the Determination of a Series of Cutters. 
 
 299. In making a set of interchangeable wheels, upon any system 
 whatever, every additional tooth changes the diameter of tho wheel 
 and the form of the acting curves, so that in order to secure absolute 
 theoretical accuracy, it would be necessary to have as many different 
 cutters as there are wheels. This is clearly out of the question, and 
 fortunately, the proportional increment, and the actual change of 
 form, become less as the wheel becomes larger; and this alteration 
 in the outline soon becomes imperceptible. Going still farther, we 
 can presently add more than one tooth without producing a sensible 
 variation in the contour. That is to say, several wheels can be cut 
 with the same cutter, without introducing a perceptible error. Pro- 
 ceeding in this way we reach the conclusion, that instead of an infi- 
 nite number of cutters, a quite limited number will suffice to cut all 
 the wheels of such a set, from the smallest up to a rack, with a suffi- 
 cient degree of accuracy for all ordinary purposes. We have then to 
 decide how many cutters shall be made, and also for what numbers of 
 teeth these should be specially adapted. The answer to the first 
 question evidently depends upon how great a deviation from absolute 
 accuracy is considered admissible. In regard to the second, it appears 
 reasonable, not to say axiomatic, that errors should be uniformly 
 distributed through the series, and that this will be effected by mak- 
 ing the greatest difference in form the same between any two consec- 
 utive cutters. 
 
WILLIS'S SEEIES GRAFT'S FORMULA. 
 
 187 
 
 300. In relation to this, Professor Willis says : 
 
 _ _it appeared worth while to investigate some rule by which 
 the necessary cutters could be determined for a set of wheels, so as to 
 incur the least possible chance of error. To this effect 
 I calculated, by a method sufficiently accurate for the 
 purpose, the following series of what may be termed 
 equidistant values of cutters ; that is, a table of cut- 
 ters so arranged, that the same difference of form ex- 
 ists between any two consecutive numbers." He gives 
 us no clue to the method by which these numbers were 
 found, nor any explanation of the sign x , which ap- 
 pears in two places. The teeth are epicycloidal (the 
 describing circle being such as to give radial flanks to 
 the wheel of 12 teeth), and have " the usual adden- 
 dum " of which the exact amount is not stated. We 
 here insert the figures as he gives them,* only num- 
 bered in the reverse order ; and they appear to have 
 been extensively adopted in practice, and that too 
 under varying conditions, which, as will presently be 
 shown would affect the values of the terms of a truly 
 equidistant series. 
 
 301. The only rule or formula for computing such 
 a series that we have met with, is given by Mr. G. B. 
 Grant, as follows : 
 
 an 
 
 z =^ 
 n = 
 
 as 
 
 n s H 
 
 z 
 
 in which 
 
 a = the number of teeth on the least wheel, usually 12 ; 
 
 " " " " " greatest " " => ; 
 
 " " cutters in a series to cut from a to z ; 
 s " " in the series, of any particular cutter ; 
 t = " " of teeth on the last wheel to be cut by s. 
 This formula is based upon the following hypothe- 
 sis : The pitch arc is greater upon the outside of the 
 wheel than on the pitch circle, by an amount which 
 in wheels of equal pitch and constant addendum, de- 
 pends upon the number of teeth ; the smaller the 
 wheel, the greater is the difference, and vice versv, 
 the arcs being equal in the case of a rack. Now selecting any two 
 
 1 
 
 12 
 
 2 
 
 x 
 
 3 
 
 13 
 
 4 
 
 14 
 
 5 
 
 .15 
 
 6 
 
 x 
 
 7 
 
 16 
 
 8 
 
 17 
 
 9 
 
 19 
 
 10 
 
 20 
 
 11 
 
 21 
 
 12 
 
 23 
 
 13 
 
 25 
 
 14 
 
 27 
 
 15 
 
 30 
 
 16 
 
 34 
 
 17 
 
 38 . 
 
 18 
 
 43 
 
 19 
 
 50 
 
 20 
 
 60 
 
 21 
 
 76 
 
 22 
 
 100 
 
 23 
 
 150 
 
 24 
 
 300 
 
 25 
 
 00 
 
 Principles of Mechanism," p. 141. 
 
188 
 
 EQUIDISTANT CUTTERS. 
 
 wheels as the least and greatest of a series, it is possible to interpolate 
 others whose numbers of teeth are such that this difference, between 
 the pitch arc as measured on the pitch circle and on the extreme 
 perimeter, shall be the same for each : and these numbers are the 
 values of t as found by Mr. Grant's formula, in which it is assumed, 
 although upon what ground is not apparent, that the consecutive 
 wheels of such a series will differ equally throughout in respect to the 
 outline of the tooth. The proper location of each cutter in its own 
 interval, is determined by doubling the number of terms in the series, 
 and taking each alternate value of t as the number of teeth for which 
 the preceding cutter is to be made exactly correct. 
 It may be admitted that tolerably close approximations, or even 
 
 FIG. 180. 
 
 Fio. 181. 
 
 correct results, may be obtained by this formula under some condi- 
 tions. But it contains no internal evidence as to when or whether 
 this will be the case -it is independent of all the elements of the 
 problem which affect the form or the length of the tooth, and should, 
 therefore, if true at all, be applicable to all systems and under all 
 circumstances. "Which would be very convenient, unquestionably ; 
 but these elements cannot be eliminated without vitiating the results 
 and destroying the "equidistant" characteristic of the series. 
 
 302. This is best illustrated by considering the question in relation 
 to the epicycloidal system, which, at least, admits of a direct and 
 exact solution. In Fig. 180, let C be the centre of the describing 
 circle, VW an arc of the pitch circle of the smallest wheel, and D 
 
EPICYCLOIDAL SYSTEM. 189 
 
 its centre ; also let AF\>Q the addendum. Then a circle through F 
 about centre D determines the arc AB of the describing circle, which 
 by rolling upon an equal arc AH of the pitch circle traces the epicy- 
 cloidal face BH si the tooth for that wheel. Again, FE perpendic- 
 ular to CD cuts off an arc AE, which by rolling upon the tangent at 
 A describes the cycloidal face EG for the tooth of a rack. 
 
 Extending the tangent line toward the right, set off upon it AT = 
 arc A M = -J- space, and draw IL, MP, respectively similar and equal 
 to GE, HB. Then regarding CD as the central plane of a series of 
 cutters, it will be seen, by comparing this figure with the following 
 one, which represents a section of a milling cutter at work, that MP 
 will be the contour of the base of the one for the smallest wheel, and 
 IL the corresponding line of the cutter for the rack. 
 
 Proceeding in a similar manner with other pitch circles, it is ob- 
 yious that the faces for all intermediate wheels will lie between MP 
 and IL, their highest points lying in a curve LP ; 
 which is shown on a larger scale in Fig. 182. Now 
 these epicycloids diverge and differ in form from each 
 other the more the farther they extend from the 
 pitch circles ; if then they be so selected as in this 
 figure that they divide the locus ZPinto equal parts, 
 the greatest variation of each from the one next in 
 order will be constant for the whole series. 
 
 And this is sufficient for our purpose, it being self-evident that the 
 errors of which the distribution is to be equalized, should be taken 
 at the maximum ; and again, it is not necessary to take into consid- 
 eration the flanks of the teeth, to which the like method is equally 
 applicable, because they vary from each other to a much less extent 
 than the faces. 
 
 303. We have, then, a key to the perfect solution of the problem. 
 The first step is to determine LP ; this is done by assuming various 
 pitch circles, and constructing the face for a tooth of each, arranging 
 these with reference to the point A as above described, until a suffi- 
 cient number of points have been located to enable the curve to be 
 properly developed and drawn. Having decided upon the number of 
 cutters in the proposed series, LP is to be divided into a correspond- 
 ing number of equal parts. In regard to the practical execution of 
 this process, it may be remarked that although the curve LP may be 
 of a complicated nature, since it has a sort of transcendental depend- 
 ence upon the epicycloid, which is itself transcendental, yet in the 
 portion of it with which we have to deal in connection with wheels 
 ranging from twelve teeth up to a rack, its curvature changes so 
 
190 EQUIDISTAKT CUTTERS. 
 
 slightly that it may without causing any appreciable error be consid- 
 ered as a circular arc. 
 
 We have next to find the radius of the wheel, whose tooth would 
 fall at each point of subdivision ; this is readily done as shown in Fig. 
 180, thus : let S be one of these points upon LP, we have but to 
 draw FS, and bisect it by a perpendicular, which latter line will cut 
 CD at 0, the centre of the required wheel. 
 
 This determines the limits between which each cutter is to be used : 
 thus, if in Fig. 182 we suppose a series of four cutters to be required, 
 PM being the face of the smallest wheel and LI that of the largest, 
 then the first cutter must cut from P to Q, the next from Q to S, 
 and so on. Evidently, each cutter should be made exactly right for 
 a wheel whose tooth would fall at the point bisecting the arc of LP 
 which limits its range of action ; so that the location of the first is 
 f ound by dividing PQ in half, and determining as above the radius 
 of the wheel corresponding to this point of division. 
 
 304. The same general mode of proceeding may be employed with 
 teeth of any form, the operation of determining points in the locus 
 LP differing in detail according to the peculiarities of each system 
 of gearing. It is apparent that this process requires us to take into 
 account in every case not only the addendum, but the path of con- 
 tact. The fact that the latter is in the epicycloidal system identical 
 with the describing circle, renders comparatively simple what in other 
 systems might be very complicated, and enables us by ordinary trigo- 
 nometrical operations to locate any required number of points in LP 
 with great accuracy. 
 
 And pursuing this line of investigation, we find that the curvature, f 
 position and magnitude of LP are affected by changes in either 
 the length of the tooth or in the diameter of the describing circle, in 
 such a manner and to such an extent as sensibly to affect in turn at 
 least the higher values of a series of equidistant cutters. 
 
 Now it costs no more to make or to use a correct series than an in- 
 correct one ; and in adopting the epicycloidal system, there is no 
 excuse for resting contented with approximations, close or other- 
 wise, since by the process above outlined, results may be obtained 
 which are exact to a single tooth. On this account, and also because 
 this system has been and is likely to be more extensively employed 
 than any other, we have made the required determinations with great 
 care, in order to reduce the magnitude, and correct the unequal dis- 
 tribution, of errors in the series in use. 
 
 305. First, taking the diametral pitch as unity, with a describing 
 circle with diameter 7i, and addendum = 1, the locus LP was 
 
e 
 ^s. 
 
 FIG. 183. 
 
 EPICYCLOIDAL SYSTEM. 191 
 
 accurately determined by calculating the positions of a great number 
 
 of points. This curve is shown, greatly enlarged, in Fig. 183, LN 
 
 in this figure corresponding to the same line in Fig. 180, being the 
 
 perpendicular let fall 
 
 from L upon PR. 
 
 The numbers upon 
 
 the horizontal ordi- 
 
 nates indicate the 
 
 diameters of the pitch 
 
 circles assumed for 
 
 the computations, or, 
 
 what is the same 
 
 thing, the numbers of 
 
 teeth. 
 
 Next taking any point thus determined, at pleasure, and assuming 
 it to lie upon a circle passing through L and P, the radius of that 
 circle was calculated which being repeated for various other points, 
 the close agreement of the radii proved that for the purpose in view 
 the curve might be regarded as a circular arc. Upon this assumption, 
 the location of any point of subdivision, with reference to CD and the 
 point F of Fig. 180, is known, and the number of teeth upon the 
 wheel from which that point would have been derived, can be calcu- 
 lated trigonometrically ; this converse process is quite different from 
 that by which a point upon the curve is found, so that the concordance 
 of the results affords a rigid test of the accuracy both of the work and 
 of the whole method. 
 
 It was not to be expected that the points of exact subdivision of 
 this curve should correspond precisely to whole numbers of teeth ; on 
 the contrary, fractional terms were to be looked for as a matter of 
 course, but since the series must in practice consist of whole numbers 
 only, the nearest integer was taken in each case. 
 
 306. It will be observed that the middle point of LP lies very close 
 to the ordinate for the wheel of 24 teeth. Therefore, in a set of 24 
 cutters, a separate one should be made for each wheel from 12 to 23 
 teeth inclusive, thus confining both the errors and the calculations to 
 the 12 subdivisions of the upper half of the curve. 
 
 A series thus determined is given in the first column of the com- 
 parative table following : in regard to which it should be stated that 
 the 24th cutter is practically used only for very large wheels, an extra 
 one being added for the rack. 
 
 When this describing circle of 74- was first introduced, a series of 
 cutters was employed, the values for which are given in the second 
 
192 
 
 EQUIDISTANT CUTTERS. 
 
 column ; these figures were copied from those of Prof. Willis, with 
 the omission of his unintelligible sign x , and the interpolation of a 
 cutter for 18 teeth. 
 
 In both cases, the cutters used for only one wheel each are correct ; 
 and the others being " equidistant " in the first series, we have for the 
 purpose of a graphical comparison set up in Fig. 184, at equal inter- 
 vals, an ordinate for 
 each cutter, of a 
 length proportional 
 to the number of 
 teeth upon the small- 
 est wheel for which it 
 is to be used, as given 
 in the first column.; 
 the curve A A thus de- 
 termined, then, rep- 
 resents the true series. 
 ^ 10< 184t The intention was to 
 
 make the second series also "equidistant," and witli the same differ- 
 ence between consecutive numbers as in the first ; accordingly the 
 curve BB is constructed by setting off on the same ordinates the cor- 
 responding numbers in the second column ; and the discrepancy 
 between these curves exhibits very clearly the inequality in the dis- 
 tribution of the errors in this series, the use of which was promptly 
 abandoned as soon as this was pointed out. 
 
 In the third column we have a series computed by Mr. Grant's 
 formula, applied to determine the values for 12 cutters to cut from 
 24 teeth up to a rack, the first 12 being made exact, as in the other 
 cases. This formula, as already stated, would give the same results 
 for any describing circle and any addendum ; but changes in these 
 will in fact sensibly affect the values of the terms in this series. For 
 example, Prof. Willis made use of a describing circle whose diameter 
 is half that of the wheel of 12 teeth ; calling this diameter, then, 6, 
 and the addendum unity as before, we find the curve LP to be differ- 
 ent from that corresponding to the larger describing circle before 
 used. Its middle point, however, still lies so close to the ordinate 
 for 24 teeth, that the upper half of the curve only need be dealt with 
 in computation, and the correct series with these data is given in the 
 fourth column. Again, taking the original describing circle of di- 
 ameter = 7J, but reducing the addendum to |, the values for the 
 series of 24 cutters appear as given in column fifth. By comparing 
 this with the first column, it will be seen that the higher values only 
 
EFFECT OF VARYING CONDITIONS. 
 
 193 
 
 are affected by this reduction of the addendum ; and it is important, 
 since a series once adopted is in the nature of things inflexible, and 
 yet it may at times be necessary to reduce the addendum for the pur- 
 pose of varying the amount of approaching or receding action, to note 
 that even these values are affected in a far less proportion than that 
 in which the difference between the consecutive cutters is reduced. 
 
 This will be clearly seen by the aid of Fig. 185, in which the curves 
 0, B, C, represent the locus LP of Fig. 180, for the first, fourth and 
 fifth columns respectively ; o, #, and c being the corresponding centres 
 of curvature. 
 
 Taking the twenty-fourth part of each of these curves as the 
 greatest difference in form between the consecutive cutters of the 
 series to which it belongs, we have the following values ; 
 
 Linear Variation. 
 
 I. 0.0156645. 
 
 IV. 0.0158606. 
 
 V. 0.0046068. 
 
 VI. 0.0147569. 
 
 307, We do not consider it worth while to discuss the determination 
 of any series consisting of a less number of cutters than 24, because 
 the tendency of modern practice is for- 
 tunately more and more toward the at- 
 tainment of the greatest accuracy con- 
 sistent with a reasonable expenditure ; 
 aud this would call for an increase rather 
 than a diminution of the numbers. We 
 therefore, give, finally, in the sixth 
 column of the table, a series of 27 cut- 
 ters especially constructed for use in 
 connection with the epicycloidal and 
 pantagraphic engines above described. 
 Of these, 14 are exactly right, there 
 being one for each wheel from 12 to 25 
 teeth inclusive ; the next 12 are com- 
 puted to cut from 26 teeth up to a 
 rack; but the 26th is used only for FIG. iss. 
 
 wheels having 322 teeth or upward, an extra cutter being added for 
 the rack. 
 
 Much smaller series have been employed, some makers contenting 
 themselves with even so few as eight cutters for all pitches ; this, as 
 an examination of Fig. 183 will show without further argument, can 
 hardly be said to give a reasonable approach to accuracy, especially 
 in the case of wheels with comparatively few teeth. Others again 
 13 
 
194 
 
 VARIOUS SERIES COMPARED. 
 
 1 g 
 * 1 
 
 NUMBERS OF TEETH CUT BY EACH COTTER. 
 
 I. 
 
 II. 
 
 III. 
 
 IV. 
 
 V. 
 
 VI. 
 
 1 
 
 12 
 
 12 
 
 12 
 
 12 
 
 12 
 
 12 
 
 2 
 
 13 
 
 13 
 
 13 
 
 13 
 
 13 
 
 13 
 
 3 
 
 14 
 
 14 
 
 14 
 
 14 
 
 14 
 
 14 
 
 4 
 
 15 
 
 15 
 
 15 
 
 15 
 
 15 
 
 15 
 
 5 
 
 16 
 
 16 
 
 16 
 
 16 
 
 16 
 
 16 
 
 6 
 
 17 
 
 17 
 
 17 
 
 17 
 
 17 
 
 17 
 
 7 
 
 18 
 
 18 
 
 18 
 
 18 
 
 18 
 
 18 
 
 8 
 
 19 
 
 19 
 
 19 
 
 19 
 
 19 
 
 19 
 
 9 
 
 20 
 
 20 
 
 20 
 
 20 
 
 20 
 
 20 
 
 10 
 
 . 21 
 
 21-22 
 
 21 
 
 21 
 
 21 
 
 21 
 
 11 
 
 22 
 
 23-24 
 
 22 
 
 22 
 
 22 
 
 22 
 
 12 
 
 23 
 
 25-26 
 
 . 23 
 
 23 
 
 23 
 
 23 
 
 13 
 
 24-25 
 
 27-29 
 
 24 26 
 
 24-26 
 
 24-25 
 
 24 
 
 14 
 
 26-28 
 
 30-33 
 
 27-29 
 
 27-29 
 
 26-28 
 
 25 
 
 15 
 
 29-32 
 
 34-37 
 
 30-32 
 
 30-32 
 
 29-32 
 
 26-27 
 
 16 
 
 33-36 
 
 38-42 
 
 83-36 
 
 33-36 
 
 33-36 
 
 28-30 
 
 17 
 
 37-41 
 
 43-49 
 
 37-41 
 
 37-41 
 
 37-41 
 
 31-34 
 
 18 
 
 42-49 
 
 50-59 
 
 42-48 
 
 42-48 
 
 42-48 
 
 35-38 
 
 19 
 
 50-59 
 
 60-75 
 
 49-58 
 
 49-58 
 
 49-58 
 
 39-44 
 
 20 
 
 60-74 
 
 76-99 
 
 59-72 
 
 59-73 
 
 59-81 
 
 45-52 
 
 21 
 
 75-99 
 
 100-149 
 
 78-96 
 
 74-97 
 
 82-109 
 
 53-63 
 
 22 
 
 100-150 
 
 150-299 
 
 97-144 
 
 98-147 
 
 110-165 
 
 64-79 
 
 23 
 
 151-302 
 
 300-co 
 
 145-288 
 
 148-296 
 
 166-334 
 
 80-106 
 
 24 
 
 303-oo 
 
 oo 
 
 289-oo 
 
 297-co 
 
 335-co 
 
 107-159 
 
 25 
 
 00 
 
 
 oo 
 
 00 
 
 00 
 
 160-321 
 
 26 
 
 
 
 
 
 
 322-00 
 
 27 
 
 
 
 
 
 
 00 
 
 have used eight for small pitches, increasing the number of cutters 
 as the pitch of the tooth increases. This practice seems to be based 
 on the idea that the actual amount of error in the form of the cutter 
 only is to be taken into account ; which may be tenable if the forma- 
 tion of the templates and cutters be dependent upon hand-and-eye op- 
 erations, but not otherwise ; since the proportionate error, and the effect 
 upon the velocity ratio, will be the same whatever the size of the tooth. 
 
CHAPTER XL 
 
 1. TWISTED SPUR GEARING. Hooke's Stepped Wheels. Twisted Wheels. Rota- 
 
 tion not due to Screw-like Action. Elimination of Sliding. Neutralization 
 of End Pressure. 
 
 2. PIN GEARING. Generation of Elementary Tooth. Derivation of Working 
 
 Tooth. Peculiarities of the Action. Backs and Pinions. Annular Wheels. 
 Determination of Angle of Action. Determination of Limiting Numbers 
 of Teeth. Wheels with Radial Planes. 
 
 3. NON-CIRCULAR SPUR GEARING. Construction of Teeth for Elliptical and Lobed 
 
 Wheels. 
 
 Twisted Spur Gearing. 
 
 308, Hooke's Stepped Wheels. Let a pair of ordinary spur wheels, 
 loose upon their shafts, be cut transversely into a number of plates. 
 Let these sections of one of the wheels be first rearranged by rotating 
 them until, as in Fig. 186, the tooth of each overlaps that of the pre- 
 ceding one by the same amount, and then firmly keyed upon the 
 shaft. In passing to the new 
 
 1M .'." 
 
 position, each plate will drive 
 that section of the other wheel 
 with which it is in gear, inde- 
 pendently of the others. Thus 
 the second series of plates will 
 necessarily be arranged in a 
 similar manner, and these 
 being now also secured upon 
 their own shaft, we have a 
 pair of the Stepped Wheels 
 first introduced by Dr. Hooke. 
 
 By this ingenious device the number of teeth is in effect increased 
 without diminishing their size. Thus, the figure shows a wheel built 
 up of four plates, or thin wheels ; supposing each to have say 20 teeth, 
 the resulting action is clearly the same as that of a single wheel of 
 80, while the fact that the acting faces lie in different planes, enables 
 
 FIG. 186. 
 
196 HOOKE'S STEPPED WHEELS. 
 
 us to retain the original pitch. The advantages are obvious ; not 
 only is the number of contact points increased, but they cross the 
 line of centres at shorter intervals ; and the action is at its best when 
 this occurs. 
 
 The teeth may have any of the forms already .described ; and the 
 extent to which they overlap is, abstractly speaking, arbitrary. But 
 clearly the best arrangement is to have the edges of the successive 
 teeth divide the pitch arc AB into equal parts, as shown in the figure. 
 There are in this case four plates, and consequently the arc AC, 
 through which the last plate is rotated from its original position, is 
 equal to J AB. And in general, letting n = No. of Plates, we shall 
 have 
 
 309. In the practical employment of wheels thus constructed there 
 is a limit to the reduction in the thickness of the plates or steps, de- 
 pending on the material used and the pressure to be transmitted, 
 since if excessively thin they will suffer from abrasion. So long as 
 actual steps of sensible thickness are used, however, the kinematic 
 action differs in no respect from that of any other spur wheels, and 
 the lines of action all lie in the planes of rotation. 
 
 If the number of plates, then, be finite, it must be comparatively 
 small ; yet if it be increased to infinity the arrangement again becomes 
 perfectly practical, but the action is modified in a new and peculiar 
 manner. 
 
 The steps now disappear entirely, as in Fig. 109 ; and the effect is 
 the same as if the original wheels had been simply twisted, as ex- 
 plained in (175). In this process the nature of the acting surfaces is 
 changed. They were in the first place cylindrical, the bases being 
 the involutes or epicycloids forming the outlines of the teeth, and 
 the rectilinear elements being parallel to the axes. Since the twist- 
 ing is uniform, these elements now become helices, all having the 
 same pitch, but obliquities differing according to the distances from 
 the axes. And the line of contact between two engaging teeth will 
 partake of the helical form, though it will not be a true helix. For 
 it is clear that the transverse sections, by successive planes, will be the 
 original tooth outlines in successive phases, and in each section there 
 will be a point of tangency, which must lie in the projection of the 
 path of contact on a transverse plane. 
 
 310. Now, in regard to the common normal at any point of con- 
 tact. Pass through the point a transverse plane, which cuts out the 
 
HOOKE'S SPIRAL GEARING NOT SCREW GEARING. 197 
 
 tooth outlines just mentioned, and draw also the common tangent of 
 the two helices which pass through the point ; these two lines determine 
 the tangent plane, and the normal must be perpendicular to both. 
 The first lies in the transverse plane, but the latter pierces that plane 
 obliquely. Consequently, the line of action can in no case lie in a 
 plane of rotation, btit will make with it an angle more or less acute. 
 In general, then, the line of action can be resolved into three com- 
 ponents, viz : 
 
 1. The component of rotation, perpendicular to the plane of the 
 axes. 
 
 2. The component of side pressure, parallel to the common perpen- 
 dicular of the axes. 
 
 3. The component of end pressure, parallel to the axes themselves. 
 
 When the point of contact lies in the plane of the axes, the second 
 of these components of course vanishes ; of which fact, as will pres- 
 ently appear, advantage may be taken in so forming the teeth that 
 there shall be no sliding between them. 
 
 311. When the wheels are thus formed by twisting, instead of with 
 successive steps of sensible thickness, the combination is known as 
 Hooke's Spiral Gearing ; and is very commonly described and classified 
 as a modification of screw gearing. 
 
 That this is an error will be, perhaps, most clearly seen from the 
 considerations, that the direction of the twist does not affect that of 
 the rotation, and its amount does not affect the velocity ratio. Ee- 
 garding the wheels as built up of exceedingly thin plates or lamina?, 
 each one of those composing the driver turns the corresponding one 
 of the follower precisely as though the thickness were sensible, not 
 only during the formative process of twisting but after. On reach- 
 ing the limit when the plates become planes and the elements of the 
 tooth surfaces become helices, the action is modified by the deflection 
 of the common normals from the planes of rotation ; but regarding 
 the driver as a screw, its endlong thrust being perpendicular to those 
 planes, may have either direction and any magnitude without affect- 
 ing the direction or velocity of the transmitted motion. Whereas in 
 screw gearing proper, this endlong thrust either lies in a plane of ro- 
 tation or has a component which does, and this component is the one 
 which produces the rotation of the follower. In that class of gearing, 
 then, the driver turning in a given direction, the follower will turn 
 one way if the driver be right handed, but the other way if it be left 
 handed ; and the screw pitch of the driver will obviously affect the 
 velocity of the imparted motion. 
 
198 PRACTICAL LIMIT TO AMOUNT OF TWIST. 
 
 312. From the very nature of the twisting process, as above ex- 
 plained, it is evident that if the screw pitch of one wheel of a pair be 
 assumed, that of the other is thereby fixed. And it may readily be 
 ascertained, because if the teeth be indefinitely increased in number 
 and diminished in size, any two which are in contact will ultimately 
 become two tangent helices lying on the surface of the original pitch 
 cylinders, and must, therefore, develope into the same right line on 
 the common tangent plane, as in Fig. 32. 
 
 Again, it makes no difference in which direction we twist the first 
 wheel ; but it follows at once from the preceding that if the pair be 
 in outside gear, the helical elements of one will be right handed, 
 those of the other left handed. In the case of internal gearing, on 
 the contrary, the helices of both wheels will be either right handed 
 or left handed, as the case may be. 
 
 313. Practical Choice of the Screw Pitch. Although, as has been 
 stated, the amount to which such wheels shall be twisted is abstractly 
 optional, with the limitation mentioned in the preceding paragraph, 
 yet in order to reduce the obliquity of action and the consequent end 
 thrust it is best to make it in practice as small as may be, consist- 
 ently with securing the advantages due to twisting them at all. 
 
 Now when, as thus far supposed, any of the ordinary forms of spur 
 teeth are used, the angle of action is greater than the pitch, so that in 
 any given transverse plane the action of one pair begins before that 
 of the preceding pair ends. Consequently if a wheel of any given 
 depth measured in the direction of the axis be tivisted through an 
 angle equal to the pitch of the teeth, all the phases of the action will 
 at all times be simultaneously presented, and there will always be a 
 point of contact in the plane of the axes. 
 
 This then may be taken as a good practical limit ; and it agrees 
 with the deduction made in (308) when the number of steps becomes 
 infinite ; for the expression there given, 
 
 
 gives 
 
 AB 
 
 ~ AB -AC 
 
 whence if n = oo, we have AC = AB. 
 314. Elimination of Sliding Friction. If the teeth be so formed that 
 
SLIDING FRACTION ELIMINATED. 
 
 199 
 
 FIG. 187. 
 
 in any one plane the contact not only begins but ends on the line of 
 centres, continuing but for a single instant ; then the point at which 
 this contact occurs, which is of course upon the surface of each pitch 
 cylinder, will be the only point of tangency between the tooth surfaces. 
 In the next instant this driving point will be found to have shifted 
 to the consecutive plane, and so on continually, thus moving endlong 
 in the common element of the pitch 
 cylinders. Under the circumstan- 
 ces there is no sliding friction, since 
 the coincident points of the acting 
 surfaces are at every instant moving 
 in the same direction and at the 
 same rate. 
 
 In order to comply with the above 
 condition, the teeth may be formed 
 as shown in Fig. 187 or as in Fig. 
 188. In the former, the outlines 
 are epicycloidal, but the describing 
 circles for the faces are smaller than 
 those used for tracing the correspond- 
 ing flanks. In the second case, sup- 
 pose the teeth originally to have been involutes, as shown in the dotted 
 lines, the flank proper, or part within the pitch circle, is still of that 
 form, but the face, or part without, is of greater curvature, though 
 it is tangent to the inner part at its intersection with the pitch circle. 
 
 In general, then, it will be 
 easily seen, it is necessary 
 that when the point of con- 
 tact is on the line of cen- 
 tres, as in either one of 
 these figures, all the con- 
 tour lines which pass through 
 that point shall be tangent 
 to each other, but the faces 
 must lie within the curves 
 which in ordinary spur gear- 
 ing would be conjugate to 
 the flanks determined on. 
 A very simple construction 
 on the epicycloidal basis is 
 shown in Fig. 189, the tops of the teeth being semicircles tangent to 
 the flanks, which are radial, at their extremities. 
 
 FIG. 188. 
 
200 
 
 END PRESSURE NEUTRALIZED. 
 
 The action of plain spur wheels thus formed would, of course, be 
 correct only at the instants here represented ; but by the simple ex- 
 pedient of twisting them, so that the action on ceasing in one plane 
 is continued in the next, these instants become consecutive, the ve- 
 locity ratio is made constant, and the rotation is transmitted by pure 
 rolling contact. On this last account, as might be expected, the 
 action of these wheels is exceedingly smooth 
 and noiseless, almost as much so, if they be 
 well made, as that of belting ; but they are 
 better suited for light work, because the pres- 
 sure is confined to a single point instead of 
 being distributed along a line. For heavy work 
 it is, therefore, preferable to employ the stepped 
 wheels, or twisted ones in which the teeth, as 
 in Fig. 186, are of the usual forms, although 
 the sliding friction still remains. 
 
 315, Neutralization of End Pressure. But the 
 end pressure due to the screw-like action will 
 exist, even after the forms of the teeth have 
 been modified as above explained. What now 
 occurs during the rotation, as will be evident 
 from a moment's study, is precisely equivalent 
 to the rolling together of two helices, one on each pitch cylinder, 
 as illustrated in Fig. 32. And this end pressure, which must be re- 
 ceived by a collar on the bearing, causes an amount of friction which 
 it is desirable to avoid. 
 
 This disadvantage may in its turn be 
 obviated by the means indicated in Fig. 
 190. Each wheel is made in two parts of 
 equal thickness, which are twisted in oppo- 
 site directions, but to the same extent. 
 The end thrust of one part, then, is ex- 
 actly counterbalanced by that of the other : FlG - m 
 and since the action now involves pure rolling contact only, this may 
 be regarded as the very perfection of geared wheel- work. 
 
 FIG. 
 
 Pin Gearing. 
 
 316, The term Epicycloidal is in its ordinary use applied only to 
 teeth all of which have both faces and flanks whose describing curves 
 are circles. 
 
 There is, however, a form of gearing, in which the teeth of one 
 
PIN GEARING. 
 
 201 
 
 wheel of a pair are true epicycloids only at a theoretical limit, upon 
 reaching which those of the other wheel become mathematical 
 points. The latter are in practice actually made cylindrical pins 
 of reasonable diameters, which fact has given rise to the name 
 Pin Gearing ; and the former in consequence deviate considerably 
 from the epicycloidal outline whence their working contours are 
 derived. 
 
 In Fig. 191, C and D being the centres of the pitch circles, let a 
 marking point be fixed at A in the circumference of the upper one. 
 Then while this point goes to E, in the progress of the rotation, it 
 will trace upon the plane of the lower circle the curve EB ; the arcs 
 AE, AB, being equal. This, evidently, is the familiar epicycloid, 
 
 FIG. 191. 
 
 FIG. 192. 
 
 generated by rolling the upper pitch circle upon the lower. Mean- 
 time, since the marking point does not change its position in the cir- 
 cumference of the upper circle, it can trace no curve at all upon its 
 plane. 
 
 Now let AF be a curve similar to BE, and imagine a pin of no 
 sensible diameter a rigid material line to be fixed at A in the 
 upper wheel. Then if the lower one turn to the right, it 
 will drive the pin before it with a constant velocity ratio, the 
 action ending at E, if the driving curve be terminated at F as here 
 shown. 
 
 317. Supposing A E to be an aliquot part of the circumference, and 
 assuming it as the pitch arc, we have only to set the pins at equal 
 

 202 FORMATION OF THE TEETH. 
 
 distances in the upper circle, and after dividing the lower one, to 
 draw through the points of division the reversed curves as shown : 
 this done, the elementary wheels are complete and capable of working 
 in either direction. From these the practical ones are derived as in 
 Fig. 192 ; the pins being made of sensible magnitude, the outlines 
 of the teeth upon the other wheel are curves parallel to the original 
 epicycloids. The diameter of the pins is usually about equal to the 
 thickness of the teeth measured on the pitch circle, the radius being, 
 therefore, one fourth of the pitch arc ; this, however, is not impera- 
 tive, and the pins are sometimes made considerably smaller. When 
 the radius has been selected, a number of arcs are described with it, 
 having their centres upon the epicycloid, and the envelope of these 
 arcs is the required contour of the working tooth. 
 
 The pins are ordinarily supported at each end, two discs beins: 
 fixed upon the shaft for the purpose, as in Fig. Ill : thus making 
 what from its form is called a lantern wheel or pinion. 
 
 318. Peculiarities of the Action. The most striking feature of wheel- 
 work of this kind is, that the action is almost wholly confined to one 
 side of the line of centres. 
 
 In the elementary form, this is a direct and obvious consequence 
 of the manner in which the tooth is generated. Thus, in Fig. 191, 
 if the curve AF drive, the action cannot begin until its root reaches 
 the point A on CD, and is entirely receding ; if on the other hand 
 this curve be driven by the pin, the action will terminate at the same 
 point A, and will be entirely approaching. 
 
 Consequently, pin-gearing is not well adapted for use in combina- 
 tions which require the same wheel both to drive and to follow. But 
 when that is not required, this peculiarity of the action is greatly in 
 its favor ; the pins of course being always given to the follower and 
 the teeth to the driver. And previously to the introduction of cut 
 gearing, it was very extensively employed, even for heavy work ; the 
 facility of forming the pins, or staves, of the lantern pinions, in the 
 lathe, especially adapting it for the construction of wooden wheels. 
 At present its use is almost exclusively confined to clock-work and 
 similar light mechanism, the pins being made of steel wire cut to the 
 proper lengths. 
 
 319. It has heretofore been taught, in all the treatises upon this, 
 subject which have come to our notice, that when the pins are made 
 of sensible diameter, as in working wheels they always must be, the 
 above-mentioned peculiarity is modified and a certain amount of ap- 
 proaching action introduced in every case. 
 
 The fact was too patent to escape notice, that the expansion of the 
 
BACKS 
 
 PIN GEAKING. 
 
 203 
 
 pin from the theoretical point into the practical circle, had the result 
 of shortening the driver's tooth and reducing the amount of receding 
 action. But while this palpable effect of the new condition upon the 
 time when the action shall end was duly recognized, no one seems 
 ever to have inquired whether the same condition might not also affect 
 the time when the action shall begin ; and it appears to have been 
 taken for granted that it would not. 
 
 Thus in Fig. 192, the pin E is just quitting contact with the tooth 
 which drives it ; and the theory as hitherto laid down is based upon 
 the assumption, that correct driving contact with the next pin A is 
 just beginning when it occupies the position here shown ; thus giving 
 an arc of approach about equal to the radius of the pin, and making 
 this as it stands an exact limiting case. 
 
 It will presently be shown that this assumption is erroneous, and 
 that the next tooth is not, in general, tangent to the pin A when in 
 
 FIG. 193. 
 
 FIG. 194. 
 
 its present position. The error itself is physically small, which may 
 account for its having remained so long undetected ; but its effect is 
 of very perceptible magnitude ; not only changing very greatly the 
 amount of approaching action, but in many cases making that action 
 absolutely negative, the first driving contact between each tooth and 
 its pin not occurring until after the latter has bodily passed the line 
 of centres. 
 
 320. Pin- Wheel and Rack. Since the pins are always given to the 
 follower, the construction of a rack will present two cases. If it is to 
 be the driver, as in Fig. 193, the elementary tooth is bounded by 
 cycloids, generated by the pitch circle of the wheel ; from which the 
 outlines of the working tooth are derived as above explained. 
 
 Pin-Rack and Wheel, In Fig. 194, the wheel is to drive, and the 
 tooth-outline is the involute of its own pitch circle, the generatrix 
 being the pitch line of the rack. It is usually stated that it is unnec- 
 
204 
 
 ANNULAR WHEELS IK PIN GEAKING. 
 
 essary to construct the derived curve, since " this process would merely 
 reproduce the same involute in a different position." * Although this 
 
 is the truth, it is not the 
 whole truth ; and the part 
 which is lacking, will be 
 found to vitiate the results 
 deduced from the part which 
 is given. 
 
 321. Inside Pin Gearing. 
 Here also we have t\vo dis- 
 tinct cases, since the annular 
 wheel may be required either 
 to drive or to follow. In 
 Fig. 195, the pinion drives, 
 and the pins are given to 
 the wheel. The outer pitch 
 circle, then, by rolling upon 
 
 the inner, generates the internal epicycloid which forms the element- 
 ary tooth of the pinion. In Fig. 196, the wheel drives, and its ele- 
 mentary tooth is the hypocycloid traced by rolling the less circle 
 within the greater ; in both cases the curves for the working teeth are 
 derived in the usual manner. If the annular driver be twice as large 
 
 195. 
 
 FIG. 196. FIG. 107. 
 
 as the pinion, the hypocycloid becomes a right line, and the process 
 of derivation gives simply another right line parallel to it. A very 
 practical construction in this case is shown in Fig. 197 ; the pinion has 
 
 * Willis. (Principles of Mechanism, p. 96.) 
 
WHAT CONDITIONS AEE PRACTICABLE. 
 
 205 
 
 but two pins, which turn in blocks sliding in the two slots which, cross- 
 ing each other at right angles, constitute the disguised annular wheel. 
 
 322. Practicability of Assumed Conditions. It is hardly necessary to 
 state that there are definite relations between the pitch arc and arc 
 of action, the diameter of the pin and the height of the tooth, such 
 that all these cannot be assumed with any certainty that the result 
 will be a practicable arrangement. The preceding figures give merely 
 an idea of the general principles and the general appearance of the 
 various modifications ; and we have now to consider more in detail 
 the processes of construction. 
 
 First, in regard to the elementary or ideal form, in which the pin 
 is a point and the tooth- 
 curve a true epicycloid. 
 Eeferring to Fig. 191, it 
 appears that if we as- 
 sumed the- pitch arc AS, 
 the greatest possible 
 height of the tooth is de- 
 termined by the intersec- 
 tion of the front and back 
 at G ; and if this height 
 be adopted, the action, 
 beginning at A, will ter- 
 minate at H, the point 
 on the upper pitch cir- 
 cumference through 
 which G must pass in its 
 rotation about D. Now 
 this point G lies on the 
 radial line which bisects 
 AB, and will, therefore, 
 fall farther to the right, 
 the larger the pitch arc. 
 Should it chance to coin- 
 cide with 7f, or in other 
 words should the point FIG. 198. 
 
 of the driver's tooth fall upon the pitch circle of the follower, the 
 proposed case is exactly a limiting one. From an examination of the 
 succeeding figures it will be seen that this is equally true when the 
 driver is a rack or an annular wheel ; and also that in order to render 
 the case practically feasible, the point G should fall within the pitch 
 circle of the follower, except when the latter is annular, in which 
 
206 
 
 LIMITING DIAMETER OF PITS. 
 
 case it should fall without. This condition being satisfied, the tooth 
 may be made pointed, or it may be topped off, at pleasure ; and 
 finally, the arc of action is determined, as in previous constructions, by 
 the intersection of the path of the highest point of the tooth, with the 
 path of contact that is to say with the pitch circle of the pin- wheel. 
 323. Second, when the pin is of sensible diameter. We give in 
 
 Fig. 198, Prof. Willis's 
 mode of determining the 
 greatest diameter which 
 can be used under any 
 assumed conditions. C and 
 D being the centres of the 
 follower and driver re- 
 spectively, suppose the 
 pitch arc, AB or AE, to 
 be assigned. Bisect AB 
 by the indefinite radial 
 line Dx, and draw the 
 chord AE, cutting Dx in 
 / ; then El is the greatest 
 radius for the pin, and 
 any less one may be used, 
 as in Fig. 192. Since / 
 is unquestionably the last 
 point of contact with the 
 pin E, this construction 
 would be correct, provided 
 that the next tooth KM 
 were now in driving con- 
 tact with the pin A. Prof. 
 Willis distinctly states that 
 it is ; * and bases this as- 
 sertion upon the previous statement f that " by the mode of its de- 
 scription the circle (of the pin) must touch the curve (of the tooth) 
 when its centre is in any point of the epicycloid." The conclusion, 
 therefore, is very plausible ; but it is not correct, because the curve 
 derived from the epicycloid AR (as was pointed out in Fig. 178), 
 consists of two branches, one of which, and a part of the other also, is 
 effaced in the very process of formation. The normal to the epicy- 
 
 Fio. 199. 
 
 * Principles of Mechanism, p. 99. 
 f " " " p. 94. 
 
LIMITING DIAMETER OF PIN. 
 
 UNIVERSITY 
 
 207 
 
 cloid at its root, is AO perpendicular to CD ; the parallel CUIM 
 at first descends, having a vertical tangent at 0, forms a cusp at a 
 point within the pin whose centre is A 9 and then rising, cuts its cir- 
 cumference at some point M, which is evidently the lowest practicable 
 point in the outline of the working tooth. Draw through M a nor- 
 mal, cutting the original epicycloid at N, and through that point de- 
 scribe an arc about D, cutting the locus of original contact in P. It 
 will now be seen that the point M cannot come into correct driving 
 contact, until N reaches the position P\ and the arc AP is the 
 measure of the error in Prof. Willis's construction ; that is to say, it 
 is the amount by which the arc of approach as determined, or rather 
 assumed by him, must be changed. 
 
 324. Absolute Determinations Impossible, It appears then that the 
 limiting diameter of the pin as found by the above construction is 
 too great. By reducing it, the tooth IL may be made enough longer 
 to continue in action until the point M comes into driving contact. 
 But the precise amount of reduction which will cause one tooth to 
 quit contact at the instant when the next one begins to drive, it is 
 impossible to determine ; because the positions of both the points M 
 and /, which fix the times of the beginning and the ending of the 
 action, depend upon the unknown magnitude, the radius of the pin. 
 We are, therefore, compelled to adopt the tentative process shown in 
 Fig. 199. Having found the point /, cio in the preceding diagram, 
 assume a radius for the pin, less than El, and continue the derived 
 curve to cut Dx in /, which will be the point of the tooth. Through 
 J 'draw a normal to the epicycloid, cutting it in 8 ; through 8 de- 
 scribe an arc about D, which will cut the upper pitch circle in T, the 
 position of the pin at the end of the action. Drawing the working 
 outline of the next tooth, we determine, as in Fig. 198, the points M, 
 N, and finally P, the position of the pin when the action begins. Now, 
 if the arc PT prove to be precisely equal to AE, we have an exact lim- 
 iting case, and the assumed radius of the pin is a maximum : if PT 
 be less than AE, the radius is too great, and must be reduced but if 
 the contrary, the case is a practicable one. 
 
 In the latter event the tooth may be topped off ; and it need hardly 
 be added that the above process enables us to determine whether the 
 case be feasible or not, if both the diameter of the pin and height of 
 the tooth are assigned ; the normal JS being drawn from the highest 
 point of the given curve.* 
 
 * Note. For details of the graphic processes of drawing these curves and the 
 normals, the reader is referred to the Appendix. 
 
208 
 
 PIN GEARING. 
 
 Limiting Numbers of Teeth and Pins. 
 
 325. When the Pin is a Mathematical Point, the determination of 
 the limiting numbers of teeth and pins is easily effected by methods 
 precisely like those used in the case of the common epicycloidal teeth. 
 Suppose the number of teeth to be assigned for the driver whose centre 
 is b in Fig. 200, and let AFbe the pitch arc. The point, 0, of the 
 
 tooth must lie on the pro- 
 longation of DL bisecting 
 AF, in such a position 
 that an arc OA of a circle 
 whose centre lies upon DA 
 produced, shall be equal to 
 the arc AF. On the tan- 
 gent at A, set off AG = 
 arc AF, and A M = J A G: 
 with centre M and radius 
 MG describe an arc cutting 
 DL produced, in ; draw 
 OA, and bisect it by a per- 
 pendicular NC, which cuts 
 DA or its prolongation in 
 C, the required centre of 
 the follower. 
 
 In this diagram, the 
 wheels evidently work in outside gear, and A (7 is a minimum. Should 
 fall upon the tangent A G, OA will coincide with that line, and 
 the follower becomes a rack : if falls above A G, the centres C and 
 D will lie on the same side of the point of contact, the follower will 
 be annular, and the radius A C thus found will be a maximum. 
 
 If, on the other hand, the number of pins be assigned for the fol- 
 lower ; then the position of is known, and also the arc A 0, whence 
 AG and AM are also determined. It is evident that OD when found 
 will bisect the chord AF, making OA = OF. Therefore, describe 
 about J-fan arc with radius MG, and about another arc with radius 
 OA ; these arcs will intersect in F, and a perpendicular to AF 
 through will cut CA or its prolongation in D, the required centre 
 of the driver, whose radius will be a minimum when, as in this figure, 
 A lies between C and D. 
 
 If the point F coincides with G, we have the limiting case when 
 the driver becomes a rack : should F fall below A G, the driver becomes 
 in its turn annular, and its radius as thus determined is a maximum. 
 
 FIG. 200. 
 
KADIAL FLAKES LIMITING EATIO OF PITCH CIRCLES. 
 
 209 
 
 326. Wheels with Kadial Planes. Since the face OF of the driver's 
 tooth in Fig. 200 is an epicycloid, it would correctly drive a flank 
 traced by the same describing circle. And this flank would be radial, 
 were we to use a pitch circle whose radius AE is twice AC. Hence 
 this driver will work not only with the pin-wheel, but with a wheel 
 of twice the diameter of the latter, furnished with twice as many 
 radial planes as there are pins : the combination presenting the ap- 
 pearance shown in Fig. 201. 
 
 In other words, the minimum 
 number of radial planes which 
 can be driven by such a wheel 
 is equal to twice the minimum 
 number of pins. When stated 
 in this way, it would appear 
 that there should always be an 
 even number of these planes. 
 But it is to be observed, that 
 in computing the least number 
 of pins, we must take the next 
 higher integer in the event of 
 a fractional result ; we may, 
 however, at once double that 
 result in order to ascertain the 
 least number of radial planes, 
 and the next higher integer is 
 not then necessarily an even number ; but on the contrary, is an 
 odd number in the majority of cases, as will be seen by reference to 
 the following tables. 
 
 327. Limiting Ratio of Pitch Circles in Inside Gear. In the use of 
 the ordinary epicycloidal teeth, or of those of involute form, the di- 
 ameters of wheels in inside gear may be made very nearly equal if the 
 teeth are short enough, as in Fig. 129 : and such combinations are 
 frequently met with in differential trains, the annular wheel often 
 having only one tooth more than the pinion. And at first glance it 
 would appear perfectly feasible to adopt such proportions in pin gear- 
 ing, whether the annular wheel drive or follow. 
 
 But in the former case, the teeth being hypocycloids of which the 
 inner pitch circle is the generatrix, will become radial when the outer 
 one is of twice its diameter ; and if this limit be passed, making the 
 pinion more than half as large as the driver, the teeth of the latter 
 will become concave : this, although geometrically satisfying the con- 
 ditions, is manifestly impracticable. 
 14 
 
 FIG. 201. 
 
210 
 
 LIMITING NUMBERS OF TEETH. 
 
 If, however, the internal wheel drive, no such difficulty is met with, 
 and it may have nearly as many teeth as there are pins in the annular 
 follower, if they be duly shortened as the number is increased. 
 
 328. When the Pin is of Sensible Diameter, the action, as has been 
 shown, begins at a different time ; and in consequence of the peculiar 
 nature of the derived curve forming the working outline, it becomes 
 in general impossible to determine the limiting numbers with precis- 
 ion. For if the diameter of the pin be assumed, then because one 
 pitch circle is the generatrix and the other is the directrix of the ele- 
 mentary tooth, the diameters of both must be known before the point 
 Mot Figs. 198 and 199, which fixes the time when the action shall 
 begin, can be found graphically or otherwise. 
 
 In one case, however, this point can be located with exactness, as 
 shown in Fig. 202 ; DD, CC, being arcs of the pitch circles of the 
 
 driver and the follower re- 
 spectively, and having the 
 same radius. Let AO be 
 the radius of the pin, which 
 is here enormously exag- 
 gerated ; CC cuts its cir- 
 cumference at N and L, 
 and DD cuts it at M and S, 
 making the arcs AN, AM, 
 equal to each other. Rolling 
 CC upon DD, the point A 
 rises in the cardioidal curve 
 AB, and when N reaches 
 M, the chord AN will 
 have the position MB, nor- 
 mal to this curve and also 
 to the derived one. 
 
 The latter is composed of the two branches OP, PE ; the second 
 one, therefore, cuts the base circle and the circumference of the pin, 
 whatever the radius of the latter, at the same point, M. Since MA = 
 NA, the points M and j^will come together at A as the rotation in- 
 dicated by the arrow progresses : and moreover, M will then be in 
 correct driving contact, the normal MB at that instant coinciding 
 with the chord AL. 
 
 329, We have thus ascertained, that when the driver and the fol- 
 lower are of equal diameters, there is no arc of approach, the action 
 beginning on the line of centres, without regard to the size of the 
 pin. "We may now assume a diameter for the latter, and by a slight 
 
 FIG. 202. 
 
DETERMINATION OF LIMITING NUMBERS. 
 
 211 
 
 modification of preceding processes we can determine the least num- 
 ber of pins and teeth that can be used. 
 
 Let us suppose the pin to be of such size that the arc NL of Fig. 
 202 shall be one half the pitch ; then in Fig 203 we shall have AO = 
 li pitch, AL = pitch, and AF 1J pitch ; and the point P, of the 
 working tooth must at the end of the action lie at the intersection of 
 the chord OA with the radial line DP which bisects AF. 
 
 Now, were it true that the arc of approach is nil in all cases, the 
 limiting numbers could be de- 
 termined. The graphic pro- 
 cess is as follows. First, if the 
 number of teeth be assigned, 
 we make the angle ADP = f 
 pitch, set off on the tangent at 
 A, a distance AG equal to the 
 pitch arc AL, make AM = J 
 A G, then with centre M and 
 radius MG describe an arc cut- 
 ting the radial line DP in P. 
 The angle GAP thus deter- 
 mined is evidently equal to 
 ACN, and, therefore, equal to 
 -f of the pitch angle of the fol- 
 lower, whence the required 
 Jninimum number of pins is 
 easily found. Second, if the 
 number of pins be assigned, 
 then PA is known, and PF 
 must be equal to it; also 'the 
 arc A F when found must be 1| times the pitch arc ; this is of a 
 known length, and having set off AH equal to it on the tangent, we 
 can find F as in previous cases, thus determining the required least 
 number of teeth, which could drive the given follower on the suppo- 
 sition that there is no approaching action. 
 
 330. We must now proceed tentatively, for this supposition is true 
 only in the one case in which there are just as many teeth as there are 
 pins. If then we assume various numbers of either, the minimum 
 values as thus determined will in general be incorrect ; but the errors 
 will diminish as the number sought approaches equality with the 
 number assumed, and when that equality is reached the result will 
 be exact. 
 
 In this way we find, by computation, 
 
 FIG. 203. 
 
212 
 
 PIN" GEARING. 
 
 Fid. 204. 
 
 Least No. of Pins for 12 Teeth 11.62. 
 Least No. of Teeth for 12 Pins = 11.77. 
 
 Twelve teeth, then, can be used to drive twelve pins, but no less 
 numbers will answer for equal wheels when the pin is of the size above 
 assumed, which is that most commonly adopted in practice. If the 
 pin be made smaller these numbers may be reduced, the limit being 
 six for each wheel when the pin becomes a point. 
 
 331, The above process, with slight and obvious modifications in the 
 
 diagram, Fig. 203, 
 would be applicable 
 in every case, were the 
 amount of approach- 
 ing action known for 
 every given ratio of 
 the pitch diameters. 
 But it is not known; 
 and further progress 
 must be made in the 
 face of additional 
 obstacles due to the perverse nature of the derived curve. Consider- 
 ing these now in reference to outside gear, we find that when the 
 pitch circles are unequal, the point M of Fig. 202, in which this curve 
 cuts the circumference of the pin, no longer falls upon the base circle 
 of the epicycloid. If the driver be the larger, the second branch PE 
 cuts the pin first, as in Fig. 204, which represents the limiting case 
 of a driving rack, subsequently intersecting the base line at some 
 point JR. The latter will come into driving contact when it has ad- 
 vanced to A, but before that occurs the normal MB will have reached 
 the position M'AB', thus bringing the point Jf into action, and giv- 
 ing a greater or less arc of approach. If, on the other hand, the fol- 
 lower be the larger, the working branch of the curve will cut the base 
 circle before it does the circumference of the pin. The former inter- 
 section is the one which would come into action on the line of centres, 
 but since it is effaced in the process of forming the actual tooth, it 
 does not come into action at all, and the latter intersection cannot, 
 evidently, begin to drive until after passing that line, thus introduc- 
 ing the phenomenon of negative approach. 
 
 The limiting case in outside gear is that of the driven pin-rack, Fig. 
 205 ; it is very true that the process of constructing the derived curve 
 " reproduces the same involute in a different position " (PE}, but it 
 also introduces a part (PO) of the other branch ; to be sure, this is 
 
METHOD OF COMPUTING TABLES. 
 
 213 
 
 effaced, but so too is a portion (PM) of what would be the acting 
 tooth ; and M cannot drive until it reaches the position B' in the 
 path of contact CC, the negative approach being, therefore, equal to 
 AB'. 
 
 332. Now, the exact mathematical determination of this lowest 
 point M 9 even when the 
 diameters of both pitch 
 circles and the pin are 
 given, is a matter of ex- 
 treme complexity, to say 
 the least of it. And were Q_ 
 it never so simple, that 
 fact would be of service 
 only in determining the 
 limiting numbers by a 
 series of approximations, 
 since when one pitch circle 
 is given the other is to be 
 found. FIG - 205 - 
 
 Still, these numbers may be ascertained with a degree of precision 
 sufficient for all practical purposes, we think, by the method which 
 we have adopted in computing the subjoined tables. This is based 
 upon the assumption that the amount of approaching action, positive 
 or negative, can be determined as accurately as necessary by grapliic 
 processes. Which will appear the more probable, when it is taken 
 into account that in the actual working tooth, the obtuse intersection 
 M will be more or less rounded off, and not sharply defined ; and the 
 effect of this will be to diminish the amount of positive, and to in- 
 crease that of negative, approach. 
 
 Accordingly, it appearing from preliminary trials that about six 
 teeth will be the least that can drive a pin-rack, an accurate diagram 
 upon a large scale (the diameter of the pinion being six feet) gave for 
 a pin of the proportions named in (329), a negative approach equal 
 
 to ~-TT Then, assuming T ^ the pitch as the actual amount, we 
 have the equation 
 
 or 
 
 tan (f- pitch) = arc (1 + T 1 ^) pitch, 
 tan 10z = arc 13z ; 
 
 which being solved by the tentative process previously described, gives 
 the limiting number, 6.44, for the pinion driving a rack. 
 
214 
 
 GEAKING. 
 
 Seven teeth, then, will drive in outside gear ; the negative approach 
 would theoretically be a little less but in view of the consideration 
 above mentioned we have not reduced the amount, but have allowed 
 -j*g- the pitch in this case also. We have previously found that 12 
 teeth will drive 12 pins, the approach being exactly zero : what re- 
 mains,, then, is to take for the drivers between 7 and 12, a negative 
 approach diminishing as the numbers of teeth increase, and to test 
 the results of computation by graphic construction. 
 
 333. Similarly, we find in the case of a rack driving a pin-wheel, a 
 positive approach of about T \- pitch; which gives 4.64 as the least 
 number of pins that can be used. As the amount of approaching 
 action here named was determined graphically for a wheel of six pins, 
 it is probably less than it would actually be at the exact limit, and 
 we have allowed the same amount in the case of a wheel of 5 pins, 
 which, on this hypothesis, can be driven by one of 110 teeth. For 
 the pin- wheels between 12 and 5, then, we have a positive approach, 
 gradually increasing from zero to y\ of the pitch. 
 
 We repeat, that no pretension to theoretical exactness is here made, 
 the object being to furnish safe rules for practical guidance in using 
 the customary proportions. We have, therefore, in apportioning the 
 positive and negative approach, made the former rather less and the 
 
 latter rather greater than the 
 precise amount would proba- 
 bly be. It may be observed, 
 that the construction of the 
 nomodont for the whole range 
 from the driving to the driven 
 rack, affords a tolerably reli- 
 able check upon the grada- 
 tion of the approaching action, 
 since any serious error would 
 be manifested by the conse- 
 quent irregularity of the 
 curve. Besides, the graphic 
 constructions have been made 
 with considerable care and 
 upon a large scale ; so that upon the whole we think it safe to say 
 that if the radius of the pin be in each case made equal to half the 
 chord of the pitch arc upon the smaller circle, the numbers here 
 given will work ; and there is a strong probability that these limits 
 cannot be practically transcended. 
 
 334. Next, in regard to inside gear. When the annular wheel 
 
 E 
 
 FlG - 206 - 
 
DETERMINATION OF LIMITING NUMBERS. 
 
 215 
 
 drives, the positive approach is still greater than with the driving 
 rack, as will be seen by comparing Fig. 204 with Fig. 206 ; the point 
 R, which will come into driving contact with N, at A upon the line 
 of centres, being in the latter case much farther from M, the lowest 
 practicable point of the working tooth. 
 
 This increase in the approaching action reaches its limit when the 
 diameter of the larger pitch circle is twice that of the smaller ; the 
 hypocycloid then becomes a right line, the derived curve becomes 
 another one parallel to it, and whatever the position or the size of the 
 pin, the common perpendicular to these two lines through its centre 
 always passes through A. In the case represented in Fig. 197, then, 
 the approaching action continues during half the revolution of the 
 
 FIG. 207. 
 
 follower, and the receding action during the other half, each being, 
 therefore, equal to the whole pitch. A greater number of pins and 
 slots may be used, but the sliding blocks must then be dispensed with, 
 and the -angle of action will be reduced, its magnitude depending 
 upon the distance from the driver's centre at which the slots terminate 
 or intersect each other ; that is, in effect, upon the length of the teeth. 
 335. On the other hand, when the pinion drives, the negative ap- 
 proach becomes greater than in the case of a pin-rack. And here too 
 an exact determination of its amount can be made when the diameter 
 of the outer circle is twice that of the inner, as in Fig. 207 ; for the 
 original tooth-outline AB is in this case also a cardioid, and the lowest 
 point M of the working tooth falls upon the pitch circle, DD y of the 
 pinion. But the driver must now make two revolutions in order to 
 have the complete cardioid traced upon its plane by a marking point 
 in the circumference of the follower, whereas when two equal wheels 
 work in outside gear but one revolution is necessary : and the same 
 
216 
 
 PIN" GEARING. 
 
 proportion holds true for the generation of any given portion of the 
 curve. Therefore, instead of coming into driving contact at A, as in 
 Fig. 202, the point M must advance twice as far before its action 
 begins ; the point B comes into the original locus of contact at B', 
 the normal MB at that instant taking the position SB' : thus fixing 
 the arc of negative approach AS, which is readily computed when 
 the size of the pin is given. 
 
 Limiting Numbers of Teeth. 
 
 D. 
 
 F. 
 
 Minimum Value, 
 
 TEETH. 
 
 PINS. 
 
 
 6.44 
 
 00 
 
 INSIDE GEAR. 
 
 7 
 
 89 
 
 D. 
 
 F. 
 
 D. 
 
 F. 
 
 8 
 
 32 
 
 
 
 
 
 
 
 TEETH. 
 
 PINS. 
 
 
 
 9 
 
 21 
 
 
 
 4 
 
 2 
 
 10 
 
 16 
 
 3 
 
 4 
 
 16 
 
 3 
 
 11 
 
 14 
 
 4 
 
 8 
 
 36 
 
 4 
 
 12 
 
 12 
 
 5 
 
 14 
 
 TEETH. 
 
 PINS. 
 
 14 
 
 11 
 
 6 
 
 35 
 
 
 
 16 
 
 10 
 
 
 
 
 
 18 
 
 9 
 
 
 ANNULAR. 
 
 
 
 
 
 MAX. VALUES. 
 
 
 21 
 
 8 
 
 
 
 
 27 
 
 7 
 
 
 37 
 
 
 
 
 
 
 PIN GEARING. 
 
 110 
 
 5 
 
 Tooth = Space. 
 
 GO 
 
 4.64 
 
 
PIN GEARING. 
 
 Limiting Numbers of Teeth. 
 
 217 
 
 D. 
 
 TEETH. 
 
 F. 
 
 PINS. 
 
 D. 
 
 TEETH. 
 
 F. 
 
 PLANES. 
 
 D. 
 
 TEETH. 
 
 F. 
 
 PINS. 
 
 2.68 
 
 00 
 
 2.68 
 
 00 
 
 2 
 
 9 
 
 3 
 
 27 
 
 3 
 
 54 
 
 
 
 
 
 4 
 
 11 
 
 4 
 
 21 
 
 F. 
 
 D. 
 
 
 
 
 
 PINS. 
 
 TEETH. 
 
 5 
 
 8 
 
 5 
 
 13 
 
 
 
 6 
 
 6 
 
 6 
 
 12 
 
 3 
 
 25 
 
 8 
 
 5 
 
 7 
 
 11 
 
 2 
 
 4 
 
 16 
 
 4 
 
 Q 
 
 10 
 
 
 
 oo 
 
 3.29 
 
 11 
 
 9 
 
 
 . 
 
 16 
 53 
 
 8 
 
 7 
 
 INSIDE GEAR. 
 MAX. VALVES. 
 
 EECESS = PITCH. 
 
 
 
 
 PIN = POINT. 
 
 00 
 
 6.58 
 
 
 336. The Path of Contact in Pin Gearing. In the elementary form, 
 the pin being a mathematical point in the circumference of the fol- 
 lower's pitch circle, that circumference is itself the locus of contact. 
 
 When the pin has sensible diameter, its centre yet lies always in 
 that circumference ; the common normals to the pin and its driving 
 tooth are chords of the circle, all passing through the point A, Fig. 
 208. Supposing the driver to turn to the right, then, we have only 
 to set off on each of these chords a distance equal to the radius of the 
 pin, measured from the circumference toward the left : the line join- 
 ing the points thus located, which is the curve called the limacon, is 
 the path of contact. 
 
 This might also be constructed from the outline of the working 
 tooth, by the process of Fig. 164 ; but the method above described is 
 more convenient ; and, as will be found more fully set forth in the 
 
218 
 
 PRACTICAL POINTS NOISE AND VIBRATION. 
 
 Appendix, it involves principles which enable us to determine with 
 greater accuracy than would otherwise he possible, certain critical 
 points not only in the path of contact itself, but in the tooth outline 
 also ; since the latter, evidently, may by reversing the construction 
 be derived from the limacon. 
 
 Some Practical Considerations. 
 
 337. Noise and Vibration. Although not coming strictly within the 
 scope of this treatise, the practical ill effects of using wheels with 
 incorrectly shaped teeth are so closely connected with the subject as 
 to demand a brief notice. 
 
 It is to be observed, then, that the noise and the vibration which 
 often attend the action of toothed gearing, especially at high speeds, arc 
 
 not necessarily identical in origin. 
 Jt is true that the causes which 
 produce noise will also produce vi- 
 bration ; but vibration may be pro- 
 duced by other causes, and may at 
 least be imagined to occur without 
 noise. 
 
 To explain; suppose two wheels 
 of perfect form and finish to gear 
 with each other, the power and the 
 resistance being absolutely uniform; 
 then, whatever the amount of back- 
 lash, there would be neither vibra- 
 tion nor noise. Now in practice, 
 this uniformity of power and re- 
 208. sistance seldom or never exists, and 
 
 the variations in speed cause the fronts, and often the backs, of the 
 teeth to strike together at short intervals. These blows cause a rat- 
 tling noise, which is worse the higher the speed, and is accom- 
 panied by vibrations due to the impact between the teeth. 
 
 The sole cause of the noise, evidently, is the existence of backlash ; 
 but even were the teeth so perfect as to have no backlash at all, these 
 irregularities in the power and the resistance would still give rise to 
 vibrations, more or less injurious according to the suddenness of the 
 changes ; they would, however, take place in quiet. 
 
 338. But again, vibratory action may result from a totally different 
 cause, namely, incorrect forms of the teeth. 
 
 To illustrate this, imagine two engaging wheels, whose teeth are, as 
 before, of perfect finish, but not of proper contour ; let the speed of 
 
EFFECTS OF WEAR. 219 
 
 the driver be absolutely uniform, and the resistance such as to keep 
 the acting outlines always in contact, so that there is none of the rat- 
 tling above mentioned. 
 
 The average velocity ratio will be correct ; if the driver has, for 
 instance, 100 teeth and the follower 50, each revolution of the former 
 will cause two revolutions of the latter. And further, T J of a revo- 
 lution of the driver will cause -^ of a revolution of the follower ; but 
 during this fractional motion the velocity ratio is not constant, 
 the follower being driven too rapidly during one part of the action, 
 and too slowly during the other part. 
 
 Thus the action of each pair of teeth, though correct as a whole, is 
 faulty in detail, being made up of two counterbalancing errors. The 
 speed of the driver is uniform, but that of the follower is fluctuating ; 
 its motion consists of a series of pulsations, not necessarily audible 
 at low speeds, though practically certain to become so at high ones. 
 But even at moderate velocities, this vibration acts injuriously upon 
 the whole mechanism ; and in many cases it is easy to see that the 
 perfection of the work done by the machine may be impaired by the 
 irregularity of its motion, no matter how slowly it runs. 
 
 339. Effect of Wear upon Teeth of Incorrect Forms. It is clearly 
 poor economy to require a machine to finish itself, even were it always 
 able and willing to do so, since in the meantime its proper work must 
 be imperfectly executed. Still it is sometimes asserted that teeth 
 originally of incorrect contour will "wear into shape" and improve 
 by use. 
 
 Now wear means abrasion ; so that the only errors which could 
 thus correct themselves, by the removal of superfluous material, must 
 be errors of excess, not of deficiency. But these are the very errors 
 which would hardly be considered admissible by the most zealous 
 defender of bad workmanship, as a little reflection will show. Let 
 us suppose that of two wheels, accurately spaced, the faces only of the 
 driver's teeth are too full, all the other curves being correct. Under 
 these circumstances, it will be seen that during the receding action, 
 the leading tooth of the driver will urge the follower too rapidly ; in 
 order to admit of this, then, considerable backlash is necessary. A 
 little space will be thus left between what should be the acting curves 
 of the following pair ; which cannot come into action until the lead- 
 ing tooth quits contact, allowing the follower for an instant to wait 
 for the driver to overtake it. Therefore, in addition to the disadvan- 
 tage of having all the work done by one tooth at a time, there will be 
 a constant clattering as the teeth come successively into driving 
 contact. 
 
220 EFFECTS OF WEAR. 
 
 If, on the other hand, the faces of the follower only are too full ; it 
 is then the last instead of the leading tooth which will have the work 
 to do, and the follower will be driven beyond the normal speed during 
 the approach. 
 
 But it will be driven most rapidly during the first part of the ap- 
 proach, its speed diminishing as the point of contact nears the line of 
 centres : and though there will not be an absolute halt or interruption 
 of the action, as in the previous case, there will still be something 
 like a blow as each tooth of the driver comes into contact, and the 
 wheels will be nearly as noisy as before. 
 
 Like reasoning applies if the faces be made correct and the flanks 
 too full ; so that practically the only errors in the forms of the teeth 
 which the worst of workmen would admit in the rudest of mechanism, 
 are errors of deficiency. These may cause less noise, but they will 
 certainly cause the pulsation above described ; and when they correct 
 themselves by wear, we shall be called upon to contemplate the phe- 
 nomenon of growth by abrasion . 
 
 340. Wearing to a Bearing. The fallacy above mentioned, if 
 crushed by theory, is pulverized by practice ; for we believe that the 
 first case has yet to be found, where teeth whose original contours were 
 not full enough, have worn into forms which made the velocity ratio 
 more nearly constant ; but every repair shop can furnish evidence 
 that in this respect such teeth have gone from bad to worse. 
 
 Nevertheless, it cannot be disputed that gear wheels, both cast and 
 cut, very frequently run better after being a while in use. In the 
 case of cast wheels, this may be due to the removal by abrasion of 
 rough and irregular spots in the metal ; but in both cases it may be 
 due to other causes. It is likely to happen, especially if the wheels 
 be of considerable thickness, that, owing to some slight error either 
 in the process of cutting or in the keying of the wheel upon the shaft, 
 some of the elements of the teeth are not exactly parallel to the axis. 
 And what is technically called " wearing to a bearing," resulting in a 
 general improvement of the action in smoothness if not in constancy 
 of the velocity ratio, consists in point of fact not in the accretion of 
 metal, where a deficiency exists, but in the removal of so much as 
 stands in the way of line contact between the teeth ; and the gain is 
 mainly due to a more equal distribution of the pressure. 
 
 341. Effects of Wear in the Bearing. In order to preserve a constant 
 velocity ratio, the teeth of a pair of wheels must be conjugate to each 
 other ; and if they be of other than involute form, it is also necessary 
 that the distance between centres shall be equal to the sum of the 
 radii of the pitch circles. But unless special provision is made for 
 
EFFECTS OF WEAR IN THE BEARINGS. 
 
 221 
 
 readjustment, which cannot always bo done, this distance is subject 
 to variation on account of wear in the bearings, so that the wheels 
 after a time will either mesh too deeply, or not deeply enough ; and 
 it is easy to see that if the line of centres be thus made too short or 
 too long by a given amount, the effect upon the velocity ratio may be 
 greater with one tooth-system than with another, although both may 
 originally work with perfect precision. 
 
 It will subsequently be shown that it is just as feasible to give a 
 practical embodiment to Sang's theory as to the epicycloidal system, 
 and to produce accurate templates for interchangeable wheels, derived 
 from any assumed rack, by automatic machinery. When this shall 
 have been done, the question will naturally arise, which of the various 
 
 MAC CORD'S ODONTOSCOPE. FIG. 209. 
 
 tooth-systems thus placed within equally easy reach will be affected 
 to the least extent by the disturbing causes above pointed out ; for 
 the one which is, will clearly be the best for many purposes. 
 
 342. The Odontoscope. In order to answer this question, it is neces- 
 sary to have some convenient means of comparing the actions of dif- 
 ferent tooth-systems under varying conditions. Neither graphical 
 nor analytical treatment appearing eligible on account of the exceed- 
 ingly intricate nature of the problems involved, the author some years 
 since devised the apparatus of which the principle is illustrated in 
 Fig. 209. This was before the introduction of automatic machinery 
 for the shaping of gear cutters, and the original object was to test the 
 accuracy of such as were made by hand. 
 
222 MAC COED'S ODCXNTOSCOPE. 
 
 For this purpose, two templates, A and B, are cut out of tolerably 
 thick sheet metal, corresponding to the teeth of a pair of wheels ; 
 these are secured to arms turning about the axes C and 2), whose 
 distance from each other is capable of adjustment. The shaft of D 
 carries a graduated limb, K, and may be slowly rotated by means of 
 the arm, L, and tangent screw, E, or any equivalent 'device ; motion 
 will thus be communicated to C, the tooth A being kept in contact 
 with B, by a light weight or spring, not shown. 
 
 J^and G are small cylindrical barrels, accurately turned to diameters 
 having the same ratio as those of the pitch circles. F is fixed on the 
 axis (7, while ft, which carries a pointer, H, turns freely on the axis 
 D, with which it is connected by a spring, not shown ; the tendency 
 being to wind up on G the fine flexible wire /, secured to both bar- 
 rels in the manner of a crossed belt. Thus F and G will always turn 
 in opposite directions, with a velocity ratio perfectly constant from 
 the nature of the connection. The velocity ratio of the motions of 
 C and D, however, is determined by the templates A and B, and will 
 not remain constant unless their contours are strictly conjugate. 
 
 The pitch circles being scribed on the templates, it will be apparent 
 that when the intersections of these circles with the contours of the 
 teeth are brought into coincidence, the centres of motion are at the 
 correct distance from each other, and the point of contact is also 
 exactly on the line CD. By turning the tangent screw in one direc- 
 tion or the other, then, we may examine the action during the arc of 
 approach or that of recess at pleasure ; if the templates are correctly 
 shaped, the limb K and the pointer ^Twill move at the same rate and 
 in the same direction, so that if the latter be set at zero on the grad- 
 uated arc, it will remain at zero throughout the action ; any inac- 
 curacy being manifested by "a movement of the pointer over the limb. 
 
 343. The sensitiveness of the instrument as designed for actual 
 work is increased by the introduction of multiplying gear between 
 the barrel or pulley G and the pointer H\ thus producing a greater 
 deflection for an error of given magnitude, and causing a very 
 minute one to proclaim its existence. Its exact locality, however, we 
 can not fix ; that, indeed, would be equivalent to determining the 
 point of contact of two tangent curves ; and if the end in view is the 
 correction of errors in the contours of the templates, the only re- 
 source is careful inspection with a magnifying glass. 
 
 But for comparative work like that previously alluded to, this in- 
 strument is well adapted. Suppose that the templates work with 
 perfect accuracy when the centres are correctly adjusted ; by means 
 of a screw, not shown, the bearing of G may now be moved either to 
 
TEETH OF NO^-CIRCULAR WHEELS. 
 
 223 
 
 the right or the left through any given distance : then let the deflec- 
 tion of the pointer be noted for, say every degree or half degree of the 
 angle of action. If this be repeated after other changes in the distance 
 CD, a comparison of these records with similar ones for a different 
 pair of templates, will furnish all needed information as to the effects 
 of wear in the bearings upon the velocity ratio in the two systems of 
 gearing represented. 
 
 Teeth of Non- Circular Wheels. 
 
 344. When two pitch curves of any form roll together about fixed 
 axes, the point of contact always lies upon the line of centres, and 
 divides it into segments inversely proportional to the angular velocities 
 at the instant. 
 
 Through this point there must pass 
 the common normal of any two con- 
 jugate tooth-curves ; these, conse- 
 quently, must satisfy the condition of 
 (181), viz.: they can be traced upon 
 the planes of rotation during the 
 action, by a marking point invariably 
 connected with a describing line which 
 moves in rolling contact with both 
 pitch curves. 
 
 The forms of non-circular wheels 
 are so numerous and diverse, that ifc 
 would be an endless task to discuss at 
 length the various possibilities rela- 
 ting to the paths of contact, limiting FIG. 210. 
 numbers of teeth, and matters of kindred nature. 
 
 But the use of such wheels is so limited that the consideration of 
 all these questions has by no means the importance or interest which 
 attach to it in relation to ordinary gearing : and we shall accordingly 
 be content with briefly explaining a purely practical method of con- 
 structing the teeth by graphic operations. 
 
 345. Laying out the Teeth of Elliptical Wheels, In Fig. 210, let AO 
 be the semi-major and 10 the semi-minor axis of a pitch ellipse. 
 Although no part of this curve is truly circular, yet circular arcs may 
 be found which agree with it so closely that for all practical purposes 
 they may be substituted for it. 
 
 Thus by trial and error we find a centre, (7, on the major axis, such 
 that an arc described about it will coincide with the ellipse in the 
 region of the vertex A, so nearly that the difference will not be per- 
 
224 ELLIPTICAL GEARING FORMS OF TEETH. 
 
 ceptiblc, even when the curve is drawn with the finest visible line. 
 The deviation will, however, become apparent if the arc be extended, 
 and it should be terminated at some point, B, a little before this limit 
 has been reached. If the closest possible approximation is desired, a 
 normal through B should pass througli C ; if on trial this prove not 
 to be the case, a minute change in the position of either B or (7, or 
 both if necessary, will cause this condition to be satisfied. On BO 
 produced, we now seek for another centre D, such that an arc BE 
 described about it through B shall sensibly coincide with the contin- 
 uation of the elliptical contour. Proceeding in this way, we shall 
 finally have a centre, H, upon the minor axis, for the arc GI, which 
 completes the quadrant of the ellipse. 
 
 For the purpose of illustration, we have in this diagram represented 
 an ellipse whose eccentricity is much greater than that of any likely 
 ever to be used as the pitch curyc of a wheel ; yet a very perfect ap- 
 proximation is made by means of the four circular arcs for each 
 quadrant ; probably three will suffice for any case to be met with in 
 practice, and in most instances two answer the purpose abundantly 
 well. 
 
 346. The next step is to subdivide the perimeter of the ellipse for 
 the location of the teeth and spaces. The outline now being com- 
 posed of circular arcs, these are rectified by Prof. Rankine's graphic 
 process. Thus in the figure, we lay off upon the tangent at G, the 
 lengths GE' 9 GT of the arcs GE, GI\ similarly upon the tangent at 
 By the lengths of BE and BA are laid off, and their sum being added 
 to I'E' already found, gives I' A' equal in length to the quadrant 
 IEA. This right line is next to be properly subdivided according to 
 the number of teeth determined on, and the points of division are 
 then transferred to the ellipse by means of Prof. Rankine's converse 
 process applied to the approximating arcs and their rectifications : 
 this spacing is indicated in the diagram by the alternating fine and 
 heavy lines on both tangents and contour. 
 
 A circle is used for generating the tooth-outlines, rolling it first 
 within the ellipse to trace the flanks, and then without to describe the 
 faces ; and thus this operation is made identical with that for circu- 
 lar wheels. The radius, AL, of this describing circle should not be 
 greater than -J- AC, which limiting value will give radial flanks to the 
 teeth whose edges cut the ellipse upon the approximating arc AB ; 
 the other flanks being concave. 
 
 347. The complete wheels in gear are represented in Fig. 211. In 
 regard to the number of teeth, it makes no difference whether it be 
 even or odd ; but in either case it is eminently desirable that the two 
 
ELLIPTICAL GEARING ARRANGEMENT OF TEETH. 
 
 225 
 
 wheels should be alike. In order to make them so, if there be an odd 
 
 number of teeth, they should be so arranged that one extremity of 
 
 either the major or the minor 
 
 axis shall bisect a tooth, the 
 
 opposite extremity bisecting a 
 
 space. If the number of teeth > //\ h 
 
 be even, then they must be so 
 
 placed that the fronts of two of 
 
 them shall cut the pitch ellipse 
 
 at opposite extremities of one of 
 
 the axes. 
 
 Now, the action of the teeth 
 on different parts of the peri- FIQ. 211. 
 
 meter cannot, under any circumstances, be equalized in all particu- 
 lars. In order to secure the same amounts of approaching and reced- 
 ing action, it would be necessary to make the faces of varying lengths ; 
 and even then the result is of doubtful utility, cincc the degree of 
 obliquity is continually changing. Consequently it is held to bo suf- 
 ficient in practice to make the. faces of uniform length ; and this 
 should be such as to ensure that in the part of the whole movement 
 in which the obliquity of action is greatest, one pair of teeth shall not 
 quit contact until the next pair has fairly come into engagement. 
 Thus the form of the blank is determined by drawing a curve par- 
 allel to the pitch ellipse at a given distance without, the bottoms of 
 the spaces are bounded by another at a given distance within, and the 
 two blanks, secured side by side, may be cut at the same time with 
 
 milling cutters precisely like those used for 
 circular wheels. 
 
 It is not necessary to furnish elliptical 
 wheels with teeth all round their periph- 
 eries, when used under circumstances which 
 permit their free foci, as in Fig. 212, to be 
 connected by a link. In this case, a few 
 teeth near the extremities of the major axis, 
 as shown, suffice to carry the wheels, past 
 the dead points, which occur when the line 
 of the link coincides with the line of cen- 
 tres. The remaining portions of each wheel 
 must be formed exactly to the contour of 
 the pitch ellipse ; these will then transmit the rotation by direct and 
 purely rolling contact, in either direction, so long as the contact 
 radius of the driver is on the increase, (88). Since while the radius 
 15 
 
 FIG. 212. 
 
226 ELLIPTICAL WHEELS INVOLUTE TEETH. 
 
 is on the decrease, the work must be done by the link, it is necessary 
 to use a sufficient number of teeth to prevent excessive obliquity of 
 action while the link is thus made the sole means of transmission. 
 
 348. Elliptical Wheels with Involute Teeth. It is worthy of notice, 
 though more as a matter of abstract interest than cf practical mo- 
 ment, that the teeth for a pair of pitch ellipses may be made in the 
 form of involutes of smaller base ellipses having the same foci. In 
 Fig. 213, let be the fixed and A the free focus of the ellipse whose 
 major axis is XX. Let J9, at any distance from C, be the fixed focus 
 of an equal and similar ellipse, the position of the major axis YY 
 being found as follows : about D describe an arc with radius = CA, 
 and about A an arc with radius CD ; these arcs intersect at B, 
 
 FIG. 213. 
 
 which is the free focus of the second ellipse. Draw CD and AB, 
 intersecting at P. The common tangent to the two ellipses, Off, 
 will also pass through P (see Appendix) ; and if we suppose it to 
 represent an inextensible band, wrapped around and secured to the 
 ellipses, it will be wound upon the right hand one if it turn about 
 C as shown by the arrow, and unwound from the other, which will 
 thus be caused to revolve about D, with a definite varying velocity 
 ratio. During this motion, a marking-point fixed at P in this band, 
 and moving with it, will trace upon the planes of rotation the invo- 
 lutes of the ellipses, KPL, OPL These curves, therefore, if used as 
 the outlines of teeth, will maintain the same velocity ratio that was 
 originally established by the band. 
 
 349. But, as will be seen by comparing this diagram with Fisr. 55, 
 
ELLIPTICAL WHEELS INVOLUTE TEETH. 227 
 
 P is also the common point, and TPT is the common tangent, of two 
 ellipses confocal with the original ones, whose major axes are WV, 
 UZ, equal to each other and to CD ; and these will roll in contact 
 about the fixed foci. 
 
 Thus four different elementary combinations are represented in this 
 figure ; we may use either 
 
 1. The levers CA, DB, connected by the link AB ; 
 
 2. The smaller ellipses, connected by the band GH\ 
 
 3. The involutes KL, 10, acting with sliding contact ; or 
 
 4. The larger ellipses, acting in pure rolling contact. Or, these 
 may be used simultaneously, the velocity ratio being the same in each 
 
 GW CV 
 at any and every instant, and varying between the limits ~7Ty9 ~7rwf- 
 
 350. The distance CD being arbitrary, it follows that for the same 
 base ellipses, an infinite number of pitch ellipses may be assigned. 
 And on the other hand, for the same pitch ellipses, the teeth may be 
 involutes of any base ellipses having the same foci. From which the 
 deduction is, that so long as the teeth of this form engage at all, they 
 will gear correctly in the sense that the action will be equivalent to 
 the rolling together of a pair of true ellipses of some form : but 
 whereas in the case of circular wheels the velocity ratio was not 
 affected by altering the distance between the centres, it is to be noted 
 that in elliptical gearing the limits between which the velocity ratio 
 varies, will be affected by such alteration. For the focal distance CA 
 is invariable, while the major axis WV is always equal to CD ; so 
 that a change in the length of the line of centres changes the eccen- 
 
 CV 
 
 tricity of the pitch ellipse and the value of r . 
 
 If then it be essential that the limits of variation should be exactly 
 maintained, the involute form of tooth possesses no advantage over 
 that previously described ; and it is hardly necessary to add that if 
 the free foci of a pair of elliptical wheels which arc provided with 
 teeth, be connected by ti link, the distance between the fixed foci must 
 be kept constant, on whatever system the teeth are laid out. If the 
 involute system be adopted, care must be taken lest the base ellipses 
 be so small as to produce excessive obliquity of action. In regard to 
 this, Prof. Kankine, to whom is due the credit of first discussing this 
 form of tooth for elliptical gearing, observes that the common tan- 
 gent should cut the fronts of at least two teeth between the base and 
 the pitch ellipse ; which, although it may not be necessary in all cases 
 in order to ensure the transmission of rotation, is unquestionably a 
 safs practical rule. 
 
228 
 
 TEETH OF LOBED WHEELS. 
 
 351. Construction of Teeth of Lobed Wheels, As above intimated, the 
 method of operation first described is perfectly general, and may be 
 employed with pitch curves of any form. The centres of the circular 
 arcs of which the contour is to be made up, may be found by assum- 
 ing a number of points, at which the curvature begins sensibly to 
 change, and drawing normals through these points (graphic processes 
 for which are described in the Appendix) ; the intersections of the 
 normals being the required centres. 
 
 But if the pitch curves be carefully drawn, it will be found that for 
 all practical purposes, these centres can be located with sufficient 
 accuracy by trial and error ; and at best the laying out of teeth for 
 such wheels requires much time, care, and patience. 
 
 If the same describing circle be used throughout, its diameter 
 should be such as to give radial flanks to the teeth in that part of the 
 pitch line where the curvature is greatest ; should other parts be very 
 much flatter the teeth may in consequence spread too rapidly at the 
 root or in the flank. This may be remedied by using different de- 
 scribing circles for the teetli in those parts, care being taken that the 
 same one be always used for the conjugate face and flank. 
 
 352. Practical Limit to the Obliquity. In any form of gearing it 
 would be very desirable to have the line of action perpendicular to 
 
 FIG. 214. 
 
 the line of centres, since then there would be no component of side 
 pressure. 
 
 In speaking of the teeth of circular wheels, the obliquity was de- 
 fined as the angle made by the common normal of the acting curves 
 with the common tangent to the pitch circles. This, however, was 
 because this tangent is then a perpendicular to thfe line of centres; 
 
LOBED WHEELS LIMIT OF OBLIQUITY. 
 
 229 
 
 and in general the obliquity is the inclination of the line of action to 
 such a perpendicular, which latter may or may not coincide with the 
 common tangent to the pitch curves. This it seldom does in the 
 class of wheels now under consideration, and in consequence a much 
 greater obliquity than would be admissible in circular gearing is often 
 unavoidable. 
 
 Thus in Fig. 214, VW, XY are arcs of the pitch curves, in con- 
 tact at P, on the line of centres CD, to which PO is perpendicular. 
 The lower wheel driving, as shown by the arrow, let E be the point 
 of tooth-contact, the line of action through which, PEN, has the 
 greatest inclination to PT, the common tangent to the pitch curves. 
 
 It will then be apparent 
 that when the angle TPO, 
 which this tangent makes 
 with the perpendicular PO 
 is greatest, the total obliquity 
 NPO will be a maximum. 
 
 The results of experience 
 show that the angle TPO FIG. 215. 
 
 should not, if it be practicable to avoid it, exceed from 25 to 30. 
 In other words, the contours of the non-circular pitch curves should 
 be such that the angle between the tangent and the radius vector 
 shall at no point be less than 60 or 65. Nor should the angle NPT 
 exceed from 15 to 20 ; thus limiting the range of the maximum 
 total obliquity, NPO, to from 40 to 50; though if the object be 
 rather the modification of motion than the performance of heavy 
 work, higher values may in extreme cases be used. 
 
 353. This limit, it will be perceived, has not been regarded in Fig. 
 215, which shows the appearance of the dissimilar pitch curves of Fig. 
 75, when furnished with teeth. 
 
 In Fig. 216, we have a pair of similar but unsymmetrical uni- 
 lobes, the pitch lines being elliptical below the horizontal centre 
 line, and of the logarithmic spiral form above, as in Fig. 72. 
 
 The lower halves of the 
 dissimilar pair shown in 
 Fig. 217 are alike, the log- 
 arithmic spiral being the 
 pitch curve ; while above 
 the line of centres, that 
 spiral is used for a part of 
 FlG> 216< the contour, the remainder 
 
 being elliptical, as in Fig. 74. But the wheels here represented pre- 
 
230 
 
 LOBED WHEELS VARIOUS EXAMPLES. 
 
 sent the peculiarity that the limits of variation in the velocity ratio, 
 
 which suddenly change when the wheels reach the position given in 
 
 the figure, are not the 
 same for the two halves 
 of the revolution. 
 
 A still greater and 
 more abrupt transition 
 is produced by the pair 
 represented in Fig. 218, 
 of which the pitch 
 curves are similar and 
 equal logarithmic 
 spirals. It is tolerably 
 evident that in this case 
 the absolute velocity 
 must be small in order 
 to avoid violent shocks 
 at the instants of the 
 change : yet wheels very 
 similar to these have 
 
 been used as for instance in the moulding machine of Gallas and 
 
 Aufderheide.* 
 
 354. Non-Circular Pin 
 
 Gearing. The labor of 
 
 laying out the teeth of 
 
 a pair of dissimilar 
 
 wheels may be abridged, 
 
 and the difficulty of 
 
 cutting them in part 
 
 avoided, by making one 
 
 of them after the man- 
 ner of a pin-wheel. The 
 
 principle of the process 
 
 (which may also be ap- 
 plied to the ellipses or 
 
 to any similar as well as 
 
 to dissimilar lobed 
 
 wheels) is precisely the FIG. sis. 
 
 same as in the case of circular gearing. That is to say, the pitch 
 
 curve, of the follower is used as the describing line, by rolling which 
 
 *For description, see "Mechanics" of February 18, 1882. 
 
PI^-GEARI^G FOR LOBED WHEELS. 
 
 231 
 
 upon the pitch curve of the driver are generated the elementary 
 
 teeth for the latter ; and the 
 
 working outlines are at a 
 
 constant normal distance 
 
 within these, equal to the 
 
 radius of the pin, as shown 
 
 in Fig. 219. 
 
 In regard to the size of the 
 pins, and their distance from 
 each other, it is sufficient to 
 say that, as the contour of 
 each wheel is regarded as 
 made up of approximating 
 circular arcs, the whole op- 
 eration is reduced to the fa- 
 miliar one of constructing epi- 
 cycloids and their parallels; 
 so that the limits in relation 
 to the pitch and the diameter 
 of the pins are readily de- 
 duced for each particular case 
 by application of the reason- 
 ing given in the discussion 
 of circular gearing of this 
 description. Though it may FIG. 219. 
 
 be well to repeat that the teeth in all cases should be as small and as 
 numerous as they can be made consistently with securing the neces- 
 sary strength. 
 
CHAPTER XII. 
 
 1. THE TEETH OF BEVEL WHEELS. General Principles of their Correct Forma- 
 
 tion by means of a Describing Cone. Tredgold's Approximate Method. De- 
 tails of correct Process. Results of the two Methods Compared, for Epicy- 
 cloidal Teeth. Inside Bevel Gearing. The Involute System. Action of 
 Bevel and Spur Wheels compared. Teeth of Conical Lobed Wheels. Methods 
 of Cutting the Teeth. Corliss's Bevel Gear Cutter. Twisted Bevel Wheels. 
 
 2. THE TEETH OF SKEW WHEELS. Theory of Willis and Rankine. Describing 
 
 Hyperboloid. Approximate Methods. Direct Construction of Teeth thus Gen- 
 crated. Fallacy of this Theory. Such Teeth Impracticable. A New Theory. 
 Oliviers's Involutes in Different Planes. The Fronts of the Teeth Single 
 Curved. They Vanish at the Gorge. The Length of the Tooth Limited. 
 Backs of Teeth Warped. The Conjugate Teeth Unlike. Action Reversed on 
 Crossing the Gorge. Twisted Skew Wheels. 
 
 3. THE TEETH OF SCREW WHEELS. Common Worm and Wheel. Construction 
 
 Referred to that of Rack and Pinion. Distinctive Features of the Action. 
 Close-fitting Tangent Screw. Practical Proportions. Sang's Theory Embodied. 
 Multiple- threaded Screw Wheels. Screw and Rack. Oblique Screw Gear- 
 ing. Construction of Teeth from Oblique Rack and Pinion. Peculiar modi- 
 fication of the Action. Close-fitting Oblique Worm. Oblique Screw and Rack. 
 Construction for Least Amount of Sliding. Resemblance to Skew Gearing. 
 Hour-glass Worm Gear. General Arrangement. Form of the Pitch Surface of 
 the Worm. Forms of the Threads. Action Confined to one Plane. Forms 
 of the Wheel-teeth. Multiple- threaded Hour-glass Worm with Face-gear 
 Wheel. Rollers Substituted for Teeth upon the Wheel. 
 
 4. THE TEETH OF FACE WHEELS. Equal Wheels with Cylindrical Pins. Equal 
 
 Wheels with Axes Perpendicular to each other. Unequal Wheels Similarly 
 Situated. Unequal Wheels with Axes in Different Planes. Miscellaneous 
 Arrangements of Face Wheels. Combination of Face and Screw Gearing. 
 Spherical Screw and Wheel. 
 
 355. In Treating cf Spur Wheels it has been convenient, since all 
 the transverse sections are alike, to consider all the motions as taking 
 place in one plane, and thus to deal with lines instead of surfaces. But 
 we must not lose sight of the fact that the pitch and describing curves, 
 and also the tooth-outlines, are but the bases of surfaces with elements 
 perpendicular to the paper, and acting in right-line contact. Thus 
 in the epicycloidal system, for instance, we must imagine a describing 
 cylinder rolling upon the pitch cylinders, the common element being 
 
GENERATION OF THE TOOTH SURFACE. 
 
 233 
 
 UNIVERSITY 
 
 FIG. 220. 
 
 the instantaneous axis, about which the describing line is revolving 
 
 at any given phase of the action. Now the teeth thus generated 
 
 must touch each other along a 
 
 right line: and this principle 
 
 is capable of a more extended 
 
 application. The very slight 
 
 modification by which spur 
 
 wheels are transformed into 
 
 bevel wheels is at once suggested 
 
 by regarding the pitch cylinder 
 
 as the limiting form of the cone, 
 
 in which the vertex is infinitely 
 
 remote ; and this leads directly 
 
 to the analogous generation of 
 
 the tookh surface by the rolling 
 
 of a describing cone in contact 
 
 with the pitch cones. 
 
 This is illustrated in Fig. 220 ; supposing the arcs AP, AS, to 
 be equal, then while the smaller cone rolls upon the larger, the ele- 
 ment CP will generate the surface CPB ; to which the plane CPA is 
 normal, since at the phase here represented, CA is the instantaneous 
 axis. This is the essential property, and is independent of the forms 
 of the bases of the cones ; but if as here shown these be circular, the 
 extremity P of the generating radius will trace a curve, BP, lying on 
 the surface of a sphere, and called, since it is described by the rolling 
 of one circle upon another (although the two are not in the same 
 plane), a spherical epicycloid ; which may be considered as the direc- 
 trix of the conical surface CPB. 
 
 In like manner, by the rolling of one cone inside of another, both 
 having circular bases, a spherical hypocycloid may be described. And 
 whatever the bases of the cones, it is evident that a surface is gener- 
 ated in this case also, to which the plane determined by the describing 
 line and the element of contact at any instant, will be normal. 
 
 It is thus readily seen that, the describing cone being tangent ex- 
 ternally to one pitch cone and internally to the other, these two sur- 
 faces will be swept up simultaneously as the rotation progresses, and 
 will at every instant have a common normal plane, cutting the plane 
 of the fixed axes of rotation always in the common element of the 
 pitch cones : they are, therefore, correct forms of tooth -surf aces, and 
 will maintain the original velocity ratio, be the same constant or 
 variable. 
 
 356. Tredgold's Approximation. Before considering in detail the 
 
234 
 
 APPROXIMATE CONSTRUCTION OF TEETH. 
 
 FIG. 221. 
 
 practical operation of laying out the teeth in accordance with the 
 exact theory as above set forth, it is proper to describe the method 
 more commonly employed, which involves possibly a little less 
 
 labor, and gives results, not 
 rigidly accurate, but suffi- 
 ciently so for many purposes 
 under ordinary conditions. 
 
 The principle of this proc- 
 ess, which, it should be ob- 
 served, is applicable only in 
 the case of circular wheels, is 
 shown in Fig. 221. Let CA, 
 CB, in the plane of the paper, 
 be the axes, and CP the com- 
 mon element, of the pitch 
 cones CPE, CPF; whose bases EP, PF, are small circles of the 
 sphere shown in dotted outline. Draw A PB perpendicular to CP; 
 then AP by revolving around CA will generate the cone PAF, and 
 BP, by revolving around CB, will generate the cone PBE : these 
 cones are tangent to the sphere, and respectively normal to the pitch 
 cones. 
 
 Now, AB is also the trace of a plane tangent to the sphere at P, 
 and tangent to both normal cones ; and, in the diagram at the right, 
 the points A', P', B', are the projections of A, P, and B, upon this 
 tangent plane, and the arcs OP'M, LP'N, are the developments of 
 the bases FP, PE. These arcs are next to be treated as though they 
 were the pitch circles of spur wheels, and teeth are to be laid out 
 upon them according to any of the systems previously explained, 
 that of pin-gearing of course excepted. 
 
 Suppose these teeth to be cut out of a thin sheet of metal, and then 
 wrapped back upon the normal cones ; their outlines are then to be 
 traced, and treated as the directrices of the conical tooth-surfaces, all 
 of whose elements, as before explained, converge to the vertex C of 
 the pitch cones. 
 
 357. The operation of finding the tooth-outlines upon the normal 
 cone may be performed graphically, as in Figs. 222 and 223. Since 
 the tooth projects beyond the pitch circle to R, the normal cone 
 must be correspondingly extended to R, which determines the ex- 
 treme diameter of the blank CTR ; and the "bottoms of the spaces 
 upon its development are limited by the circle whose radius is A'S', 
 to which AS' is, of course, equal. The points R, P, and S, in re- 
 volving about AC } describe circles which in the side view appear as 
 
APPROXIMATE CONSTRUCTION OF TEETH. 
 
 235 
 
 the right lines R T, PF, SCr ; these are seen in their true form and 
 size in the end view, Fig. 223, which it is necessary to construct in 
 order to draw the side view. Obviously, the length of the arc which 
 measures the breadth of the tooth at the top, at the bottom, or on 
 the pitch circle, will be the same in the end view as in the develop- 
 ment, the chord being a little less in the former. The same holds 
 true in regard to any intermediate circles which may be drawn, and 
 thus the outlines of the teeth in the end view may be determined with 
 any required degree of precision ; after which they are projected to 
 the side view, the tops to the line RT, the bottoms to the line SG, 
 etc., in the usual manner. 
 
 Since all the elements of the tooth run to the vertex of the 
 
 
 FIG. 223. 
 
 pitch cone, it follows that if it be limited at the smaller end by 
 another normal cone, as in the figure, its outline there will be of the 
 same form as that of the outer end ; and the mode of drawing it is 
 sufficiently indicated without further explanation. 
 
 358, The shaping of the blank by the addition of a frustum of the 
 normal cone TAR, is the readiest and best means of giving a present- 
 able finish to the teeth, which would be weak at the extreme points, 
 and project beyond each other in a very unsightly manner, were the 
 larger end of the wheel terminated by a simple transverse plane. 
 
 And the assumption upon which this method of laying out the 
 
23G 
 
 EXACT CONSTRUCTION OF TEETH. 
 
 teeth rests, is that the trace upon this normal cone, of the surface 
 generated by rolling a describing cone upon the pitch cone, will upon 
 development become a true epicycloid, or at least not sensibly differ 
 from it within the limits made use of for the teeth. Let us first, 
 then, see how the exact trace of this surface upon the normal cone 
 may be determined. 
 
 359. Construction of the Correct Tooth- Outline. In Fig. 224, let CPF 
 be the pitch cone, GAH the normal cone, indefinitely extended ; 
 and let CL be the axis of a describing cone, ICK, which is 
 tangent to the pitch cone along the element CP, and intersects 
 the normal cone in the curve PON. Making a projection upon a 
 plane perpendicular to CA 9 the circle FP is seen in its true size as 
 F'TP, the curve of intersection appearing as P'RN'S. 
 
 Now taking CPI as the describing element, it is clear that if it 
 
 FIG. 224. 
 
 revolve about CL, the other cones remaining fixed, it will pierce the 
 normal cone, at every instant, in some point of the curve PON. 
 From this curve, then, the required trace may be found as follows. 
 Suppose the pitch cone fco turn as shown by the arrow, through an 
 angle measured by the arc P'a. The describing element CP will 
 thus be caused to revolve about CL, through an angle which, the 
 velocity ratio being known, can be readily ascertained ; and this will 
 enable us to fix the point #, in which at that time it will pierce the 
 normal cone. And the line la will be a portion of the required trace. 
 Let the rotation progress until a, reaches the position c ; then since la 
 moves with the pitch cone, it will at that instant appear as ec ; and 
 the angular motion of the describing element meantime being known, 
 we can find the point h upon the curve P'RN', in which it will then 
 
THE RESULTS COMPARED. 
 
 237 
 
 pierce the normal cone, and extend the trace, as ced : which operation, 
 repeated at short intervals, will enable us to construct the true form 
 of the tooth-outline with any stipulated degree of accuracy, within 
 the scope of graphic operations. 
 
 By proceeding in like manner with the describing cone internally 
 tangent to the pitch cone, we may complete the tooth by finding the 
 trace of the flank-surface. 
 
 360. Results of the Two Methods Compared. In Fig. 225, the full 
 outline represents the tooth of a wheel laid out by the method above 
 explained ; and the dotted 
 curves are the forms of the 
 faces as constructed by Tred- 
 gold's process. 
 
 The wheel selected for this 
 comparison is one of a pair 
 of mitre-wheels, as those are 
 technically called which are 
 of equal diameters, and have 
 their axes perpendicular to 
 each other; the diameter of 
 the base of the pitch cone 
 being 30 inches, the number 
 of teeth 24. 
 
 In the application of Tred- 
 gold's process, we are at lib- 
 erty to assume a describing 
 circle which will make the 
 flanks of the developed teeth 
 radial, its diameter being, 
 therefore, equal to the slant 
 height of the normal cone, 
 and this was done in laying out the tooth here shown. On wrapping 
 the developed sheet back upon the cone, then, these radial lines will 
 become elements of that surface ; so that the flanks of the teeth thus 
 constructed will be radial planes. 
 
 But a straight line can be generated by the rolling of a circle, only 
 when it rolls within another circle of double its diameter, in its own 
 plane. No circular cone, then, can generate a perfect plane by roll- 
 ing within another one. However, if the angle at the vertex of the 
 describing cone be half that at the vertex of the pitch cone, the sur- 
 face swept up will curve so slightly near the first element of contact, 
 that for all practical purposes the flanks may be regarded as truly 
 
 FIG. 225. 
 
238 INSIDE BEVEL GEAR INVOLUTE TEETH. 
 
 plane surfaces. But now, the same describing cone must be used for 
 the face- surf aces ; and, as shown in the figure, the discrepancy between 
 the results of the two methods is quite decided, and sufficient to 
 affect materially the constancy of the velocity ratio if the approximate 
 form be adopted. 
 
 361. Bevel Wheels in Inside Gear. r ln laying out the teeth of a hol- 
 low bevel wheel, by Tredgold's process, it is apparent that the size of 
 the describing circles must be fixed with due regard to the limits 
 deduced for the case of annular spur gearing, in order that there may 
 be no interference upon the development of the normal cone. The 
 spherical epicycloids and hypocycloids, it is true, diifer from their 
 plane namesakes in this, that they are not capable of two generations ; 
 so that the peculiar feature of double tangency between the face and 
 the flank of a pair of engaging teeth does not exist in inside bevel 
 gearing. But it is none the less evident that if the outlines of the 
 teeth upon the developed sheet be such as to interfere with each other, 
 the wheels constructed from them by this process will not work. 
 
 From this consideration we deduce, as a safe practical rule in se- 
 lecting the describing cones v/hen laying out the teeth by the exact 
 method, that the diameters of their bases should not be greater than 
 those of the describing circles which can be employed in the approxi- 
 mate method ; the circumferences of these bases being tangent to 
 those of the pitch cones. 
 
 A pitch cone rolling in contact with a plane disk (see Fig. 81), 
 presents, upon the application of Tredgold's process, a case analogous 
 to that of a rack and wheel in spur gear, and requires no special 
 notice. In the use of the exact method, the details of the operation 
 are substantially the same as already explained ; the vertex of the 
 cone normal to the plane disk being infinitely remote, that cone be- 
 comes a cylinder, but no additional difficulties result from the intro- 
 duction of this new feature. 
 
 362. Bevel Wheels with Involute Teeth. When the developed bases 
 of the normal cones are taken as pitch circles, teeth may be laid out 
 upon them by the involute as well as by the epicycloidal system, and 
 the wheels thus made will work as well as the others. The surfaces 
 of their teeth are obviously approximations to those which would be 
 generated by the rolling of a plane in contact with two base cones. 
 Such a rolling plane must, clearly, cut the plane of the fixed axes 
 always in the same line, and this line will bo the common element of 
 two pitch cones whose diameters will have the same ratio as those of 
 the base cones. And, taking this radius of the rolling plane (whose 
 motion is, during the rotation of the cones, one of revolution about 
 
ACTION OF SPUR AND BEVEL WHEELS COMPARED. 239 
 
 an axis perpendicular to the plane, passing through the common 
 vertex), as a describing line, we may follow its movements, and by 
 finding the points in which at given intervals it pierces the normal 
 cones, determine the exact tooth-outline with as great facility and 
 precision as in the preceding case. 
 
 This tooth- surf ace may be also generated in another manner ; sup- 
 pose the pitch cone to be cut from vertex to base along an element, 
 and the surface to be unrolled into a plane ; this plane will, during 
 the process, be always tangent to the cone, the c#t edge will sweep 
 up the surface under consideration, and, since it is of constant length, 
 its extremity will trace a curve which may be properly called a spher- 
 ical involute. 
 
 It is not necessary to go into a detailed examination of the form 
 of the surface which by rolling in contact with the pitch cones would 
 generate teeth of the forms here spoken of, since the method above 
 given is much more simple. The close analogy between the plane 
 and the spherical involute, however, is sufficient to make it apparent 
 that such a conical surface, whose spherical directrix will have a 
 corresponding analogy to the logarithmic spiral, is capable of thus de- 
 scribing these identical teeth, which consequently form no exception 
 to the general law. 
 
 363. Action of Bevel and Spur Wheels Compared. It is true of bevel 
 as well as of spur gearing, that the smaller and more numerous the 
 teeth, the better will be the action, because there will be less of both 
 sliding and obliquity. 
 
 Now experience shows that, other things being equal, a pair of 
 bevel wheels will run more smoothly, and in a general way operate 
 more satisfactorily, than a pair of spur wheels of the same numbers 
 of teeth. The reason of this will be seen very clearly by reference to 
 Fig. 221, if we imagine the outer ends of the teeth to be bounded by 
 the surface of the sphere, and confine our attention to the action 
 upon each other of the outlines thus formed. The point of contact, 
 throughout the engagement of any one pair of teeth, will always lie 
 in or near to the plane AB 9 perpendicular to CP, upon which are 
 developed the normal cones, which latter in their turn do not, within 
 the limits of the height of the teeth, deviate greatly from the spheri- 
 cal surface. The result of this is, that although the velocity ratio is 
 determined by the lengths of the perpendiculars PG, PH, the dura- 
 tion of the action is very nearly the same as it would be were the 
 developed teeth to work together in their own plane about the centres 
 A', B'. The action of these wheels, then, in so far as it is affected 
 by the numbers of the teeth, is substantially the same as that of spur 
 
240 COKICAL LOBED WHEELS. 
 
 wheels of the same pitch, with PA, PB, for the radii of the pitch 
 circles. Or, in general, the action of a bevel wheel of a given num- 
 ber of teeth, is in this respect equivalent to that of a spur wheel hav- 
 ing a number greater in the proportion of the slant height cf the 
 normal cone to the radius of its base, or of the slant height of the 
 pitch cone to its altitude. 
 
 364. Teeth of Conical Lobed Wheels. If a cone of any form be inter- 
 sected by a sphere whose centre is at the vertex, every point of the in- 
 tersection is equally distant from that centre ; let a describing cone be 
 rolled upon the first one, as in Fig. 220, then the extremity of any one 
 of the elements of the rolling cone will trace upon the sphere a curve, 
 which may be considered as the directrix of the conical surface genera- 
 ted by the element itself ; and that surface will, in accordance with 
 the general principle, be of the form required for the tooth, the first 
 or base cone being regarded as a pitch surface ; and, without going 
 into details, it is apparent that the determination of the spherical 
 directrices may be effected by graphic means. 
 
 Otherwise ; let a plane be passed, say perpendicular to the fixed 
 axis of rotation, or to the line of symmetry, if there be one, thus 
 forming a base for the pitch cone. Inasmuch as its contour may be 
 practically made up of approximating circular arcs, the pitch cone 
 may be considered as made up of portions of various circular cones ; 
 and these may be treated as previously explained, by either the exact 
 or the approximate method. The normal cones, upon which the 
 traces of the teeth are thus obtained, will not have a common vertex ; 
 but since each is to be dealt with separately, this is of no consequence. 
 
 There is no instance to be cited of the actual employment of these 
 non-circular conical wheels ; which indeed have not been described 
 by any previous writer. An objection might be urged on account of 
 the difficulty of making them : which, however, does not lessen the 
 abstract interest attaching to them as a new class of kinematic com- 
 binations. The above suffices to show that the graphic operations 
 relating to them are sufficiently simple in detail, although tedious by 
 reason of their number ; and it may be questioned whether they 
 would really prove more difficult to make than the skew wheels 
 which are occasionally met with. 
 
 365. Methods of Cutting the Teeth. Since the teeth taper from end 
 to end, the outlines of the transverse sections are continually chang- 
 ing ; it is therefore, impossible to cut them with perfect accuracy by 
 means of a milling cutter of the usual form. Nevertheless, this is 
 the method almost universally adopted when they are cut at all. In 
 order to distribute the unavoidable errors as uniformly as may be, it 
 
METHODS OF CUTTING BEVEL TEETH. 241 
 
 is customary to make the outline of the cutter agree in form with the 
 cross section of the tooth at the middle of its length ; its travel is 
 directed along the line joining the face and flank of the tooth, that 
 is to say along the element of the pitch cone, first on one side of the 
 space, and then on the other, the thickness of the cutter being less 
 than the breadth of the space. 
 
 The effect of this is, that at the larger end of the tooth the flank 
 is too full and the face not full enough, while at the smaller end these 
 errors are reversed ; besides, the tooth surfaces thus cut are not conical, 
 but cylindrical, all the elements being parallel to the line along which 
 the travel of the cutter is directed. These surfaces, consequently, 
 will not work in true line contact, except at the one instant when the 
 junctions of faces and flanks reach the plane of the axes, until they 
 have "worn to a bearing" by use ; but neither then nor thereafter 
 will they work correctly. 
 
 366. Corliss's Bevel Gear Cutting Engine. There are, however, ways 
 and means of accomplishing better results than this; and to the in- 
 genuity and enterprise of Mr. George H. Corliss, of Providence, R. L, 
 is due the production of an engine, capable of doing the work 
 with absolute theoretical precision. In this admirable machine the 
 milling cutter plays no part, but the teeth are planed out, element 
 by element. 
 
 Without going into details, it will be understood, first, that the 
 cutting point of the tool is made to travel always in a right line pass- 
 ing through the vertex of the pitch cone ; it must, therefore, at each 
 cut plane out an element of a conical surface. Second, that the 
 motions are so controlled with reference to a guide template, that a 
 line drawn from the vertex through the cutting point shall always 
 touch the outline of the template, which is, then, the directrix of 
 this conical surface. 
 
 Now, all that is necessary to the production of a perfect wheel, is 
 to provide this machine with a template whose outline accurately 
 conforms to that of a transverse section of a tooth. This, it should 
 also be stated, is practically laid out upon a larger scale than if the 
 outer end of the tooth were to be drawn in the usual manner, because 
 it is placed in the engine considerably farther than the base of the 
 blank from the vertex of the pitch cone ; thus reducing the propor- 
 tional magnitude of any unavoidable errors in the graphic processes. 
 
 367. Twisted Bevel Wheels. The feasibility of placing upon the 
 same shafts a series of bevel wheels cut from the same cones, but in 
 different phases of their action, is self-evident. The advantages of 
 such stepped wheels would be exactly the same as in the similar ar- 
 
 16 
 
242 
 
 TWISTED BEVEL WHEELS. 
 
 rangement of spur gearing ; but there are practical drawbacks which, 
 it may be admitted, would prevent their employment. But the pro- 
 cess of twisting, by which the spur wheels are transformed into 
 Hooke's spiral wheels, may be equally well applied to conical ones ; 
 and to the form of bevel gearing thus produced no reasonable objection 
 can be made, except the difficulty of making it ; and this, as will 
 presently appear, is not insuperable, nor so great as might be supposed. 
 In Fig. 226, it is evident that if each plate of the vertical pitch 
 cone be twisted through the same angle, it will drive the correspond- 
 ing plate of the inclined cone through an angle depending upon the 
 relative diameters ; so that if these plates be indefinitely increased in 
 
 number, the rectilinear elements AB, 
 EF 9 will become conical helices, as 
 shown. 
 
 The graphical construction pre- 
 sents no difficulty ; we have in each 
 case a uniform rotation, combined 
 with a uniform advance along an 
 clement. Consequently since if the 
 amount of twist upon one cone be 
 assumed that on the other is known, 
 both curves are readily drawn ; they 
 move in perfect rolling contact, and 
 develcpe upon the common tangent 
 plane into the same Archimedean 
 spiral. 
 
 Just as with the spur wheels, any 
 form of tooth which will work cor- 
 rectly before twisting will do so after- 
 ward ; and if the faces be purposely 
 made not full enough to be conjugate to the engaging flanks, there 
 will be at each instant but a single point of tangency, which will 
 alway lie in the plane of the axes, and the wheels will run in pure 
 rolling contact. 
 
 Now, although such teeth as these cannot be made by the use of a 
 milling cutter, it will be apparent that if the blank be placed in the 
 Corliss machine above described, and given a motion of uniform rota- 
 tion during the advance of the cutting point, they can be planed out 
 as easily and as accurately as any others. 
 
 The Teeth of Skew Wheels. 
 368, The Problem of Determining the Forms of Teeth which shall work 
 
 FIG. 226. 
 
TEETH OF SKEW WHEELS ANALOGICAL THEOKY. 243 
 
 in right-line contact while transmitting rotation about fixed axes, pre- 
 sents itself in a new form when those axes lie in different planes. 
 
 All wheels with teeth which act in this manner must ultimately 
 reduce to pitch surfaces whose class depends upon the relative posi- 
 tions of the axes ; these will be tangent to each other along a right 
 line, and move in contact of which the sliding, if there be any, is also 
 along the common element. A third surface of the same class being 
 placed in contact with these along the same line, will move in contact 
 of the same nature with either or both ; and if one pitch surface 
 drive the other, the motion of the third is the same whether we 
 regard it as derived from the first or the second. When the velocity 
 ratio is constant, these will be surfaces of revolution cylinders, cones, 
 or hyperboloids ; and in either case all three rotate in contact about 
 fixed axes. 
 
 During such rotation, when the axes are parallel, an element of the 
 third cylinder, by its motion relatively to the pitch cylinders, simul- 
 taneously generates the conjugate surfaces for the teeth of spur 
 wheels. Now the cylinder is the limiting form assumed by the cone 
 when the vertex is infinitely distant ; and accordingly, when the axes 
 intersect, the third cone in like manner describes the surfaces for the 
 teeth of bevel wheels. 
 
 Again, the cone is but the special case of the hyperboloid in which 
 the generatrix approaches infinitely near to the axis, and by pursuing 
 the analogy we should reach the deduction that an element of the 
 third hyperboloid, if taken as a describing line, will generate the con- 
 jugate tooth-surfaces for skew wheels. 
 
 369. This conclusion is plausible, and has received the indorsement 
 of high authorities. Thus, Prof. Willis says : * 
 
 " The surfaces adapted for teeth in the case of rolling hyperboloids 
 might be obtained in a manner similar to those of rolling cones, by 
 taking an intermediate describing hyperboloid ; but it does not appear 
 possible to derive from this any rules sufficiently simple for appli- 
 cation." 
 
 He adds that a sufficiently close approximation may be made by 
 drawing two cones normal to the hyperboloidal frustum selected, de- 
 veloping these, and after laying out teeth, as in Tredgold's method for 
 bevel wheels, wrapping them back in their proper relative positions. 
 The forms and proportions of the teeth traced upon the larger and 
 the smaller cone respectively, are presumably the same ; and the tooth- 
 surface is composed of right lines joining the corresponding points of 
 
 * Principles of Mechanism, p. 151. 
 
244 
 
 TEETH OF SKEW WHEELS. 
 
 their contours after the developments are restored to their original 
 conical forms, in due relation to each other as determined by the gen- 
 eratrix of the pitch hy- 
 perboloid. 
 
 Prof. Kankine,* again, 
 remarks: " The surf aces 
 of the teeth of a skew- 
 bevel wheel belong, like 
 its pitch surface, to the 
 hyperboloidal class, and 
 may be conceived to be 
 generated by the motion 
 of a straight line which, 
 in each of its successive 
 positions, coincides with 
 the line of contact of a 
 tooth with the corre- 
 sponding tooth of an- 
 other wheel. Those 
 surfaces may also be con- 
 ceived to be traced by 
 the rolling of a hyper- 
 boloidal roller upon the 
 hyperboloidal pitch 
 surface." 
 
 And he in turn pro- 
 ceeds to describe at 
 some length a process 
 of constructing the teeth 
 which, although differ- 
 ing in detail from that 
 above explained, gives 
 nearly, and in some 
 cases precisely, the same 
 results ; both are, then, 
 to be regarded as ap- 
 proximations more or 
 less close. 
 
 370. Direct Process of 
 Construction. But, as- 
 suming that the surfaces generated by the use of an auxiliary hyper- 
 
 Fic. 227. 
 
 Machinery and Mill- Work, p. 146. 
 
DIRECT PROCESS OF CONSTRUCTION. 
 
 245 
 
 boloid are correct, it is nearly, if not quite, as easy to construct them 
 directly as in the exact method for bevel wheels ; supposing the pitch 
 surface and the generating surface to rotate in contact, the axes being 
 fixed, we have simply to follow the movements of the describing line, 
 and find the points in which it pierces the normal cones at various 
 phases of the action. 
 
 In Fig. 227, V, Y are the vertices of the cones normal to the verti- 
 cal pitch surface, and the inclined describing hyperboloid, touching 
 the other internally, is shown in position for generating the flank of 
 the tooth. 
 
 For convenience, the velocity ratio of these two hyperboloids is 
 taken as 2 to 1 ; hence a rotation of the vertical surface through the 
 angle BVP, will cause the other to turn through the angle BCO 
 twice as great. The describing line, which was originally at AS, will 
 then have the position 720, piercing the upper normal cone in some 
 point $, which being found, occupies a known position in relation to 
 the element VP ; and a point similarly situated with reference to the 
 element VB, will evidently lie in the trace of the required flank upon 
 that cone. By repeating this process any desired number of points in 
 the curve may be located ; and applying it to the lower normal cone, 
 we determine the outline of the smaller end of the tooth. Or by ex- 
 tending the describing line, and finding where it pierces the plane 
 of the gorge circle, we may in a similar manner map out the trace 
 upon that plane. 
 
 371. Next, the describing surface being placed in external contact 
 along the same line, AB, and the normal 
 cone being sufficiently extended, the face 
 of the tooth is constructed in a similar 
 manner, needing no illustration. The 
 method of finding the point in which the 
 describing line pierces the normal cone is 
 shown in Fig. 228. V being the vertex of 
 the cone, AB its base, and MN the line, 
 draw through Fa line, VO, either parallel 
 to or intersecting MN; this line pierces the 
 plane of the cone's base in 0, and MN 
 pierces it in P. Therefore the plane deter- 
 mined by MN and V.O cuts the plane AB 
 in the line OP, which, produced if neces- 
 sary, cuts the circumference of the base in 
 the point L ; now drawing VL, that element 
 of the cone cuts the given line MN'm S, which is the required point. 
 
 M 
 
 FIG. 228. 
 
TEETH NOT SYMMETRICAL. 
 
 372. The Teeth not Symmetrical. It will be perceived that the trace 
 of the tooth upon the normal cone is derived from the intersection of 
 that cone with the describing surface, just as it was in constructing 
 the correct teeth for bevel wheels. In that case the axes of these two 
 surfaces were in the same plane, which divided the curve of intersec- 
 tion symmetrically. But this is not so with the hyperboloid, and in 
 Fig. 227 that curve will not be symmetrical with reference to the ele- 
 ment VB of the normal cone. The result of this is, that the traces of 
 the two flank surfaces, derived from the parts of the curve on opposite 
 sides of the point B, will not be similar. The same holds true in 
 regard to the face surfaces, so that the trace of the complete tooth upon 
 the normal cone will not be symmetrical to a radius, as it is in the 
 constructions of Prof. Willis and Prof. Eankine. 
 
 This may be seen from another point of view, thus : The inclina- 
 tions of the describing line to the elements of the normal cone will 
 not. vary in the same manner nor in the same degree in the opposite 
 directions of its motion from the position AB. The difference of form 
 is not great for the flanks, but is more conspicuous in the faces, as 
 might be expected from their greater length and more rapid changes 
 of curvature. 
 
 373. Determination of Height of Tooth. In Fig. 229, let D be 
 the axis of a wheel, perpendicular to the paper. LBO the base 
 
 of the normal cone, 
 AB the generatrix 
 of the pitch surface, 
 BT the intersection 
 of the normal cone 
 with the describing 
 hyperboloid, from 
 which is derived the 
 trace BFG of the face 
 13 of the tooth. The ac- 
 tion of this face begins 
 at B, and BT is the locus of contact ; hence if the angle of recess, 
 BDP, be given, we draw through P a curve similar to BG, cutting 
 BT\i\ E, which will be the highest point of the face. Conversely, 
 if the height BF of the tooth be assumed, an arc through F about 
 the centre D cuts BT in E, and FDE or its equal BDP will be the 
 angle of action between this face and the engaging flank of the other 
 wheel. 
 
 Let C be the axis of that second wheel, also perpendicular to the 
 paper ; HK the generatrix (coincident with AB when the wheels are 
 
 FIG. 229. 
 
HEIGHT OF TOOTH AtfD FORM OF BLANK. 247 
 
 in position) ; HU the trace, on the second normal cone, of the 
 same describing hyperboloid that was used for the face of the first 
 wheel : and HN t\\Q trace of the flank. Now the turning of the first 
 wheel through the given angle, BDP, causes the describing surface to 
 turn through a known angle ; which enables us to find the point J in 
 HU, corresponding to that angular motion. Then an arc about 
 centre C through this point /, cuts HN in R, which limits the acting 
 depth of the flank. Or, otherwise ; the given rotation of the first 
 wheel will drive the second through a known angle HCS ; then a curve 
 through S, similar to HN, cuts ITU in. /; and thus the acting depth 
 is found without reference to the rotation of the describing surface. 
 
 The same process is to be repeated with respect to the face of the 
 second wheel and the flank of the first, which completes this op- 
 eration and determines the whole angle of action for each wheel. 
 But the flank-curves are to be continued beyond the points thus 
 found, because, as in any other form of gearing, clearing spaces must 
 be provided in each wheel, for the passage of the teeth of the other. 
 
 374, Determination of the Form of the Blank. When the height of 
 the tooth has thus been ascertained, or assumed, the position of the 
 describing line which passes through the outer point of its face is defi- 
 nitely fixed. Now, this element of the tooth-surface, in revolving 
 around the fixed axis of the wheel, generates a new hyperboloid, and 
 the blank should be made in the form of a frustum of that surface, 
 limited by the cones normal to the pitch hyperboloid. For otherwise 
 the teeth will begin and end their action at a single point, instead of 
 along a line : the obvious result being that the points, especially of 
 the follower's teeth, will suffer from abrasion. 
 
 If the wheels are located in the immediate neighborhood of the 
 gorge planes, the true hyperbolic outline must be carefully followed 
 in making the blank, in order to avoid this danger. But if not, the 
 hyperbola becomes so flat at a short distance from the vertex, that it 
 will ordinarily suffice in practice to use a frustum of a cone tangent 
 to this external hyperboloid, at the middle of the length of the tooth 
 as measured along its outer element. 
 
 375. This is illustrated in Fig. 230 ; the dotted curve EF is the 
 outline of the pitch surface, to which the cones whose vertices are 
 V and Y are normal, and ON is the outline of the external hyper- 
 boloid, generated by revolving the describing line which passes 
 through N the highest point of the tooth, about the vertical axis VX. 
 NM being the length of the outer hyperbola to be used, let this be 
 bisected at R, at which point draw a tangent to the curve ; this will 
 intersect the axis at X, the vertex of the tangent cone. 
 
248 
 
 FORMS OF THE COMPLETE TEETH. 
 
 The method of drawing the tangent to the hyperbola is also shown. 
 Since the position of the generatrix of the outer surface with respect 
 to the axis is known from the antecedent constructions, it will, when 
 parallel to the paper as in the diagram, take a known direction, AB, 
 asymptotic to the hyperbola ON. The companion generatrix then 
 becomes the other asymptote AL ; now draw through R a parallel to 
 AB, cutting AL in P, and on AL set off PC = AP, then RC is the 
 
 required tangent. 
 
 376. The forms of the 
 teeth for a pair of wheels, 
 laid out by the processes 
 above described, are shown 
 in Figs. 231 and 232, the 
 wheels being seen from 
 the smaller end; the long 
 tooth 011 the right, in the 
 upper part of each figure, 
 is continued till cut by the 
 gorge plane, the shorter 
 teeth on the left being lim- 
 ited by an inner normal cone 
 as in Fig. 230. In the 
 lower part of each figure 
 are shown the common ele- 
 ment of the pitch surfaces, 
 the trace of one side of the 
 tooth upon the outer normal 
 cone, and the parts of the 
 
 FlG . 030. curves of intersection used 
 
 in the construction. The lines employed in the determination of the 
 other side of the tooth are omitted in order to prevent confusion ; 
 but it is proper to state that these figures are copied from drawings 
 executed with great care upon a large scale, in exact accordance with 
 the methods above set forth. The conditions assumed are, that the 
 projections of the axes upon a plane parallel to both, shall cross each 
 other at an angle of 60, as in Fig. 97, and that the wheels shall have 
 eighteen and twenty-seven teeth respectively. 
 
 It is to be noted that in consequence of the peculiar relation 
 between the pitch and describing surfaces, not only are the fronts and 
 backs unlike, but the traces on the various normal cones are also dis- 
 similar, the whole form of the tooth changing with the distance from 
 the gorge plane. 
 
THE ANALOGICAL THEORY WRONG. 249 
 
 377. Fallacy of the Preceding Construction. The correctness of the 
 above method of generating the conjugate tooth-surfaces for skew 
 
 FIG. 231. 
 
 wheels has hitherto passed unchallenged ; a fact perhaps not to be 
 
 wondered at, so plausible 
 
 is the analogical reasoning 
 
 by which the deduction is 
 
 reached. 
 
 Either one is unquestion- 
 ably correct in relation to 
 the describing hyperbo- 
 loid. Let the upper and 
 lower bases of the latter be 
 two discs fixed upon one 
 axis, and in these let a 
 single wire be secured, to 
 represent the describing 
 line. If this axis be prop- 
 erly placed in relation to 
 either pitch surface, it will FIG. 232. 
 
 be correctly driven by the corresponding tooth-surface acting against 
 
250 
 
 ROLLING OF OtfE IIYPERBOLOID UPOK ANOTHER. 
 
 the wire, after the manner of a pin-wheel in spur gearing. And by 
 hypothesis the engaging surfaces have at every instant in common 
 this describing line, by which they were simultaneously swept up 
 during the action. 
 
 So far, well ; but one thing more is necessary this common line 
 must be a line of tangency ; and it appears to have been assumed that 
 this is the case, probably because up to this point the analogy is perfect. 
 Every link in the chain has stood the test ; now let us examine the hook. 
 378. It is necessary, first, to gain a clear idea of what occurs when 
 one hyperboloid rolls around another which is stationary ; a question 
 
 not previously considered. 
 
 If the two surfaces shown 
 in Fig. 233 turn as shown by 
 the arrows (the axes being 
 for the moment regarded as 
 fixed), let AL, A I be the 
 linear motions of the points 
 which fall together at A ; 
 these will have the same com- 
 ponent, AK y perpendicular to 
 the common element, AB, 
 and IL represents the amount 
 of sliding between the two 
 hyperboloid s along that ele- 
 ment. 
 
 Now suppose the motion 
 of the inclined surface to be 
 suddenly arrested, and the 
 axis of the vertical one to be 
 at the same instant released 
 from its bearings, its rotation 
 continuing. The latter must 
 then revolve around the axis 
 of the former in the direction 
 opposite to that in which the 
 former was turning, and with 
 the same angular velocity. 
 The point A of the vertical hyperboloid will then have a component 
 motion, AL, due to its original rotation, and another, AH, equal and 
 opposite to AT, due to the revolution. 
 
 The resultant of these is A 0, in BA produced, and equal to IL. 
 This is as it should be, evidently, since the relative sliding of the two 
 
 FIG. 233. 
 
THE ANALOGICAL THEORY WRONG. 
 
 surfaces must remain the same ; but, moreover, the motion of every 
 point in or connected with the travelling hyperboloid must have a 
 component equal and parallel to AO. The only other motion of 
 which any such point is capable is one of revolution about AB, and 
 since the angular velocity must be the same for every point, it will 
 suffice to find it for one. 
 
 The point E has a component motion, EP, equal and opposite to 
 AL, due to the rotation ; and another due to the revolution, in the 
 direction of AH, but greater, in the proportion of DE to DA. The 
 resultant is DT, the tangent to the helix which the point E is at the 
 instant describing. Therefore OT, perpendicular to AB, is the linear 
 velocity of the rotation of this point about the common element of the 
 
 OT 
 
 two surfaces, and -r-= is the angular velocity for every point in the 
 AJl/ 
 
 rolling hyperboloid. 
 
 379. Precisely the same result has already been reached in a differ- 
 ent manner, as will be seen by reference to Fig. 25, and the accompa- 
 nying text. The nature of the action is most explicitly stated by 
 Prof. Rankine, thus * : *' If one of those bodies is fixed, and the other 
 made to roll upon it, they continue to touch each other in a straight 
 line, which is the instantaneous axis of the rolling body, and the rota- 
 tion about that instantaneous axis is accompanied by a sliding motion 
 along the same axis so as to give, as the resultant compound motion, 
 a helical motion about the instantaneous axis." 
 
 If, then, we take any element of the rolling hyperboloid as a describ- 
 ing line, it is at any instant in the act of generating a helicoid, which 
 will be tangent all along this line to the tooth-surface generated by a 
 continuation of the rolling. Therefore the plane tangent to the latter 
 surface at any point, is determined by the describing line itself, and 
 the tangent to the helix at that point constructed as above. 
 
 380. Now in generating the conjugate tooth-surfaces for a pair of 
 wheels, the describing hyperboloid rolls upon the outside of one pitch 
 surface, and upon the inside of the other, as in Fig. 234. The direc- 
 tion of the rotation about the instantaneous axis is of course the same 
 in both cases, but that of the sliding is not ; in the rolling upon the 
 inclined pitch surface the sliding is represented by AO, while in the 
 rolling within the vertical surface it is represented by AP, in the op- 
 posite direction. 
 
 In Fig. 235 let the vertical line AB represent the common element 
 of the pitch surfaces or instantaneous axis, and OE the describing 
 
 * Machinery and Millwork, p. 71. 
 
252 
 
 THE CONJUGATE SURFACES INTERSECT. 
 
 line, the rotation being indicated by the arrows. The tangent to the 
 helix described at the instant by the point P will lie in the plane tan- 
 gent to the cylinder upon which that helix lies, and in the horizontal 
 projection will have the direction P'l' perpendicular to P'A'. Let 
 the motion parallel to AB be upward, then PI will be the vertical 
 projection of this tangent. Let MN be a horizontal plane ; it cuts 
 PI at L, and OE at 0, giving O'L' as the trace upon MN of the 
 
 M 
 
 Fro. 234. 
 
 FIG. 235. 
 
 plane tangent to the helicoid at P. If, on the other hand, P have a 
 downward motion parallel to AB, PI' will be horizontal and P//the 
 vertical projection of the tangent to the helix, which pierces MN at 
 R, giving O'R as the trace of this tangent plane, which cuts the first 
 one in the line OE. A vertical plane through the moving line will 
 evidently also contain the tangents to the helices described by the 
 point E of that line, which is at the least distance from AB. At this 
 point, then, and at no other, will the two helicoids have a common 
 tangent plane. 
 
 381. Consequently the hook breaks, the chain is useless, and these 
 teeth will not work at all. The common line of the engaging face 
 and flank is a line not of tangency, but of intersection, and these sur- 
 faces, simultaneously swept into existence by the auxiliary hyperboloid, 
 are instantaneously swept out again by each other. That this would 
 occur in the special case in which the pitch hyperboloids retain their 
 limiting forms, the one being a cone and the other a plane disc or 
 
A KEW THEORY OLIVIER'S INVOLUTES. 
 
 253 
 
 rather annulus, as in Fig. 101, was pointed out by the author in a 
 paper published some years since ; * but not at the time of that writing, 
 nor until the present, was it perceived that the analogy fails in gen- 
 eral, and that this whole method of constructing teeth for skew wheels 
 is radically wrong. 
 
 382. A Practicable Method. A different line of procedure must, 
 therefore, be sought ; and a basis of operations which will lead to 
 reasonable results is found in a peculiar property of the involute, first 
 pointed out by M. Theodore Olivier. 
 
 In Fig. 236, AB is the common tangent of the two circles whose 
 centres are C' and />, and is, therefore, the locus of contact of the 
 involutes O'P, PH y shown in contact at P. Let the first circle be 
 now revolved on the line AB, 
 as on a hinge, until its centre 
 reaches the position C 9 the 
 circle then appearing as an 
 ellipse, and the tooth taking 
 the position OP. The axes 
 will then lie in different 
 planes, but it is evident that 
 OP will still drive PH exactly 
 as before, AB being the locus 
 of contact, since it is the inter- 
 section of the planes of the base 
 circles, whatever their inclina- 
 tion to each other. 
 
 Therefore the action of the 
 fronts of the teeth is not af- 
 fected by the change in the 
 position of the originally par- 
 allel axes ; but with regard to 
 the backs the case is quite different. Their locus of contact, when the 
 circles were in the same plane, was the other common tangent EF 9 
 which cuts AB at G, and it will readily be seen that the two points of 
 the backs of the teeth which would meet at G in the first place, will 
 do so still, but no other points of those curves will come into contact 
 at all. Consequently, regarding the wheels as of no sensible thick- 
 ness, we perceive that either of them can drive continuously in only 
 one direction. 
 
 383. Of this combination Prof. Willis remarks, f 
 
 * Scientific American Supplement, Nos. 174, 176 and 178. 
 f Principles of Mechanism, p. 152. 
 
 FIG. 236. 
 
254 
 
 A NEW THEOKY OLIVIEK'S INVOLUTES. 
 
 "Involute wheels, therefore, maybe employed to communicate a 
 constant velocity ratio between axes that are inclined at any angle to 
 each other, but which do not meet. But the demonstration supposes 
 the wheels to be very thin, since they coincide with the planes that 
 intersect, and the invariable points of contact are situated in this 
 line of intersection. The edge of one of the wheels must be in prac- 
 tice rounded so that it may touch the other teeth in a point only." 
 
 This, however, is not the only expedient for securing substance or 
 body for the teeth, although it is strictly true if both the curves are 
 involutes. But, retaining one of them, it is possible so to modify the 
 other, when sensible thickness is given to the wheels, that the locus 
 of contact shall not coincide ^vith the intersection of the two planes, 
 and to form teeth which shall work together in contact along a right 
 line. 
 
 384. In Fig. 237, let AB, the common tangent to the bases of the 
 
 w 
 
 FIG. 237. 
 
 two cones, be parallel to the common perpendicular of their axes, 
 whose extremities, V and W, are the vertices. In the tangent plane 
 thus determined, draw EF parallel to AB ; it will be tangent to the 
 two circles cut from the cones by planes parallel to their bases, and 
 dividing proportionally both the axes and the elements. 
 
 The involutes PG, PH, of the bases, will work together as in Fig. 
 236 ; so too will those of the other sections, shown in contact at R. 
 These curves may be considered as traced, each in its own plane, by 
 marking points at P and R, drawn along by two cords, AB and EF, 
 while the cones turn as indicated by the arrows ; in which motion 
 the linear velocities of the marking points are directly proportional 
 
A HEW THEORY OLIVIER'S INVOLUTES. 255 
 
 to the radii of the bases of either frustum, and these to their distances 
 from the vertex, measured either on the axis or on an element. 
 
 Draw PR and produce it to cut FTFin 0; then if P move to a 
 new position /or L, the corresponding position of R will be M or N, 
 such that 
 
 AV - BW - 
 
 EV ~~ FW'~~ RO* 
 
 that is to say, the point and the two marking points will always 
 lie in one straight line. 
 
 385. The Tooth-surface a Single-curved One. Consequently any 
 number of tracing points may be placed between P and 0, each in 
 like manner generating a pair of involutes which will work together 
 as in the preceding figure ; the point itself describes two circles, 
 which may be regarded as the involutes of their centres Fand W. 
 And the straight lines joining the corresponding points of the invo- 
 lutes belonging to either cone constitute a surface ; which may also 
 be generated by sliding a right line upon any three of these involutes 
 as directrices. 
 
 In the latter mode of generation, every point of the right line is at 
 any instant moving in the same direction, because the tangents to all 
 the involutes are parallel : but the upper end is moving faster than 
 the lower end. Therefore, the consecutive elements intersect each 
 other two and two, and the surface is not a warped one, but single 
 curved, and the plane tangent to it at any point is tangent to it all 
 along a right line. 
 
 But it does not follow, nor is it true, that the two surfaces thus 
 simultaneously generated by the line PO will be tangent to each 
 other. The plane tangent to either at any point is determined by 
 the rectilinear element, and the tangent to the involute, which pass 
 through that point. But the two involutes generated as above lie in 
 different planes, their tangents have different directions, and in con- 
 sequence the plane tangent to one surface is not tangent to the other. 
 Retaining one of them, however, it is possible to construct another 
 which shall be tangent to it in each of the different positions, by a 
 process analogous to that of deriving a conjugate tooth from* a given 
 form. 
 
 386. Adaptation to Hyperboloidal Wheels. The application of this 
 in the case of hyperboloidal wheels will be understood by reference to 
 Fig. 233 ; if through C and D two lines be drawn parallel to the 
 common element AB, these lines, by revolving about the vertical and 
 inclined axes respectively, will generate two cones situated relatively 
 
256 
 
 ADAPTATION- TO SKEW WHEELS. 
 
 to each other as in Fig. 237, and the tooth-surfaces for the wheels 
 may be formed as above explained. Those for the first wheel, being 
 composed of true involutes, may be constructed without reference to 
 the second, as shown in Fig. 238, which is drawn without regard to 
 practical conditions or proportions, for the purpose of exhibiting dis- 
 tinctly some peculiarities which must be taken into account in laying 
 out such teeth for actual use. 
 Let KIM be the upper base and fiOFihe gorge circle of the hyper- 
 
 Z X Y 
 
 FJGS. 238 and 239. 
 
 boloid, whose generatrix, OM, is parallel to the vertical plane, K8, con- 
 taining'the axis C. Through M draw a perpendicular to OM, pierc- 
 ing this plane at A' ; then the circle about C through A 1 will be the 
 base of the cone. On A 'M produced take any point P, and let PO 
 be the position of the generatrix of the tooth-surface at the beginning 
 of the action. The wheel turning as shown by the arrow, let the 
 action continue until a marking point at P reaches A', tracing the 
 involute, PA, whose position at the end is P'A. Any other point R 
 
TEETH VANISH AT THE GORGE. 257 
 
 on P Owill meantime trace another involute, RB, whose final position 
 is R'B', while will trace the arc 00' ; and the final position of the 
 generatrix is OB 1 A'. When P has advanced to M, R will have 
 reached N, and OM is, it will be noted, tangent to all the involutes 
 traced by points upon PO. 
 
 387. The Teeth Vanish at the Gorge Plane. In Fig. 239, MO 
 is the generatrix, and SVFX the meridian outline, of the pitch hy- 
 perboloid. 
 
 Now, considering first the part which lies above the gorge plane 
 EF, we perceive in Fig. 238 that the outer points of the teeth, as P, 
 R, lie in the line PO, whose least distance from the axis is CG, and 
 G is below the plane EF. Similarly, the lowest points, A', B', lie in 
 a line, A'O, of which the point, If, nearest to the axis, lies above that 
 plane. These two lines generate two other hyperboloids, whose 
 meridian outlines in Fig. 239 are respectively TWFtind. MHF. 
 
 The teeth then vanish at the gorge circle, but if the generatrix be 
 continued below it, they will reappear ; and their highest and lowest 
 points will also lie upon an exterior and an interior hyperboloid. It 
 is here to be noted, that it is not necessary thus to prolong the same 
 generatrix ; but if it be done, as in these diagrams, the interior hy- 
 perbola, FGZ, will be a continuation of the exterior one, TWF, and 
 FY an extension of MHF. In this case it is also necessary to make 
 the angle POM in Fig. 238 equal to the angle MO A, if it be required 
 that the teeth of the upper and lower portions shall be symmetrically 
 disposed, so that for example Q'J' may be equal, similar, and opposite 
 to P'A'. Such symmetry, though not essential, is obviously desira- 
 ble if the wheels are to be used in double pairs (161) ; but the 
 angles just mentioned need riot be made equal ; and if not, the result 
 Avill be that in Fig. 229, although TWF and .FG^will be respect- 
 ively similar to FY and FHM, they will be parts of different hyper- 
 bolas. 
 
 388. Limit to the Prolongation of the Tooth. It is to be understood 
 that in Fig. 238, as in previous cases, the vertical plane containing 
 the axis C is parallel to the axis of the engaging wheel, the point 
 lying upon the common perpendicular. If then the angle MO A' be 
 assumed, the point A' in which the generatrix of the tooth-surface 
 pierces that plane limits the distance of the upper base from the gorge 
 circle. For the transverse plane through that point cuts from the 
 fundamental cone a circle to which PMA' is tangent, so that A' is the 
 root of the involute traced by the marking point in that plane ; and a 
 marking point in any higher plane, in the prolongation of OP, would 
 advance beyond the point of tangency between its path and the corre- 
 
 17 
 
258 
 
 CONSTRUCTION OF BACK OF TOOTH. 
 
 spending section of the cone, and thus begin to trace the reverse 
 branch of the involute of that circle, which is clearly inadmissible. 
 Conversely, if the distance from the gorge be assumed, the point A' 
 is thereby determined, which in turn fixes the greatest possible value 
 of the angle A'OM. 
 
 "We have thus far referred only to the lowest acting points of the 
 teeth which lie upon the interior hyperboloid in Fig. 239. But it 
 may be necessary to cut more deeply into the blank in order to pro- 
 vide clearing spaces for the passage of the teeth of the engaging wheel. 
 The involutes are then to be continued as far as required below the 
 last acting points ; should the root of the involute have been already 
 reached in any transverse plane, as at A' in Fig. 238, the continuation 
 in that plane will be a radial line. 
 
 389. Construction of the Back of the Tooth. The complete working 
 teeth must be provided with backs as well as fronts ; and we have now 
 to consider the results of this operation. In Fig. 240 we have a 
 
 FIG. 240. 
 
 single tooth upon a larger scale, the curves A'P', B'R, correspond- 
 ing to those similarly lettered in Fig. 238. For our present purpose 
 we may disregard the clearing space, and suppose the depths deter- 
 mined by the points A', B', to be sufficient. Considering A'P', then, as 
 the front of the tooth, its back is to be a similar involute in the same 
 plane, and for illustration we will assume the pitch to be such as to 
 make the tooth pointed, as shown ; then the radius through P' will 
 bisect the arc A'D, which measures the thickness of the tooth on the 
 circle through A. 
 
 Make the angle OCG equal to the angle A CD ; then the arc QG is 
 the thickness of the tooth at the gorge circle, and the lowest (acting) 
 
CONSTRUCTION OF BACK OF TOOTH. 259 
 
 points of the backs will lie upon the right line GD. Thus, a circle 
 through B', about C ', cuts GD in E, and B'E is the thickness of the 
 tooth in the plane containing the involute B'R. The radius which 
 bisects B'E, cuts B'R at K, through which point passes the reverse 
 involute, EK, or back of the tooth in that plane ; in alike manner we 
 determine the section HLF of the complete tooth by a plane still 
 nearer the gorge, and so on indefinitely. 
 
 390. The Extension of the Tooth Practically Limited in Both Direc- 
 tions. It will be observed that the points of these sections become 
 more and more obtuse, and the depths less and less, as we approach 
 the gorge plane. And in that plane, bisecting OGby the radius CJ, 
 the front of the tooth is the arc OJ, and its back is the arc GJ "; these 
 being, as previously suggested, the involutes of the centre, C. 
 
 If then we make the section of the tooth by the plane farthest from 
 the gorge pointed, we perceive that the fronts of the sections nearer 
 to the gorge cannot be extended to the full heights, as /, R, found 
 as in Fig. 238, since they are intersected by the backs before those 
 heights are reached. And a sufficient number of these intersections, 
 as L, K y being determined, a line, JLKP', is drawn through them, 
 which, by revolving round the axis, (7, will generate a surface which is 
 evidently the correct blank for the wheel upon the above supposition. 
 
 The meridian outline of this surface is the curve TVF, in Fig. 239, 
 which, it is to be noted, intersects the pitch hyperboloid at V. But 
 OM, of Figs. 238 and 239, the generatrix of that hyperboloid, is also 
 a line of the front tooth-surface. This is now cut off at V by the 
 transverse plane through V, and it is clear that between this plane 
 and the gorge the teeth cannot possibly engage with each other. 
 Practically, then, they should terminate still farther from the gorge 
 circle. 
 
 391. Suppose now, however, that the pitch is so much increased 
 that the section of which B'R' is the front may be carried up to the 
 full height and made pointed at R. The reverse involute through R 
 will cut the circle through B' at a point JV; the angle OCG must 
 then be made equal to that measured by the arc B'N 9 and a straight 
 line from this new position of G, through N, of which NQ is a por- 
 tion, will contain the lowest acting points of the backs under the new 
 condition, as did GD under the old one. Below R the state of 
 things will be analogous to that which previously existed below P' 
 and we may proceed to construct as before a curve corresponding to 
 JLCP', and to ascertain the limit beyond which the teeth cannot be 
 continued in the direction of the gorge. 
 
 But above R we shall have, in the first place, a right line, R'P 1 , 
 
2GO ACTIOK OF BACK OF TOOTH. 
 
 which is, of course, a portion of P'O' of Fig. 238, the generatrix of 
 the exterior hyperboloid, TWF, of Fig. 239, which is now the form of 
 the blank. In the next place the tooth above R will not be pointed, 
 but will have a sensible thickness at the top. Thus, the arc A'Q 
 being made equal in circular measure to B'N and QT similar and 
 opposite to A P', the tooth outline on the upper plane is bounded at 
 the top by the arc P'T. Other planes perpendicular to the axis may 
 be passed at any convenient intervals, and the arc limiting the section 
 of the tooth at the top being found in like manner for each, the line 
 R'Tis determined, which is the outer edge of the back of the tooth, 
 and lies, of course, upon the hyperboloid generated by R'P' . It is 
 not a straight line, but in most, if not all, practical cases, its curvature 
 within the limits employed will be very slight. 
 
 392. Nature and Action of the Back of the Tooth. Evidently, when 
 we thus assume any plane as the lower base of the frustum to be used 
 for a wheel, we need not make the section of the tooth by that plane 
 pointed ; for instance, we might have made B'M greater, so that the 
 lower, as well as the upper, section would have been blunted. This, 
 however, would not in the least affect the mode of proceeding, nor the 
 nature of the surface which forms the back of the tooth. This sur- 
 face can not be generated as was that forming the front, as will be 
 seen from the consideration that the tangents to the involutes at D, 
 E, and F, in Fig. 240, are not parallel. Therefore it is not of single 
 curvature ; it is not of double curvature, for it contains the right line 
 GD ; it is, then, a warped surface, and generated by sliding the right 
 line upon any three of the involutes as directrices. 
 
 Nor does it work in right-line contact with the engaging tooth of 
 the other wheel. Supposing the latter for the moment to be of the 
 same nature, it will be seen, by reference to Fig. 236, that the only 
 point of contact between the involutes of any two transverse sections 
 of the fundamental cones which have a common tangent (which may 
 be called conjugate sections), is found by revolving either of those 
 circles about that line until it lies in the plane of the other, and then 
 drawing the line of centres, which will cut the tangent at the point 
 in question. 
 
 Were the wheels made up of laminae, each acting upon its conjugate 
 only in this manner, we should have a series of such contacts, the first 
 occurring in the planes nearest the gorge, the last in those most 
 remote ; the teeth touching each other in a single point, which would 
 travel endlong, in a line parallel and near to the common element of 
 the pitch hyperboloids ; very much as in the case of twisted spur 
 or bevel wheels. 
 
BACK OF CONJUGATE TOOTH. 261 
 
 393. Peculiar Action of the Backs of the Teeth. This, however, is 
 not the case, as the supposed laminae are not capable of such independ- 
 ent action. Let a conjugate pair at any instant touch each other as 
 above ; then at the next instant, the consecutive point of one section 
 will come into contact with the engaging tooth at a point not situated 
 in the conjugate plane but in a plane beyond. But since the above 
 described contacts between conjugate planes occur at successive 
 instants, we have in effect that successive phases* of the action are 
 simultaneously represented, the result being that the two teeth at any 
 given instant touch each other along a curved line of limited length ; 
 each point of which during the action travels along the tooth from 
 end to end in the manner previously explained. 
 
 394. Construction of the Back of the Conjugate Tooth. Under these 
 circumstances, if the back of the tooth of one wheel be made up of 
 the reverse involutes, as at first supposed, it is not demonstrable that 
 the back of the engaging tooth should be, but its proper form may be 
 determined n,s follows : pass any plane perpendicular to the axis of 
 the second wheel, and upon it draw the outline of the section cut by 
 it from the assumed tooth of the first. Rotate this plane through a 
 small angle, and turn the first wheel through the corresponding 
 angle, determined by the given velocity ratio. The outline of the new 
 section of the given tooth is now to be traced upon the plane, and the 
 process repeated until tho section of the conjugate tooth, which, of 
 course, is the envelope of the various curves thus traced upon the perpen- 
 dicular plane, is mapped out with the desired degree of accuracy. A 
 new transverse plane is now to be passed, and the same series of opera- 
 tions again performed ; and this once more repeated, gives us finally 
 three directrices, upon which a right line being made to slide will 
 generate the back of the required conjugate tooth. 
 
 395. Construction of the Front of the Conjugate Tooth. The front 
 also of this conjugate tooth, as stated in (385), will be different from 
 that of the first wheel, whose generation was illustrated in Fig. 238. 
 Referring to that diagram, it is seen that in every position of the gen- 
 eratrix, the tangents to all the involutes are parallel to OM ; and they 
 are all parallel to the gorge plane. Therefore OM is the trace upon 
 the gorge plane of the plane tangent to the tooth-surface at any 
 phase of the action ; the latter plane turning upon OM as upon a 
 hinge, during the rotation of the wheel. And in each of its positions 
 it will cut a right line from any transverse plane of the second wheel, 
 which, meantime rotating about its own axis, will thus have traced 
 upon it a series of lines, to which the front outline of the section of 
 the conjugate tooth by that plane mnst be tangent. 
 
262 
 
 FROKT OF CONJUGATE TOOTH. 
 
 Thus, in Fig. 241 O'C' is the vertical axis of the first wheel, corre- 
 sponding to C of Fig. 238, O'M' corresponds to OM of the same 
 diagram, O'D' is the axis of the second wheel, and the right-hand 
 portion of this figure is a projection upon a plane perpendicular to 
 the inclined axis ; in which PABisihe path of a marking-point which 
 generates an involute pertaining to the first wheel, this path corre- 
 sponding to PMA' of Fig. 238. The arrow indicating the direction of 
 the rotation, P is already the lowest possible point of the acting face 
 of the tooth of the second wheel, and PM is the position of the plane 
 tangent to the tooth of the other at this phase. When P has moved 
 to A this plane will have the position A M, and the radius DP will be 
 found at DE, the arc PE being equal to PA. So when P reaches B, 
 the plane will appear as BM, and DF will be the new position of DP, 
 the arc PF being equal to PB, Thus the outline of the tooth being 
 
 M 
 
 FIG. 241. 
 
 tangent to the trace of the plane in each of its successive positions, 
 we are enabled to map out its form as shown at FB. 
 
 Evidently this curve is not a true involute, which could be produced 
 only on the supposition that the tangent plane should always be par- 
 allel to AM ; but in most practical cases the angle PMB will be small, 
 and the deviation of the tooth-outlines from the involute form will be 
 comparatively slight. 
 
 396. Different Actions on Opposite Sides of the Gorge. A singular fact 
 in relation to the teeth thus constructed is that on opposite sides of the 
 gorge plane the fronts and backs are transposed, as will be seen by refer- 
 ence to Fig. 238 ; so that if a double pair of such wheels be employed, 
 turning in a given direction, the work of one pair will be done by the 
 single curved surfaces of the teeth, that of the other pair by the 
 warped surfaces, the distribution being reversed if the directions of 
 the rotations be changed. 
 
APPEARANCE OF COMPLETE WHEEL. 
 
 263 
 
 Due attention should be paid to this point if a single pair only is to 
 be used, and unless the arrangement of other parts of the machine 
 prevents, the wheels should be placed on that side of the gorge plane 
 which will secure the advantage of having the acting surfaces of the 
 teeth touch each other throughout their length. 
 
 397. Practically, then, this construction enables us to make a pair 
 of skew wheels which in one direction of the driver's rotation will 
 work in right-line contact. The location of the upper bases of the 
 hyperboloidal frusta being arbitrary, they should be placed as far as 
 
 TJ 
 
 :^> 
 
 FIG. 242. 
 
 possible from the gorge planes, in order to reduce the transverse 
 obliquity to the lowest limit. 
 
 The blanks should, of course, be terminated by cones normal to the 
 pitch surfaces, as previously explained ; the traces of the teeth upon 
 these cones are most readily found by passing a series of transverse 
 planes, and drawing in each the outlines of the tooth ; which will 
 intersect the circles cut from the cones in points of the required 
 curves. 
 
264 THE ANALOGY CORRECTED AND COMPLETED. 
 
 The appearance of the complete wheel as thus laid out is shown in 
 Fig. 242, which represents the larger of a pair having respectively 48 
 and 32 teeth, the angle between the projections of the axes being 60. 
 On the right is shown the development of the tooth-outline on the 
 outer normal cone of the larger wheel, the corresponding development 
 for the smaller one being given on the left. These outlines, it will 
 be observed, are by no means symmetrical, owing to the transverse 
 obliquity of the teeth, although in one wheel the sections by planes 
 perpendicular to the axis may be symmetrical with respect to the cen- 
 tral radii. 
 
 398. It will be apparent on reflection that the analogy between 
 conical and hyperboloidal wheels, in order to be perfect, requires that 
 the teeth of the latter should vanish at the gorge, as do those of the 
 former at the vertex. It is true that models have been made, as, for 
 instance, those by Schroeder of Darmstadt, in which teeth of sensible 
 magnitude arc given to wheels whose mid-planes coincide with the 
 gorge circles of hyperboloids, whose elements are represented by 
 wires passing through the teeth ; it is also true that these wheels 
 work, transmitting the rotation with perfect constancy of velocity 
 ratio. 
 
 But it does not follow that these are true skew teeth, nor that they 
 work in right-line contact at all on either side. These wheels, as 
 made, are thin ; and, as will be seen presently, screw teeth formed 
 upon pitch cylinders tangent to these hyperboloids at the gorge 
 circles will curve so little in the small portion used, that it would be 
 difficult by mere inspection to detect the curvature or to ascertain 
 whether contact existed in more than one point. And in fact such 
 teeth are very often, if not always, actually made by means of a mill- 
 ing cutter travelling in the direction of the tangent to the helix at the 
 mid-plane of the wheel, without rotating the blank during the 
 operation. 
 
 But these wheels cannot be made of any considerable length in the 
 direction of the axis, and the suspicion of their identity with those in 
 the models above mentioned has yet to be removed by the production 
 of a pair in which straight-line teeth extend past the gorge circle's 
 from end to end of long hyperboloids. 
 
 399. Skew-bevel wheels are not often met with in practice. The 
 usual expedient, when two axes lie in different planes, is to introduce 
 a counter-shaft, whose axis intersects both the others, and to use two 
 pairs of bevel wheels. And when they are at a great distance from 
 each other this may be unavoidable ; but if they be not, there can be 
 no question that the loss of power due to the imperfect rolling of the 
 
TWISTED SKEW GEABING. 
 
 265 
 
 
 \ 
 
 pitch surfaces, and the transverse obliquity of the teeth, with a pair 
 of properly constructed skew wheels would in many cases be less than 
 that incurred when the arrangement above mentioned is adopted ; to 
 say nothing of the superiority of the single pair in respect to neatness, 
 lightness, and compactness. 
 
 400. Twisted Skew Wheels. If we suppose a pair of tangent hyper- 
 boloids to be made up of a series of transverse laminae, it is clear that 
 by twisting them uniformly, as in 
 Fig. 243, the rectilinear elements 
 will be changed into hyperboloidal 
 helices, while the surfaces still touch 
 each other along a right line as 
 before. And had teeth been added 
 previously to the twisting, then, as 
 was the case with the spur and the 
 bevel wheels, these teeth would con- 
 tinue to act with the proper velocity 
 ratio. . 
 
 This at first glance appears a useless 
 addition to an already ample degree 
 of complexity, but upon closer exam- 
 ination it will be seen that such twisted 
 teeth can actually be made more easily 
 than straight ones. The hyperbo- 
 loidal helix can be traced by suppos- 
 ing a marking point to travel uni- 
 formly along an element, while the 
 surface turns uniformly upon its axis. 
 And if the teeth begin and end their 
 contact upon that line, as in Dr. 
 Hooke's spiral wheels, this will be the 
 only line of their surfaces whose form is of essential importance. 
 
 Practically, therefore, it is requisite merely to arrange proper mech- 
 anism for simultaneously moving a milling cutter along the line of an 
 element of the pitch surface, and rotating the blank upon its axis, 
 both motions to be uniform.* 
 
 The same mode of proceeding holds good when, as in Pig. 101, the 
 hyperboloids retain the limiting forms of a cone and a plane, for if 
 we imagine the disc to be made up of as many concentric rings as 
 there are laminse in the cone, each one will be driven round by the 
 twisting of the latter, so that the original line of tangency on the 
 
 * Scientific American Supplement, No. 178. 
 
 \ 
 
 FIG. 243. 
 
266 
 
 TEETH OF SCREW WHEELS. 
 
 plane will be distorted into a curve of a spiral form, which may also 
 be traced as above described by the uniform motion of a marking 
 point along the right line of contact, while the disc rotates uniformly 
 about its axis. 
 
 The teeth thus formed, it is evident, will have but a single point of 
 contact, which will travel along the common element of the pitch 
 surfaces, just as in the case of other twisted wheels ; but this is suffi- 
 cient to make the action continuous, if the amount of twist in the 
 length of the tooth be a little greater than the pitch ; and the action 
 will be peculiarly smooth, since the amount of sliding friction will be 
 no greater than that between the pitch surfaces. 
 
 The Teeth of Screw Wheels. 
 
 401. The most common example of Screw Gearing is the arrangement 
 familiarly known as the Endless Screw, or Worm and Wheel. In this 
 case the axes are situated in planes which are perpendicular to each 
 other, and the nature of the action will be readily seen by inspection 
 of Fig. 244. being the centre of a pitch circle, and TT the pitch 
 
 line of a rack, let 
 teeth be constructed 
 of any of the forms 
 proper for spur 
 gearing. In the 
 plane of the paper 
 draw any line DD 
 parallel to TT, and 
 taking it as an axis, 
 let the outline of 
 the rack be made 
 the meridian sec- 
 tion of a screw 
 whose pitch is equal to that of the rack teeth. This screw is still a 
 rack, and if moved endlong will turn the wheel ; but if instead of this 
 the screw itself be turned, the effect will be precisely the same. For, 
 supposing the wheel to be very thin, its tooth is confined between the 
 threads of the screw, all of whose meridian sections are alike, but each 
 successive one is in advance of the preceding, so that when the screw 
 has made one revolution, the wheel-tooth must have been driven 
 through an angle measured by the pitch arc. In short, the screw is a 
 rack which advances by rotation ; and this is the fundamental princi- 
 ple of all screw gearing, with the exception of one combination, which 
 will be described hereafter. 
 
 244. 
 
NATURE OF THE ACTION. 
 
 267 
 
 402. Distinctive Peculiarities of the Action. The line TT, by revolv- 
 ing about the axis DD, generates the pitch cylinder of the screw, 
 which is tangent to that of the wheel at the point A. 
 
 Three characteristic features distinguish the action from that of 
 twisted gearing, viz. : 
 
 1. The velocity ratio is independent of the relative diameters of the 
 pitch cylinders, and depends wholly upon the screw pitch. 
 
 2. The directional relation depends upon the direction of the twist ; 
 the screw, turning in a given direction, will drive the wheel one way 
 if right-handed, the other way if left-handed. 
 
 3. The rotation of the wheel is caused solely by the end thrust of 
 the screw. 
 
 403. Wheels with Similar Transverse Sections. In giving sensible 
 thickness to the wheel, we may proceed as follows : the elements of 
 the two pitch cylinders which pass through A determine the common 
 tangent plane represented by MN, in Fig. 245. The screw helix 
 through A will develope upon this plane into a right line, which, when 
 the plane is wrapped upon the 
 
 pitch cylinder of the wheel, will 
 become another helix lying on 
 that surface ; these two helices 
 will be either both right-handed 
 or both left-handed. Through 
 each point in the outline of the 
 wheel-tooth, already laid out, 
 draw a helix of the same pitch ; 
 we shall thus have a wheel pre- 
 cisely like one of the twisted pair 
 shown in Fig. 109, all the trans- 
 verse sections being similar. The thickness of this wheel is to be de- 
 termined only by considerations relating to the pressure to be trans- 
 mitted and the strength of the material; it has no bearing upon the 
 kinematic action, since at any instant each tooth touches the engaging 
 screw-thread only in a single point in the original transverse plane 
 through the axis of the screw. Therefore, this form of wheel is open 
 to the objection that the wear will be comparatively rapid, being con- 
 fined to one line upon each thread and each tooth. 
 
 404. Close-fitting Tangent Screws. A worm wheel to which this 
 objection does not apply can be practically made in this manner : an 
 exact copy of the screw in steel is notched and hardened so as to be- 
 come a cutting tool, which is used to finish the teeth, usually roughly 
 cut upon the blank with an ordinary milling cutter. The cutting 
 
 M 
 
268 CLOSE-FITTING TANGENT SCREW. 
 
 screw is often made to drive the worm wheel during this operation ; 
 but a better plan is to have the wheel -blank driven at the proper 
 speed by independent means. 
 
 When this method is adopted it is necessary, after taking one cut, 
 to press the axes nearer together, and then take a lighter finishing cut. 
 Therefore, an involute wheel-tooth, working with a straight-sided slo- 
 ping rack tooth, is to be preferred, because this change in the position 
 of the axis does not affect the velocity ratio. 
 
 In this manner a perfect worm wheel is practically made with great 
 facility. The accurate delineation of it is more difficult and tedious, 
 but it can be made by a process illustrated on the right in Fig. 244, 
 where is shown a section by a plane through the axis of the wheel per- 
 pendicular to that of the worm. LL is the mid-plane of the wheel, 
 in which the teeth were first laid out, as above explained. 
 
 405. If now we pass any plane as JV parallel to LL, it will cut from 
 the screw a curved line ; this being taken as a rack tooth, the form of 
 the wheel tooth which will work with it may be determined by the 
 process of Fig. 162, and this will be the outline of the wheel tooth in 
 that plane. Another parallel plane at M will give a different section 
 of the screw, from which we derive, as before, the conjugate form of 
 the wheel tooth, and this may be repeated as many times as is deemed 
 necessary. 
 
 The blank for the wheel is usually of the form shown ; the line OR 
 describes a cone, from which the parallel planes, M and N, cut circles, 
 and the intersections of these circles with the outlines of the wheel- 
 tooth in these planes will be points in the visible contour of the tooth. 
 
 406. Superior Action of the Close-fitting Screw. Not only are all 
 meridian sections of the screw alike, but all sections by planes parallel 
 to and equidistant from its axis are alike. The whole screw being a 
 rack which advances by rotation, it is clear that at each instant there 
 will be points of contact not only in the plane LL, but in other con- 
 secutive planes as N, M, etc. These points constitute a line of contact 
 which, though not a true helix, will evidently be a line of double cur- 
 vature of kindred nature, and during the rotation it will travel along 
 the wheel-tooth from the point toward the root. Between each tooth 
 and its thread, then, we have contact along a line, and the wear is 
 distribute^, over a surface. 
 
 407. Practical Proportions. Abstractly considered, both the diameter 
 of the worm and the number of teeth in the wheel are optional. But 
 it is found that in practice the results are not satisfactory if the wheel 
 has less than from twenty-five to thirty teeth ; with a straight-sided 
 rack and involute wheel-tooth, the obliquity being 15, thirty-six teeth 
 
PEACTICAL PKOPOKTIOKS. 269 
 
 will give a total angle of action greater than twice the pitch, with the 
 arc of recess one and a half times as great as that of approach, which 
 is all that could be desired. This is, of course, for a single-threaded 
 worm ; in regard to its diameter, a simple and a good practical rule 
 is to make the radius of the blank about twice the pitch. If the wheel 
 be a simple twisted one with all its transverse sections alike, its blank 
 will be a cylinder, whose thickness should be from two and a half to 
 three times the pitch ; if it be cut by the screw, and of the form 
 shown in Fig. 244, the angle ROP may be made from 60 to 90, the 
 thickness being as just given. 
 
 408. Physical Embodiment of Sang's Theory. The employment of 
 the screw to cut its own wheel at once suggests the formation, in the 
 same manner, of guide templates suitable for use in connection with 
 the pantagraphic cutter engine of Pratt & Whitney. In cutting such 
 a template the screw practically and automatically executes the process 
 of finding the form of a tooth conjugate to that of a given rack, as 
 illustrated in Fig. 162. 
 
 Now if, as in Fig. 244, we use a straight-sided sloping rack tooth, 
 the result will be the formation of a series of involute templates ; if 
 cycloidal arcs be substituted for straight lines, as in Fig. 120, we shall 
 have a set of templates for epicycloidal teeth with a constant describ- 
 ing circle. But the outline of the screw-thread may be made of any 
 other reasonable form ; and according to Sang's Theory (283), if it be 
 bounded by any four similar and equal curves in alternate reversion, 
 the series of templates produced will be interchangeable, and thus 
 by means of the pantagraphic engine, the cutters for wheels upon any 
 desired basis or system may be readily and accurately duplicated. 
 
 409. Multiple-threaded Screw Wheels. Thus far we have supposed 
 the screw to be single-threaded, with a pitch equal to that of the fun- 
 damental rack tooth. 
 
 Now the helical pitch may be doubled, 
 as shown in Fig. 246 ; this will double 
 the angular velocity of the wheel, but 
 if no other change be made, the alternate 
 teeth only will come into action. This 
 difficulty is obviated by making the screw 
 double-threaded, as in Fig. 247 ; which 
 at the same time reduces by one half the 
 pressure upon each tooth. 
 
 In like manner we may make the heli- 
 cal pitch three, four, or any whole num- 
 ber of times as great as the tooth-pitch, increasing the number of 
 
270 
 
 MULTIPLE-THKEADED SCEEW WHEELS. 
 
 threads accordingly, and taking care to make the diameter great 
 
 enough to avoid excessive obliquity of action. 
 And in this way we may give to the screw as many threads as there 
 
 are teeth upon the wheel, or even more ; the combination then having 
 
 but slight resemblance to the single-threaded endless screw, as will be 
 
 seen by referring to Fig. 110, which repre- 
 sents a pair of screw wheels properly so 
 called. 
 
 When the numbers of the threads and 
 teeth are equal, Prof. Willis states that the 
 two wheels may be made exactly alike ; * 
 this we imagine to be a mere slip of the 
 pen, since that eminent writer was the 
 first to point out the true construction, 
 which requires the section of the screw 
 thread to be a rack tooth, although it need 
 FIG. 247. not be disputed that the difference would 
 
 not be conspicuous, nor that if they were exactly alike they would 
 
 engage and transmit rotation, but with a slight fluctuation in the 
 
 velocity ratio. 
 
 410. Screw and Rack. There is no limit to the increase in the num- 
 ber of teeth upon the worm wheel, and if it be made infinite, the 
 wheel becomes a rack, which if cut by the screw itself, will be iden- 
 tical with a portion of an ordinary nut. 
 
 Now the exterior surface of the screw and the interior surface of the 
 nut are precisely the same ; and this affords an illustration of the 
 extreme case in deriving the conjugate to a given rack tooth, men- 
 tioned in (282), for if we split the nut and screw longitudinally 
 through the axis, the meridian sections will correspond to the two 
 conjugate racks shown in Fig. 163, being exactly converse to each 
 other. 
 
 Oblique Screw Gearing. 
 
 411. The axis of the screw thus far has been supposed to lie in the 
 plane of rotation of the wheel. This, however, is not essential, for 
 even if it cross that plane obliquely, rotation can still be transmitted 
 with a constant velocity ratio, by the end thrust of the screw. 
 
 And in this new relative position of the axes, as before, the screw 
 may have two, three, or any number of threads, and thus we pass 
 from a simple endless screw to the disguised forms of oblique screw 
 wheels. 
 
 Principles of Mechanism, p. 163. 
 
OBLIQUE RACK AKD WHEEL. 
 
 271 
 
 The fundamental principle remains unchanged, however ; the screw 
 is still a rack which advances hy rotation, and the first step in the con- 
 struction is to determine the form of its conjugate tooth with refer- 
 ence to the new conditions. 
 
 412. The Oblique Eack and Wheel. That this is the case, may be 
 perhaps most readily seen by first considering the things which may 
 be accomplished by the rack and wheel alone. 
 
 Ordinarily, as is well known, the rack travels in the plane of rota- 
 tion of the wheel. But this, again, is not a matter of necessity ; 
 without the slightest change in the forms of the teeth, it may be made 
 to travel obliquely across that plane, the velocity ratio remaining ab- 
 solutely constant, although its 
 value will be changed. 
 
 This is illustrated by Fig. 
 248 ; if we suppose a rack to be 
 made by cutting teeth across a 
 broad rectangular plate, MN, 
 indicated by the dotted lines, it 
 will at once be seen that it can- 
 not only move from right to 
 left, causing the wheel to turn 
 as usual, but is also free to slide Mr 
 in the direction of the axis, and 
 that it may receive both motions 
 at once. 
 
 If now a strip be cut diagon- 
 ally from this broad rack, and 
 made to travel by guide rollers, 
 as shown, or by any other means, 
 in the direction of that diag- 
 onal, the effect is precisely the FIG. 248. 
 same. The action between the teeth is unchanged ; but assigning 
 a definite linear velocity to the rack in the new direction, that motion 
 may be resolved into two components, one lying in the plane of ro- 
 tation, the other perpendicular to it. The latter does not affect the 
 wheel, but the former does, and causes it to rotate ; and the linear 
 velocity of the pitch circumference is equal in magnitude to this effec- 
 tive component. 
 
 It will readily be seen that another rack may be cut upon the back 
 of this one, with teeth perpendicular to its sides, which may engage 
 with another wheel in the ordinary manner ; and thus we have a new 
 means of transmitting a limited rotation with a constant velocity 
 
272 
 
 OBLIQUE SCEEW GEARING. 
 
 ratio between two axes in different planes, by the use of common 
 spur gearing only. 
 
 413. The resolution above mentioned is represented in Fig. 248 ; 
 the obliquity of the rack's travel, when assigned, gives the direction 
 of the resultant, and if the component in the plane of rotation be 
 made equal to the pitch arc, the magnitude of the resultant deter- 
 mines the distance through which the rack will advance while the 
 wheel turns through the pitch angle. 
 
 Let us assume this distance as the helical pitch in constructing 
 from the rack an oblique single-threaded worm, as shown in Fig. 249, 
 
 \ 
 
 K 
 
 PIG. 249. 
 
 A being its pitch cylinder, B that of the wheel, P their point of tan- 
 gency. Let P be also the present point of contact between a thread 
 and a tooth, as shown at P\ below; when cut by the plane LM, normal 
 to the axis of the ivheel, let the section of the thread have the form of 
 the rack tooth in the preceding figure, and that of the wheel- tooth be 
 conjugate to it as before. 
 
 As the screw rotates there will always be a sectiornof its thread sim- 
 ilar to this, similarly situated with regard to its axis. This will 
 
Oy THE 
 
 ^UNIVERSITY 
 
 ALL TRANSVERSE SECTIONS OF WHEEL 
 
 travel along with uniform speed, as indicated by the 
 advancing in one rotation to the new position, 0. 
 
 Now in order that the velocity ratio may remain strictly constant, 
 this travelling section of the thread must be always acting against a 
 tooth outline of the same form. Consequently, in every transverse 
 section of the wheel, the teeth will be bounded ~by similar curves, 
 although, as will subsequently appear, they will not necessarily be of 
 uniform height. 
 
 414. We have now to consider the twist of the wheel itself, which 
 depends upon the same principles as in the common arrangement. 
 Thus, suppose the thread and the tooth in contact at P, to be grad- 
 ually reduced in size; they will ultimately bedome two helical lines, 
 lying one upon each pitch cylinder, still tangent to each other, and 
 therefore developing upon the common tangent plane into the same 
 straight line. Thus, if the pitch of the screw-helix be given, that of 
 the other is found as in Fig. 245, and is'- the same for all the helices 
 of the teeth of the wheel, which, if cut from a cylindrical blank, will 
 be a simple twisted one, as in Fig, 110, with all its transverse sections 
 alike. 
 
 415. Careful consideration of the action will show that the surfaces 
 generated as above explained are precisely such as the screw would 
 cut for itself under the assumed conditions, and thus confirm the 
 previous statement (401) that the proper forms of the tooth in this, 
 as well as in the common arrangement of screw gearing with axes in 
 planes mutually perpendicular, are to be determined by the principles 
 which apply to the case of a wheel working with a rack, and not with 
 another wheel of any finite radius whatsoever ; notwithstanding the 
 fact that the latter construction is the one given by Prof. Rankine.* 
 We would not be understood to assert that two oblique screw-wheels 
 will absolutely refuse either to engage or to work with each other 
 because otherwise fashioned, but that the method of construction 
 here set forth is the only one by which perfect theoretical precision 
 can be attained. 
 
 416. Peculiar Features of the Action. The teeth of the wheel in 
 Fig. 249 are sections by successive transverse planes through R, P, ; 
 their conjugates, R, P', 0', being cut from the screw by the same 
 planes. 
 
 We now observe that one rotation of the worm will drive the wheel 
 through more than the original pitch angle, although the helical 
 pitch is equal to the diagonal pitch of the rack in the preceding 
 
 * Machinery and Mill- Work, p. 163. 
 
274 
 
 OBLIQUE SCREW GEARISTG. 
 
 figure. In that case, the rack-tooth was always acting against a sur- 
 face whose elements were right lines perpendicular to the plane of 
 rotation. But the worm acts against helices which cross the plane of 
 rotation obliquely, and with different degrees of obliquity, for they all 
 have the same pitch, but lie at different distances from the axis. 
 Since, however, the velocity ratio is constant, we may confine our 
 attention to the helices upon the pitch cylinders, and study their 
 action as represented in the development of these surfaces upon the 
 common tangent plane. 
 
 Thus, in Fig. 249, having drawn PK, perpendicular to PO and 
 equal to the circumference of the pitch cylinder A, OK is the devel- 
 oped helix through 0, and PE is the length of the new pitch arc for 
 the wheel. If then a complete wheel is to be made, the proportions 
 must be such that PJZis an aliquot part of the pitch circumference. 
 
 417. And all this accords precisely with the fundamental principles 
 relating to the composition and resolution of motions. In Fig. 250 
 CCbG the elements of the pitch cylinders of the worm 
 and the wheel respectively, inter- 
 secting at P ; PO the pitch of the 
 worm, PK perpendicular to PO, 
 and equal to its pitch circumfer- 
 ence, and OK the developed helix, 
 all as in the preceding figure ; then 
 PM parallel to OK is the developed 
 helix through P, which point we 
 will assume as before to be the pres- 
 ent point of contact between a 
 tooth and a thread. By the turn- 
 ing of the worm, the point P 
 virtually advances uniformly in the 
 direction PD, going in one revolu- 
 tion to the new position 0, while 
 meantime the coincident point P 
 of the wheel must move in the di- 
 rection PE. Regarding PM as a 
 helix of the wheel, and supposing the screw to be pushed endlong in 
 the manner of a rack, we observe that PO may be resolved into the 
 components PM, PE. Of these, the first is simply the sliding compo- 
 nent and ineffective ; but PE is the one which must represent the rota- 
 tion of the wheel, due to the supposed rectilinear motion of the screw. 
 When, however, we suppose the worm to have a motion of rotation 
 only, in the direction shown by the arrow in Fig. 249, let PK repre- 
 
 FIG. 250. 
 
ANALYSIS OF THE ACTIOK. 275 
 
 sent the linear motion of its driving point P, which acts against the 
 helix PM of the wheel. The normal and tangential components of 
 PK are respectively PG perpendicular to PM, and PJ coincident 
 with it. The resultant motion of the point P of the wheel must be 
 in the direction PF, perpendicular to (7(7, and of such magnitude, PE, 
 as to have the same normal component, PG. 
 
 The same method mighfc have been used in relation to the recti- 
 linear motion PO of the screw when used as a rack, and with concor- 
 dant results, since PG is the normal component of that motion also. 
 
 418. In all the foregoing the diameter of the pitch cylinder of the 
 worm was assumed at pleasure, which it may be, subject to the condi- 
 tion above pointed out, that PE must be an aliquot part of the pitch 
 circumference of the wheel. 
 
 It is perfectly possible, then, that a double as well as a single thread 
 might be made upon a pitch cylinder of given size. Supposing this 
 to be desired, and also that the subdivision of the given wheel should 
 remain unchanged, it is then evident that in Fig. 250 we must double 
 PE, the pitch arc, and not PO, the helical pitch of the screw, which 
 will now become PN, determined by producing KFto cut DD in N. 
 We thus determine new helices, KN, PL, with reference to which the 
 motion PK is to be resolved as before, PH being the normal or effect- 
 ive, jP/the tangential or sliding component. 
 
 419. The diagram, Fig. 250, is drawn without regard to practical 
 proportions, the conditions being selected with a view to illustrating 
 one particular in which oblique screw gearing may, in some cases, 
 differ from the ordinary arrangement. We have seen that when the 
 axes lie in planes perpendicular to each other, the helices on the worm 
 and the wheel, whatever the number of threads and teeth, are either 
 both right-handed or both left-handed (403). 
 
 Now in Fig. 250, both KO and KN will form right-handed helices 
 upon the pitch cylinder of the screw ; but on wrapping the tangent 
 plane down upon the wheel, KN will become a right-handed, KO a 
 left-handed, helix. Evidently there is an intermediate position in 
 which the developed worm-helix will be parallel to CC, and will, 
 therefore become a rectilinear element of the wheel's pitch cylinder. 
 In that case the screw will work with a common spur-wheel ; and the 
 proportions which must obtain in order to secure this result are shown 
 in Fig. 251. If the pitch arc, PE, and the obliquity, CPD, are both 
 assigned, the pitch and circumference of the screw are determined 
 by drawing through- E a perpendicular to PE, cutting DD at 0, and 
 ^^perpendicular to DD at K. If the pitch arc, PE, and the circum- 
 ference PK are assigned, we first describe an arc about P with radius 
 
276 CLOSE-PITTING OBLIQUE SCREW. 
 
 PE\ draw KE tangent to it, and also PD perpendicular to PK. 
 Then KE produced cuts PD in 0, determining the pitch PO and 
 the obliquity OPE. 
 
 420. Close-fitting Oblique Tangent-screw. The 
 oblique worm may be used to cut its own wheel, 
 just as in the common arrangement ; and the 
 wheel itself, instead of being cut from a plain 
 cylindrical blank, may, for the sake of a neater 
 appearance, conform somewhat to the curvature 
 of the screw, and be terminated by conical frusta 
 instead of transverse planes. The form of the 
 blank is fixed by very simple considerations ; 
 regarding the screw as a thread wrapped around 
 a cylindrical core, we perceive that the tops of 
 the wheel-teeth must just go clear of that core 
 during the rotation. Now imagine the whole screw to revolve about 
 the axis of the wheel ; then the element of the core which lies nearest 
 to that axis will generate a hyperboloid, and if the blank for the 
 wheel be turned to this exact size, the tops of the teeth will just 
 touch the core. It should therefore be made a little smaller. 
 
 It will be seen that the screw still travels obliquely across the plane of 
 rotation, so that the transverse sections of the teeth are all of the same 
 form, contact existing between each thread and its tooth at a single 
 point only ; whence the only advantage gained by the adoption of this 
 form of blank lies in the fact that the teeth become higher as they 
 recede from the mid-plane, and, therefore, continue longer in action. 
 421. Oblique Screw and Rack. Further, it will be seen that the di- 
 ameter of the oblique worm-wheel may be increased at pleasure until 
 it eventually loses its curvature, and assumes the form of a rack. 
 Under these circumstances Prof. Rankine's instructions,* if we inter- 
 pret them correctly, are to the effect that the normal section of the 
 worm-thread should be that of the tooth of a wheel working with a 
 rack, the tooth-outline of the latter to be adopted as the normal sec- 
 tion of the rack to work with this worm ; and he gives specific direc- 
 tions for finding the radius of that wheel in any given case. 
 
 Without disputing that the thread and tooth thus formed will 
 work, we would remark that it is not the only nor yet the best 
 manner of forming them. The tooth-surfaces of the rack must be 
 made up of parallel rectilinear elements, and since in the case of a 
 rack and wheel the teeth have but a single point of contact, the 
 
 * Machinery and Mill-Work, p. 290. 
 
OBLIQUE SCREW AND RACK. 277 
 
 tooth-surface of the rack will, in Prof. Rankine's construction, touch 
 each other in that point only. Which would hold true at the limit 
 when the obliquity vanishes ; whereas it is perfectly patent that the 
 rack may then be a part of a nut, and its whole surface in contact 
 with that of the screw-thread, whatever the form of the latter. This 
 superficial contact is not attainable when the rack travels obliquely, 
 it is true ; but line-contact between the acting surfaces may be secured 
 with a screw-thread 'of any reasonable form, as will appear from the 
 following considerations. 
 
 422. Taking a common triangular-threaded screw in illustration, 
 we have, in Fig. 252, a meridian section on the left, and an outside 
 view on the right, the axis being parallel to the paper. The. apparent 
 contour of the completed screw 
 
 is the trace of its projecting 
 cylinder, whose elements are 
 perpendicular to the paper, and 
 tangent to the surface. If then 
 a rack be made, as shown in 
 section below, its teeth bounded 
 
 by the same outlines and their elements also perpendicular to the 
 paper, these teeth will always touch the screw along the line of its 
 visible contour, whether the rack be made to travel endlong by turn- 
 ing the screw, or to slide transversely, or both. And if these actions 
 occur simultaneously, we have the oblique motion desired. 
 
 Line-contact is thus obtained, and the normal section of the rack- 
 tooth is determined by merely drawing the screw, in its simplest 
 position. But this is not the only form which will effect the same 
 result ; for in whatever direction we chose to look at the screw, if the 
 space between the threads be visible at all, the line of apparent con- 
 tour will give the normal section of a tooth for a rack which will 
 work in line-contact with it, cither longitudinally or obliquely. 
 
 423. Among so many forms, the selection of the best may be safely 
 left to the screw itself, which will assert its preference if given the 
 opportunity. This may be afforded by allowing it to cut its own 
 rack, the latter being moved by independent means m the required 
 direction, and at the same speed as that which it is eventually to re- 
 ceive from the screw. 
 
 It is clear that the amount of metal removed will be the least possi- 
 ble ; whence the deduction that the normal section of the rack-tooth 
 will be the visible contour of the space between the screw-threads 
 when viewed from the direction which will give the greatest apparent 
 breadth to that space. 
 
278 
 
 MINIMUM AMOUNT OF SLIDING. 
 
 What that precise direction is, may depend somewhat upon the 
 meridian section of the screw, but probably will not vary appreciably 
 from that of the tangent to the helix of mean obliquity. 
 
 It will be seen by reference to Fig. 251, that if the pitch, diameter, 
 and inclination of the screw to the line of travel be properly propor- 
 tioned, the rack may be made of the usual form, that is, with the 
 teeth cut transversely across its face. But whatever the arrangement, 
 proportions, or form, the kinematic action consists of sliding contact 
 pure and simple ; there is not, as has been sometimes erroneously 
 stated, the slightest admixture of rolling contact, or of anything even 
 
 distantly resembling it. 
 
 424. To Secure the Least 
 Amount of Sliding, Although 
 the velocity ratio depends upon 
 the pitch and not upon the 
 size of the screw, yet it is evi- 
 dent that for a given pair of 
 axes, and a given velocity ratio, 
 there must be some definite ratio 
 between the diameters of the 
 pitch cylinders, which will in- 
 volve less sliding than any other. 
 "What this proportion is, may 
 be thus deduced. If upon the 
 given axes we construct a pair 
 of rolling hyperboloids with 
 the assigned velocity ratio, these 
 surfaces will work in contact 
 with no sliding other than that 
 along the common element. 
 This element passes through 
 the common point of the gorge 
 circles, and lies in a plane tan- 
 gent at that point not only to both hyperboloids, but to their inscribed 
 tangent cylinders of which the gorge circles are the bases. 
 
 Now, if these be taken as the pitch cylinders of the screw wheels, 
 and the common element as the developed helix, the amount of slid- 
 ing will be the same as that between the two hyperboloids when rota- 
 tating w'ith the given velocity ratio. 
 
 And this amount is then a minimum. In Fig. 253, let PI repre- 
 sent the motion of the point P of the inclined surface ; if the ver- 
 tical one be driven by it, we find in the usual manner the motion of 
 
 FIGS. 253 and 254. 
 
 v- 
 
KESEMBLANCE TO SKEW WHEELS. 279 
 
 the coincident point to be PL, and that the sliding is represented by 
 NM on the line of contact. PI remaining unchanged, let the verti- 
 cal hyperboloid be compelled to revolve, by independent means, faster 
 than the other should drive it ; and let PK be the new velocity of 
 its point P. The sliding component along PB now becomes PO, 
 which is greater than PN '; and there will be also a sliding perpen- 
 dicular to the common element, represented by GH. A similar 
 result would follow, were PI to bo increased whilo PL remained the 
 same ; whence it appears that the sliding is least when the velocity 
 ratio is that for which the hyperboloids were constructed ; and this 
 holds true of the tangent cylinders, the obliquity of the helices being 
 determined as above. 
 
 425. The sliding action represented by GH cannot actually occur 
 when teeth are used ; but the linear motions PI and PK can still be 
 retained in the case of screw gearing, as shown in Fig. 254, by mak- 
 ing the developed helix PB parallel to KI\ all the sliding must 
 necessarily be in this direction, and its amount OM is, of course, 
 greater than before. This proceeding, it will be observed, is merely 
 the converse of that explained in connection with Fig. 250, the results 
 being exactly concordant. 
 
 If the axes lie in planes perpendicular to each other, the hyperbo- 
 loids have two common elements, either of which may be taken as the 
 developed helix, according to the directional relation desired ; both 
 wheels will be right-handed in one case, and left-handed in the other. 
 
 426. Resemblance to Skew Wheels. The appearance of screw gear- 
 ing thus constructed is quite unlike that of the familiar worm and 
 wheel. In that combination the wheel cannot move the screw, nor 
 is it usually desirable that it should, since in most ea^es the latter is 
 required not only to turn the wheel, but to hold it in any given posi- 
 tion. Eut in the arrangement now under consideration either wheel 
 may be used as the driver ; it is quite apparent that a single-threaded 
 worm made in the manner above described could not work unless it 
 were of great length and its axis nearly parallel to that of the wheel, 
 but that this construction is adapted only for the teeth of screw wheel- 
 work properly so called ; and equally evident that for this it is the 
 best. 
 
 The mid-planes of these wheels are the gorge circles of the tangent 
 hyperboloids ; the pitch helices are tangent to the elements of those 
 surfaces ; and from the considerations presented in (420), it will be 
 seen that both blanks may be made of hyperboloidal outline. From 
 all which, there results a resemblance between these and skew wheels 
 sufficiently close to be misleading. 
 
280 
 
 HOUR-GLASS WORM GEAR. 
 
 This resemblance, however, is superficial ; the teeth are made up 
 of cylindrical helices, touch each other in a single point only, can be 
 extended only to a short distance either way without ceasing to 
 engage, and their acting sections are respectively those of a spur wheel 
 and a rack. 
 
 And the one whose section corresponds to that of the rack-tooth 
 may be made of any length, and if desired may be used as a rack, 
 driving the other wheel not by rotation, but by moving longitudinally, 
 whatever the inclination of the axes or the number of threads. 
 
 Hindley's Screw, or Hour-glass Worm. 
 
 427. A Worm may be so shaped as to conform to the curvature of 
 
 IO the wheel, assuming a figure 
 
 somewhat like that of an hour- 
 glass ; in Fig. 255, the complete 
 worm is represented on the left, 
 and on' the right is given a sec- 
 tion through its axis by a plane 
 perpendicular to that of the 
 wheel. The surface of the 
 thread is of a complicated and 
 peculiar nature, but practically 
 it is very easily made, thus : a 
 tool, with a cutting point of the 
 contour of the wheel's tooth, is 
 so clamped to a disc that its 
 upper surface lies in the meridian plane of the worm, and both the 
 disc and the worm blank are driven by intermediate gearing at their 
 proper relative velocities. 
 
 In this manner the screw was made by Hindley, who first intro- 
 duced it. For some reason it has never come into general use, 
 although in smoothness and steadiness of action it would appear to 
 be superior to the common form of tangent-screw. 
 
 The worm itself may now be formed into a cutter, and made to 
 finish its own wheel in the usual manner ; which is the method 
 adopted by Messrs. Clem & Morse of Philadelphia, Penn., who have 
 recently constructed a very elegant engine specially designed for cut- 
 ting this description of screw gearing. 
 
 428. The outline of the pitch surface of this worm is an arc of the 
 pitch circle of the wheel. Upon this suface the helix EFP, in Fig. 
 256, is traced by a point which moves about the centre C through the 
 arc EP, while the worm makes one revolution, both motions being 
 
 FIG. 255. 
 
FORM OF PITCH SURFACE. 
 
 281 
 
 uniform in velocity. The longitudinal advance, UV, is not equal to 
 EP ; it is not uniform, nor is it the same for successive convolutions. 
 
 The projection of this helix upon a plane, OC, perpendicular to DD, 
 as shown at the right, is of a spiral form, resembling that of the hy- 
 perboloidal helix in Fig. 243 ; to which this curve is very similar. 
 Other points in the outline of the tooth, either within or without the 
 pitch circle, also dc- N 
 
 scribe curves of the 
 same nature, lying 
 upon surfaces whose 
 meridian outlines are 
 circular arcs ; these 
 curves are easily 
 drawn, and their 
 envelope is the visi- 
 ble contour of the 
 worm. 
 
 The form of the 
 threads is, abstractly, 
 arbitrary ; their me- 
 ridian sections are FIG. 256. 
 exactly converse to those of the wheel-teeth, the spaces between 
 which, as seen in Fig. 255, they fit and fill entirely, throughout the 
 action. Thus there is no relative motion in the manner of a rack or 
 otherwise, but the worm is locked into the wheel, and can move only 
 by revolving about one axis or the other. A good and simple practi- 
 cal form for the teeth of the wheel is that given in Fig. 255, the sides 
 being straight sloping lines, equally inclined to the radius of symme- 
 try ; the amount of inclination is fixed by the consideration that, in 
 the extreme position of the action, the outer side should lie in a line, 
 AB, perpendicular to DD, in order that the worm may be readily 
 disengaged from the wheel. 
 
 429. The Action Confined to the Mid-plane of the Wheel. The ad- 
 vantage of the hour-glass worm lies in the fact just stated, that the 
 whole side of every thread, in the meridian plane, is always in con- 
 tact with the adjacent tooth. 
 
 It has been asserted that the teeth of the wheel, when cut by the 
 worm, touch the threads of the latter at all points ; * or, in other 
 words, that the whole surfaces are in contact. 
 
 Either one of two considerations is sufficient to show that this is 
 
 "Mechanics " for January 14, 1882. 
 
282 THE ACTION CONFINED TO ONE PLANE. 
 
 impossible. Let any side plane be passed, parallel to the axis of the 
 worm and perpendicular to that of the wheel ; then, in the first place, 
 the sections of the successive threads will not be similar ; and, in the 
 second place, the section of the pitch surface of the worm will not be 
 a circle, whereas the teeth of the wheel travel in circular paths. 
 Therefore, such superficial contact is attainable only when the radii 
 of these paths become infinite, in which event the pitch surface of 
 the worm becomes a cylinder, the worm itself a common screw, and 
 the wheel a portion of a nut ; but, in general, the action is confined 
 to the central plane of the wheel. 
 
 430. The Pitch Surface of the Wheel may to some extent conform to 
 the curvature of the worm, as. shown in section in the right-hand 
 part of Fig. 256. Any side plane, L M, parallel, to the mid-plane, NR, 
 cuts the base, Iff, of the worm's pitch surface in a point L, which 
 when revolved about the axis of the wheel into the gorge plane 00, 
 takes the position L', thus giving a point in the contour L P'S. 
 
 By a similar process, reducing the radius VP of the gorge of the 
 worm, and increasing the radius CP, according to the height of the 
 tooth, we may determine the outline of the blank. This conforma- 
 tion of the wheel does not, it must be noted, secure any additional 
 contact, or in any way affect the action, but merely gives a neater 
 finish than if the blank were made a plain cylinder. 
 
 431. The Teeth of the Wheel are automatically shaped by the cutting 
 worm with perfect ease; but the accurate delineation of the form 
 thus determined is a rather complicated and tedious matter. Still, if 
 necessary, it can be accomplished in the following manner. 
 
 When the outline of the blank has been, found, as above, it will be 
 seen that any transverse plane will cut from it a circle, which will 
 limit the height of the teeth. The same plane will cut from the 
 threads of the screw a series of sections of varying form, each lying 
 a little in advance of the meridian section of the same thread. The 
 sections of the wheel-teeth by the same plane must lie in the spaces 
 between those of the screw-threads ; and since each revolution of the 
 worm turns the wheel one tooth ahead, it follows that each tooth- 
 section must take in succession all the different positions thus deter- 
 mined. We may, then, proceed thus : lay a piece of tracing paper 
 over the drawing, and secure it by a pin fixed at the centre of the 
 wheel. Upon this, trace the sections of two consecutive threads, and 
 also the circular arc bounding the top of the tooth in this plane. 
 Then rotate the paper about the central pin through the pitch angle, 
 again trace the sections of the two adjacent threads, and so on until 
 all have been traced ; the clear space within the lines thus drawn 
 
WORM WHEEL WITH ROLLERS. 
 
 283 
 
 will be the required section of the tooth by the given plane. Abso- 
 lute certainty of definition will require the screw to be placed in sev- 
 eral positions, cut in each by the same plane, and the above process to 
 be repeated for each, using the same tracing throughout ; because the 
 sections of these peculiar screw-threads in the different phases of rota- 
 tion are dissimilar. And finally, this whole operation must also be 
 repeated with several different planes, the number depending upon 
 the degree of accuracy aimed at : all of which we leave the reader to 
 execute at leisure. 
 
 432. Multiple-threaded Hour-glass Worm. Though we are not aware 
 that this form of worm has ever been made with two or more threads, 
 there seems to be no abstract objection to increasing the number. 
 Also it is clearly possible, by making the screw pitch sufficiently 
 large, to use the wheel 
 
 as the driver : in which 
 case the teeth might pre- 
 ferably be made in the 
 form of turned pins, set 
 into the periphery of a 
 cylinder as in face gear- 
 ing. Evidently a greater 
 number of these pins, or 
 teeth, Avould be simul- 
 taneously engaged than 
 in the common form of 
 screw wheels, which 
 would certainly tend to 
 increase the steadiness of 
 the motion ; but the 
 whole action is confined, 
 as before, to the meridian 
 plane of the worm per- 
 pendicular to the axis of 
 the wheel. 
 
 433. Rollers Substitu- 
 ted for Teeth. With the purpose of reducing the sliding friction as 
 much as possible, the singular device illustrated in Fig. 257 has been 
 proposed. 
 
 The meridian section of the screw-thread is bounded by right lines 
 at right angles to each other, and the worm is made long enough to 
 embrace more than one quarter of the circumference of a wheel, the 
 number of whose teeth must be a multiple of four ; for this wheel is 
 
 FIG. 257. 
 
284: 
 
 FACE GEARING. 
 
 substituted a frame carrying rollers, arranged in the manner shown 
 in the figure. 
 
 The object of diminishing the friction is certainly accomplished ; 
 but, on the other hand, only one thread of the worm is in action at 
 once, during the greater part of the time. And that one has at no 
 instant more than a single driving point ; for, since the warped sur- 
 face of the screw cannot be placed in right-line contact with a surface 
 of single curvature, these rollers cannot be made cylindrical, as they 
 have sometimes been represented ; but their contours must be slightly 
 convex curves tangent to the meridian sections of the threads. 
 
 Taking these drawbacks into consideration, it would appear that 
 this arrangement is, from a practical point of view, more curious 
 than useful ; although it might serve a purpose in very light-running 
 mechanism intended rather for the modification of motion than for 
 the transmission of power. 
 
 The Teeth of Face-Wheels. 
 
 434. Let two Wheels, exactly alike, whose teeth are cylindrical pins 
 fixed in the faces of circular discs, be so 
 placed that each axis lies in a plane perpen- 
 dicular to the other, at a distance from it 
 equal to the diameter of the pins, as in Fig. 
 258. 
 
 Under these circumstances, the angle 
 FDE will always be equal to the angle 
 HCG, as long as the pin E of the driver A 
 is in contact with the pin G of the follower 
 B : the velocity ratio is, therefore, perfectly 
 constant. 
 
 The length of the pin E must be such 
 that the next pin / of the other wheel shall 
 not catch upon its end in going into gear. 
 And it will also be noted that although the 
 pin 7", at the instant when the next pin K 
 of the driver begins to act upon it, may also 
 touch the pin E upon the back, it cannot 
 continue to do so. It is not possible, there- 
 fore, even theoretically to secure entire free- 
 FIG. ass. dom from backlash. 
 
 The maximum length of one pin having been ascertained, it is of 
 course the same for all. And it is next to be observed that if the 
 number be increased this length must be diminished ; also, that there 
 
FACE GEARING. 
 
 285 
 
 will in every case be a limit beyond which the number cannot be in- 
 creased without at the same time diminishing the diameter, and in 
 consequence the distance between the axes. 
 
 Ultimately, then, the axes will intersect at right angles, and the 
 pins will become consecutive points in the circumferences of two 
 equal circles rolling together like the bases of the pitch cones of a 
 pair of mitre-wheels. In other words, there are no pitch surfaces, 
 these degenerating into lines, and the elementary teeth into points. 
 
 435. Equal Wheels with. Axes at Right Angles. The preceding is 
 the simplest form of face gearing, but it cannot be used if the axes 
 intersect, as is often necessary. When they do, however, the teeth 
 of one wheel may still be made cylindrical : ( 
 
 those of the other being solids of revolution, 
 whose outlines may be determined as follows. 
 
 In Fig. 259, let the two wheels be of 
 equal size, their axes perpendicular to each 
 other. Lot E be a pin of no sensible di- 
 ameter, fixed in the wheel A, and F a simi- 
 lar one fixed in the wheel B, the distance 
 between the two being arbitrary. Let the 
 wheels turn through equal angles as indica- 
 ted by the arrows ; the pins will come into 
 the new positions G, H, and /, K, and in 
 the meantime their common perpendicular 
 will continually change both in position and 
 in magnitude. 
 
 But it is always easily determined ; and 
 it is evident that if we now make F the 
 axis of a surface of revolution, the radius of 
 each transverse section being this common 
 perpendicular, we shall have the form of a 
 pin or tooth for B, which will work with 
 the pin E of no sensible diameter, the velocity ratio being constant 
 throughout. 
 
 This process is more fully illustrated in Fig. 260, in which portions 
 of the pitch circles only are shown. The action is readily traced : 
 the curve El will be generated while the pins ^and F traverse the 
 equal arcs EP, FO, and the curve LG, while they move through 
 the arcs PG, OR, which are also equal to each other ; the ordinates 
 x, z, being found by placing the pins in the intermediate positions 
 shown, any required number may be determined, in a similar 
 manner. 
 
 FIG. 
 
286 
 
 FACE GEARING. 
 
 436. Upon assuming a sensible diameter for the pin of the upper 
 wheel, we proceed to derive the outline of the working tooth of the 
 other, precisely as was done in the case of pin gearing, as shown on 
 
 the right in the figure ; drawing a curve at a constant normal dis- 
 tance from LG, equal to the radius of the pin. 
 
 Since on the equal circles AA, BB, we have the arcs EP = FO = 
 F'O', and PG = OR = O'E', it follows that the arcs EF 1 , P0\ GR, 
 etc., are also equal to each other. Whence, EP being equal to PG, 
 we find that EH is greater than GK\ similarly x is greater than z, 
 and, in general, all the ordinates of El are greater than the corre- 
 sponding ordinates of GL, the difference diminishing as we descend, 
 until finally IM is equal to LN. 
 
 Consequently, the working tooth for the lower wheel must be de- 
 
 Fio. 261. 
 
 rived from the curve GL and not from El ; and in order to secure 
 receding instead of approaching action, the cylindrical pins must be 
 given to the driver, and not to the follower as in pin gearing. 
 
 437. Unequal Wheels with Axes at Right Angles. In Fig. 261, let the 
 cylindrical pins be given to the smaller wheel, of which AP'A is the 
 pitch circle. Assuming any point E upon this circumference, draw 
 
FACE GEARIKG. 
 
 287 
 
 through it a parallel to CD, cutting the larger pitch circle BPB in F. 
 Then the arc EP' is greater than the arc FP ; therefore, if we make 
 FO equal to EP', a parallel to CD through will cut the tangent at 
 P' in sorgo point Ton the right of P' ; and IM will be the lowest 
 ordinate of the curve El, which, as in the preceding figure, will be 
 generated while a pin at E traverses the arc EP' and one at F moves 
 to 0. Also P' Twill be equal to LN 9 the lowest, and evidently the 
 least, ordinate of the curve LG generated while the pins traverse the 
 equal arcs OR and P'G. The whole process, including the determi- 
 nation of the intermediate ordinates, is the same as before' ; but since 
 the lease of the two elementary surfaces must necessarily be employed, 
 we perceive that the working tooth in this case must be derived from 
 El instead of LG, and also that the cylindrical pins must be given 
 to the follower, as in pin gearing, and not to the driver, in order that 
 the action may be receding. 
 
 438. As the diameter cf the driver becomes greater its curvature 
 becomes less, and at the limit will disappear ; the combination then 
 becoming identical with that of a rack driving a pin-wheel. The 
 curves El and GL will in that case evidently be similar and equal, 
 each being the cycloid of which AA is the generating circle. 
 
 There is, in fact, a close analogy between pin-gearing and face- 
 gearing ; for if in the latter we suppose the axes to be parallel, the 
 teeth of one wheel will project radially from the curved surface of a 
 cylinder, and their contours will be precisely like those of a spur- 
 wheel working with the round staves of a trundle, or lantern pinion. 
 
 439. But the cylindrical pins may be given to the larger of the two 
 unequal wheels. The 
 
 case is then very nearly 
 the converse of the pre- 
 
 : the rotations 
 
 the same di- 
 rection, the pointed 
 tooth, as seen in Fig. 
 262, appears on the op- 
 posite side of the plane 
 of the axes, and if the 
 action is to be receding, 
 the larger wheel, with the cylindrical pins, must drive. 
 
 The greater the difference between the diameters of the pitch 
 circles, the shorter will be the teeth of the follower for a given arc of 
 action. And, in consequence of this, the size of the driver cannot 
 be indefinitely increased, since upon reaching the limit, its pins 
 
 A 
 
 ceding one 
 being in 
 
FACE GEARING. 
 
 would move in a plane parallel to the face of the follower, and the 
 tooth of the latter would reduce to a ssries of concentric circles lying 
 in the same plane. 
 
 440. Unequal Wheels with Axes in Different Planes. Precisely the 
 same method of construction is applicable also when each axis lies in 
 a plane perpendicular to the other, although they do not meet, and 
 whether the wheels be equal or unequal in size. Thus in Fig. 263, 
 
 c 
 
 K 
 
 supposing the cylindrical pins to be assigned to the smaller wheel, the 
 axis E of one of them is placed arbitrarily with reference to the axis 
 F of a tooth of the other, whose contour is derived from a curve El, 
 determined as in Fig. 260. It is clear that under some conditions 
 this elementary tooth may be poiifted, as it was in Fig. 261 ; and in 
 
 Fig. 264 the condi- 
 tions are so selected 
 that this is the case ; 
 but, as before, the cy- 
 lindrical pins being 
 given to the large 
 wheel, this pointed 
 tooth makes its ap- 
 pearance on the other 
 side of the axis, the 
 direction of the rota- 
 tion being unchanged. 
 Hence we perceive that in order to secure receding action, the larger 
 wheel must drive, no matter to which one the cylindrical pins are 
 assigned. 
 
 Had the wheels been made of the same size, but with the least dis- 
 tance between the axes greater than the diameter of the cylindrical 
 
 FIG. 
 
TEETH OF FACE WHEELS. 289 
 
 pin, the same process would have enabled us to determine the form 
 of the working tooth for the other wheel : in short, the construction 
 is general for the relative positions of the axes above mentioned, and 
 Fig. 258 represents only a special case. 
 
 441. Miscellaneous Arrangements of Face Gearing. It is not essential 
 that either the pins or the teeth, because they are turned in the lathe, 
 should be inserted into plane surfaces ; and we append a few exam- 
 ples in which they are otherwise arranged. 
 
 Let the axes in Fig. 265 intersect at any angle ; let the velocity 
 ratio be given, and suppose the 
 cylindrical pins *to be assigned to 
 the wheel with the vertical axis. 
 
 Draw AB, dividing the angle 
 DAG according to the velocity 
 ratio as in bevel gearing. Through 
 any point E of AB draw a verti- 
 cal line as the axis of a pin ; and 
 also a horizontal line cutting the 
 inclined axis in C ; then EC by FIG. 205. 
 
 revolving around A C will generate the cone ECF. The teeth of the 
 inclined wheel are to be solids of revolution, whose axes will evidently 
 be elements of this cone ; and they may be fixed in the surface of 
 another cone OPL, normal to ECF. 
 
 A pin of the vertical wheel is shown at E in contact with such a 
 tooth, whose contour may be determined as follows : First, suppose 
 the cylindrical pin to have no sensible diameter ; then if the vertical 
 wheel turn through a given angle, the other will also turn through 
 an angle which is known, because the circumferential velocities of 
 the circles which roll in contact must be equal. 
 
 Consequently, the relative positions of the axes of the given pin 
 and the required tooth may be determined for any phase of the 
 action, and their common perpendicular at that instant may be 
 found. 
 
 Having repeated this process as many times as may be deemed nec- 
 essary, these common perpendiculars will obviously be the radii of the 
 transverse sections of the required tooth for a pin of no sensible diam- 
 eter. From these the meridian outline can be constructed, and the 
 contour of the working tooth is derived from this in the usual manner 
 by assigning any radius at pleasure to the cylindrical pin. 
 
 442. A modification of this arrangement is shown in Fig. 266, the pins 
 being fixed radially in the periphery of the cylinder with the horizontal 
 axis ; but the process of constructing the tooth is the same as before. 
 
 19 
 
290 
 
 MISCELLANEOUS ARRANGEMENTS. 
 
 FIG. 266. 
 
 From the employment of the conical surfaces, which afford the most 
 convenient and natural means of supporting the teeth of one wheel 
 
 of a pair, these combinations 
 have been erroneously supposed 
 to contain the germ from which 
 bevel gearing was developed. 
 
 But a moment's study of 
 these figures will show that 
 the principle of rolling cones, 
 upon which bevel gear wholly 
 depends, is not here involved 
 in any way whatever ; the fun- 
 damental idea appears to have 
 been that of causing the teeth 
 or pins to present themselves to 
 
 each other, at the instant of passing the plane of the axes, in the same 
 relative position as though the axes were parallel. 
 
 But, as shown in Fig. 267, the same methods and processes are ap- 
 plicable also to the construction of what may be called bevel face- 
 gearing, the axes of the pins coinciding with the elements of one 
 pitch cone, and projecting nor- 
 mally from the surface of the 
 other. 
 
 Face gearing is for general 
 purposes practically obsolete ; 
 hence it has not been deemed 
 necessary to discuss in relation 
 to it the questions of limiting 
 lengths or numbers of teeth, 
 angles of action, and the like. 
 For models and light machinery 
 it is sometimes employed, on ac- 
 count of the facility of forming the teeth in the lathe ; a special con- 
 struction would in any event be almost necessary for each individual 
 case, and this can readily be made by the aid of the preceding expla- 
 nations. 
 
 443. Screw and Face Gearing Combined. An example has already 
 been given, in Fig. 257, of the use of rollers for reducing the friction 
 which attends the action of the worm and wheel. In that case the 
 screw was of the hour-glass form ; but the same expedient has been 
 employed in connection with, the common variety, as for instance in 
 Van der Mark's hoisting apparatus ; the axes of the rollers projecting 
 
SCREW FACE GEAR. 
 
 291 
 
 radially from the periphery of a cylinder, and their outlines being 
 the same as those of the spur-teeth, conjugate to the rack-teeth cut 
 from the screw by a meridian plane. Since the teeth of the wheel, 
 whether they be rollers, capable of rotating, or not, are solids of rev- 
 olution, this arrangement combines the peculiarities of both screw 
 and face gearing. 
 
 In Fig. 268 is represented another combination of essentially the 
 same nature. Upon the upper 
 face of the disc B is formed a 
 spirally coiled rib, constituting 
 what may be properly called a 
 plane or face screw. 
 
 The section of this rib by a 
 plane normal to the axis of the 
 wheel C should, as in the figure, 
 have the form of a rack-tooth, 
 the conjugate to which is the 
 meridian section of the teeth of C. 
 Thus, the distance between the 
 corresponding edges of the adja- 
 cent coils being equal to the 
 pitch, the action is precisely 
 equivalent to that of a rack and 
 spur-wheel. 
 
 444. This arrangement has been 
 erroneously represented as having 
 the rib coiled in the form of an FIG. 268. 
 
 Archimedean spiral, and the axis of B lying in the same plane with 
 the axes of the teeth of (7. 
 
 But the radiant of that curve does not cut the coils normally ; and 
 in order to secure perfect kinematic action, the spirals, as shown in 
 the figure, should be involutes of a circle whose circumference is 
 equal to the tooth -pitch, and the axes of the teeth of C must lie in a 
 plane parallel to the axis of B at a distance from it equal to the 
 radius of that circle. * 
 
 This is, of course, upon the supposition that, as in the case selected 
 for illustration, the spiral expands at the rate of one tooth-pitch in 
 each convolution, thus making the action equivalent to that of a com- 
 mon single-threaded worm. We may, however, double or treble the 
 rate of expansion, increasing the angular velocity of the face-wheel 
 in the same proportion, and introducing a corresponding number of 
 intermediate ribs or coils ; if this be done, the diameter of the base 
 
292 
 
 SPHERICAL SCREW AKD WHEEL. 
 
 circle of the involutes must be also doubled or trebled, as the case may 
 be, and the position of the plane of action changed accordingly. 
 
 445. Spherical Screw and Wheel. A curious, and we believe a novel, 
 modification of the preceding arrangement is shown in Fig. 269, the 
 rib being coiled upon the concave surface of a hollow sphere, forming 
 what may be called a spherical screw ; the construction is as follows. 
 
 The spherical involute OFF is generated by a point in the plane 
 ACB, while the plane rolls around the cone ACKto which it is tun- 
 gent. Since the element of contact is the instantaneous axis, the 
 tangent to the curve at any point is perpendicular to the rolling 
 
 FIG. 269. 
 
 plane ; which latter cuts from the sphere a great circle tangent to the 
 base of the cone. Therefore, the points G, F, E, etc., of the curve, 
 will in revolving about the horizontal axis of the cone, come normally 
 into the plane ACB, at points in the circumference of the great circle 
 cut by it from the sphere. 
 
 Or otherwise : let the cone ^IC/Tand a disc ACB move in rolling 
 contact about fixed axes ; then while the' former makes one revolu- 
 tion to the right, the latter will turn to the left through an arc PO 
 equal to the circumference of the cone's base AK, and if the sphere 
 
SPHERICAL SCREW AtfD WHEEL. 293 
 
 revolve with the cone, the point P will trace upon its surface the 
 same curve PFO in a reverse order. 
 
 If then a groove of this form be cut in the concave surface of a por- 
 tion of a spherical shell, and 0, P, Q, etc., represent equidistant pins 
 projecting from the edge of the disc A CB, it is clear that a rotation 
 of the shell about the horizontal axis will impart to the disc a rota- 
 tion about an axis perpendicular to its plane, with a constant velocity 
 ratio. 
 
 446. Let us now take the circumference of the disc as the pitch 
 circle of a wheel, and lay out upon it the form of a tooth suitable for 
 driving in its own plane a pinion in inside gear, as shown in the 
 upper diagram in the figure. It will then be apparent from what 
 precedes that this tooth outline will be the normal section of the 
 thread of the spherical screw, whose surface it will sweep up if the 
 disc roll around the cone as before ; and that this screw will correctly 
 drive the internal pinion if its teeth be made solids of revolution as 
 shown in the cut. The axis of the screw being horizontal, the axis, 
 XX, of the pinion, is not vertical, but perpendicular to the plane 
 ACS. 
 
 It is hardly necessary to point out, finally, that the pinion may be 
 placed in outside gear, and the screw cut upon the convex surface of 
 the sphere. In short, the construction is a general one, the face- 
 screw and wheel being but the special case in which the radius of the 
 sphere becomes infinite and its surface degenerates into a plane. 
 
APPENDIX. 
 
 PROBLEMS RELATING TO CIRCULAR ARCS. 
 
 1. In Graphic Operations there is frequent occasion to rectify circu- 
 lar arcs, and to lay off arcs of given linear values, either upon circles 
 of given radii or subtending given angles. 
 
 In such cases the following constructions, which we borrow from 
 Prof. Rankine, will be found extremely useful and convenient ; the 
 results being obtained much more expeditiously than by calculation, 
 and with a degree of accuracy amply sufficient for all ordinary pur- 
 poses. 
 
 L To lay off on a right line a distance approximately equal in length 
 to a given circular arc. 
 
 Let AB, Fig. 1, be the given arc, and A Hits tangent at A. Draw 
 the chord BA, and produce it; bisect AB in D, and set off AE equal 
 
 FIG. 1. 
 
 to AD. About E as a centre, with EB as radius, describe an arc cut- 
 ting AH'm F\ then AF = arc AB, very nearly. 
 
 II. To lay off on a given circle an arc approximately equal in length 
 to a given right line. 
 
 In Fig. 2, let AB, the given line, be tangent at A to the given 
 
29G APPENDIX. 
 
 circle. Set off AD = \AB, and about D as a centre, with radius 
 DB = | AB, describe an arc cutting the given circle in F. Then 
 arc AF AB, very nearly. 
 
 III. To find the radius of a circular arc which shall subtend a given 
 angle, and he approximately equal in length to a given right line. 
 
 Let AB, Fig. 3, be the given right line. Draw AG perpendicular 
 to AB ; and also AH, making the angle BAR 
 equal to half the given angle. Set off AD = 
 J AB, and about centre D, with radius DB = 
 } AB, describe an arc cutting AH in F. 
 Bisect AF by the perpendicular EC, which 
 will cut AG in C, the centre of the required 
 arc AF. 
 
 2. Amount of the Error in the above Proc- 
 . It is stated by Prof. Eankinc that in 
 
 D NE the application of either of these rules the 
 
 F: - 3t straight line is a little less than the arc. 
 
 The magnitude of the error is given as about 7 J part of an arc 
 
 of 60 ; but it varies as the fourth power of the subtended angle, and 
 
 may be reduced to any desired extent by subdivision. Thus, for an 
 
 arc of 30, the error will be T J T x -j 1 ^ = TI io ; and for one of 20, 
 
 only Tffcj- x A- = Tsto- 
 
 Practically, therefore, if the given or required arc subtends an angle 
 of over GO , subdivision should be resorted to in applying either proc- 
 ess. 
 
 The first two rules are applicable also in many cases to other curves 
 than circles, provided that the change of curvature in the part to be 
 dealt with be small and gradual. 
 
 GRAPHIC CONSTRUCTIONS RELATING TO TANGENT LINES. 
 
 3. The Determination of a Tangent to any curve with absolute pre- 
 cision, must of course depend upon a previous investigation of its 
 mathematical properties. 
 
 But a sufficient degree of accuracy for most practical purposes can 
 be attained by purely graphic methods, very simple, and quite inde- 
 pendent of the special nature of the curve. 
 
 The most direct and expeditious of these consists in finding by 
 trial the centre and radius of a circular arc which shall sensibly 
 coincide with the given curve in the immediate vicinity of the point 
 of tangency. This is more particularly eligible when the change of 
 curvature in that part is small and gradual, as in Fig. 4. 
 
APPENDIX. 
 
 297 
 
 
 P 
 FIG- * 
 
 If it be desired to draw the tangent at a given point P, we find by 
 trial the centre C, as above, and the required line will be perpendicu- 
 lar to CP. 
 
 If it be required to draw a tangent in a given direction, or through 
 a given point not upon the curve, the 
 line is drawn mechanically with the aid 
 of a ruler, which can be done as accu- 
 ately as a line can be drawn through two 
 given points. Then in order to locate 
 the point of tangency, we find the centre 
 C as before, and draw the perpendicular 
 CP, cutting the tangent in the required 
 point. 
 
 4. This tentative method is not very 
 reliable in respect to the exac. location of the centre of curvature, 
 especially if the curve be flat ; in which case a considerable variation 
 in the radius may. be made without sensibly affecting the curvature 
 of the circular arc. But the direction of the radius of curvature is 
 thus ascertained with considerable precision, as may readily be veri- 
 fied by seeking for two other centres, D and E> of circular arcs which 
 shall lie respectively within and without the given curve, and deviate 
 from it equally on opposite sides of the given or apparent point of 
 tangency. If the manipulation be made with care, these three cen- 
 tres will be found to lie very nearly, as they should, in one straight 
 line. And this fact may be practically utilized in dealing with a very 
 flat curve ; in which case, should the centre C fall at an inconvenient 
 distance, the centre D of the inner circle may be used instead with 
 equal confidence. 
 
 5. The follow- 
 ing constructions 
 may sometimes 
 be preferred, al- 
 though they are 
 less direct and 
 more laborious. 
 
 "In Fig. 5, de- 
 scribe with any 
 convenient radius 
 a circular arc MN 
 about P, the point upon the curve XL at which the tangent is to be 
 drawn. From various points, as A, B, etc., taken at pleasure upon 
 KL, draw through P right lines cutting MN in a, b, etc. From 
 
 FIG. s. 
 
298 
 
 APPENDIX. 
 
 \a 
 
 \ 
 
 K 
 
 \ 
 
 these intersections set off, on these lines, distances equal to the corre- 
 sponding chords of the curve measured from P, and in the same di- 
 rection ; thus, for instance, aa! is laid off to the right, and equal to 
 PA ; while bb', equal to PB, is laid off to the left, and so on. 
 Through the points d, V, etc., thus located, draw the auxiliary 
 curve R OP, cutting MNin ; then OP is the required tangent. 
 
 Should the intersection at be too acute, the distances aa, IV, 
 etc., may be made twice as great as the chords, thus changing the 
 
 direction of the auxiliary curve. 
 
 6. A similar principle is involved 
 in the process for finding the 
 
 \ point of contact when the tangent is 
 
 \ \ 9 It given, as in Fig. 6. Draw any num- 
 
 ber of chords parallel to the tangent, 
 and, through their opposite extremi- 
 o? ~~ ties, draw in opposite directions a 
 series of parallel lines, which may or 
 may not be perpendicular to the tan- 
 gent. Upon each ordinate set off 
 from the tangent a distance equal to 
 , the corresponding chord or some mul- 
 ^ tiple thereof, and through the points 
 Fro. 6. thus determined draw the auxiliary 
 
 curve, which will cut the given one at the required point of tangency. 
 
 NORMALS. 
 
 7. For the General Problem of drawing a normal to a given curve 
 from a given point without, no solution, by graphic or any other 
 means, has yet been discovered. 
 
 When the given point lies upon the curve, the mathematical prop- 
 erties of the latter may or may not be such as to enable us to draw 
 the normal by a direct and independent construction. But, in gen- 
 eral, the graphic operation in this case depends upon the previous 
 determination of the tangent, by one or other of the methods above 
 described. 
 
 THE ELLIPSE. 
 
 8. First Method. About the centre (7, Fig. 7, describe two circles 
 whose diameters are respectively equal to the major and minor axes, 
 ^4 5 and MN. 
 
 Draw any radius CE at pleasure, cutting the inner circle in D and 
 the outer one in E. Draw through D a parallel to one axis, and 
 
APPENDIX. 
 
 299 
 
 through E a parallel to the other : the intersection of these lilies 
 will be a point upon the ellipse. 
 
 This is the most accurate of all methods of constructing this curve 
 by points, all the intersections, D, E, 0, being right angles. 
 
 To draio the tangent at a given point. By reversing the above proc- 
 ess, we find that the given point P would have been determined by 
 the radius CGH. At G and H, draw tangents to the two circles, 
 cutting the axes produced, in L and R respectively : then LR is the 
 required tangent at P. 
 
 Conversely, to find the point of contact : if the tangent be given, pro- 
 
 Fio. 7. 
 
 duce it to cut MN produced in L, and AB produced, in R. From 
 L draw a tangent to the inner circle, from R a tangent to the outer 
 one, and from C draw a perpendicular to these lines, thus determin- 
 ing G and //; draw through G a parallel to AB, and through If a 
 parallel to MN '; these will intersect in P, the point sought. 
 
 9. Second Method. Let C, Fig. 8, be the centre, AB'the major 
 axis, DE the minor axis. About D with radius A C describe an arc 
 cutting AB in the foci F, F'. About F as a centre, describe an in- 
 definite arc with any radius FH, greater than AF and less than BF. 
 About F' describe another arc, with a radius F'G = AB FH \ 
 
300 
 
 APPENDIX. 
 
 this arc will cut the one first drawn in 0, #', two points of the re- 
 quired curve. 
 
 This method is not eligible for the ordinary purposes of the 
 
 FIG. 8. 
 
 draughtsman, since it involves much more labor than the preceding 
 one, and the intersections of the arcs are in many cases too acute to 
 be reliable. But it is of interest as depending upon the property of 
 the curve that the sum of the focal distances is the same for every 
 point upon it, and equal to the major axis ; thus, 
 
 pp> = OF + OF' = AF + AF' = BF + BF' = AB. 
 
 10, Third Method. Upon this property also depends the operation 
 
 of drawing the "gardener's 
 ellipse" by the aid of a 
 string and two pins. 
 
 Let two fine pins be fixed 
 in the drawing board at F 
 and F' ; around these pass 
 a loop of waxed sewing silk, 
 of which the total length is 
 AB + FF' : if this loop be 
 kept constantly taut by a 
 pencil P, the latter in mov- 
 
 produce FO and F'O, and 
 
 FIG. 9. 
 
 ing will trace the curve. 
 
 To draw a tangent at any point, as 
 bisect the exterior angles FOI, F'OK, by the line TT. To draw 
 the normal at the same point, bisect the angles I OK, FOF', by the 
 line NN. Obviously, the axes cut the curve normally at their ex- 
 tremities A, B, D, E. 
 
APPENDIX. 
 
 301 
 
 11. Fourth Method. Fig. 9 illustrates the principle of a common 
 elliptographic trammel. The three points. P, M, N, are in one right 
 line, the distance PM being equal to CD, and PN equal to CA. The 
 point M being then kept always upon the line of the major axis, and 
 the point N upon the line of the minor axis, the point P will at all 
 times lie upon the ellipse. 
 
 This method is extremely convenient when no great precision is 
 required, the three points being selected upon the graduated edge of 
 a scale, or marked upon the edge of a smoothly cut strip of paper. 
 
 To draiv the normal at P : at M draw a perpendicular to AB, at N 
 a perpendicular to DE ; these perpendiculars intersect at 0, and OP 
 is the normal required. For MO and NO are the planes normal to 
 the paths of the two moving points, and is consequently the in- 
 stantaneous axis of the whole line NP. 
 
 FIG. 10. 
 
 12. Fifth Method. To inscribe an ellipse in a given rectangle, 
 Fig. 10. Join the middle points of the opposite sides by the right 
 lines AB, EF \ these will 'be the axes, and intersect in the centre (7. 
 Divide the semi-minor axis CF, and the half side GF of the rectan- 
 gle, into the same number of equal or proportional parts ; through 
 the points of subdivision on CF, draw right lines from B, and pro- 
 duce them to intersect the lines drawn from A to the corresponding 
 points on GF; these intersections will lie upon the ellipse. 
 
 Or, divide the semi-major axis A C and the half side AH in like 
 proportion, and proceed in a similar manner, the two series of inter- 
 secting lines converging in the extremities E and Fof the minor axis. 
 
 13. The same process is applicable, when it is required to inscribe 
 the ellipse in any given parallelogram, as shown in Fig. 11. But in 
 
302 
 
 APPENDIX. 
 
 this case AB, EF, will not be the axes. They are, however, conju- 
 gate to each other, for each is parallel to the tangents at the extrem- 
 ities of the other : and since the parallelogram can always he con- 
 structed if AB and EF are given, we have thus a simple and ready 
 method of constructing the ellipse upon any pair of conjugate diameters. 
 
 PIG. 11. 
 
 In order to determine the directions of the axes, describe about the 
 centre, (7, a circle upon either of the given conjugate diameters, as 
 EF: its circumference cuts the ellipse in P, and the supplementary 
 chords, PE 9 PF, are parallel to the axes. 
 
 14. Sixth Method. -To construct the ellipse by means of ordinates 
 of the circle. In Fig. 12, let it be required to draw the ellipse of 
 which A is the semi-major and A E the semi-minor axis. Describe a 
 circle with radius CR = AE, and divide CP and OA into any number 
 of proportional parts. At each point of .subdivision, erect a perpen- 
 dicular to OA, equal to the ordinate of the circle at the corresponding 
 
 FIG. 12. 
 
 point of CP ; the curve OE thus determined is the required ellipse. 
 
 Or CR may be made equal to the semi-major axis BF ; then if the 
 radius CP and the semi-minor axis BN be similarly subdivided, the 
 
APPENDIX. 
 
 303 
 
 T 
 
 FIG. 13. 
 
 ordinates of the circle will be equal to the corresponding ordinates 
 of the ellipse NF. 
 
 15. In a given ellipse, to find the conjugate to a given diameter. 
 Let PO, Fig. 13, be the given diameter. Draw any chord EF parallel 
 to PO, and bisect it, then the required conjugate diameter JfJ^passes 
 through the point of bisection, and TT, the tangent at P, is parallel 
 to MN. 
 
 Otherwise : draw the chord EF 
 parallel to PO, and also the diam- 
 eter EG : then MN and TT are 
 parallel to the supplementary chord 
 Fff. 
 
 16. To draw the tangents to an 
 ellipse from' a point without. Let 
 
 A A, Fig. 14, be the major axis, C and the foci, P the given point. 
 About P describe an arc through the nearer focus C ; intersect this 
 at M and N by another arc whose centre is 0, with a radius ON equal 
 to A A. Draw OM and ON", cutting the ellipse in G and H ; then 
 PG and P^Twill be the required tangents. For, 
 
 OH + HN = A A = 0//+ HC-, . . CJV = HIT ; 
 
 also, PO = PN, therefore, PH is perpendicular to ON, and bisects 
 the angle CHN. 
 
 Similarly, PG is perpendicular to CM, and bisects the angle CGM. 
 
 17. In Fig. 14, draw PC, PO; then the angles CPG, OPH, will 
 
 be equal. . For MN is a 
 common chord of the cir- 
 cular arcs whose centres 
 are and P, therefore, 
 PO is perpendicular to 
 MN; also, PH\s perpen- 
 dicular to CN; whence 
 the angles OPH, CNM, 
 are equal. 
 
 But CNM, in the cir- 
 cumference, is one half of 
 CPM, at the centre, which 
 again is equal to CPG ; 
 therefore, OPH = CPG. 
 
 18. In Chapter XL 
 
 i4 - (348), the argument in 
 
 connection witli Fig. 213 depends upon the following demonstration. 
 
304 
 
 APPENDIX. 
 
 FIG. 15. 
 
 Let the foci C, E, of tlie ellipse whose major axis is RR, Fig. 15, 
 be so situated with respect to the foci D, F, of the equal and similar 
 ellipse whose major axis is BB, that the right lines CD, EF, which 
 intersect at P, shall be equal to each other ; then the common tan- 
 gents to the two ellipses, HK and GL, will also intersect in P. 
 
 For, first, joining 
 DE, the triangles 
 DEF, DEC, have 
 two sides of one re- 
 spectively equal to 
 two sides of the 
 other, and the third 
 side is common, 
 whence the angle at 
 F is equal to the 
 angle at C\ and in 
 the triangles DPF, 
 CPE, the angles at 
 P are equal, whence 
 the angles PDF, PEC, are also equal : and since EC DF, we have 
 also DP = PE, and FP = PC. 
 
 Next, draw TT bisecting the angles DPE and FPC; by the pre- 
 ceding construction (16) draw PG tangent to one ellipse, and find the 
 point of tangency G ; draw DG, and produce it to cut TT in A ; draw 
 AE cutting the other ellipse in H, and join HP. 
 
 Then the triangles DP A, EPA, have the angles at P equal, the 
 side PD = side PE, and the side PA common j the triangles are 
 therefore similar, and we have 
 
 ALP = AEP ; 
 but PDF = PEC, 
 .-. ADF=AEC: 
 
 consequently, since the 
 ellipses are similar, HE 
 = GD, and the tri- 
 angles PHE, PGD, 
 having the two sides and 
 the included angle of 
 one equal to the two D 
 sides and the included 
 angle of the other, are 
 similar and equal; whence PH is equal to PG and tangent to the 
 

 ttp* 
 
 \^ V OF THE 
 
 UNIVERSITY 
 
 APPENDIX. 
 
 ellipse at H, and also the angle GPD is equal 
 HPE. 
 
 Now draw PK tangent to the ellipse at K\ then (17) the angle 
 FPK is equal to the angle GPD, and, therefore, to the angle HPE. 
 But FE is a right line by hypothesis, therefore, HPK\s also a right 
 line and tangent to both ellipses. In like manner it may be shown 
 that PL, tangent to the ellipse on the right, is a prolongation of PG. 
 
 It will also be observed that if xx be an arc of. an ellipse whose 
 foci are D, F, the major axis being equal to DP + PF, and yy an 
 arc of a similar and equal one whose foci are C, E, then TT will be 
 tangent at P to both those ellipses. 
 
 THE PARABOLA. 
 
 19. The Parabola, Fig. 1C, is a curve every point of which is equally 
 
 FIO. ir. 
 
 distant from a given point F, called the focus, and a given right line 
 DD 9 called the directrix. Hence the axis OFC is perpendicular to 
 DD, and the vertex V lies at the middle point of PC. Draw any 
 parallel to DD, as RLP ; then the points in which this line is cut by 
 an arc described about F, with a radius equal to LC 9 will lie upon, 
 the curve. 
 
 To draw a tangent at any point, as P : Draw FP, and also PA 
 perpendicular to DD : then the required tangent bisects the angle 
 FPA. 
 
 Otherwise: let R be the given point. Draw RL perpendicular to 
 the axis, and on the axis make VM = L V ; then MR is the required 
 tangent. 
 
 Second Method. The parabola may be traced mechanically, as 
 
 shown in Fig. 17. Let DD be a ruler fixed to the drawing-board. 
 
 Let a fine thread whose length is equal to LA be fixed at one end to 
 
 the point Lon the vertical side of the right-angled triangle, and at the 
 
 20 
 
306 
 
 APPENDIX. 
 
 other end to the drawing-board at F, the focus of the required curve. 
 Then by sliding the triangle along the ruler, keeping the thread taut 
 by a pencil P which always touches the side A L of the triangle, the 
 A 7*3 T, motion of the pencil will 
 
 be so controlled as to 
 trace a parabola, since 
 PF is always equal to 
 PA. 
 
 20. Third Method. 
 In Fig. 18, let V be the 
 J- vertex, VO the axis, and 
 P a point in the required 
 curve. 
 
 Draw PO, VA, per- 
 pendicular to the axis 
 and equal to each other ; 
 divide VA in any man- 
 ner, and through the 
 points of division draw parallels to VO : divide AP in like propor- 
 tion, and join the points of division with the vertex V. The inter- 
 sections of the lines thus drawn through corresponding points upon 
 VA and AP will lie upon the parabola VP. 
 21. Fourth Method. IK Fig. 19, let Fbe the vertex, VO the axis, 
 
 FIG. 18. 
 
 FIG. 19. 
 
 and P a point through which the curve is to pass. 
 
 Draw PR perpendicular to VO, and make RO = PO. On the. 
 axis set off VL VO ; draw PL and RL, divide them into the 
 same number of equal parts, number the points of division in oppo- 
 site directions, and join the points correspondingly numbered, as 
 
APPENDIX. 
 
 307 
 
 1, 1, 2, 2, etc. : the lines thus drawn will be tangents to the required 
 curve. 
 
 To find the point of tangency on any one of these lines, for instance 
 1, 1. This line cuts the axis at N ; set off on the axis, VK = VN, 
 and draw KB perpendicular to V0 9 cutting 1, 1, in the required 
 point B. 
 
 THE HYPERBOLA. 
 
 22. The Hyperbola is a plane curve generated by the motion of a 
 point subject to the condition that the difference of its distances from 
 two fixed points called the foci shall always be equal to a given line, 
 whose length must be less than the distance between the foci. 
 
 In Fig. 20, set off OF = CF r , and let F, F', be the foci : set off 
 also CA = OB, and let AB be the given constant difference. It is 
 evident that 
 
 FB - BF' = AF' - AF= AB ; 
 
 therefore A and B satisfy the conditions and are points on the curve. 
 With any radius FO greater than FB, describe an arc about F as a 
 centre ; then about F' describe, 
 with a radius F'O FO 
 AB, another arc, which will 
 cut the one first drawn in two 
 points 0, 0', of the required 
 hyperbola. Since with the 
 same radii, arcs may be de- 
 scribed about the other foci, 
 it follows that the curve is 
 composed of two equal and op- 
 posite branches ; and since 
 FO may be of any length, 
 these branches are infinite. 
 The point C is called the cen 
 tre, and the line AB the Fl - 20 - 
 
 major axis, to which the minor axis is perpendicular. 
 
 To dmiv a tangent at any point of the hyperbola, as P : draw PF 9 
 PF', and bisect the angle between them. 
 
 Otherwise thus: describe a circle upon AB as a diameter, let fall 
 PG perpendicular upon the major axis, and from its foot G draw 
 GK tangent to that circle. Find the point of tangency K 9 and draw 
 KL perpendicular to AB and cutting it in L ; then LP is the re- 
 quired tangent. 
 
308 
 
 APPENDIX. 
 
 FIG. 21. 
 
 By the converse operation we may find the point of contact if the 
 tangent be given. 
 
 To find the asymptotes. From either focus, as F, draw tangents to 
 the circle described upon AB : find the points of tangency R and 8, 
 then CR and CS are the asymptotes. 
 
 23. Second Method. In Fig. 21, let C be the centre, AB the 
 
 major axis, and a point 
 through which the hyper- 
 bola is to pass. 
 
 Describe upon AB the 
 semicircle ADB, and draw 
 DCE perpendicular to AB, 
 also DT tangent to the semi- 
 circle. Draw OS perpen- 
 dicular to DE, and with 
 centre C and radius CL = 
 OS, describe an arc cutting 
 DT in L ; on SO set off 
 SG = DL, then CG is an 
 asymptote to the curve. 
 
 Draw any line JT7 parallel to AB, and produce it ; set off DM on the 
 
 tangent, equal to KI, then on KI produced set off KN = CM, and 
 
 JVwill be a point on the curve. 
 
 To find the focus. Produce GC to cut the semi-circumference in 
 
 R, at which point draw a perpendicular to GR ; which will cut the 
 
 major axis in the focus F. 
 
 To draw a tangent at any point, P. Draw PV parallel to the 
 
 asymptote HC, cutting the other 
 
 asymptote GR in V. On GR set off 
 
 VU = CV, and draw PU, which will 
 
 be the tangent required. 
 
 24, Third Method. Given, the 
 asymptotes and one point in the curve. 
 The construction depends upon the 
 property that if any line be drawn 
 cutting both asymptotes, the parts in- 
 tercepted between each of those lines 
 and the curve are equal. 
 
 In Fig. 22, let CR, CS, be the 
 asymptotes, P the given point. 
 Through P draw any line at pleasure, F io. 22. 
 
 as EF, and make FG = EP then G will lie upon the curve. Any 
 
APPENDIX. 
 
 309 
 
 point thus found may then be treated in like manner, thus : drawing 
 HGK, make XL = JIG, then L is a point upon the hyperbola ; and 
 so on. 
 
 To find the vertex. Set off upon the asymptotes any equal distances 
 CM, CN-, draw MN, and bisect it by a perpendicular CO, which will 
 be in the direction of the major axis, and cut the curve at the ver- 
 tex F. 
 
 25, Fourth Method. Given, in Fig. 23, the major axis AB, 'and P 
 a point in the required curve. 
 
 Draw PO, BE, perpendic- 
 ular to AB and equal to each 
 other, and join PE. Divide 
 PO and PE into the same 
 number of equal parts, num- 
 bering the points of division 
 from P upon each line. From 
 A draw lines to the points 
 upon PO, and from B, lines 
 to the points upon PE ; the 
 intersections of the lines thus drawn to corresponding points of divi- 
 sion, as for instance, Al, Bl, will lie upon the hyperbola required. 
 
 THE SPIRAL OF ARCHIMEDES. 
 
 26. A Spiral is a plane curve traced by a marking point which moves 
 along a right line, while at the same time the right line revolves about 
 one of its points as a fixed centre. 
 
 This fixed centre is called the pole, and a right line drawn from it 
 to any point of the curve is called a radiant, or radius vector. Sup- 
 posing the angular velocity of the revolution to be uniform, the linear 
 motion of the marking point along the radius vector may be governed 
 by any law at pleasure, and thus an infinite variety of spirals may be 
 produced. 
 
 In Fig. 24 both motions are uniform, and the resulting curve is the 
 well-known spiral of Archimedes, also called the equable spiral, be- 
 cause the rate of expansion is constant, so that the distance between 
 any two consecutive coils, measured on a radiant, is the same. 
 
 If this distance or rate of expansion be given : with it as radius, 
 describe a circle about the pole as centre, and divide its circum- 
 ference into any number of equal parts by radial lines ; divide the 
 given distance into the same number of equal parts, and set out from 
 the pole P, upon consecutive radii, as 1, II, III, IV, distances 
 
310 
 
 APPENDIX. 
 
 equal to one, two, three, etc., of these subdivisions : the spiral is 
 then drawn through the points thus determined. 
 
 If any two radiants de given : 
 divide the included angle into 
 any number of equal parts, 
 and the difference between the 
 given radiants into the same 
 number ; then set off on the 
 first dividing radial line, a dis- 
 tance equal to the least radiant 
 JI plus one of the subdivisions of 
 the difference, and so on, eacli 
 successive radiant being greater 
 than the preceding, by one of 
 these subdivisions. 
 
 To draw a tangent at any 
 point. The tangent to this 
 spiral can readily be drawn by 
 geometrical construction ; for 
 the motion of the tracing point 
 is the resultant of two known 
 components. The direction of 
 tho circular motion is that of 
 the tangent to the circle through 
 the given point, of which the pole is the centre, and its magnitude 
 may be represented by the circumference of that circle. While mak- 
 ing one revolution, the tracing point also travels along the radiant 
 through a given distance, which is the other component ; and the re- 
 sultant, representing both in magnitude and direction the actual 
 motion of the point at the instant, is the required tangent to the spiral 
 path. It is not necessary to set off the entire circumference in making 
 this construction ; should this be inconveniently large, we may use 
 one-half, one- third, or any other fraction, reducing the radial compo- 
 nent in the same proportion. 
 
 27, In Fig. 24 PA is the zero line from which the circular divisions 
 are reckoned in constructing the curve in the first manner above ex- 
 plained. And as the length of this radiant is zero, the circum- 
 ference of the circle described with it as radius is also zero, but 
 the other, or radial component in ihe determination of the tan- 
 gent is constant ; the zero line, therefore, is tangent to the curve at 
 the pole. This is called the axis of the spiral ; and it will be noted 
 that 
 
 FIG. 24. 
 
APPENDIX. 
 
 3U 
 
 1. The length of the radiant varies directly as the angle of rotation 
 
 from this fixed axis. 
 
 2. The lengths of successive radiants which include equal angles, hav- 
 
 ing a constant difference, form a series in arithmetical progres- 
 sion. 
 
 3. This spiral consists of two infinite branches, curving in opposite 
 
 directions, and symmetrically placed with reference to a line 
 passing through the pole, perpendicular to the axis. 
 
 THE RECIPROCAL SPIRAL. 
 
 28. This is the exact converse of the Archimedean spiral, the 
 lengths of the radiants varying inversely as the angle of rotation from 
 a fixed axis. 
 
 In Fig. 25 describe about the pole B a circle with any convenient 
 radius, and beginning at the axis BA, divide it into any number of 
 
 n 
 
 FIG. 25. 
 
 qual parts at the points /, 77, 777, etc. On the radii through these 
 points set off in their order distances measuring respectively 1, }, J, 
 J, etc., by any scale of equal parts. Draw also lines from B, making 
 with BA angles equal to |, J, \, etc., of the angle ABI '; and set off 
 
312 
 
 APPENDIX. 
 
 on them by the same scale distances measuring 4, 3, 2, etc. ; the 
 spiral is then drawn through the points thus determined. 
 
 This curve, evidently, makes an infinite number of convolutions 
 about the pole, which it continually approaches, but never reaches. 
 
 THE LOGARITHMIC SPIRAL. 
 
 29. This curve presents to the Archimedean spiral the contrast that 
 the successive radiants which include equal angles form a series in 
 geometrical progression : each being greater or less than the preceding 
 in a certain constant ratio, instead of by a constant distance. 
 
 Hence the radiant which bisects the angle between two others is a 
 mean proportional between them ; thus in Fig. 26, if the angles APK, 
 KPH, HPC, be equal, we have 
 
 AP iPKi: PK-.PH. 
 
 PK iPH'.'.PHi PC, and so on. 
 
 Had the radiants AP, PH, and their included angle, then, been 
 given, the intermediate point K would have been found by bisecting 
 
 the given angle, and 
 setting off PK, a 
 mean proportional 
 between the two 
 given radiants. 
 
 If the given radi- 
 ants lie in the same 
 straight line, as^4P, 
 PB, the construc- 
 tion is the same 
 this angle of 180 is 
 bisected by CP per- 
 pendicular to AB, 
 and describing on 
 AB a semicircle, 
 cutting this perpendicular in C, we have CP* AP. PB, as re- 
 quired, therefore (7 is a point upon the spiral. Toward A set off 
 PD = PC, and describe a semicircle on DB as a diameter, cutting 
 CP in K Bisect the angle CPB, and on the bisector set off PF = 
 PE, then F is also a point on the curve. 
 
 In like manner the spiral may be extended as far as desired : thus, 
 drawing the chord CB, bisect it by a perpendicular cutting CP in O, 
 and about describe a semicircle passing through C and B ; this 
 
 Fl ' ?6 - 
 
APPENDIX. 
 
 313 
 
 semicircle will cut CP produced in G, which will lie upon the spiral, 
 since by this construction PB? CP. CG. 
 
 30. In view of the practical application of curves of a spiral form 
 in the construction of cams for communicating definite motion to one 
 piece by the rotation of another, it is of interest to note that by set- 
 ting up the successive radiants as equidistant ordinates, any spiral 
 may be transformed into a curve capable of transmitting motion with 
 corresponding changes in velocity, while the driver moves in a right 
 line. And conversely, any curve may be transformed into a spiral 
 possessing analogous mechanical properties, by setting out its equi- 
 distant ordinates as the radiants, taking care that the successive ones 
 include equal angles. 
 
 THE HELIX. 
 
 31. The Helix is a curve traced upon the surface of a cylinder of 
 revolution by a point which moves uniformly around the axis, and at 
 the same time travels uniformly in a direction parallel to the axis ; 
 the rates of the two motions being entirely independent of each other. 
 
 Thus in Fig. 27, let the relative motions be such that the marking 
 
 point shall traverse the distance A 1 M while going once around the 
 cylinder ; then in going half way around, it will travel half as far ; in 
 going one quarter around, or through the arc AB, it will travel one 
 quarter as far, or through a distance equal to AL, which will, evi- 
 dently, bring it to the position B'. Intermediate points are readily 
 found by subdividing the arc AB and the distance AL into the same 
 number of equal parts at 1, 2, 3, etc., and projecting the former 
 points of subdivision to lines drawn through the latter, perpendicular 
 to the axis, as shown. 
 
 If the cylinder be cut along the line A'M, and unrolled into a 
 plane, it will develope into a rectangular sheet whose length is equal 
 
314 
 
 APPENDIX. 
 
 to the circumference, and the helix will develope into the diagonal of 
 this rectangle. 
 
 To draw the tangent at any point, as B'. The curve pierces the 
 base of the cylinder at A 1 , corresponding to A in the end view. Pro- 
 ject B' to B, and perpendicular to the radius BC, set off BJ equal to 
 the arc BA. Project J to T' upon NA' produced, then B'T' is the 
 required tangent. 
 
 By projecting J to J' upon a perpendicular to the axis through the 
 given point B', it will be readily perceived that the same result would 
 have been reached by compounding the two motions of the point, the 
 resultant being the tangent. 
 
 A curve analogous to the helix may also be traced upon the surface 
 of a cone, or of a hyperboloid, by a point moving uniformly along an 
 element, while the surface at the same time rotates uniformly about 
 its axis. (For illustration t)f the conical helix, see Fig. 226 ; the hy- 
 perboloidal helix is represented in Fig. 243.) 
 
 DRAWING OF ROLLED CURVES. 
 
 32. In Fig. 28 A A is a curved ruler fixed to the drawing-board, and 
 BB is a free one rolling along it. Let a pencil be fixed to and carried 
 
 D 
 
 FIG. 28. 
 
 by the latter, either in the contact edge, as at D, or at any distance 
 from it, as at E. Since P, the present point of contact, is the in- 
 stantaneous axis, the motion of D is in the direction DF, perpendic- 
 ular to DP, the contact radius. DF is, therefore, tangent to the path 
 
APPENDIX. 
 
 315 
 
 of D, traced as the ruler BB rolls ; but it is also tangent to the cir- 
 cular arc whose centre is D and radius PD, consequently the path of 
 D is also tangent to that arc. 
 
 Let the arcs PC, Po, of BB, be equal to the arcs PC', Po', of AA, 
 then cD will be contact radius when c reaches c', and oD when o 
 reaches o'. If, then, we describe with these radii circular arcs about 
 c' and o', the curve traced by D will be tangent to those arcs ; and 
 that traced by E will be tangent to arcs about the same centres with 
 aE and oE as radii. 
 
 Curves thus described by points carried by one line which rolls 
 upon another are called rolled curves, roulettes, or epitrochoids; and 
 the drawing of a series of tangent arcs as above explained is the read- 
 iest and most reliable known method of laying them out. 
 
 The line which carries the tracing point is called the generatrix or 
 describing line, and the one in contact with which it rolls is called the 
 directrix or base line ; either of these may be straight, or both may 
 be curved. 
 
 THE CYCLOID. 
 
 33, This curve is traced by a point in the circumference of a circle 
 which rolls upon its tangent. 
 
 In Fig. 29 find Aa', the length of a convenient fraction Aa of the 
 circumference ; step this oif upon the tangent the required number of 
 
 E 
 
 FIG. 29. 
 
 times, making A E equal to the semicircumference. Divide each into 
 the same number of equal parts, draw chords from P to the points of 
 division on the semicircle, with which as radii, strike arcs about the 
 corresponding points on AE\ the cycloid is tangent to all these arcs. 
 
316 
 
 APPENDIX. 
 
 To find points on the curve : "When aC becomes contact radius, it 
 has the position a'R, perpendicular to AE. The angle aCP remains 
 unchanged ; therefore, make a'RL equal to it ; then RL is the gener- 
 ating radius in its new position, and L is a point on the cycloid. 
 Also, a'L, the instantaneous radius, is normal, and a perpendicular 
 to it is tangent to the curve at L. 
 
 Conversely : Let be any point on the curve ; about this as a 
 centre describe an arc with radius equal to CP. This arc cuts CD, 
 the path of the centre of the rolling circle, in S ; then 08 is the gen- 
 erating radius ; Sb r , perpendicular to AE, is the contact radius, and 
 V.O is normal to the cycloid. 
 
 THE EPICYCLOID. 
 
 34. The describing circle, in Fig. 30, rolls on the outside of another 
 whose centre is G. 
 
 Draw the CQmmon tangent at A, set off upon it the length of A a 
 
 FIG. 30. 
 
 (any convenient fraction of semicircumference AP) and find the arc 
 of the base circle equal to that length. Step this oif as above, mak- 
 ing AE = semicircumference A P. The curve is drawn by tangent 
 arcs in the same manner as the cycloid. 
 
APPENDIX. 
 
 317 
 
 The path of the centre of the describing circle is in this case another 
 circle whose centre is G ; and the contact radii a'R, b'S, are prolon- 
 gations of the radii Ga', Gb', of the base circle ; which slightly mod- 
 ifies the processes of finding the point of the curve corresponding to 
 a given point of contact and the converse. 
 
 THE HYPOCYCLOID. 
 
 35. Traced, as shown in Fig. 31, by a point in the circumference of 
 a circle rolling inside another. 
 
 The construction is in all respects the same as in the case of the 
 
 FIG. 31. 
 
 epicycloid, and the diagrams being lettered to correspond throughout, 
 no further explanation is needed. 
 
 In all three of these curves, if the rolling continue beyond E, anew 
 branch springs up, which is of course perfectly symmetrical with EL. 
 
 It is to be particularly noted that these branches are tangent to ED, 
 and therefore to each other at E. 
 
 THE INVOLUTE OF THE CIRCLE. 
 
 36. This is in a manner the converse of the cycloid, being gener- 
 ated by the rolling of a tangent right line upon a circle. Thus in 
 Fig. 32, the ruler, carrying in the line of its edge the pencil P 9 while 
 rolling around the cylinder describes the curve in question. 
 
 It may also be regarded as generated by unwinding an inextensible 
 fine thread from a cylinder : the thread being always taut arid always 
 tangent to the circle, its length is equal to that of the arc from which 
 it was unwound ; thus, beginning at 0, make the tangents AP, BE, 
 DF, respectively equal to the arcs OA, OAB, OBD, and so on ; then 
 
318 
 
 APPENDIX. 
 
 the curve passes through the extremities P, E, F 9 G, etc., of these 
 tangents. 
 
 The tangent OG, then, will be equal to the circumference of the 
 circle, and if the unwinding be continued, the result will be the for- 
 mation of a spiral, the distance between, the successive convolutions, 
 
 measured on the 
 tangent to the base 
 circle, as for in- 
 stance PH 9 being 
 constant and equal 
 to the circumfer- 
 ence. 
 
 Considering it as 
 traced by the ruler 
 as in the figure, it 
 will be seen that as 
 the point of contact 
 A is the instantane- 
 ous axis, the edge 
 AP is normal to 
 the curve. This 
 l>eing true for all 
 positions of the 
 ruler, we have the 
 simple construction 
 Fl - 32 - that the normal at 
 
 any point of the curve is tangent to the base circle. 
 
 If the rolling of the ruler continue in the direction of the arrow, it 
 is evident that after P reaches a new branch will be formed as shown 
 by the dotted line ; the two branches being tangent to each other, 
 and to the radius CO at its extremity. 
 
 THE EPITROCHOID. 
 
 37. Although the term epitrochoidal is used in a general sense, in- 
 cluding all rolled curves, yet custom sanctions also a special sense, 
 and the curve traced by the rolling of one circle upon another, when 
 the marking point is not situated upon the circumference, is the one 
 ordinarily meant when " The Epitrochoid " simply is mentioned with 
 no qualifying word in connection with it. 
 
 In Fig. 33, if the circle whose centre is C, roll upon the circle whose 
 centre is D, carrying the marking point P situated without the cir- 
 cumference, it describes the looped curve PLE, called the curtate 
 
APPENDIX. 
 
 319 
 
 epitrochoid. If the tracing point be situated at F, within the circum- 
 ference, the resulting waved curve FJOTis called the prolate epitro- 
 choid. 
 
 Since the rolling circle measures itself off upon the base circle as in 
 the preceding cases, the position of the generating radius can always 
 be found as in the construction of the epicycloid, and its length being 
 constant, points on the curve are readily found ; and the instantane- 
 ous radius being always normal to the epitrochoid, the tangent at any 
 
 FIG. 33. 
 
 point may be drawn with the same facility. For example, let it be 
 required to draw the tangent at L ; with radius equal to CP describe 
 an arc cutting the path of the centre in R, draw R G, cutting the base 
 circle in a' : then a'R is contact radius, RL is generating radius, a'L 
 is the normal, and a perpendicular to it is the required tangent. 
 
 DOUBLE GENERATION OF THE EPICYCLOID. 
 
 38. By way of distinction, the curve traced as in Fig. 30, by the 
 rolling of one circle upon another in external contact, is called an ex- 
 ternal epicycloid. But if the contact be internal, the curve traced by 
 the rolling of the larger upon the smaller is called an internal epicy- 
 
320 
 
 APPENDIX. 
 
 cloid ; and in Fig. 34, the circle whose centre is B, rolling npon the 
 fixed circle whose centre is D, and carrying the marking point F, thus 
 describes this curve, of which FL is a portion. 
 
 Now let the same point F be carried by the circle whose centre is 
 C, whose diameter FA is equal to the difference between the diame- 
 ters FG and A G of the other circles : it will then trace the same 
 path FL. 
 
 First, let the three centres, C, B, D, the two points of contact A 
 and G, and the tracing point F, lie in one straight line FG. Then 
 through A draw EAH in any direction at pleasure, and draw .F/par- 
 
 Fia. 34. 
 
 allel to it. Then the three chords EA, AH, FI, subtending equal 
 angles in the three circles, are proportional to the radii ; therefore, 
 
 EA + AH = FS + SI. 
 
 Also, the triangles AHG, FIG, are similar. 
 
 Set off the arc GHK = arc GL Draw the chord KM =HA, and 
 prolong it to L, making ML = AE. Draw the diameter MDT 9 and 
 a parallel to it through L, cutting KT produced in the point R. 
 Then the triangles MKT, LKR, are respectively similar and equal to 
 the triangles AHG, FIG. 
 
 Now KD bisects MT in D, and when produced will bisect LR in 
 P ; therefore, the circle round LKR will be tangent at If to the circle 
 round J/LKT and AHG. Consequently, if the circle whose centre is 
 
APPENDIX. 321 
 
 B rolls upon the circle whose centre is D until J reaches K, its centre 
 will then be at P, and KL will be the new position of the chord IF ; 
 the point ^meantime tracing the internal epicycloid FL. 
 
 39. Produce TM to 0, making MN = NO = AC. Then the 
 circle whose centre is .2V" and radius NM 9 will pass through L, and the 
 arcs OL, FE, will be equal ; because M^L = AE, and the angles CAE 9 
 NML, are equal. 
 
 The point Fwill, therefore, reach L, if it first describe the arc FE 
 about centre C, and then the arc EL about centre D. And it will 
 be perceived that this is equivalent to the rolling of the circle whose 
 centre is C, upon the circle whose centre is D, if it be proved that the 
 arcs AM, OL, are equal. 
 
 In order to do this, we have, first, 
 
 BD = DP AG NL, by hypothesis, 
 and 
 
 LP ND, by construction ; 
 
 . . the angles ONL, MDP, are equal. 
 Draw BI; then since arc KHG = arc GI, we have, second, 
 
 KDG iIBG :: BG :DG; 
 
 but 
 
 IBG = 2 (IFG) = 2 (CAE) = 2 (NML) = ONL = MDP, 
 , also 
 
 KDG = ADP. 
 Whence 
 
 ADP : MDP ::BG : DG ; 
 
 .-. ADP - MDP :MDP :BG - DG :DG, 
 or 
 
 ADM : MDP :: BD : DG, 
 
 ADM : ONL : : ON : AD 
 
 arc AM = arc OL = arc FE. 
 
 Q. E. D. 
 
 40. Although the epicycloids thus traced by the rolling of the two 
 circles upon the same base circle are identical, it is not to be assumed 
 that the epitrochoids generated by marking points not in the circum- 
 ferences of the describing circles will be the same. 
 
 On the contrary, they will be quite different, as shown in the dia- 
 gram. If the generating radius BF be extended to W, the latter 
 point will trace the internal epitrochoid WZ, daring the generation 
 21 
 
322 
 
 APPENDIX. 
 
 of the internal epicycloid FL, the final position of the generating 
 radius being PLZ. On the other hand, prolonging the radius CF 
 to the same point W, we perceive that during the generation 
 
 of the external epi- 
 A cycloid FL by the 
 
 r o 1 1 i n g of t h e 
 smaller circle, the 
 external epitrochoid 
 WX will be de- 
 scribed, and NLX 
 will bs the final po- 
 sition, of the gene- 
 rating radius C W. 
 41. Every inter- 
 nal epicycloid, 
 tl\en, is also an ex- 
 ternal one ; and it 
 may be remarked 
 that the latter 
 mode of generation 
 is usually more 
 convenient in prac- 
 tical execution. 
 
 Similarly, every hypocycloid is capable of two generations. Thus, 
 if in Fig. 34 we take FIG as the fixed base circle, the hypocycloid 
 A Y will be traced by the point A, whether it be carried by the semi- 
 circumference FSA of the smaller circle, which is equal to FY, or by 
 the semicircumference GHA of the larger, which is equal to GIY. 
 
 PARALLEL CURVES. 
 
 42. Parallel Curves are those whose normal distance from each other 
 is everywhere the same. If one curve and the length of the normal 
 be given, the other is readily mapped out by merely describing any 
 number of circular arcs with their centres upon the first curve, and a 
 radius equal to the normal : the envelope of these arcs is the parallel 
 curve. 
 
 At first thought it is natural to suppose that two curves thus related 
 will be similar in form, like two concentric circles. And this will 
 really be the case, if the derived curve be exterior to the first. But if 
 it lie within, that is, upon the concave side of the fundamental curve, 
 quite curious and unexpected results may arise, of which Figs. 35 and 
 36 are sufficiently remarkable illustrations. 
 
 Fro. 35. 
 
APPENDIX. 
 
 323 
 
 In the former, the original curve is the parabola A VB, of which 
 VC is the axis and the focus. 
 
 The two dotted curves resemble it in form, but though both are 
 symmetrical, neither is a true parabola. The normal distance Vffis 
 less than VO ; but the result of assuming a greater one is the forma- 
 tion of the figure of which the construction is shown, and its resem- 
 blance to the original curve is at least not striking. 
 
 In Fig. 36^the ellipse of which the axes are AB, DE, is the funda- 
 mental curve ; and as before, when the normal distance is small, the 
 parallel curves are somewhat similar to it, although neither is a true 
 ellipse. But again, upon increasing the normal, the derived curve 
 loses all resemblance to the original, and developes the four-cusped 
 
 FIG. 36. 
 
 figure shown within, symmetrical about the centre, and also with re- 
 spect to the axes of the ellipse. 
 
 THE LIMACON AND THE PARALLEL TO THE EPICYCLOID. 
 
 43. Mention has previously been made of these curves, whose pecu- 
 liarities, which were shown to have direct practical bearing upon the 
 true theory of pin gearing, merit for that reason further examination. 
 
 In Fig. 37 the epicycloid PBE is traced by the rolling of the 
 circle from C to D, and the extremity / of the normal PI meantime 
 traces the parallel curve IOHF, consisting of two branches. 
 
 It is worthy of note that this curve will always exhibit these two 
 branches, however small the normal distance chosen may be; which 
 will be readily seen if we suppose it to be traced in the opposite direc- 
 
324 
 
 APPENDIX. 
 
 tion by the rolling of the circle from D to C, in which case the initial 
 motion of the circle having the direction of the arrow at E, and the 
 point E being the centre of rotation, it is apparent that the initial 
 motion of the extremity of the normal, be that line long or short, will 
 have the direction of the arrow at F ; so that under no circumstances 
 will there be an interior curve similar to the original, as was the case 
 with the ellipse and the parabola. 
 
 Beginning at F, then, this curve descends to some point H, and 
 then begins to rise, the normal taking successively the positions EF, 
 WU, SH. The fact that there will be a cusp at the lowest point, is 
 
 FIG. 37. 
 
 shown by the consideration that were there either a wave or a loop, 
 there would also be a tangent in some direction nearly coincident with 
 US, the impossibility of which is perfectly obvious. 
 
 44. Now again beginning at P : the rolling of the circle is a com- 
 pound motion, consisting of a rotation around the travelling centre (7, 
 and a revolution around the fixed centre G. We may then suppose 
 these two motions to take place separately, in succession. 
 
 Thus if the generating circle be first turned through the angle 
 PCR, and then be made to revolve about G through an angle meas- 
 ured by an arc A of the base circle equal to PR, the tracing point 
 
APPENDIX. 325 
 
 P will have reached the position B on the epicycloid, and OB, the 
 normal to that curve, will be the new position of the chord AR. 
 
 "VVe may reverse this process ; if, for instance, we select any point 8 
 on the epicycloid, and describe a circle about G, cutting the genera- 
 trix at J, then AJ is the chord which, by the preceding operation, 
 will become the normal ST, and the arc PJ will be equal to the 
 arc AT. 
 
 45. A series of chords in the generating circle, drawn through the 
 point A, then, are the lines which will eventually become normals to 
 the epicycloid. If now we set oif upon each of these chords, from the 
 circumference toward A, a distance equal to P/, as er, JK, etc., the 
 points thus located will determine a new curve IAKL, called the 
 Limacon. 
 
 And just as points upon the epicycloid are derived as above from the 
 upper extremities of these chords, so points upon the parallel curve 
 are derived from their lower extremities. Or in other words, as points 
 of the former are derived from points upon the generating circle, so 
 points' on the latter are derived from points upon the limacon. For 
 instance, the point g of the epicycloid would be determined by rotating 
 P about C to e, then revolving the whole generatrix about G, the 
 angle being measured by the arc AZ equal to Pe : during this revolu- 
 tion the point r of the limacon goes through the same angle, giving 
 the position^ of a point upon the parallel curve, which lies upon gZ, 
 the final position of eA. 
 
 By the aid of the limacon, then, we may determine the direction of 
 a normal at any point of the parallel curve : let p, for instance, be the 
 given point; we first describe an area bout G through p, cutting the 
 limacon at r, then draw Ar and produce it to cut the generating 
 circle at e, and finally draw through e an arc cutting the epicycloid in 
 g} then pg is the required normal. 
 
 46. To Draw a Normal to a Given Epicycloid from a Given Point 
 Without. It may be pointed out that the foregoing indicates a method, 
 circuitous, it is true, and probably more curious than useful, of graph- 
 ically solving the problem just enunciated. 
 
 Supposing the epicycloid given as in the figure, and any pointy 
 assigned, the normal distance of this point from the curve, even if not 
 given, may practically be ascertained with great accuracy by drawing 
 a circle about p as a centre, tangent to the curve ; for the eye is capa- 
 ble of appreciating the fact of tangency with extreme nicety, if the 
 lines be fine, although wholly unable to locate the point of contact. 
 The radius of this tangent circle then is used in the construction of 
 the limacon, by means of which, as above, the normal is drawn. It 
 
326 APPENDIX. 
 
 may be added, that should the intersection of the epicycloid at g, by 
 the arc through e, in this construction, be too acute, a better determina- 
 tion can be made as follows : about p y wijth radius rA, describe an arc 
 cutting the base circle in Z 9 or set off the arc AZ equal to Pe ; GZ 
 produced will then be the corresponding contact radius, thus locating 
 the position of the centre of the generatrix ; an arc about this centre 
 with radius CP will then cut the arc eg less acutely than that arc 
 cuts the epicycloid ; and if gZ be found to pass, as it should, though 
 the given point p, the determination may be accepted as at least accu- 
 rate enough for all practical purposes, if any such there be. 
 
 47. By the aid of the limacon, however, we can determine with ab- 
 solute precision the location of the point of cuspidation H, and of the 
 points and U at which the parallel curve crosses the base circle, as 
 well as the directions of the normals at these points. 
 
 In order to do this we will first examine the limacon itself more par- 
 ticularly. If we suppose CJ to be a crank turning about C as a fixed 
 centre, and JK to be a rod jointed to it at J, and capable of sliding 
 endwise through a socket pivoted so as to rotate freely about the fixed 
 centre A ; then if JK be equal to PI, it is clear that a pencil fixed at 
 the end A' of this rod will mechanically trace the limacon during the 
 rotation of the crank. 
 
 Now, the motions of the points J and A being always known, the 
 instantaneous axis of the rod can be found for any given position, and 
 we are thus enabled to draw the tangent and normal to this curve at 
 any point with geometrical accuracy. The motion of J being perpen- 
 dicular to CJ, the plane normal to the path of this point is CJ itself ; 
 the motion of the point A of the rod /JTat the instant is in the direc- 
 tion JA, therefore, AQ perpendicular to JK is the plane normal to 
 its path. But since both A and / lie in the circumference of the 
 circle whose centre is (7, the intersection Q of these normal planes, 
 that is, the instantaneous axis, will also lie in that circumference 
 throughout the action, and will at any given instant be diametrically 
 opposite to the position of the crank at that instant. 
 
 48. To draw the tangent to the limacon at any point. In illustra- 
 tion of the above, let it be required to draw the tangent to the given 
 limacon at the point M. Draw through A the chord MA and produce 
 it to cut the generating circle in N, and draw the diameter Nb ; then 
 ~b is the instantaneous axis of NM y IM is normal, and a perpendicular 
 to it is tangent to the curve at M as required. 
 
 49. Now when, as in this figure, the normal distance PI is less than 
 PA, it will be perceived that the limacon must cross the base circle 
 of the epicycloid at A ; and since the final chord AL lies outside of 
 
APPENDIX. 327 
 
 that base circle, the limacon must also cross its circumference again 
 at some point M between A and L. And further, there must be some 
 point K, intermediate between A and M, nearer to G than any other 
 point of the curve ; at this point, therefore, the limacon must be tan- 
 gent to a circle whose centre is G. 
 
 Obviously, the point of the parallel curve is derived from the 
 point A of the limacon ; and making AR equal to IP, and the arc 
 A equal to the arc PR, we may find the direction of the normal 
 OB either by describing about G an arc through R cutting the 
 epicycloid in B, or by producing GO to cut CD, the path of the 
 centre, in X, describing the generating circle in that position, and 
 making the chord OB = AR = PI. Also, the intersection U is de- 
 rived from the point M. We here observe that the chords MA, AN, 
 of the two* circles which are tangent at A, lie in one right line ; there- 
 fore, the triangles A CN, A GM, are similar, and we have the known 
 magnitude MN divided at A into segments directly proportional to 
 the given radii AC, AG. The arc PN being thus determined, we 
 lay off the arc A V equal to it, then with centre F and radius equal 
 to AM describe an arc cutting the base circle in U, at which point 
 UVWis the normal. 
 
 50. Finally, the point of cuspidation H is derived from the lowest 
 point K of the limacon. And this point must be so situated that GK 
 produced shall pass through the instantaneous axis Q of the rod KAJ 
 when the limacon is mechanically traced as above described. 
 
 But Q is then diametrically opposite to J, and P is diametrically 
 opposite to A ; therefore, PQ is parallel to A J, and consequently to 
 AKi which is a prolongation of JA ; and, moreover, PQ is equal 
 to AJ. 
 
 Then from similar triangles, A KG, PQG, 
 
 AK AG AK AG 
 
 __ TCT n on r*r* " 
 
 PQ ~ PG' AJ ~ PG ' 
 
 The magnitude of A J being thus determined, the arc AT is made 
 equal to the arc PJ, whence the position of the contact radius TY 
 and the normal 2 J S to the epicycloid are found as before, and 8TH 
 being made equal to JAK, the point in question is located with 
 geometrical precision. 
 
 FINIS. 
 
INDEX 
 
 A. 
 
 PAGE 
 
 Action, arc and angle of 94 
 
 " line of 43 
 
 " approaching and receding 96 
 
 " of spur and bevel wheels compared 239 
 
 Angular velocity 4 
 
 " " parallelogram of 69 
 
 Annular wheels epicycloidal teeth 105 
 
 " " involute teeth 148 
 
 " " in pin gearing 204 
 
 Approximate forms for teeth of spur wheels 172 
 
 bevel " 233 
 
 Arbitrary proportions for teeth of spur wheels 103 
 
 Axis of rotation, instantaneous 14 
 
 B. 
 
 Backlash 107 
 
 Band motions 39 
 
 Bevel wheels, pitch surfaces 64 
 
 " " in double pairs 66 
 
 "insidegear 238 
 
 " " teeth of, approximate forms 233 
 
 " " " " correct " 236 
 
 " " " " involute " . 238 
 
 C. 
 
 Centres, line of 37 
 
 Circular pitch 94 
 
 Classification of gearing 89 
 
 Clearance.. 96 
 
 Close-fitting tangent screws 267 
 
 Composition and resolution of motions 12 
 
 " of rotation and translation 19 
 
 Computation of limiting numbers of teeth 128 
 
330 IOTEX. 
 
 PAGE 
 
 Condition of compulsory rotation 41 
 
 " " constant velocity ratio 41 
 
 Condition of rolling contact 42 
 
 Conical lobed wheels 74 
 
 Conjugate teeth 167 
 
 Connected points, motion of 13 
 
 Constructive mechanism 2 
 
 Contact, rolling, sliding, and mixed ' 30, 35 
 
 Contact motions 40 
 
 " " rate of sliding in 42 
 
 Continuous motion 5 
 
 Contraction of angles 51 
 
 Cutter engine, pantagraphic 182 
 
 Cutters, series of equidistant . 186 
 
 Cycle of motions 6 
 
 D. 
 
 Dead points 38 
 
 Degenerated hyperboloids 85 
 
 Describing circle 93 
 
 " " intermediate, limiting diameter of 106 
 
 " " interior and exterior, limiting diameter of 107 
 
 Determination of path of contact 170 
 
 " "a series of cutters 186 
 
 Diametral pitch 176 
 
 Direction 3 
 
 Directional relation .' 38, 42 
 
 Dissimilar lobed wheels 54, 61 
 
 Double contact of teeth in inside gear 107 
 
 Driver and follower. . 8 
 
 E. 
 
 Elementary combinations 7 
 
 Ellipses, rolling 51 
 
 " spherical 70 
 
 Elliptical bevel wheels 72 
 
 " gearing, construction of teeth 223 
 
 involute teeth 226 
 
 pulleys 226 
 
 Epicycloid and involute compared 163 
 
 spherical 233 
 
 Epicycloidal milling engine 178 
 
 Equidistant gear cutters 186, 194 
 
 F. 
 Face and flank. . 
 
INDEX. 331 
 
 PAGE 
 
 Face gearing 287 
 
 " " and screw gearing combined 290 
 
 Fallacy of Willis's and Rankine's theory of skew teeth 250 
 
 Friction gearing 46 
 
 " " grooved 46 
 
 " bevel 65 
 
 " " skew 87 
 
 G. 
 
 Gear cutters, equidistant series of 186, 194 
 
 " " manufacture of 178, 181 
 
 Gearing classified 88, 92 
 
 Geometrical method of investigation 11 
 
 Graphic representation of motion 10 
 
 n. 
 
 
 
 Hindley's screw, or hour-glass worm 280 
 
 Hollow cones, double pair 68 
 
 Hooke's stepped wheels 195 
 
 " spiral gearing 197 
 
 Hyperboloids of revolution 77 
 
 " in internal contact 81 
 
 " one rolling upon another which is fixed 250 
 
 I. 
 
 Inside gearing 105, 148, 204 
 
 Instantaneous axis of rotation 14 
 
 Interchangeable spur wheels ' 100 
 
 lobed " 58 
 
 Intermittent motion 5 
 
 Intermediate describing circle 106 
 
 Irregular lobed wheels 63 
 
 Involute, first generation 144 
 
 second " 165 
 
 spherical 239, 292 
 
 teeth ._ 143 
 
 ' ' with epicycloidal extension 153 
 
 " " original pitch circle 146 
 
 " proper obliquity 146 
 
 " max. pitch for given obliquity 151 
 
 *' compared with epicycloidal 163 
 
 " for elliptical wheels 226 
 
 " Olivier's, in different planes 253 
 
 L. 
 Least amount of sliding in screw gearing 278 
 
332 IKDEX. 
 
 PAGE 
 
 Limiting numbers of teeth, epicycloidal 114, 128, 134 
 
 " " involute 153,162 
 
 " " " fl in pin gearing 208,216 
 
 " diameter of pin " " " 206 
 
 Line of action 43 
 
 " "centres 37 
 
 Links 8 
 
 Link motion, velocity ratio in 36 
 
 Lobed wheels 49, 230 
 
 " " derived from ellipse 52 
 
 " " pin gearing for 231 
 
 Logarithmic spiral 47 
 
 Low-numbered pinions Ill, 149, 160 
 
 M. 
 
 Machine 1, 2, 3 
 
 Mechanism, pure and constructive 1,2 
 
 " train of 8 
 
 Mechanical movements 6 
 
 Milling engine, epicycloidal 178 
 
 Mixed contact 31, 35 
 
 Momentary constancy of velocity ratio 38 
 
 Motion and rest 3 
 
 " composition and resolution of 12 
 
 " continuous, intermittent, reciprocating 5 
 
 " graphic representation of 10 
 
 " modes of transmitting 8 
 
 " modification of 8 
 
 " of connected points 13 
 
 " of a rigid body in space 25 
 
 " of translation 18 
 
 " phases and cycles of 6 
 
 Motive power 1 
 
 N. 
 
 Noise and vibration of incorrect teeth 218 
 
 Nomodont 139 
 
 Non-circular cones " 75 
 
 " wheels 49, 230 
 
 Normal component 13 
 
 0. 
 
 Oblique rack and wheel 271 
 
 " screw gearing 272 
 
 " " and rack . 376 
 
333 
 
 PAGE 
 
 Obliquity, transverse, in skew gearing 82 
 
 " proper, of involute teeth 146 
 
 Odontograph, Robinson's templet 175 
 
 Willis's 173 
 
 Odontoscope, Mac Cord's 221 
 
 Olivir's involutes, in different planes 253 
 
 Original pitch circle 146 
 
 P. 
 
 Pantagraphic cutter engine 182 
 
 Parallelogram of motions 12 
 
 " " angular velocities 69 
 
 Parallelopipedon of motions 12 
 
 Path 3 
 
 < ' of contact 151, 170 
 
 " "in pin gearing 217 
 
 Phases and cycles of motion 6 
 
 Pinions, low-numbered Ill, 149, 160 
 
 Pin gearing 200 
 
 " " for lobed wheels 231 
 
 Pitch, circular 94 
 
 " diametral 1 76 
 
 Proper obliquity of involute teeth 146 
 
 R. 
 
 Rack and wheel ! 102, 147 
 
 " " " oblique 271 
 
 " " " in pin gearing 203 
 
 " " screw 270 
 
 Radial planes, wheels with 209 
 
 Rate of sliding in contact motions 42 
 
 Ratio, velocity 9, 36 
 
 Receding and approaching action 96 
 
 Reciprocating motion 5 
 
 Relation, directional 38, 42 
 
 Resolution of motions 12, 20' 
 
 Rest, absolute and relative 3 
 
 Resultant 12 
 
 " motion of a rigid body 25 
 
 Revolution and rotation 4 
 
 Robinson's odontograph 175 
 
 Rolling contact 32, 45 
 
 " " condition of 42 
 
 " cones 64 
 
 " " in double pairs. . . 66, 68 
 
 " " with varying velocity ratio 69 
 
 1 < ellipses 51 
 
334 INDEX. 
 
 PAGE 
 
 Rolling hyperboloids 78 
 
 " " double tangency of 85 
 
 S. 
 
 
 
 Sang's theory of the teeth of wheels 168 
 
 " " practical embodiment of 269 
 
 Screw gearing 197, 266 
 
 " " combined with face gearing 291 
 
 " " distinguished from twisted gearing 91 
 
 " hour-glass 280 
 
 " " least amount of sliding in 278 
 
 " multiple-threaded 269 
 
 " " oblique 270 
 
 " " practical proportions o*f 268 
 
 Skew gearing, pitch surfaces 78 
 
 " " teeth of 242 
 
 " " transverse obliquity 82 
 
 " " resemblance of to screw gearing 279 
 
 Sliding, rate of in contact motions 42 
 
 Sliding contact 80 
 
 Spherical epicycloid 233 
 
 " involute 239, 292 
 
 " screw and wheel *. 292 
 
 T. 
 
 Tables of equidistant cutters 194 
 
 " 4< limiting numbers of teeth, epicycloidal 134, 135 
 
 " " " " " " involute 162,163 
 
 " " " " " " pin gearing 216,217 
 
 " " " " " " computation of 128 
 
 Tangential component 13 
 
 Tangent screws, close-fitting 267 
 
 Teeth of bevel wheels, approximate 233 
 
 " " " " correct 236 
 
 " "face " 284 
 
 " " lobed " 223,228 
 
 " " " " conical 240 
 
 " " screw " ^ 266 
 
 " "skew " 242 
 
 " " " " new theory 253 
 
 " " spur " conjugate 168 
 
 " approximate 172 
 
 " " " " epicycloidal 92 
 
 " " " " involute t 143 
 
 " " " " Sang's theory 168 
 
 Three-leaved pinion 113, 117 
 
INDEX. 335 
 
 PAGE 
 
 Transformation of rolling curves 51 
 
 Translation .'. , 18 
 
 Transmitting motion, modes of 8 
 
 Transverse obliquity in skew gearing 82 
 
 Twisted spur wheels (Hooke's spiral gearing) 195 
 
 " bevel " 241 
 
 " skew " 265 
 
 Two-leaved pinion, external 116 
 
 " " internal.. . 122 
 
 U. 
 
 Uniform periodic motion 6 
 
 Use of low-numbered pinions 110, 149 
 
 Unsymraetrical teeth 171 
 
 V. 
 
 Velocity 3 
 
 Velocity ratio 9 
 
 " " in band motions 39 
 
 " " " contact motions 40 
 
 " ''link " 36 
 
 " " condition of a constant 41 
 
 " " momentarily constant 38 
 
 " " varying 47 
 
 " " of rolling hyperboloids 79 
 
 W. 
 
 Wear, effects of on incorrect teeth 219 
 
 " in bearings, effects of 220 
 
 Wearing to a bearing 220 
 
 Wheels with radial planes 209 
 
 Willis's odontograph 173 
 
Table of Cutters for Teeth of Gear Wheels, 
 
 MADE BY 
 
 THE PKATT & WHITNEY COMPANY, 
 
 HARTFORD, CONN., U. S. A. 
 
 All Gears of the same pitch cut with our Cutters are perfectly interchangeable. 
 
 Diameter of 
 Cutters. 
 
 Diametral 
 Pitch. 
 
 Price of 
 
 Cutters. 
 
 Size of Hole in 
 Cutters. 
 
 SET OF 24 CUTTERS. 
 
 For each pitch coarser than 10. 
 
 5 inches. 
 
 1* 
 
 $25 00 
 
 11 inches. 
 
 No. 1 cuts 12 T 
 
 41 
 
 2 
 
 20 00 
 
 a 
 
 No. 2 " 13 
 
 4 
 
 2? 
 
 18 00 
 
 a 
 
 No. 3 " 14 
 
 31 " 
 
 3 
 
 15 00 
 
 * 
 
 No. 4 " 15 
 
 'Si 
 
 Si 
 
 12 00 
 
 1 
 
 No. 5 " 16 
 
 31 
 
 4 
 
 9 00 
 
 i 
 
 No. 6 " 17 
 
 3^ " 
 
 5 
 
 7 00 
 
 " " 
 
 No. 7 " 18 
 
 3 
 
 6 
 
 6 00 
 
 
 
 No. .8 " 19 
 
 21' " 
 
 7 
 
 5 00 
 
 
 
 No. 9 " 20 
 
 21 " 
 
 8 
 
 4 50 
 
 i 
 
 No. 10 " 21 to 22 
 
 2| " 
 
 9 
 
 4 00 
 
 < 
 
 No. 11 " 23 "24 
 
 OL 
 
 10 
 
 3 50 
 
 
 
 No. 12 " 25 " 26 
 
 2f 
 
 12 
 
 3 50 
 
 ( H 
 
 No. 13 27 " 29 
 
 21 
 
 14 
 
 3 50 
 
 tt 
 
 No. 14 " 30 " 33 
 
 2k " 
 
 16 
 
 3 00 
 
 " " 
 
 No. 15 " 34 " 37 
 
 2 
 
 18 
 
 3 00 
 
 " 
 
 No. 16 " 38 " 42 
 
 H " 
 
 20 
 
 3 00 
 
 " 
 
 No. 17 " 43 " 49 
 
 Ht 
 
 22 
 
 3 00 
 
 ( 
 
 No. 18 " 50 " 59 
 
 11 
 
 24 
 
 3 00 
 
 
 
 No. 19 " 60 " 75 
 
 11 
 
 26 
 
 3 00 
 
 u 
 
 No. 20 " 76 " 99 
 
 11 < 
 
 28 
 
 3 00 
 
 < 
 
 No. 21 " 100 " 149 
 
 H " 
 
 30 
 
 3 00 
 
 ( 
 
 No. 22 " 150 " 299 
 
 11 " 
 
 32 
 
 3 00 
 
 ( 
 
 No. 23 " 300 Rack. 
 
 
 
 
 
 No. 24 " Rack. 
 
 The cutters are made for diametral pitches. By diametral pitch is meant the 
 number of teeth per inch in the diameter of the gear at pitch line. Two pitches 
 should always be added to this diameter in preparing a gear for cutting. For ex- 
 ample : a gear of 80 teeth, 8 to the inch, diametral pitch, would be 10 inches on 
 pitch circle, but the gear should be turned 10$ (or i). The teeth should always be 
 cut two pitches deep beside clearance. 
 
 The cutters are made for a clearance of -^ of the depth of the tooth ; example : 
 8 to the inch has a clearance of fi L 4 ; therefore the tooth should be cut two pitches 
 (i) and t deep. The gears must be set to run with this clearance to give the 
 oest results. 
 
 In ordering bevel gear cutters, give the diameter of gear at outside pitch line, 
 and number of teeth, also the width of face. For the present all cutters are made 
 to order 
 
UNIVERSITY OF CALIFORNIA LIBRARY 
 
 THIS BOOK IS DUE ON THE LAST DATE 
 STAMPED BELOW 
 
 SEP 18 1916 
 
 0CT23 1319 
 
 ton? 
 
 MAY } 
 
 30m-6,'14 
 
YC 12713