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 UTY OF CALIFORNIA 
 
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A TREATISE ON 
 
 GEAR WHEELS, 
 
 BY GEORGE B. GRANT. 
 h 
 
 
 51 TD 
 
 PUBLISHED BY 
 
 GEORGE B. GRANT, 
 
 LEXINGTON, MASS. 
 PHILADELPHIA, PA. 
 
 SIXTH EDITION 
 
 George B. Grant, 
 
 86 Seneca St. 
 Cleveland, - Ohio. 
 
PRICK, clotln and. gilt, $1.OO post-paid. / 
 paper covers, .60 " 
 
 
 AQE.NTS WANTKD. 
 
 Any active and intelligent man can make money selling this book. I allow a liberal dis- 
 count and take back unsold copies. Write for circular of " terms to agents." 
 
 Contributed to ttx 
 
 AMERICAN MACHINIST, 
 Kor 1890. 
 
 COPIED from the American Machinist, and published in full by the " English Mechanic and 
 World of Science," and by the " Mechanical World." 
 
 ADOPTED and used as a text or reference book by Michigan and Cornell Universities, Rose 
 Polytechnic Institute, and by many schools of mechanics and drafting. - 
 
 Copyright 1893. 
 By GEORGE B. GRANT. 
 
 A Working Course of Study. 
 
 It is not necessary that the student, especially if he is a workman, should 
 learn all that is taught in this book, for it contains much that is not only difficult 
 but also of minor practical importance. 
 
 The beginner is therefore advised to master only the following sections : 
 
 i, 2, 7 to 15, 22, 25, 31, 32 of the general theory; 
 
 -35 to 47 of the spur gear ; 
 
 53 to 64 of the involute tooth ; 
 
 76, 77, 80 to 83, 89 of the cycloidal tooth ; 
 
 91, 95, 97 of the pin tooth ; 
 
 98, 99, in, 113 to 119 of spiral and worm gears; 
 
 154, 157, 158, 161 to 169 of the bevel gear. 
 
 These include not half of the whole matter, but, knowing this much well, 
 the student has a good outline knowledge of the whole, and he can then take 
 the balance at leisure. 
 
A TREATISE ON 
 
 GEAR WHKELS 
 
 1. THEORY OK TOOTH ACTION. 
 
 1 . INTRODUCTORY. 
 
 The present object is practical, to reach 
 and interest the man that makes the thing 
 written of ; the machinist or the millwright 
 that makes the gear wheel, or the drafts- 
 man or foreman that directs the work, and 
 to teach him not only how to make it, but 
 what it is that he makes. 
 
 To most mechanics a gear is a gear. 
 
 " A yellow primrose by the shore, 
 A yellow primrose was, to him, 
 And it was nothing more ; " 
 
 and, in fact, the gear is often a gear and 
 nothing more, sometimes barely that. 
 
 But, if the mechanic will look beyond the 
 tips of his fingers, he will find that it can 
 
 be something more ; that it is one of the 
 most interesting objects in the field of scien- 
 tific research, and not the simplest one ; that 
 it has received the attention of many cele- 
 brated mathematicians and engineers ; and 
 that the study of its features will not only 
 add to his practical knowledge, but also to 
 his entertainment. There is an element in 
 mathematics, and in its near relative, theoreti- 
 cal mechanics, that possesses an educating 
 and disciplining value beyond any capacity 
 for earning present money. The thinking, 
 inquisitive student of the day is the success- 
 ful engineer or manufacturer of the future. 
 
 2. METHOD. 
 
 The method will be fitted to the object, and 
 will be as simple and direct as possible. It is 
 not possible to treat all the items in simple 
 every-day fashion, by plain graphical or arith- 
 metical methods, but where there is a choice 
 the path that is the plainest to the average 
 intelligent and educated mechanic will be 
 chosen. 
 
 A thousand pages could be filled with the 
 subject and not exhaust anything but the 
 reader thereof, but what is written should 
 receive and deserve attention, and must be 
 condensed within such reasonable limits, that 
 it shall not call for more time and labor than 
 
 its limited application will warrant. Demon- 
 strations and controversies will be avoided, 
 and the matter will be confined as far as is 
 possible to plain statements of facts, with 
 illustrations. The simplest diagram is often 
 a better teacher than a page of description. 
 First, we shall study the odontoid or pure 
 tooth curve as applied to spur gears, then 
 we shall consider the involute, cycloid, and 
 pin tooth, special forms in which it is found 
 in practice ; then the modifications of the 
 spur gear, known as the spiral gear, and the 
 elliptic gear ; then the bevel gear, and lastly 
 the skew bevel gear. 
 
Literature. 
 
 3. PARTICULARLY IMPORTANT. 
 
 Begin at the beginning. 
 
 The natural tendency is too often to skip 
 first principles, and begin with more ad- 
 vanced and interesting matter, and the result 
 is a trashy knowledge that stands on no 
 foundation and is soon lost. When a fact 
 is learned by rote it may be remembered, 
 but when it follows naturally upon some 
 simple principle it cannot be forgotten. 
 
 Therefore the student is urged to begin 
 with and pay close attention to the odontoid 
 or pure tooth curve, before going on to its 
 
 special applications, for the apparently dry 
 and trivial matter relating to it is really the 
 foundation of the whole subject. 
 
 The usual course is to begin at once with 
 the cycloidal tooth, to hurry over the in- 
 volute tooth, and then, if there is room, it 
 is stated that such curves are particular 
 forms of some confused and indefinite general 
 curve. Our course will be to study the unde- 
 fined tooth curve first, and then take up 
 its particular cases. 
 
 4. LITERATURE. 
 
 It is impracticable to acknowledge all the 
 sources from which information has been 
 drawn, but it is in order to briefly mention 
 the principal works devoted to the subject. 
 
 Professor Herrmann's section of Professor 
 Weisbach's "Mechanics of Engineering and 
 Machinery" is the most important work that 
 can be named in this connection. It treats 
 of much besides the teeth of gears, but its 
 treatment of that branch is particularly 
 valuable. It is not easy reading. Wiley, 
 $5.00. 
 
 Professor Willis' "Principles of Mechan- 
 ism " is a celebrated book, now many years 
 behind the age, but it is, nevertheless, of the 
 greatest value and interest in this matter. To 
 Willis we are indebted for many of the most 
 important additions to our knowledge of 
 theoretical and practical mechanism. Long- 
 mans, $7.50. Out of print. 
 
 Professor Rankine's "Machinery and Mill- 
 work " should not be neglected by the 
 student, for, although it is the dryest of 
 books, its value is as great as its reputation. 
 Griffin, $5.00. 
 
 Professor MacCord's "Kinematics" is a 
 work that abounds in novelties, and is writ- 
 ten in an attractive style. It contains many 
 errors, and some hobbies, and needs a thorough 
 revision, but the student cannot afford to 
 avoid it, or even to slight it. Wiley, $5.00. 
 
 Mr. Beale's "Practical Treatise on Gear- 
 ing " is really practical. Many of the so-call- 
 ed "practical " books are neither practical or 
 theoretical, but we have in this small book 
 a collection of workable information that 
 
 should be within the reach of every man 
 who pretends to be a machinist. We have 
 drawn from it, by permission, particularly 
 with regard to spiral and worm gears. 
 Mr. Beale's experimental work, in connection 
 with the spiral gear, has been of great 
 service. The Brown & Sharpe Mfg. Co., 
 cloth $1.00, paper 75c. 
 
 Professor Reuleaux's " Konstrukteur " is a 
 justly celebrated work in the German lan- 
 guage. A translation of it is now being 
 published in an American periodical Me- 
 chanics. 
 
 Professor Klein, the translator of Herr- 
 mann's work, has lately published the " Ele- 
 ments of Machine Design," a collection of 
 practical examples, with illustrations. J. F. 
 Klein, Bethlehem, Pa., $6.00. 
 
 "Mill Gearing," by Thomas Box, is a 
 practical work by an engineer, and from it 
 we have drawn much of our matter on the 
 cloudy subject of the strength and horse- 
 power of gearing. Spon, $3.00. 
 
 "Elementary Mechanism," by Professors 
 Stahl and Woods, is a recent work of general 
 merit. It is well designed as a text book, 
 and treats the subject in a simple and in- 
 teresting manner. Van Nostrand, $2.00. 
 
 In addition to the above works, reference 
 may be made to numerous articles to be 
 found in periodicals, notably in the " Ameri- 
 can Machinist," the "Scientific American 
 Supplement," the "Journal of the Franklin 
 Institute," " Mechanics," and the " Transac- 
 tions of the American Society of Mechanical 
 Engineers." 
 
General Theory . 
 
 5. KINEMATICS. 
 
 This, the science of pure mechanism, re- 
 lates exclusively to the constrained and 
 geometric motions of mechanism, and it 
 has nothing to do with questions of force, 
 weight, velocity, temperature, elasticity, 
 etc. The path of a cannon ball is not with- 
 in the field of kinematics, because it depends 
 upon time and force. A belt and pulley are 
 kinematic agents, because the contact be- 
 tween them can be assumed to be definite, 
 
 and the action is therefore geometric, but 
 the slipping and stretching of the belt is not 
 kinematic. The action of gear teeth upon 
 each other is purely kinematic, but we can- 
 not consider whether the material is wood, or 
 steel, or wax, whether the gears are lifting 
 one pound or a ton, or whether they are run- 
 ning at one revolution per second or one per 
 day. 
 
 6. ODONTICS. 
 
 The name "odontics' may be selected 
 for that limited but important branch of 
 kinematics that is concerned with the trans- 
 mission of continuous motion from one 
 body to another by means of projecting 
 teeth. 
 
 Even this restricted corner of the whole 
 subject is too large for the present purpose, 
 for it covers much that cannot be considered 
 within our set limits, and gear wheels must, 
 therefore, be defined as devices for trans- 
 mitting continuous motion from one fixed 
 axis to another by means of engaging teeth. 
 
 Thus confined, gear wheels may be con- 
 veniently divided into three general classes. 
 
 Skew bevel gears, transmitting motion be- 
 tween axes not in the same plane. 
 
 Bevel gears, transmitting motion between 
 intersecting axes. 
 
 Spur gears, transmitting motion between 
 parallel axes. 
 
 The last two classes are particular cases of 
 the first; for, if the shafts may be askew at 
 any distance, that distance may be zero, and 
 if they intersect at any point, that point may 
 be at infinity. 
 
 It would be scientifically more correct to 
 first develop the skew bevel gear, and from 
 that proceed to the bevel and spur gear, but 
 practical clearness and convenience is often 
 more to be admired than strict accuracy, 
 and, as the true path is difficult to follow, we 
 shall enter in the rear, and consider the spur 
 gear first. 
 
 Odontics does not properly include the 
 consideration of questions of strength, pow- 
 er and friction, but we must admit certain 
 important items in that direction. 
 
 7. PITCH 
 
 The fixed axes are connected with each 
 other by imaginary surfaces called "axoids," 
 or pitch surfaces, touching each other along 
 a single straight line. We must imagine 
 that the pitch surfaces roll on each other 
 without slipping, as if adhering by friction. 
 The whole object of odontics is to provide 
 these imaginary surfaces wilh teeth, by 
 
 SURFACES. 
 
 which they can take advantage of the 
 strength of their material and transmit 
 power that is as definite as the geometric 
 motion. 
 
 The pitch surface of the skew bevel gear 
 is the hyperboloid of revolution, which be- 
 comes a cone when the axes intersect, and a 
 cylinder when the axes are parallel. 
 
 8. NORMAL SURFACES. 
 
 An important adjunct of the pitch surface I For the skew bevel gear there does not 
 
 is the normal surface, or surface that is 
 everywhere at right angles to both pitch sur- 
 faces of a pair of axes, and upon which the 
 action of the teeth on each other may best 
 be studied. 
 
 appear to be any normal surface. For the 
 bevel gear the normal surface is a sphere, 
 and for the spur gear the sphere becomes a 
 plane. 
 
General Theory. 
 
 9. UNCERTAINTIES. 
 
 The theory of tooth action is not yet full 
 and definite in all its parts, for there are 
 some disputed points, and some confusion 
 and clashing of rules and systems. This is 
 
 particularly the case with the theory of spiral 
 and skew bevel teeth, for much of the work 
 that has been done is clearly wrong, and 
 there is little that has been definitely decided. 
 
 10. PITCH CYLINDERS. 
 
 Fig. 1 
 
 'I'itch cylinder* 
 
 Two cylinders, A and B, Fig. 1, that will 
 roll on each other, will transmit rotary mo- 
 tion from one of the fixed parallel axes c and 
 G to the other, if their surfaces are provided 
 with engaging projections. 
 
 When these projections are so small that 
 they are imperceptible, the motion is said to 
 be transmitted by friction, and it is prac- 
 tically uniform. But when they are of 
 large size, and readily observed, the motion, 
 
 although it is unchanged in nature, is said to 
 be transmitted by direct pressure, and it is 
 irregular unless the acting surfaces of the 
 projections are carefully shaped to produce 
 an even motion. 
 
 The whole object of odontics is to so shape 
 these large projections or teeth that they 
 shall transmit the same uniform motion be- 
 tween the rotating cylinders, as would be 
 apparently transmitted by friction. 
 
 These cylinders are imaginary in actual 
 practice, although they are one of the 
 principal elements of the theory, and they 
 are called the axoids, or pitch cylinders of 
 the gears. 
 
 The normal surface (8) of the spur gear is 
 a plane, and, as all sections by normal sur- 
 faces are alike, we can study the action on 
 a plane figure easier than in the solid body 
 of the gear. 
 
 Fig. 2. 
 
 Tooth action 
 
 11. THE LAW OP TOOTH CONTACT. 
 
 The common normal to the tooth curves must 
 pass through the pitch point. 
 
 That is, in Fig. 2, if the tooth curves OD 
 and o d are to transmit the same motion 
 between the pitch lines pi and PL as 
 would be transmitted by frictional contact 
 at the pitch point 0, they must be so shaped 
 that their common normal Op at their com- 
 mon point p shall pass through that pitch 
 point. 
 
 Conversely, if the tooth curves are so 
 shaped that their common normal always 
 passes through the pitch point, they will 
 
 "With the above conditions given we can 
 deduce the following law: 
 
 transmit the required uniform motion. 
 
 12. THE ODONTOID. 
 
 This universal law enables us to define the 
 "odontoid," or pure tooth curve, for the 
 contact of the pitch lines at the pitch point 
 is continuous and progressive, and, if the 
 tooth curves are to transmit the same motion, 
 their normals must be arranged in a contin- 
 
 uous and progressive manner. The normals 
 nl, as in Fig. 3, must be arranged without 
 a break or a crossing, not only springing 
 from the odontoid at consecutive points, but 
 intersecting the pitch line at consecutive 
 points. This arrangement may be called 
 
General Theory. 
 
 Fig. 3. 
 
 Odontoid* 
 
 "consecutive," and the definition is not a 
 law by itself, but an expression of the given 
 universal law. 
 
 It is seen that the odontoid is inseparably 
 connected with its pitch line, and that the 
 same curve may be an odontoid with re- 
 spect to one pitch line, and not with respect 
 to some other. The curve Fig. 4 is an 
 odontoid with respect to the pitch Mne pi, 
 
 but not with respect to the pitch line pF be- 
 yond the point p at which the normal is tan- 
 gent to that pitch line. 
 
 The odontoid, so far as defined, is not a 
 definite thing, and, for practical purposes, it 
 must be given some particular shape. It 
 may be involute or cycloidal, or of other 
 form, but must always have normals ar- 
 ranged in consecutive order. 
 
 13. THE LINE OF ACTION. 
 
 As the tooth curves od and OD, Fig. 5, 
 work together, the point of contact will 
 travel along a line Op W called the "line of 
 action." 
 
 There is a definite relation between the 
 odontoid and the line of action, so that, if 
 either one is given, the other is fixed. If 
 the odontoid OD is given, with its pitch 
 line PL, the line of action is determined 
 without reference to the pitch .line pi or its 
 odontoid; and, conversely, if the pitch line 
 and line of action are given, the odontoid to 
 correspond is determined. 
 
 Fig. 
 
 Line of 
 action. 
 
 14. INTERCHANGEABLE ODONTOIDS. 
 
 This feature leads at once to the broad 
 and useful fact that all odontoids, on pitch 
 lines of all sizes, that are formed from the 
 same line of action, will work together inter- 
 changeably, any one working with any other. 
 
 Therefore, to produce an interchangeable 
 set of odontoids we can choose any one line 
 of action, and form any desired number of 
 them from it. 
 
 15. INTERNAL CONTACT. 
 
 The pitch lines of Fig. 5 curve in opposite 
 directions, and the contact is said to be "ex- 
 ternal." But the principles involved are in- 
 dependent of the direction of the pitch lines, 
 and they may curve in the same direction, as 
 in Fig. 6, in " internal" contact. 
 
 Tooth contact is between lines only, there 
 being no theoretical need of a solid material 
 on either side of the line, so that either side 
 
 Fig. 0. 
 
 Interna.1 action 
 
Cusps and Terminah 
 
 of the tooth may be chosen as the practical 
 working side. 
 
 Therefore the internal gear is precisely like 
 the external gear of the same pitch diameter, | 
 working on the same lines of action, so far j 
 as the odontoids are concerned, as illustrated 
 by Fig. 7. 
 
 Internal and 
 external teeth 
 
 Fig. 8 
 
 16. THE CUSP AND INTERFERENCE. 
 
 When, as in Fig. 8, the pitch circle p Ms 
 so small with respect to the line of action 
 C' C" W, that two tangent circles G' c' and 
 C" c" can be drawn to the line of action from 
 the center G of the pitch line, we shall have 
 a troublesome convolution in m the resulting 
 flank curve o d. This convolution will be 
 formed of two cusps, a first cusp c' on the 
 inner tangent arc, the "base circle" C' c', 
 and a second cusp c" on the outer tangent 
 arc G" c". 
 
 This happens with any form of odontoid, 
 although sometimes in disguised form, and 
 creates a practical difficulty that can be 
 avoided only by stopping the tooth curve at 
 the first cusp c'. 
 
 Furthermore, any odontoid OD that is to 
 work with the odontoid o d, must be cut off j 
 at the point k on the "limit line" C' k 
 through the point C' from the center c. 
 
 If the odontoids, when the pitch line is so 
 small that the cusps occur, are not cut off as 
 
 required, the action will still be mathemati- 
 cally perfect, but, as the contact changes at a 
 cusp, from one side of the curve to the other, 
 the action is no longer practicable with solid 
 teeth. The curves will cross each other, 
 and there will be an interference. 
 
 17. THE SMALLEST PITCH CIRCLE. 
 
 To determine the smallest pitch circle that 
 can be used, and avoid the cusps altogether, 
 find by trial the point C, Fig. 9, from which 
 but one tangent arc C 1 c' can be drawn to the 
 line of action C' W. This point will be 
 the center of the smallest pitch circle, and 
 all points outside of it will avoid interference, 
 while all inside of it will be subject to it. 
 
 Fig 
 
 18. THE TERMINAL POESTT, 
 
 When a tangent arc can be drawn, from 
 the pitch point as a center, to the line of 
 action at any point T, except the vertex W, 
 Fig. 10, there will be a corresponding cross- 
 ing of the normals to the odontoid commenc- 
 ing at the point t, and a termination of the 
 action when the point t reaches the point T. 
 
 As the action approaches the terminal 
 point T there will be two points of action, 
 
 Fig. 10. 
 
 o 
 
 Terminal point 
 
Secondary Action. 
 
 since the odontoid crosses the line of action 
 at two points--one point of direct and ordi- 
 nary action at 8, and another point of retro- 
 grade and unusual action at F. These two 
 points of action will come together at T, the 
 odontoid will leave the line of action, and all 
 
 tooth action will then cease. The retro- 
 grade action is theoretically and actually 
 correct, but it is so oblique that it is of 
 no practical value, and therefore the odon- 
 toid may as well be cut off at its terminal 
 point t. 
 
 19. SPEED OF THE POINT OP ACTION. 
 
 Lay off 8, Fig. 5, to represent the speed 
 of the pitch lines, and draw 8 A at right 
 angles with the common normal p. Draw 
 *p C tangent to the line of action at the point 
 of action p. 
 
 Lay off p B equal to A, and draw B C 
 at right angles to B. Then p G will be 
 the speed of the point of action along the 
 line of action. 
 
 When the line of action is a circle the 
 angle 8 A is always equal to the angle 
 B p C, and therefore the speed of the point 
 of action is uniform, and equal to that of the 
 pitch lines. 
 
 If the line of action is a straight line the 
 angle B p G will be constant always zero 
 and therefore the speed of the point of action 
 will be uniform and always equal to A. 
 
 20. NATURE OF THE TOOTH ACTION. 
 
 The nature of the action may be deter- 
 mined by a study of the normal intersections; 
 the intersections of the normals with the 
 odontoid being at uniform distances apart, 
 their intersections with the pitch lines will 
 indicate the action of the teeth. If the nor- 
 
 mal intersections, as in Fig. 3, are quite regu- 
 lar, the action of the teeth will be smooth 
 and regular, while if they are crowded with- 
 in a narrow space the action of the tooth will 
 be crowded and jerky. 
 
 21. THE SECONDARY LINE OF ACTION. 
 
 From the universal law of tooth contact 
 stated in (11) we can reason that any 
 point on the tooth curve is in position for 
 contact whenever its normal passes through 
 the pitch point 0, and therefore that the 
 point will then be upon a line of action. 
 
 In Fig. 11 the normal to the point p must 
 cross the pitch line twice at a primary in- 
 tersection a, and at a secondary intersection 
 b, and therefore there will be a point of 
 action on a primary line of action M' at q, 
 when the curve has moved so that the pri- 
 mary point of intersection a is at the pitch 
 point 0, and a point of action w on a second- 
 ary line of action, when the secondary point 
 of intersection b has reached the pitch point. 
 
 Therefore there will generally be not only 
 the primary line of action q M or O q' M', 
 but also a secondary line w T or w' Y 1 . 
 
 The secondary line of action must have the 
 same property as the first, as a locus of con- 
 tact, and therefore if we can so arrange two 
 pitch lines with their odontoids that their 
 secondary lines of action coincide, there will 
 be secondary contact between the odontoids. 
 
 Fig 
 
 When it so happens that both primary and 
 secondary lines coincide, we shall have 
 double contact. Two points of contact will 
 exist at the same time, one on the primary 
 and the other on the secondary line of action. 
 
 The secondary lines of action cannot be 
 made to coincide when the contact is exter- 
 nal, but when it is internal they sometimes 
 can be, so that the matter has an application 
 to internal gears. 
 
Interchangeable Tooth. 
 
 It is to be noticed that the primary line is 
 independent of the pitch line, while the sec- 
 ondary is dependent upon it. 
 
 Secondary contact is an interesting feature 
 of tooth action, but it is of small importance, 
 and has been studied but little. 
 
 22. THE INTERCHANGEABLE TOOTH. 
 
 The simple odontoid so far studied is the 
 perfect solution of the problem from a 
 mathematical point of view, for it will trans- 
 mit the required uniform motion as long as 
 it remains in working contact. But from a 
 mechanical point of view it is still incom- 
 plete, as it works in but one direction, 
 through but a limited distance, and, although 
 the odontoids are interchangeable, the gears 
 are not. 
 
 In order that the gears shall be fully in- 
 terchangeable, it is necessary that the teeth 
 shall have both faces and flanks, and that the 
 line of action for the face shall be equal to 
 that for the flank; that is, the tooth must 
 have an odontoid -on each side of the pitch 
 line, the face o d. Fig. 12, outside, and the 
 flank o d' inside of it, and the line of action 
 I a for the faces must be like the line of 
 action I a' for the flanks. If so made, any 
 gear will work with any other, without re- 
 gard to the diameters of the pitch lines. 
 
 But such a gear will run in but one direc- 
 tion, and to make it double-acting it must 
 have odontoids facing both ways, as in Fig. 
 IS. Gears so made will be both double-act- 
 ing and interchangeable, and it is not neces- 
 sary that both sides of the tooth shall be 
 alike. 
 
 Again, the unsymmetrical gear of Fig. 13 
 fails when it is turned over, upside down, 
 for then the unlike odontoids come together, 
 and, to avoid this last difficulty, all four of 
 the lines of action must be alike, producing 
 the complete and practically perfect tooth of 
 Fig. 14. 
 
 We can therefore define the completely in- 
 terchangeable tooth, as the tooth that is 
 formed from four like lines of action. 
 
 Unsynt-inetrical teeth 
 
 Fig. 14. 
 
 Complete teeth 
 
 &. INTERCHANGEABLE RACK TOOTH. 
 
 When the pitch line is a circle the flanks of 
 the tooth are not like the faces, but when it 
 is a straight line there is no distinction be- 
 
 tween face and flank. We then have the im- 
 portant practical fact that the four odontoid^ 
 of the interchangeable rack tooth are alike. 
 
Construction by Points. 
 
 24. CONSTRUCTION BY POINTS. 
 
 When we have an odontoid and its pitch 
 
 line given, it is a very simple matter to con- 
 struct either the line of action or the conju- 
 gate odontoid for any other pitch line. 
 
 We know, for example, the odontoid p, 
 Fig. 15, on the pitch line p I, and it is re- 
 quired to construct an odontoid on the pitch 
 line P L that is conjugate to it. 
 
 As the odontoid is given we know or can 
 construct its normals. Construct the normal 
 p a from any chosen point p, draw the radial 
 line da C, lay off A equal to a 0, draw the 
 radial line A C, lay off the angle NAD 
 equal to the angle n a d, lay off P A equal 
 to p a, and P will be a point in the required 
 conjugate odontoid 8 P. P A will be a 
 normal to the curve. Construct a number of 
 points by this process, and draw the required 
 curve through them. The tangents * t and 
 8 T make equal angles with the pitch lines, 
 so that the required curve can often be fully 
 determined by drawing its tangent and one 
 or two points. 
 
 To construct the line of action, make the 
 angle m e equal to the angle n a d, 
 and lay off q equal to p a. The point 
 q is on a circle from either p or P drawn 
 from the centers (7, and is the point at 
 which p and P will coincide when the two 
 curves are in working contact, the normals 
 p a and P A then coinciding with the 
 radiant q. 
 
 Fig. 15. 
 
 Construrtio 
 by points 
 
 When the line of action alone is given, the 
 odontoids for given pitch lines are fully de- 
 termined, but there seems to be no simple 
 graphical method for constructing them ex- 
 cept for special cases. They can be obtained 
 by the use of the calculus (33), or drawn by 
 the integrating instrument of (34). 
 
 The two tooth curves thus constructed are 
 paired, and are said to be " con jugate'* to 
 each other. 
 
 25. THE ABC OP ACTION. 
 
 The action between two teeth commences 
 and ends at the intersections m and N of the 
 line of action with the addendum lines of the 
 two gears, a I and A L, Fig. 16. The arc 
 of action is the distance a b on the pitch line 
 that is passed over by the tooth while it is in 
 action. 
 
 The arc a passed over while the point of 
 contact is approaching the pitch point, is 
 called the arc of approach, and b, that 
 passed over while the action is receding from 
 that point, is the arc of recess. 
 
 With a given line of action the arcs of ap- 
 proach and recess can be controlled by the 
 addenda. If it is desirable to have a great 
 recess and a small approach, the addendum 
 
 of the gear that acts as a driver is to be in- 
 creased. When there is a limit line (16), it 
 limits the addendum and the arc of action. 
 
10 
 
 Molding Processes. ^ 
 
 26. OBLIQUITY 
 
 When a pair of teeth bear upon each 
 other, the direction of the force exerted be- 
 tween them is that of the common normal 
 Op, Fig. 17, and passes through the pitch 
 point 0. Except when the point of contact 
 Is at the pilch point the direction of the 
 pressure will deviate from the normal to the 
 line of centers by the angle of obliquity 
 Z Op, and with many forms of teeth the 
 angle is never zero. 
 
 The force exerted between two teeth at 
 their point of contact is found by laying off 
 the tangential force //with which the driv- 
 ing gear D is turning, and drawing the line 
 H V parallel to the line of centers, to find 
 the force V P K. It is proportional to 
 ihe secant of the angle of obliquity, and in- 
 creases rapidly with that angle. 
 
 The chief influence of the obliquity is 
 upon the friction between the teeth, and con- 
 sequent inefficiency of the gear, and upon 
 the destruction by wearing. It is par- 
 ticularly important upon the approaching 
 action, and a gear that is otherwise perfect 
 may be inoperative on account of excessive 
 obliquity. 
 
 Although the direct pressure of the teeth 
 upon each other at their point of contact 
 
 OF THE ACTION 
 K 
 
 .17. 
 
 will vary with the obliquity, the tangential 
 force exerted to turn the gear is always 
 uniform. Leaving friction out of the calcu- 
 lation, the two gears of a pair always turn 
 with the same force at their pitch lines. 
 
 The obliquity of the action has an effect 
 upon the direction and amount of the 
 pressure of the gear upon its shaft bearing, 
 but the usual variation is of little conse- 
 quence. 
 
 It is desirable that the pressure between 
 the teeth should be as uniform as possible, 
 not only in amount, but in direction, and 
 excessive obliquity is to be carefully avoided. 
 
 27. CONSTRUCTION BY MOLDING. 
 
 The mode of action of the conjugate teeth 
 upon each other, suggests a process by which 
 a given tooth can be made to form its conju- 
 gate by the process of molding. 
 
 The given tooth, all of its normal sections 
 being of some odontoidal form, is made 
 of some hard substance, while the blank in 
 which the conjugate teeth are to be formed 
 is made of some plastic material. The shafts 
 of the two wheels are given, by any means, 
 the same motions as if their pitch surfaces 
 were rolled together. The hard tooth will 
 then mold the soft tooth into the true conju- 
 gate shape. 
 
 It matters not what shape is given the 
 molding tooth, if its sections are all odon- 
 toidal, and a twisted or irregular shape will 
 be as serviceable as the common straight tooth. 
 
 This process is continually in operation be- 
 tween a pair of newly cut teeth, or between 
 rough cast teeth, until the badly matched 
 surfaces have been worn to a better fit, but 
 it is too slow for ordinary purposes, and is 
 of little practical value. 
 
 Gears can be formed by this process, by 
 rolling a steel forming gear against a white 
 hot blank, but the process can hardly be 
 called practical. 
 
 28. MOLDING PLANING PROCESS. 
 
 Although the described molding process 
 is of limited practical value, having but one 
 direct application, it leads to a process of 
 great value when the tooth is straight or of 
 
 such a shape that it can be followed by a 
 
 planing tool, its normal sections being alike. 
 
 The originating tooth is fixed in the shape 
 
 of a steel cutting tool C, Fig. 18, which is 
 

 Planing Processes. 
 
 \ \ 
 
 rapidly reciprocated in guides G, in 
 the direction of the length of the 
 tooth, as the two pitch wheels A and 
 B are rolled together. Although the 
 tool has but a single cutting edge, its 
 motion makes it ihe equivalent, of the 
 molding tooth, and it will plane out 
 the conjugate tooth D by a process 
 that is the equivalent of the more 
 general molding process. 
 
 A simple graphical method is 
 founded upon this molding process, 
 the shaping tool taking the form of a 
 thin template (7, Fig. 19, that is re- 
 peatedly scribed about as the pitch 
 wheels are rolled together, the marks 
 combining to form the conjugate 
 tooth curves D. 
 
 This mechanical process has the 
 decided advantage over the procets 
 of construction by points (24), that 
 the tooth is formed with a correct 
 fillet (44), and is much stronger. 
 The dotted lines show the tooth that 
 would be constructed by points. 
 
 The only practicable method for 
 forming the line of action when this 
 method is used is by observing and 
 marking a number of points of con- 
 tact between the teeth. This method 
 is applicable to all possible forms of 
 spur teeth, either straight, twisted or 
 spiral. It can be practically applied 
 only to the octoid form of bevel tooth. 
 
 On account of the fillet (44) that 
 is formed by this process, the tooth 
 space cannot be used with a mating 
 gear having more teeth than that of 
 the forming gear, although it belongs 
 to the same interchangeable set. The 
 tooth space of the figure will not run 
 with a tooth on a pitch line larger 
 than the pitch line A. 
 
 Therefore the rack tooth must be 
 useu as the forming tooih, to allow of 
 the use of all gears of the set up to 
 the rack. Gears of the set thus formed 
 will not work with internal gears. 
 
 Hlolding planing 
 method 
 
 fttcipro eating 
 Too/ >- 
 
 JFig. 18. 
 
 phical molding 
 method 
 
 29 LINEAR PLANING PROCESS. 
 
 A second planing process, quite 
 distinct from the molding process of 
 (27), is founded upon the fact that 
 the tooth curves are in contact at a 
 single point which has a progressive 
 motion along the line of action. 
 
 Therefore if a single cutting point 
 p, Fig. 20, is caused to travel along 
 the line of action with the proper 
 speed relatively, to the speed of the 
 piich line, it will trim the tooth out- 
 line to the proper odontoidal shape. 
 
 The figure shows the application to 
 
 Fig. 20. 
 
 fn^olu-^-e To 
 
 Linear plan ing 
 met 
 
 ' 
 
 OF THZ 
 
12 
 
 Particular Forms. 
 
 the involute tooth, the path of the cutting 
 point being the straight line I a, and its speed 
 being the speed of the base line b I. 
 
 When the cutting point follows the circu- 
 lar line of action with a speed equal to that 
 of the pitch line, it will plane out the 
 cycloid al tooth curve. 
 
 This process is applicable to all possible 
 forms of gear teeth, either spur or bevel, in 
 either external or internal contact. 
 
 When the curvature of the odontoid will 
 permit, the milling cutter may take the place 
 of the planing tool, and is the equivalent of it. 
 
 ). THE BACK ORIGINATOR. 
 
 The molding planing process of (28) sup- 
 plies a means for easily and accurately pro 
 ducing an interchangeable set of gears or 
 cutters for gears, and it is best applied by 
 means of the rack tooth as the originator. 
 All four curves of the rack tooth being alike, 
 the tooth is easily formed, particularly for 
 the involute or the segmental systems, and it 
 is a matter of less consequence that the curves 
 
 shall be of some particular form, if care is 
 taken that it is odontoidal. 
 
 It has been taught, and it is therefore some- 
 times considered, that any " four similar and 
 equal lines in alternate reversion" will an- 
 swer the purpose, but it is necessary that the 
 four similar curves shall be odontoids. Four 
 circular arcs, with centers on the pitch line, will 
 answer the definition, but are not odontoids. 
 
 31. PARTICULAR FORMS OF THE ODONTOID. 
 
 The odontoid, as so far examined, is un- 
 defined except as to one feature of the ar- 
 rangement of its normals, and to bring it 
 into practical use it is necessary to give it 
 some definite shape. This is most easily ac- 
 complished by choosing some simple curve 
 for the rack odontoid, and from that making 
 an interchangeable set. A more correct but 
 much more difficult method would be to 
 choose some definite line of action, and from 
 that derive the odontoids. 
 
 If the rack odontoids are straight lines, 
 Fig. 21, the common involute tooth system 
 will be produced, and the line of action will 
 be a straight line at right angles with the 
 rack odontoid. For bevel teeth, as will be 
 shown, the straight line odontoid produces 
 the octoid tooth system, while to produce the 
 involute system it is necessary to define the 
 line of action as a straight line, and derive 
 the system from that. 
 
 If the rack odontoids are cycloids, as in 
 Fig. 22, the resulting tooth system will be 
 ihe cycloidal, commonly misnamed the 
 " epicycloidal " system. The line of action 
 will be a circle equal to the roller of the 
 cycloid. 
 
 If the rack odontoids are segments of cir- 
 cles from centers not on the pitch line, but 
 inside of it, as in Fig. 23, the tooth system 
 
 Fig 
 
 Segmental 
 
 Fig. 23. 
 
Rolled Curve Theory. 
 
 will be the segmental, and its line of action 
 will be the loop of the " Conchoid of Nico- 
 medes." 
 
 If we choose a parabola for the rack tooth, 
 as in Fig. 24, the parabolic system will be 
 formed with its peculiar "hour glass" line 
 of action. 
 
 Only three of these tooth systems are in 
 actual use, the involute and the cycloidal for 
 spur gears, and the octoid for bevel gears 
 only, and we will therefore confine the ap- 
 plication of the theory to them. 
 
 Only one of the sys'ems in common use 
 for spur gears, the involute, should be in use 
 at all, and we will pay principal attention to 
 that. 
 
 Parabolic 
 
 Fig 
 
 The segmental system would be superior 
 to the cycloidal, and in many cases to the in- 
 volute; but as there is already one system too 
 many, we will not attempt to add another. 
 
 32. THE ROLLED CURVE THEORY. 
 
 If any curve R, Fig. 25, is rolled on any 
 pitch curve p I, a point p in the former will 
 trace out on the plane of the latter a curve 
 s p z, called a rolled curve. 
 
 The line p q, from the tracing point p to 
 the point of contact q, is a normal to the 
 curve * p z, and, as all the normals are ar- 
 ranged in "consecutive" oraer, that curve 
 must be an odontoid. The converse of this 
 statement is also true, that all odontoids are 
 rolled curves ; but the fact is general^ . ery 
 far fetched and of no practical importance. 
 
 It is also a property of all such curves 
 that are rolled on different pitch lines, that 
 they are interchangeable. 
 
 This accidental and occasionally useful 
 feature of the rolled curve has generally 
 been made to serve as a basis for the general 
 theory of the gear tooth curve, and it is re- 
 sponsible for the usually clumsy and limited 
 treatment of that theory. The general law 
 is simple enough to define, but it is so diffi- 
 cult to apply, that but one tooth curve, the 
 cycloidal, which happens to have the circle 
 for a roller, can be intelligently handled 
 by it, and the natural result is, that that 
 curve has received the bulk of the atten- 
 tion. 
 
 For example, the simplest and best of 
 
 Rolled curve 
 
 Fig. 25. 
 
 all the odontoids, the involute, is entirely 
 beyond its reach, because its roller is the 
 logarithmic spiral, a transcendental curve 
 that can be reached only by the higher mathe- 
 matics. 
 
 No tooth curve, which, like the involute, 
 crosses the pitch line at any angle but a 
 right angle, can be traced by a point in a 
 simple curve. The tracing point must be 
 the pole of a spiral, and therefore the trac- 
 ing of such a curve is a mechanical impossi- 
 bility. A practicable rolled odontoid must 
 cross the pitch line at right angles. 
 
 To use the rolled curve theory as a base of 
 operations will confine the discussion to the 
 cycloidal tooth, for the involute can only be 
 reached by abandoning its true logarithmic 
 roller, and taking advantage of one of its 
 peculiar properties, and the segmental, 
 sinusoidal, parabolic, and pin tooth, as well 
 as most other available odontoids, cannot be 
 discussed at all. 
 
 33. MATHEMATICAL RELATION OP ODONTOID AND LINE OF ACTION. 
 
 In Fig. 26 the odontoid on the pitch line by the relations P T = p t = y, and T 8 
 p I is connected with the line of action I a, j t = x, where P 8 is the normal to the 
 
14 
 
 Mathematical Relation. 
 
 odontoid at the point P, T 8 is a tangent to 
 the pitch line at the intersection of the nor- 
 mal, and P TMsa normal to the tangent. 
 
 When any odontoid is given by its equa- 
 tion, that of the line of action can be found 
 by a process of differentiation, and when 
 the line of action is given by its equation, 
 that of the odontoid can be found by a 
 process of integration. 
 
 These processes, for the general case where 
 the pitch line is curved, are quite intricate, 
 Imt when the pitch line is a straight line, 
 they are simple, and may be worked as 
 follows. 
 
 To get the equation of -the line of action 
 from that of the given rack odontoid, ar- 
 Tange the equation of the odontoid in the 
 form x f(y), and put its differential co- 
 efficient - equal to . Thus, the equation 
 d y x 
 
 of the straight rack odontoid of the involute 
 system is y x tan. A, from which 
 <lx 1 y x 
 
 Tj ^ 't^TA ' "* =tST^i 1S 
 
 the equation of the straight line of action at 
 right angles to the odontoid. Again, the 
 equation of the cycloid being x = vi r. sin.- 1 
 x = ver. sin. -' 
 
 Fig. 2V. 
 
 and x a -\-y a 2ry is the equation of the 
 circular line of action. 
 
 To get the equation of the odontoid when 
 that of the line of action is given, arrange 
 the equation of the line of action in the form 
 
 =f(y) f put it equal to - - , and inte- 
 grate. T.hus, the equation of the straight 
 line of action being 
 
 x 
 
 lan. A ' 
 we have 
 
 y 1 _ dx 
 
 x tan. A ~ dy' 
 
 and y x tan. A is the equation of the 
 straight odontoid at right angles to the line 
 of action. Again, the equation of the circu- 
 lar line of action being x* -|- y 2 = 2ry, we 
 have 
 
 and x = 
 cycloidal odontoid. 
 
 34. THE ODONTOIDAL 
 
 The form <5f the odontoid to correspond to 
 & 'given line of action and a given pitch line 
 can be determined only by the integral cal- 
 culus (33), it evidently being impossible to 
 contrive a general graphical or algebraic 
 method. 
 
 But it can be directly drawn by an instru- 
 ment, the principle of which is analogous to 
 that of the well-known polar planimeter for 
 Integrating surfaces. 
 
 The bar jR, Fig. 27, moves at right angles 
 to the line of centers, and it moves the pitch 
 wheel A, with the same speed at the pitch 
 line. The bar G has a point p, that is 
 confined to move in the given line of action 
 Op W, and it is so guided that it always 
 passes through the pitch point 0. 
 
 The two bars bear upon each other by 
 friction, and we must suppose that there 
 5s no other friction to oppose the motion of 
 the bar C. 
 
 Odontoidal Integrate? 
 
 Then the pointy will trace out the odontoid 
 spzupon the pitch wheel A, or upon any 
 other pitch wheel B rolling with the bar R 
 on either side of it. 
 
THE SPUR. GEAR IN GENERAL. 
 
 35. THE CIRCULAR PITCH. 
 
 The distance a 0, Fig. 14, covered by each 
 tooth upon the pitch circle, is commonly 
 called the "circular pitch," and often the 
 "circumferential pitch." The term "pitch 
 arc" is the most appropriate but is not in 
 common use. 
 
 This was formerly the measurement by 
 which the size of the tooth was always 
 stated, a tooth being said to be of a certain 
 " pitch," and all of its other dimensions 
 being expressed in terms of that unit, but it 
 is fast being replaced, and should be entirely 
 replaced, by the more convenient "diametral 
 pitch " unit. 
 
 The circumference of a circle is measured 
 in terms of its diameter by means of an in- 
 commensurable fractional number 3.14159, 
 called TT (pi), and, therefore, if the tooth is 
 measured upon the arc of the circle by means 
 of the circular pitch, one of two inconveni- 
 ences must be tolerated. Either the pitch 
 must be an inconvenient fraction, or else the 
 pitch diameter must be as inconvenient, for 
 the gear cannot have a fractional number of 
 teeth. The fractional calculations are so 
 clumsy that a table of pitch diameters cor- 
 responding to given numbers of teeth should 
 be used, and errors in the laying out of the 
 work are of constant occurrence. 
 
 Again, outside of the liability of error in 
 making calculations, the circular pitch sys- 
 tem is a constant source of error in the hands 
 of lazy or incompetent draftsmen or work- 
 men, for there is a constant temptation, 
 often yielded to, to force the clumsy figures 
 a little to produce some desired result. For 
 example, a millwright has to make a gear of 
 fourteen inches pitch diameter with fourteen 
 teeth. He finds by the usual computation 
 that the circular pitch is 3.14 inches, and, as 
 his odontograph has a table for three-inch 
 pitch, he uses that with the remark that it is 
 "near enough," laying the blame on the 
 odontograph or on the iron founder if the 
 resulting gear roars. His next order is for a 
 
 gear of one-inch pitch to match others in 
 use, and to be fourteen and a half inches 
 diameter. The circumference of the pitch 
 line is 45.53 inches, and he has his choice be- 
 tween 45 and 46 teeth, both wrong. Per- 
 haps the most frequent cause of error is 
 that the workman is apt to apply a rule 
 directly to the teeth of a gear he is about to 
 repair or match, to get the circular pitch, 
 and the result is more likely to be wrong than 
 right. 
 
 The best plan when using this unit is to 
 get convenient pitch diameters and let the 
 pitch come as it will, provided that gears 
 that work together are*of the same pitch, and 
 that is simply a roundabout way of using the 
 diametral pitch unit. 
 
 When tbe circular pitch must be used the 
 following table will greatly assist the work 
 and save calculation. For example, the 
 pitch diameter of a gear of three-quarter-inch 
 pitch and 37 teeth is three-quarters the tabu- 
 lar number 11.78, or 8.84 inches. 
 
 PITCH DIAMETERS. 
 FOR ON/5 INCH CIRCULAR PITCH. 
 
 FOR ANY OTHER PITCH MULTIPLY BY THAT PITCH. 
 
 T. P. D. 
 
 T. P. D. 
 
 T. P. D. 
 
 T. P. D. 
 
 10 
 
 8.18 
 
 33 
 
 10.50 
 
 58 
 
 17.83 
 
 79 
 
 25.15 
 
 11 
 
 3.50 
 
 34 
 
 10.82 
 
 57 
 
 18.14 
 
 80 
 
 25.47 
 
 12 
 
 3 H2 
 
 35 
 
 11.14 
 
 58 
 
 18.46 
 
 81 
 
 25.79 
 
 13 
 
 4.14 
 
 36 
 
 11.46 
 
 59 
 
 18.78 
 
 82 
 
 26.10 
 
 14 
 
 4.46 
 
 37 
 
 11.78 
 
 60 
 
 19.10 
 
 83 
 
 26.42 
 
 15 
 
 4.78 
 
 38 
 
 12.10 
 
 61 
 
 19.42 
 
 84 
 
 26.74 
 
 16 
 
 5 09 
 
 39 
 
 12.42 
 
 62 
 
 19.74 
 
 85 
 
 27.06 
 
 17 
 
 5.41 
 
 40 
 
 12.73 
 
 63 
 
 20.06 
 
 86 
 
 27.38 
 
 18 
 
 5.73 
 
 41 
 
 13.05 
 
 64 
 
 20.37 
 
 87 
 
 27 70 
 
 19 
 
 6.05 
 
 42 
 
 13.37 
 
 65 
 
 20.69 
 
 88 
 
 28.01 
 
 20 
 
 6.37 
 
 43 
 
 13.69 
 
 66 
 
 21.01 
 
 89 
 
 2.S.33 
 
 21 
 
 6.69 
 
 44 
 
 14.00 
 
 67 
 
 21.33 
 
 90 
 
 28.65 
 
 22 
 
 7.00 
 
 45 
 
 14.33 
 
 68 
 
 21.65 
 
 91 
 
 28 97 
 
 m 
 
 7.32 
 
 46 
 
 14 64 
 
 69 
 
 21.97 
 
 92 
 
 29.29 
 
 24 
 
 7.64 
 
 47 
 
 14.96 
 
 70 
 
 22.28 
 
 93 
 
 29.60 
 
 25 
 
 7.96 
 
 48 
 
 15.28 
 
 71 
 
 22 60 
 
 94 
 
 29.92 
 
 26 
 
 8.28 
 
 49 
 
 15.60 
 
 72 
 
 22.92 
 
 95 
 
 30.24 
 
 27 
 
 8.60 
 
 50 
 
 15.92 
 
 73 
 
 23.24 
 
 96 
 
 30.56 
 
 28 
 
 8.91 
 
 51 
 
 16.24 
 
 74 
 
 23.56 
 
 97 
 
 30.88 
 
 29 
 
 9.23 
 
 52 16.55 
 
 75 
 
 23.88 
 
 98 
 
 31.20 
 
 30 
 
 9.55 
 
 53 16.87 
 
 76 
 
 24.19 
 
 99 
 
 31.52 
 
 31 
 
 0.87 
 
 54 17.19 
 
 77 
 
 24.51 
 
 100 
 
 31.83 
 
 32 
 
 1 
 
 10.19 
 
 55 j 17.51 
 
 78 
 
 24. 83 
 
 
 
 36. THE DIAMETRAL PITCH. 
 
 This is not a measurement, but a ratio 
 or proportion. It is the number of teeth in 
 the gear divided by the pitch diameter of the 
 
 gear. Thus, a gear of 48 teeth and 12 inches 
 pitch diameter is of 4 pitch. The advantages 
 of the diametral pitch unit are so apparent 
 
1C 
 
 Pitches and Addendum. 
 
 that it is fast displacing the circular pitch 
 unit, and has almost entirely displaced it for 
 cut gearing. It is so simple that a table of 
 pitch diameters is entirely useless, although 
 such useless tables have been published. 
 
 The diametral pitch is sometimes defined as 
 the number of teeth in a gear of one inch 
 diameter. It is a common, but bad practice, 
 to designate diametral pitches by numbers, 
 as No. 4, No. 16, etc. 
 
 
 Diametral 
 
 Circular 
 
 Circular 
 
 Diametral 
 
 37. RELATION OF PITCH UNITS. 
 
 Pitch. 
 
 Pitch. 
 
 Pitch. 
 
 Pitch. 
 
 The product of the circular pitch by the 
 
 2 
 
 1.571 inch 
 
 2 
 
 1-571 
 
 diametral pitch is the constant number 
 
 
 1.396. 
 1.257 
 
 
 1 IS76 
 1 795 
 
 3.1416, so that if one is given the other is 
 
 2% 
 
 1J42 
 
 i4 
 
 
 easily calculated. 
 
 3 
 
 1.047 
 .898 
 
 ? 
 
 2 094 
 2 185 
 
 The following tables of equivalent pitches 
 
 4 
 
 5 
 
 .785 
 
 K 
 
 2 2f<5 
 2.394 
 
 will be convenient in this connection. 
 
 6 
 
 '.524 
 
 y. 
 
 2^513 
 
 
 7 
 
 .449 
 
 3 ff 
 
 2.646 
 
 
 8 
 
 .393 
 
 IT& 
 
 2.793 
 
 
 9 
 
 .349 
 
 IT'S 
 
 2.957 
 
 
 10 
 
 .314 
 
 1 
 
 3.142 
 
 . 
 
 11 
 
 .288 
 
 15 
 
 3 351 
 
 
 12 
 
 .262 
 
 4? 
 
 3 590 
 
 38. ACTUAL SIZES. 
 
 14 
 
 !224 
 
 n 
 
 3 '.867 
 
 
 16 
 
 .196 
 
 A^ 
 
 4.189 
 
 Figs. 28 and 29 show the actual sizes of 
 
 18 
 
 .175 
 
 \b 
 
 4.570 
 
 standard teeth of the usual diametral pitches, 
 
 20 
 22 
 
 .157 
 .143 
 
 A 
 
 5.027 
 5. 88 
 
 and give a better idea of the actual teeth than 
 
 24 
 26 
 
 .131 
 .121 
 
 8 
 
 6.283 
 
 7 181 
 
 can be given by any possible description. 
 
 28 
 
 .112 
 
 5 
 
 8 378 
 
 They are printed from cut teeth, and may be 
 
 30 
 32 
 
 .105 
 
 .098 
 
 il 
 
 10 Of>3 
 12.566 
 
 depended upon as accurate. 
 
 36 
 40 
 
 .087 
 .079 
 
 ft 
 
 16.755 
 25.133 
 
 
 48 
 
 .065 
 
 T 1 5 
 
 50.266 
 
 
 
 
 
 
 39. ADDENDUM AND DEDENDUM. 
 
 The tooth is limited in length by the circle j 
 a I, Fig. 30, called the addendum line, and 
 drawn outside the pitch line at a given 
 distance, called the addendum. Its depth is 
 also limited by aline r I, called the dedendum 
 or root line, drawn at a given distance inside 
 of the pitch line. 
 
 The addendum and the dedendum are 
 both arbitrary distances, but, for convenience 
 in computation, they are nxed at simple 
 fractions of the unit of pitch that is in use. 
 When the circular pitch is used the ad> 
 dendum is one-third of the circular pitch. 
 
 When the diametral pitch unit is used 
 the addendum is one divided by the pitch. 
 
 It is customary to make the addendum and 
 the dedeodum the same, except in certain 
 cases where some special requirement is to 
 be satisfied. 
 
 Fig. 30. 
 
 #1 
 
 Addenda 
 Clearance 
 Backlash 
 
Actual Sizes. 
 
 17 
 
18 
 
 Actual Sizes. 
 
 2 Pitch. 
 
 2 I Pitch. 
 
 3 Pitch. 
 
 Fig. 29. 
 
Items of Construction. 
 
 19 
 
 40. THE CLEARANCE. 
 
 To allow for the inevitable inaccuracies of 
 workmanship, especially on cast gearing, it 
 is customary to carry the tooth space slightly 
 below the root line to the clearance line c I, 
 Fig. 30. 
 
 The clearance, or distance of the clearance 
 line inside of the root line, is arbitrary, but it 
 is convenient and customary to make it one> 
 eighth of the addendum. 
 
 41. THE BACK-LASH. 
 
 When rough wooden cogs or cast teeth are 
 used, the irregularities of the surface, and 
 inaccuracies of the shape and spacing of 
 the teeth, require that they should not pre- 
 tend to fit closely, but that they should clear 
 each other by an amount b, Fig. 30, called 
 the back-lash. 
 
 The amount of the back-lash is arbitrary, 
 
 but it is a good plan to make it about equal 
 to the clearance, one-eighth of the addendum. 
 Skillfully made teeth will require less, 
 back-lash than roughly shaped teeth, and". 
 properly cut teeth should require no back- 
 lash at all. Involute teeth require less back- 
 lash than cycloidal teeth. 
 
 42. THE STANDARD TOOTH. 
 
 The tooth must be composed of odontoids, 
 preferably of odontoids of which the proper- 
 ties are well known, and an advantage is 
 gained if it is still further confined to a par- 
 ticular value of that odontoid. If the teeth 
 are to be drawn by an odontograph some 
 standard must be fixed upon, since the 
 
 method will cover but one proportion of tooth.. 
 For example, the standard involute tooth 
 is that having its line of action inclined at an 
 angle of obliquity of fifteen degrees. For 
 the cycloidal system the standard agreed upon 
 is the tooth having radial flanks on a gear of 
 twelve teeth. 
 
 43 . ODONTOGR APHS . 
 
 The construction of the tooth is generally 
 not simply accomplished by graphical means, 
 as it is generally required to find points in 
 the curve and then find centers for circular 
 arcs that will approximate to the curve thus 
 laid out. 
 
 It is sometimes attempted to construct the 
 curve by some handy method or empirical 
 
 rule, but such methods are generally worth- 
 less. 
 
 An odontograph is a method or an instru- 
 ment for simplifying the construction of the 
 curve, generally by finding centers for ap- 
 proximating circular arcs without first find- 
 ing points on the curve, and those in use will 
 be described. 
 
 44. THE 
 
 When the teeth are laid out by theory 
 there will be a portion of the tooth space at 
 the bottom that is never occupied by the 
 mating tooth. Fig. 31 shows a ten-toothed 
 pinion tooth and space with a rack tooth in 
 three of its positions in it, showing the un- 
 used portion by the heavy dotted line. 
 
 If this unused space is filled in by a 
 " fillet "/ the tooth will be strengthened just 
 where it needs it the most, at the root. 
 
 The fillet is dependent on the mating tooth, 
 and is therefore not a fixed feature of the 
 tooth. If a gear is to work in an inter- 
 changeable set, it may at some time work 
 with a rack, and therefore its fillet should be 
 fitted to the rack ; but if it is to work only 
 
 FILLET. 
 
 with some one gear it may be fitted to that. 
 The light dotted line shows the fillet that 
 would be adapted to a ten-toothed mate. 
 The fillet to match an internal gear tooth 
 would be even smaller than that made by the 
 rack. 
 
 31. 
 
 The fillet 
 
20 
 
 Equidistant Series. 
 
 When the tooth is formed by the molding 
 process of (27), or by the equivalent planing 
 process of (28), the fillet will be correctly 
 formed by the shaping tool, but not so when 
 the linear process of (29) is used. When the 
 tooth is drawn by theory or by an odonto- 
 graph the fillet must be drawn in, and can be 
 
 most easily determined by making a mating 
 tooth of paper, and trying it in several posi- 
 tions in the tooth space, as in the figure. 
 
 Except on gears of very few teeth the 
 strength gained will not warrant the trouble 
 of constructing the fillet. 
 
 45. THE EQUIDISTANT SERIES. 
 
 When arranging an odontograph for 
 drafting teeth, or a set of cutters for cutting 
 them, we must make one sizing value do 
 duty for an interval of several teeth, for it is 
 impracticable to use different values for two 
 or three hundred different numbers of teeth. 
 The object of the equidistant series is to so 
 place these intervals that the necessary errors 
 are evenly distributed, each sizing value 
 being made to do duty for several numbers 
 each way from the number to which it is 
 fitted, and being no more inaccurate than any 
 other for the extreme numbers that it is 
 forced to cover. 
 
 This series is readily computed for any 
 case that may arise, and with a degree of ac- 
 curacy that is well within the requirements 
 ^>f practice; by the formula 
 
 . as 
 , *-. + - 
 
 in which a is the first and z is the last tooth 
 of the interchangeable series to be covered; 
 n is the number of intervals in the series, and 
 * is the number in the series of any interval 
 of which the last tooth t is required. 
 
 For example, it is required to compute the 
 series here used for the cycloidal odonto- 
 graph, having twelve tabular numbers to 
 cover from twelve teeth to a rack. 
 
 Putting a = 12, z = infinity, and n = 12, 
 the formula becomes 
 
 12 X 12 12 X 12 144 
 
 t = 
 
 12 s-f-0 ~ 12 
 
 and then, by putting successively equal to 
 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12, we get 
 the series of last teeth, 13 r ' T , 14|, 16, 18, 20|, 
 24, 28f, 36, 48, 72, 144, and infinity. These 
 give the required equidistant series of inter- 
 vals. 
 
 12 
 
 13 to 14, 
 15 to 16, 
 17 to 18, 
 19 to 21, 
 22 to 24, 
 
 25 to 29 
 30 to 36, 
 37 to 48, 
 49 to 72, 
 73 to 144, 
 145 to a rack ; 
 
 and the method is as easily applied to any 
 other practical example. 
 
 This formula and method is independent of 
 the form and of the length of the tooth, and 
 therefore is applicable to all systems under 
 all circumstances. This is proper and con- 
 venient, for these elements can be eliminated 
 without vitiating the results or destroying the 
 "equidistant" characteristic of the series. 
 The formula is an approximation based upon 
 an assumption, but nothing more convenient 
 or more accurate has so far been devised by 
 laboriously considering all the petty elements 
 involved. 
 
 The sizing value, or number for which the 
 tabular number is computed, or the cutter is 
 accurately shaped, can best be placed, not at 
 the center of the interval, but by considering 
 the interval as a small series of two intervals, 
 and adopting the intermediate value. The 
 sizing value for the interval from c to d is 
 
 given by the formula 
 
 2 cd 
 
 Thus, the sizing value for the interval 
 from 37 to 48 teeth should be 41.8, and that 
 for the interval from 145 to a rack should be 
 290. 
 
 It is sometimes the practice to size the cut- 
 ter for the lowest number in its interval, on 
 the ground that a tooth that is considerably 
 too much curved is better than one that is 
 even a little too flat. This makes the last 
 tooth of the interval much more inaccurate 
 than if the medium number was used. 
 
Friction of Approach. 
 
 46. THE HUNTING COG. 
 
 It is customary to make a pair of cast gears 
 with incommensurable numbers of teeth so 
 that each tooth of each gear will work with 
 all the teeth of the other gear. If a pair of 
 equal gears have twenty teeth each, each 
 tooth will work with the same mating tooth 
 all the time; but if one gear has twenty and 
 the other twenty-one teeth, or any two num- 
 bers not having a common divisor, each tooth 
 will work with all the mating teeth one after 
 the other. 
 
 The object is to secure an even wearing 
 action; each tooth will have to work with 
 many other teeth, and the supposition is that 
 
 all the teeth will eventually and mysteriously 
 be worn to some indefinite but true shape. 
 
 It would seem to be the better practice to 
 have each tooth work with as few teeth as 
 possible, for if it is out of shape it will dam- 
 age all teeth that it works with, and the 
 damage should be confined within as narrow 
 limits as possible. If a bad tooth works with 
 a good one it will ruin it, and if it works 
 with a dozen it will ruin all of them. It is 
 the better plan to have all the teeth as near 
 perfect as possible, and to correct all evident 
 imperfections as soon as discovered. 
 
 47. THE MORTISE WHEEL. 
 
 Another venerable relic of the last century 
 is the "mortise" gear, Fig. 32, having 
 wooden teeth set in a cored rim, in which 
 they are driven and keyed. 
 
 Where a gear is subjected to sudden strains 
 and great shocks, the mortise wheel is better, 
 and works with less noise than a poor cast 
 gear, and will carry as much as or more 
 power at a high speed with a greater dura- 
 bility. But in no case is it the equal of a 
 properly cut gear, while its cost is about as 
 great. 
 
 In times when large gears could not be cut, 
 and when the cast tooth was not even ap- 
 proximately of the proper shape, the mortise 
 wheel had its place, but now that the large 
 cut gear can be obtained the mortise gear 
 should be dropped and forgotten. 
 
 Mortise wheel 
 
 Fig. 32. 
 
 48. 
 
 FRICTION OF APPROACH. 
 
 When the point of action between two 
 teeth is approaching the pitch point, that is, 
 when the action is approaching, the friction 
 between the two tooth surfaces is greater than 
 when the action is receding. This extra fric- 
 tion is always present, but is most trouble- 
 some when the surfaces are very rough, as on 
 cast teeth, giving little trouble when the teeth 
 are properly shaped and well cut. When the 
 roller pin gear (93) is used, the friction 
 between the teeth is rolling friction, and is 
 no greater on the approach than on the recess. 
 
 tion of 
 roach 
 
 rig. 33. 
 
22 
 
 Efficiency. 
 
 The difference in the friction is probably 
 due to the difference in the direction of the 
 pressure between the small inequalities to 
 which all friction is due. When the gear D, 
 Fig. 33, is the driver, the action between the 
 teeth is receding, and the inequalities lift over 
 each other easily, while if F is the driver, 
 the action is approaching, and the inequalities 
 tend to jam together. 
 
 In the exaggerated case illustrated, it is plain 
 that the teeth are so locked together that ap- 
 proaching action is impossible, while it is 
 equally plain that motion in the other direc- 
 tion is easy. The same action takes place in 
 a lesser degree with the small inequalities of 
 ordinary rough surfaces. 
 
 The action of the common friction pawl r 
 which works freely in one direction and jam& 
 hard in the other, is upon the same principle. 
 A weight may be easily dragged over a rough 
 surface that it could not be pushed over by a 
 force that is not parallel to the surface. 
 
 The extra friction of approaching action 
 can be avoided by giving the driver the long- 
 est face. When the driver has faces only, 
 and the follower has only flanks, the action is 
 particularly smooth. 
 
 Teeth that are subject to excessive maxi- 
 mum obliquity, such as cycloidal teeth, should 
 not be selected for rough cast gearing, for it 
 is the maximum rather than the average obli- 
 quity that has the greatest influence. 
 
 49. EFFICIENCY OF GEAR TEETH. 
 
 Much has been written, but very little has 
 been done to determine the efficiency of the 
 teeth of gearing in the transmission of power, 
 and therefore but little of a definite nature 
 can be said. The question is mostly a prac- 
 tical one, and should be settled by experi- 
 ment rather than by analysis. 
 
 The only known experiments upon the fric- 
 tion of spur gear teeth are the Sellers experi- 
 ments, more fully detailed in (112), and but. 
 one of these relates to the spur gear. From 
 that one it is known that a gear of twelve 
 teeth, two pitch, working in a gear of thirty- 
 nine teeth, has an efficiency varying from 
 ninety per centum at a slow speed to ninety- 
 nine per centum at a high speed. That is, 
 an average of five per centum of the power 
 received is wasted by friction at the teeth and 
 shaft bearings. This result is probably a 
 close approximation to that for any ordinary 
 practical case. 
 
 Although theory can do nothing to de- 
 cide such a question as this, it can do much 
 to indicate probable results. 
 
 If a pair of involute teeth, for example, 
 move over a certain distance, w, either way 
 from the pitch point, the distance being mea- 
 sured on the pitch line, they will do work that 
 is theoretically determined by the formula : 
 
 / P k h 
 work done = - . -= =- w 9 
 
 & K fl 
 
 in which / is the coeflBcient of friction, P is 
 
 the pressure, and k and h are the pitch radii 
 of the gears. The positive sign is to be used 
 for gears in external, and the negative sign, 
 for those in internal contact. 
 
 The loss by friction, as shown by the for- 
 mula, decreases directly as the diameters in- 
 crease, the proportion of the diameters being 
 constant. 
 
 The loss increases rapidly with the distance 
 of the point of action from the pitch point 
 When the contact is at the pitch point the 
 teeth do not slide on each other, and there is 
 no loss, but away from that point the loss is 
 as the square of the distance in this case, and 
 in a still greater proportion in the case of the 
 cycloidal tooth. Therefore a short arc of 
 action tends to improve the efficiency. 
 
 It has been satisfactorily determined that 
 the loss is greater during the approaching 
 than during the receding action. This is not 
 shown by the formula, but it may be laid to 
 a variation in the coefficient /. 
 
 The formula shows that the loss is inde- 
 pendent of the width or face of the gear, 
 and therefore strength can be increased by 
 widening the face, without increasing the 
 friction. 
 
 If the work of internal gearing is com- 
 pared with that of external gearing of the 
 same sizes, the losses are in the proportion, 
 k h 
 k-\-h' 
 
Strength. 
 
 23 
 
 BO that the internal gear is much the more 
 economical, particularly when the gear and 
 pinion are nearly of the same size. If the 
 gear is twice the size of the pinion the loss 
 is but one-third of the loss when both gears 
 are external. 
 
 Small improvement can DC effected, by put- 
 ting a small pinion inside rather than outside 
 of a large gear. A six-inch pinion working 
 with a six-foot gear has but 1.18 times the 
 loss by the same gears, when the gear is in- 
 ternal. 
 
 Theoretical efficiency is discussed at great 
 length in the Journal of the Franklin Insti- 
 tute, for May, 1887: Also by Reuleaux, and 
 again by Lanza, in the Transactions of the 
 
 American Society of Mechanical Engineers 
 for 1887, and the discussion has been carried 
 far enough. 
 
 A series of experiments with gear teeth oi 
 various sizes and forms, of various metals, 
 would add greatly to our knowledge of this 
 important matter. 
 
 A true determination of the efficiency of 
 the rough cast gear, as compared with that of 
 the cut gear, would tend to discourage the 
 use of the former for the transmission of 
 power, for experiment would undoubtedly 
 show that the power wasted by the cast gear 
 would soon pay the difference in cost of the 
 better article. 
 
 50. STRENGTH OF A TOOTH. 
 
 The strength of a tooth is the still load it will 
 carry, suspended from its point, and is to be 
 carefully distinguished from the horse-power, 
 or the load the gear will carry in motion. 
 
 The strength of a substance is not a fixed 
 element, but will vary with different samples, 
 and with the same sample under different 
 circumstances ; allowance must be made for 
 the amount of service the sample has seen, 
 concealed defects must be provided against, 
 and therefore nothing but an actual test will 
 surely determine its character. 
 
 Although no possible rule can be depended 
 upon, the ultimate or breaking strength of a 
 standard cast-iron tooth, having an addendum 
 about equal to a third of the circular pitch, 
 will average about three thousand five hun- 
 dred pounds multiplied by the face of the 
 gear and again by the circular pitch, both 
 in inches. 
 
 But a tooth should never be forced up to 
 its ultimate strength, and the best practice is 
 to give it only about one-tenth of the load it 
 might possibly bear, so that the following 
 rule should be used : Multiply three hundred 
 and fifty pounds by the face of the gear, and 
 again by the circular pitch, both in inches, 
 and the product will be the safe working 
 load of one tooth. 
 
 Example : A cast-iron gear of one inch 
 pitch, and two inches face, will safely lift 
 350 X 2 x 1 = 700 pounds, although it 
 would probably lift 7,000 pounds. 
 
 When there are two teeth always in work- 
 ing contact, it is safe to allow double the 
 load, but care must be taken that both teeth 
 are always in full contact. 
 
 A hard wood mortised cog has about one- 
 third of the strength of a cast-iron tooth : 
 steel has double the strength ; wrought-iron 
 is not quite as strong. 
 
 A small pinion generally has teeth that are 
 weak at the roots, and then it will increase 
 the strength to shroud the gear up to its 
 pitch line, but shrouding will not strengthen 
 a tooth that spreads towards its base, like an 
 involute tooth, and when the face of the 
 gear is wide compared with the length of the 
 tooth the shroud is of /ittle assistance. 
 
 It does not increase the strength of a tooth 
 to double its pitch, for when the pitch is 
 increased the length is also increased, and the 
 strength is still in direct proportion to the 
 circular pitch, wnile the increase has reduced 
 the number of teeth fu contact at a time. 
 
 Cut gears and cast gears are about equal 
 as to actual strength, with the advantages in 
 favor of the cut gear, that hidden d 3fects are 
 likely to be discovered, and that it is not as 
 liable to undue strains on account of defective 
 shape. 
 
 The rules for strength must not be used for 
 gears running at any considerable speed, for 
 they are intended only for slow service, as in 
 cranes, heavy elevators, power punches, etc, 
 
J-f or se- Power. 
 
 f 
 
 51. HORSE-POWER OP CAST GEARS. 
 
 The horse-po\ver of a gear is the amount 
 of power it may be depended upon to carry 
 in continual service. 
 
 It is very well settled that continual strains 
 and impact will change the nature of the 
 metal, rendering it more brittle, so that a 
 tooth that is perfectly reliable when new 
 may be worthless when it has seen some years 
 of service. This cause of deterioration is 
 particularly potent in the case of rough cast 
 teeth, for they can only approximate to 
 true shape required to transmit a uniform 
 speed, and the continual impact from shocks 
 and rapid variations in the power carried 
 must and does destroy the strength of the 
 metal. 
 
 There are about as many rules for com- 
 puting the power of a gear as there are 
 manufacturers of gears, each foundryman 
 having a rule, the only good one, which he 
 has found in some book, and with which he 
 will figure the power down to so many 
 horses and hundredths of a horse as con- 
 fidently as he will count the teeth or weigh 
 the casting. 
 
 Even among the standard writers on en- 
 gineering subjects the agreement is no bet- 
 ter, as shown by Cooper's collection of 
 twenty-four rules from many different wri- 
 ters, applied to the single case of a five-foot 
 gear. See the "Journal of the Franklin 
 Institute" for July, 1879. For the single 
 case over twenty different results were ob- 
 tained, ranging from forty-six to three- 
 hundred horse-power, and proving conclu- 
 sively that the exact object sought is not to be 
 obtained by calculation. 
 
 This variety is very convenient, for it is 
 always possible to fit a desired power to 
 a given gear, and if a badly designed gear 
 should break, it is a simple matter to find a 
 rule to prove that it was just right, and must 
 have met with some accident. 
 
 Although no rule can be called reliable, 
 the one that appears to be the best is that 
 given by Box, in his Treatise on Mill Gear- 
 ing. Box's rule, which is based on many 
 actual cases, and which gives among the 
 lowest, and therefore the safest results, is by 
 the formula: 
 
 12 c 2 / *J~dn 
 
 Horse-power of a cast gear = THA^ 
 
 J ,UUO 
 
 in which c is the circular pitch, /is the face, 
 the M is the diameter, all in inches, and n is the 
 number of revolutions per minute. 
 
 Example : A gear of two feet diameter, 
 four inches face, two inches pitch, running 
 at one hundred revolutions per minute, will 
 transmit 
 
 12 X 2 X 2:x 4 X A/ 24 x 100 
 
 1,000 
 
 = 9.4 h. p. 
 
 For bevel gears, take the diameter and 
 pitch at the middle of the face. 
 
 It is perfectly allowable, although it is not 
 good practice, to depend upon the gear for 
 from three to six times-the calculated power, 
 if it is new, well made, and runs without 
 being subjected* to sudden shocks and varia- 
 tions of load. 
 
 The influence of impact and continued 
 service will be appreciated when it is con- 
 sidered that the gear in the example, which 
 will carry 9.4 horse-power, will carry seventy 
 horse-power if impact is ignored, and the 
 ultimate strength of the metal is the only 
 dependence. 
 
 A mortise gear, with wooden cogs, will 
 carry as much as, or more than a rough cast- 
 iron gear will carry, although its strength is 
 much inferior. The elasticity of the wood 
 allows it to spring and stand a shock 
 that would break a more brittle tooth of 
 much greater strength. And, for the same 
 reason, a gear will last longer in a yielding 
 wooden frame than it will in a rigid iron 
 frame. 
 
 52. HORSE-POWER OP CUT GEARS. 
 
 We know a little, and have to guess the 
 rest, as to the power of a cast gear, but with 
 respect to that of a cut gear we are not as 
 well posted, for there are no experimental 
 
 data upon which a reliable rule can be 
 founded. 
 
 Admitting, as we must, that impact is 
 the chief cause of the deterioration of the 
 
The Involute Tooth. 
 
 cast gear, we are at liberty to assume that a 
 properly cut and smoothly running cut gear 
 
 ris much more reliable. 
 
 No definite rule is possible, but we can 
 safely assume that a cut gear will carry at 
 least three times as much power as can be 
 trusted to a cast gear of the same size. 
 i The great reliance of those who claim that 
 a cast gear is superior to a cut gear is upon 
 
 , the hard scale with which the cast tooth is 
 covered. This scale is not over one-hun- 
 dredth of an inch thick, is rapidly worn 
 
 away, and is of no account whatever. From 
 that point of view it is difficult to explain 
 why a wooden tooth will outwear an iron one, 
 I although it is softer than the softest cut iron. 
 Assuming that a cut gear is about three 
 ' times as reliable as a cast gear, we can com- 
 pute its power by the formula : 
 
 d n 
 
 c'f 
 I Horse-power of a cut gear = - 
 
 in which c is the circular pitch, / is the face, 
 and d is the pitch diameter, all in inches, and 
 n is the number of revolutions per minute. 
 
 3. THE IKVOLUTE SYSTEM. 
 
 53. THE INVOLUTE TOOTH. 
 
 The simplest and best tooth curve, theo- 
 retic-ally, as well as the one in greatest prac- 
 tical use for cut gearing, is the involute. 
 
 The involute tooth system is based on the 
 
 ' straight rack odontoid, (31) and Fig. 21, and 
 
 \ it is illustrated by Fig. 34. If the four odon- 
 
 ! toids of the rack outline are equally inclined 
 
 i to the pitch line, the resulting tooth system 
 
 will be completely interchangeable; but if, 
 
 as in Fig. 35, the face and flank are inclined 
 
 at different angles of obliquity, T S K and 
 
 T S K', the system is not interchangeable, 
 
 although otherwise perfect. 
 
 The rack odontoid cannot have a corner or 
 change of direction anywhere except at the 
 pitch line, without causing a break in the 
 line of action. 
 
 As the normals p q are parallel, the line of 
 action is a straight line W O W at right 
 angles to the rack odontoid. The inter- 
 changeable line of action is continued in a 
 straight line on both sides of the pitch line, 
 bui the non-interchangeable line changes di- 
 rection at that line. 
 
 In accordance with the universal custom 
 we will consider that the involute tooth is 
 'always interchangeable, having a single angle 
 ot obliquity. 
 
 Z/ic involute tooth 
 interchangeable 
 
 Fiff. 34. 
 
26 
 
 Involute Interference. 
 
 54. THE CUSP. 
 
 As a circle t c, Fig. 34, can always be drawn 
 tangent to the line of action at an interfer- 
 ence point i, from the center b of any pitch line 
 B, there will always be a cusp in the curve at 
 the point c (16), and at that point the working 
 part of the curve must stop. The working 
 part of the rack tooth must end at the limit 
 line i L through the interference point i. 
 
 The working curves of any two teeth that 
 work with each other must each end at the 
 line drawn through the interference point of 
 the other, Fig. 43, being limited by limit 
 lines 1 1 and L L. 
 
 r The second branch c m' of the curve is 
 equal to the first branch c m, but is re- 
 versed in direction. The second cusp is at 
 infinity, and therefore has no practical ex- 
 istence. 
 
 The tangent circle i c, through the inter- 
 ference point and the cusp, is called the 
 "base line." 
 
 It is customary to continue the flank of the 
 tooth inside the base line by a straight radial 
 line, as far as may be necessary to allow the 
 mating gear to pass. 
 
 55. INTERFERENCE. 
 
 When the point of the tooth is continued 
 beyond the limit line it will interfere with and 
 cut away a portion of the working curve of 
 the mating tooth. Fig. 36 shows a rack tooth 
 working with the tooth of a small pinion, and 
 cutting out its working curve. 
 
 This cut is not confined to the flank, but 
 extends across the pitch line into the face, as 
 shown by the line qmn. The rack tooth of 
 the figure will not work with the pinion tooth 
 unless it is cut off at the limit line 1 1 through 
 the interference point i. 
 
 The mathematical action still continues, 
 and the figure shows the rack tooth in. action 
 at k with the second branch of the curve. 
 
 Effect of Interference 
 
 . 36. 
 
 56. ADJUSTABILITY. 
 
 An interesting and in many cases a valua- 
 ble feature of the involute curve, and one 
 that is confined to it, is the fact that its posi- 
 tion as a whole with regard to the mating 
 curve is adjustable. 
 
 Two involutes, each with its base line, will 
 work together in perfect tooth contact when 
 they are moved with respect to each other, 
 as long as they touch at all. The lines of 
 action and the pitch lines will shift as the 
 curves are moved, and will accommodate 
 themselves to the varying position of the 
 base lines. 
 
 But this valuable feature of the involute 
 curve is not always available, and involute 
 gears are not, as commonly supposed, neces- 
 
 sarily adjustable, for the conditions are often 
 such that the teeth will fail to act when the 
 centers are moved, except within very narrow 
 limits. Care must be taken that the arc of 
 action is not so reduced by separating the 
 centers of the gears that it is less than the cir- 
 cular pitch, for the former arc is variable and 
 the latter is fixed. Care must also be taken 
 that the working curve is not pushed over the 
 limit line when the centers are drawn to- 
 gether. 
 
 In any limiting case, such as in Fig. 43, the 
 centers are not adjustable. The gears of the 
 standard set are either not adjustable at all 
 or are so within very narrow limits, on ac- 
 count of the correction for interference. 
 
Involute Construction. 
 
 27 
 
 57. CONSTRUCTING THE INYOLUTE BY POINTS. 
 
 The simple involute curve can be con- 
 structed by points by the general method of 
 (24), but it is much better to take advantage 
 of the property that it is an involute of its 
 base circle, and construct it by the rectifica- 
 tion of that circle. 
 
 As in Fig. 37 any convenient small dis- 
 tance A G is taken on the dividers, and the 
 points on the curve located by stepping 
 along the circle and its tangent from any 
 given point to any desired point. 
 
 This method is so accurate, if care is taken 
 to step accurately on the line, that the curve 
 seldom needs correction; but, when great ac- 
 curacy is required, correction can be applied 
 at the rate of one- thousandth of an inch to 
 the step, if the length of the step is regulated 
 by the diameter of the circle according to the 
 following table: 
 
 Diameter of Circle : 
 
 12345 678 9 10 11 12 
 Length of Step : 
 .17 .26 .37 .46 .53 .60 .67 .73 .76 .79 .82 .84 
 
 For example: If the circle of Fig. 37 is 
 
 Construction 
 points 
 
 Fig. 37. 
 
 four inches in diameter, and the dividers are 
 set to .46 inch, the true curve, A b' d', will be 
 outside of the constructed curve A b d by .002 
 inch at b and .005 inch at d. 
 
 From the table we can form the handy and 
 sufficiently accurate rule that the length of 
 the slep should be about one-tenth of the di- 
 ameter of the circle, for a correction of about 
 one-thousandth of an inch per step. 
 
 Having thus found several points of the in- 
 volute, we can draw it in by hand, or by con- 
 structing a template, or by finding centers 
 from which approximately accurate circular 
 arcs can be drawn. 
 
 58. THE STANDARD INVOLUTE TOOTH. 
 
 The tooth that is selected for general use, 
 and the one that is the best for all except a 
 few special cases and limiting cases, is the in- 
 terchangeable tooth having an angle of ob- 
 liquity of fifteen degrees, an addendum of 
 one-third the circular pitch, or one divided by 
 the diametral pitch, and a clearance of one- 
 -eighth of the addendum. 
 
 The standard to which involute cutters are 
 made is slightly different, having an angle of 
 14 28' 40", the sine of which is one-quarter, 
 .and a clearance of one-twentieth of the circu- 
 lar pitch. 
 
 If the obliquity is 15 the smallest possible 
 pair of equal gears have 11.72 teeth, and 
 
 therefore 12 is the smallest gear of the inter- 
 changeable set. 
 
 The base distance, the distance of the base 
 line inside of the pitch line, is about one-tifty- 
 ninth of the pitch diameter, and one-sixtieth 
 is a convenient fraction for practical use. 
 
 The limit points of the whole set must be 
 determined by that of the twelve-toothed 
 gear, for any gear of the set may be required 
 to work with that one, and the working curve 
 of each tooth must end at the point thus de- 
 termined. As the limit point is always in- 
 side of the addendum line there must always 
 be a false extension on the tooth, the point 
 being rounded over outside of the limit point. 
 
 59. THE INVOLUTE ODONTOGRAPH. 
 
 As the base line must always be drawn, it 
 is advisable, to save work, to locate the cen- 
 ters of the approximate circular arcs upon 
 that line. It is also necessary that the points 
 -of the teeth shall be rounded over, to avoid 
 
 interference. These requirements made it 
 impracticable to compute the positions of the 
 centers, and an empirical rule had to be adopt- 
 ed instead. 
 Teeth were carefully drawn by the stepping 
 
Ten arid Eleven Involute Teetli. 
 
 method of (57) on a very large scale, one- 
 quarter pitch, giving a tooth eight inches in 
 length. These teeth were corrected for inter- 
 ference by giving them epicycloidal points 
 that would clear the radial flanks of the 
 twelve-toothed pinion. 
 
 Then the proper centers on the base line 
 were determined by repeated trials, and tooth 
 curves obtained that would agree with the 
 true involute up to the limit point, and still 
 
 clear the corrected point. The odontograph 
 table is a record of these radii, which are be- 
 lieved to be as nearly coyrect as the given 
 : conditions will permit. 
 
 It was found that separate curves were, 
 required for face and flank up to thirty-six 
 teeth, but that one curve would answer for 
 teeth beyond. 
 
 It was found necessary to devise a separate 
 method for drafting the rack tooth. 
 
 60. TEN AND ELEVEN TEETH. 
 
 Theoretically the twelve-toothed pinion is 
 the smallest standard gear that will have an 
 arc of action as great as the circular pitch, 
 but ten and eleven teeth may be used with 
 an error that is not practically noticeable. 
 Fig. 38 shows a pair of ten-toothed gears in 
 
 action. They can be in correct action only 
 when the point of contact is between the two 
 interference points i and J, but they will be 
 in practical contact for a greater and suffi- 
 cient distance 
 
 Fig. 3S. 
 
 OdontoyrapJiic pair 
 
 61. A BAD RULE. 
 
 There is a simple and worthless rule for 
 involute teeth that deserves notice only be- 
 cause it is considerably in use. 
 
 It constructs the whole tooth curve, face and 
 flank, for all numbers of leeth, as a single 
 
 arc from a center on the base line, and with a 
 radius equal to one-quarter of the pitch radius, 
 Fig. 39. 
 
 This is wonderfully convenient, but the 
 convenience is purchased at the expense of 
 
The Involute Odontograph. 
 
 29 
 
 'ordinary accuracy, for the rule is not even 
 approximately correct. It is handy, and 
 nothing else. 
 
 Figs. 38 and 40 show the kind of teeth 
 that are constructed by this rule on gears of 
 ten and twelve teeth, where its error is the 
 greatest, and it is reasonable that the invo- 
 lute tooth should not be in great favor with 
 those who have been taught to draw it thus. 
 
 The error gradually decreases, until, for 
 more than thirty teeth, it is tolerably correct, 
 but it gives the rack with the straight, uncor- 
 rected working face that would interfere, as 
 shown at g, Fig. 40. 
 
 As it is tolerable only for thirty or more 
 
 teeth, and not good then, it may well be 
 dropped altogether. 
 
 bad rule 
 
 Figl 39. 
 
 62. USING THE INVOLUTE ODONTOGRAPH. 
 
 INVOLUTE ODONTOGRAPH. 
 STANDARD INTERCHANGEABLE TOOTH, CENTERS ON 
 
 (For Table of Pitch Diameters see 35.) b } j 
 
 
 Divide by the 
 Diametral Pitch. 
 
 Multiply by the 
 Circular Pitch. 
 
 Teeth. 
 
 Face 
 
 Flank 
 
 Face 
 
 Flank 
 
 
 Radius. 
 
 Radius. 
 
 Radius. 
 
 Radius. 
 
 10 
 
 2.28 
 
 .69- 
 
 .73 
 
 .22 
 
 11 
 
 2.40 
 
 .83 
 
 .76 
 
 .27 
 
 12 
 
 2 51 
 
 .96 
 
 .80 
 
 .31 
 
 13 
 
 2.62 
 
 .09 
 
 .83 
 
 .34 
 
 14 
 
 2.72 
 
 22 
 
 .87 
 
 .39 
 
 15 
 
 2,82 
 
 .34 
 
 .90 
 
 .43 
 
 16 
 
 2.92 
 
 .46 
 
 .93 
 
 .47 
 
 17 
 
 3.02 
 
 .58 
 
 .96 
 
 .50 
 
 18 
 
 3.12 
 
 .69 
 
 .99 
 
 .54 
 
 -19 
 
 3.22 
 
 .79 
 
 1.03 
 
 .57 
 
 20 
 
 3.32 
 
 .89 
 
 .06 
 
 .60 
 
 21 
 
 3.41 
 
 .98 
 
 1 09 
 
 .63 
 
 22 
 
 3.49 
 
 2.06 
 
 1 11 
 
 .66 
 
 23 
 
 3.57 
 
 2 15 
 
 1.13 
 
 .69 
 
 24 
 
 3.64 
 
 2.24 
 
 1.16 
 
 .71 
 
 25 
 
 3.71 
 
 2.33 
 
 1 18 
 
 .74 
 
 26 
 
 3.78 
 
 2.42 
 
 1 20 
 
 .77 
 
 27 
 
 3.85 
 
 2.50 
 
 .23 
 
 .80 
 
 28 
 
 3 92 
 
 2.59 
 
 25 
 
 .82 
 
 29 
 
 3 99 
 
 2.6T 
 
 .27 
 
 .85 
 
 80 
 
 4.06 
 
 2.76 
 
 .29 
 
 .88 
 
 81 
 
 4.13 
 
 2.85 
 
 .31 
 
 .91 
 
 82 
 
 4.20 
 
 2.93 
 
 .34 
 
 .93 
 
 33 
 
 4 27. 
 
 3 01 
 
 36 
 
 .96 
 
 34 
 
 4.33 
 
 3 09 
 
 .38 
 
 .99 
 
 35 
 
 4 89 
 
 3 16 
 
 .39 
 
 1.01 
 
 36 
 
 4.45 
 
 3 23 
 
 .41 
 
 1.03 
 
 8740 
 
 4.20 
 
 1.34 
 
 41-45 
 
 4.63 
 
 1.48 
 
 46-51 
 
 5 06 
 
 1.61 
 
 62-60 
 
 5 74 
 
 1.83 
 
 61-70 
 
 6 52 
 
 2.07 
 
 71-90 
 
 7.72 
 
 246 
 
 91120 
 
 9.78 
 
 3.11 
 
 121-180 
 
 13.38 
 
 4.26 
 
 
 - r= ^JL^A l-o 
 
 \ Fat, f?^ fJ ( .^)L_ 
 
30 
 
 The Involute Odontograph. 
 
 To draft the tooth lay off the pitch, ad- 
 dendum, root, and clearance lines, and space 
 the pitch line for the teeth, as in Fig. 40. 
 
 Draw the base line one-sixtieth of the pitch 
 diameter inside the pitch line. 
 
 Take the tabular face radius on the divid- 
 ers, after multiplying or dividing it as re- 
 quired by the table, and draw in all the faces 
 
 from the pitch line to the addendum line 
 from centers on the base line. 
 
 Set the dividers to the tabular flank radius, 
 and draw in all the flanks from the pitch line 
 to the base line. 
 
 Draw straight radial flanks from the base 
 line to the root line, and round them into the 
 clearance line. 
 
 Fig. 40. 
 
 \. SPECIAL RULE FOR THE RACK. 
 
 Draw the sides of the rack tooth, Fig. 40, 
 as straight lines inclined to the line of centers 
 c c at an angle of fifteen degrees, best 
 found by quartering the angle of sixty de- 
 grees, 
 
 Draw the outer half a b of the face, one- 
 
 quarter of the whole length of the tooth, 
 from a center on the pitch line, and with a 
 radius of 
 
 2.10 inches divided by the diametral pitch. 
 .67 inches multiplied by the circular pitch. 
 
 64. DRAFTING INTERNAL GEARS. 
 
 When the internal gear is to be drawn, the 
 odontograph should be used as if the gear 
 was an ordinary external gear. See Fig. 41. 
 
 But care must be taken that the tooth of 
 the gear is cut off at the limit line drawn 
 through the interference point * of the pin- 
 ion. The point of the tooth may be left off 
 
 altogether or rounded over to get the appear- 
 | ance of a long tooth. 
 
 The pinion tooth need not be carried in to 
 the usual root line, but, as in the figure, may 
 just clear the truncated tooth of the gear. 
 
 The curves of the internal tooth and of its 
 pinion may best be drawn in by points (57), 
 
The Involute Odontograph. 
 
 31 
 
 for the odomographic corrected tooth is not 
 as well adapted to the place as the true tooth, 
 and no correction for interference is needed 
 on the points of the pinion teeth or on the 
 flanks of those of the gear. 
 
 Care must be taken that the internal teeth 
 do not interfere by the point a striking the 
 point b, as they will if the pitch diameters 
 are too nearly of the same size. 
 
 Internal involutes 
 
 65. INVOLUTE GEARS FOR GIVEN OBLIQUITY AND ADDENDA. 
 
 When the obliquity and addenda, as well 
 as the pitch diameter and number of teeth in 
 a gear are given, as is generally the case, we 
 can proceed to draft the complete gear as 
 follows: 
 
 Draw the pitch line p I, Fig. 42, the ad- 
 dendum line a I, the root line r I, and the 
 clearance line c I, as given. Draw the line of 
 action I a at the given obliquity W Z = K. 
 Draw the base line b I tangent to the line of 
 action. Find the interference point * by bi- 
 secting the chord t. 
 
 Draw the involutes i a m and t" a" in", 
 and a a" will be the maximum arc of ac- 
 tion. 
 
 If the given arc of action a a' is not great- 
 er than the maximum arc, the pitch line is 
 to be spaced and the tooth curves drawn in 
 from the base line to the addendum line. 
 
 These tooth curves, when small, are best 
 
 drawn as circular arcs from centers on or 
 
 | near the b#se line, one center x for the flank 
 
 from the base line to the pitch line, and 
 
 another center y for the face from the pitch 
 
 line to the addendum line. One involute 
 
 i a m should be carefully constructed by 
 
 ! points, and then the required centers can be 
 
 1 found by trial. One center and arc will 
 
 ! often answer for the whole curve, and it is 
 
 only when great accuracy is required that 
 
 more than two centers will be necessary. 
 
 Continue the flanks of the teeth toward 
 center by straight radial lines, and round 
 j these lines into the clearance line. 
 
 If the interference point for the gear that 
 
 the gear being drawn is to work with is at I, 
 
 i within the addendum line, the limit line 1 1 
 
 i must be drawn through it, and the points of 
 
32 
 
 Involute Special Cases. 
 
 Fig. 42. 
 
 Given obliquity and addendum 
 
 the teeth outside of this limit must be slightly 
 rounded over, to avoid interference (55). 
 If a fillet / is desirable, to strengthen the 
 
 tooth, it can be drawn in by the method 
 
 of (44). 
 
 66. INVOLUTE GEARS FOR GIVEN NUM- ' 
 BERS OF TEETH. 
 
 When the numbers of teeth and the pitch 
 lines are the only given details, the shape and 
 action of the tooth depends upon the obli- 
 quity, and the action will fail if the angle is 
 too small. The principal object is to deter- 
 mine the least possible angle that is permitted 
 by the given pitch diameters and numbers of 
 teeth. 
 
 Draw the pitch lines P L and p I, Fig. 43, 
 lay off the given pitch arc, as a straight line 
 c d or C D, at right angles to the line of 
 centers, and draw the line C d or c D. Then 
 the required line pf action will be I a pass- 
 ing through at right angles to c D or G d. 
 The complete teeth can then be drawn in as 
 previously directed. 
 
 In this case, the obliquity WO Z being the 
 least possible, the limit lines and the adden- 
 dum lines must coincide, but the addenda 
 may be reduced by increasing the angle. 
 
 Fig. 43. 
 
 la 
 
 Given numbers of teeth 
 
Limiting Involute Teeth. 
 
 33 
 
 67. INVOLUTE GEARS FOR GIVEN OBLIQUITY. 
 
 When the pitch diameters and the obliquity 
 are the only given details, the lines C I and 
 c i, Fig. 43, drawn from the centers at right 
 angles to the line of action, will determine 
 the limit lines. The maximum arc of action 
 a a' may be found either by drawing the 
 involutes i a and la', or by continuing the 
 line G I to the line c d, and measuring the 
 
 required distance c d. Any arc of action less 
 than a a' may be used. 
 
 The drawings should always be made to 
 a scale of one tooth to the inch radius, so 
 that the pitch arc will be 2*. If the scale 
 is one tooth to the inch of diameter, the 
 pitch arc will be ?r. 
 
 68. INVOLUTE GEARS WITH LESS THAN FIVE EQUAL TEETH. 
 
 The "method of Fig. 43 and (66) will be 
 found to apply to any given numbers of 
 teeth not less than five, and to fail, if either 
 gear has but three or but four teeth. Any 
 external gear of five or more teeth will work 
 with any external gear of five or more teeth, 
 and with an internal gear of any number of 
 teeth unless stopped by internal interfer- 
 ence (64). 
 
 For example, if a pair having four and five 
 teeth, Fig. 44, is tried, the four-toothed 
 pinion will fail, because its tooth will come 
 to a point upon the line of action before it 
 has passed over the required pitch arc. The 
 difficulty cannot be remedied by increasing 
 the obliquity, for an angle that would allow 
 the four-toothed pinion to act would also 
 cause the five-toothed pinion to fail. 
 
 The practical limit is five teeth, but the 
 mathematical limit is the pair having the 
 fractional number 4. 62 teeth, Fig. 45. 
 
 The four-toothed pinion will not work with 
 any external gear, not even with a rack, but 
 it will work with an internal gear that has 
 ' about ten thousand teeth, and is practically a 
 rack. It will work with any internal gear 
 having less than ten thousand teeth, and Fig. 
 46 shows it working with an internal gear of 
 six teeth. Internal interference will prevent 
 its working with an internal gear of five 
 teeth. 
 
 The three-toothed pinion has no practical 
 action. It has a mathematical action with in- 
 ternal gears of 3.56 or less teeth, as shown 
 by Fig. 47, but as its limit is less than four, it 
 cannot work with any whole number. The 
 figure shows the interference at a. 
 
 The extreme mathematical limit may be 
 said to be the gear of 2.70 teeth, which has a 
 
 theoretical action with an internal gear of the 
 same size, coinciding with it. 
 
 4.69 X 4.62 limit 'for equal teeth 
 
 Fig. 45. 
 
 
 
Limiting Involute Teeth. 
 
 Fig 
 
 w 
 
 69. INVOLUTE GEARS WITH LESS THAN 
 FIVE UNEQUAL TEETH. 
 
 If we drop the condition that the pitch 
 line must be equally divided into tooth and 
 space arcs, we can make gears of three and of 
 four teeth work with external gears by the 
 method of (65). The failing case of Fig. 44 
 may be corrected by widening the failing 
 tooth until it acts, and narrowing the other 
 tooth to correspond, as shown in broken lines. 
 
 In this way a four-toothed pinion will 
 work with any number of teeth not less than 
 5.57, at which limit both gears have pointed 
 teeth, as in Fig. 48. 
 
 The three-toothed pinion will work with 
 any gear having 10.17 or more teeth. Fig. 49 
 shows the 3x10.17 limiting pair, and Fig. 50 
 shows the three-toothed pinion working 
 with an internal gear of five teeth. It will 
 not work with an internal gear of four teeth, 
 on account of internal interference, and there- 
 fore the combination shown by Fig. 50 may 
 be said to be the least possible symmetrical in- 
 volute pair. 
 
 A gear of 2.70 teeth will work with a rack, 
 but there seems to be no way to make a 
 pinion of two teeth work under any circum- 
 stances. 
 
 Fig. 47 
 
 5.57 teeth 
 
 4 teeth 
 
 Fig. 48. 
 
Limiting Involute Teeth. 
 
 35 
 
 Fig. 49. 
 
 3 X 10. 17 
 Unequal teetU 
 
 Fig. 50 
 
 70. THE MATHEMATICAL LIMITS. 
 
 The above results for low numbered pinions 
 can be obtained by graphical means, but that 
 method is not accurate enough to determine 
 the limits with great precision, and in any 
 case is tedious and laborious. 
 
 The mathematical process is not particu- 
 larly difficult, and consists in repeated trials 
 with given formulae. 
 
 To determine the obliquity at which a 
 limiting pinion will be pointed on the line of 
 action, for tooth equal to space, we use the 
 formulae : 
 
 . . 27T M 
 
 tan. h = 
 
 M+n 90 ~ 
 
 in which n is the given number of teeth in 
 the pointed gear, Fig. 51, M is the number 
 in the gear having the radius M, and h is 
 the angle c I. Knowing n, we assume a 
 value for M, and from that find a value 
 for h by means of the first formula. This 
 value of 7i, tried in the second formula, will 
 give an error. A second assumption for M 
 will give a second error, and if the two 
 errors are not too great a comparison will 
 nearly locate the true value of M. 
 
 Knowing n and M, we find the obliquity 
 from 
 
 tan. K = 
 
 M+n 
 
 pointed pinion. 
 
 Fig. 51. 
 
 In this way the following values were de- 
 termined : 
 
 n 
 
 M 
 
 K 
 
 2.695 
 
 1.26 
 
 57 49! 
 
 3. 
 
 1.51 
 
 54 20' 
 
 4. 
 
 2.86 
 
 42 29' 
 
 4.62 
 
 4.62 
 
 34 11' 
 
 5. 
 
 6.75 
 
 28 8' 
 
 5.58 
 
 00 
 
 
 
 Having determined the obliquity for the 
 pointed pinion, we can determine the least 
 number of teeth it will work with by means 
 of the following formula : 
 
 180 n 90 
 Angle B = -=? tan. K == -4- K 
 
 IT JV -/V 
 
 tan. B = -== tan. K-\- tan. K 
 in which N is the required least number. 
 
 
 
 
36 
 
 Limiting Involute Teeth. 
 
 In this way it was found that a gear of four 
 teeth will not work with a rack, but will work 
 with an internal gear having a number of 
 teeth not easily calculated with existing loga- 
 rithmic tables, but which is approximately 
 ten thousand. Also that a pinion of three j 
 teeth will not work with an internal gear 
 having more than 3.56 teeth. 
 
 For unequal teeth we can use the formulae, 
 
 2 TT n 
 
 tan. 7i = 
 
 tan. H = 
 
 in which N and n are the numbers of teeth 
 in the pair of pointed gears. By these form- 
 following results were determined, 
 
 N K 
 
 oo 
 
 n 
 
 2.695 
 3. 
 
 4. 
 4.62 
 
 10.17 
 5.57 
 4*62 
 
 25 27' 
 33 17 
 34 11 
 
 71. MINIMUM NUMBERS FOR UNSYMMETRICAL TEETH. 
 
 If we drop the condition that the fronts 
 and backs of the teeth shall be alike we have 
 an unimportant case that is similar to that 
 already studied, but much more intricate. 
 
 If we carry this case to its extreme, and 
 adopt single acting teeth, we have no mini- 
 mum numbers at all, for any two numbers 
 of teeth will then work together. Fig. 52 
 shows one tooth working with three teeth, 
 and any other combination can be obtained. 
 The minimum obliquity for a given pair is 
 obtained, as in (66), by laying off the known 
 pitch arc, G D, at right angles to G c, and 
 drawing the line of action at right angles 
 to the line D c. The obliquity is also given 
 by the formula : 
 
 2 7T 
 
 tan. K = -=-. , 
 
 N+n ' 
 
 in which n and N are the numbers of teeth. 
 When the obliquity is as great as is often 
 
 Fig. 52. 
 
 Unsymrnetrical 
 teeth 
 
 the case for very low numbers of teeth the 
 action may be impracticable on account of 
 the great friction of approach (48). The 
 gears of Fig. 52 will not drive each other on 
 the approach, unless the tooth surfaces are 
 very smooth,and the power transmitted is 
 almost nothing. 
 
 72. MINIMUM NUMBERS FOR GIVEN ARC OF RECESS. 
 
 It has generally been assumed, although 
 no good reason for the assumption has ever 
 been given, that the minimum numbers of 
 teeth occur when the tooth of one of the 
 gears, Fig. 53, is pointed at the interference 
 point /, and at the same time has passed 
 over an arc of recess a that is a given part 
 of the whole pitch arc a' a. 
 
 The solution is simple enough, graphically 
 by repeated trials, or by a formula that can 
 be applied directly without the usual process 
 by trial and error. 
 
 But, as involute teeth have a uniform ob- 
 liquity, there is no necessity for assuming 
 
 Fig. 
 
 definite arc of recess, and the condition on 
 
Involute Efficiency. 
 
 37 
 
 which the problem is based is unwarranted. I spur gears, in either external or internal con- 
 No real limit is reached, and the matter is I tact, in the Journal of the Franklin Institute 
 not worth examination at any length. The for Feb., 1888, and it has received more atten- 
 problem is investigated, for both bevel and I tion than its slight importance entitles it to. 
 
 . EFFICIENCY OF INVOLUTE TEETH. 
 
 But little can be said in addition to the 
 matter in (49), for both forms of teeth in 
 common use are substantially equal with re- 
 spect to the transmission of power. 
 
 From the formula of (49), which is the 
 formula for the involute tooth, it is seen that 
 the loss from friction is entirely independent 
 of the obliquity, and, therefore, all systems of 
 involute teeth are independent of the ob- 
 liquity in this respect. This is contrary to j 
 
 the accepted idea that a great efficiency re- 
 quires a small obliquity. 
 
 It has been stated on high authority that 
 the involute tooth is inferior to the cycloidal 
 tooth in efficiency, but the statement is not 
 true. The difference in efficiency is minute, 
 a small fraction of one per centum, but what 
 | little difference there is is always in favor of 
 the involute tooth. 
 
 74. OBLIQUITY AND PRESSURE. 
 
 The involute tooth action is in the direction 
 of the line of action, and the obliquity is 
 a constant angle. It is variable only when 
 the shaft center distance is varied. 
 
 As the pressure is always equal to the 
 product of the tangential force at the pitch 
 line multiplied by the secant of the obliquity, 
 
 (26), it is constant for the involute tooth. 
 Involute teeth, therefore, have a steady ac- 
 tion that is not possessed by other forms; 
 particularly by forms which, like the cy- 
 cloidal, have a pressure and an obliquity that 
 varies between great extremes. 
 
 75. THE ROLLER OF THE INVOLUTE. 
 
 The involute odontoid, like all possible 
 odontoids, can be formed by a tracing point 
 in a curve that is rolled on the pitch line, and 
 this roller is the logarithmic spiral with the 
 tracing point at its pole, (32). 
 
 This feature is, however, more curious than 
 useful, and it is not of the slightest im- 
 portance in the study of the curve. Neither 
 is the operation of rolling the involute me- 
 chanically possible, for the logarithmic roller 
 has an infinite number of convolutions about 
 
 its pole, and the tracing point would never 
 reach the pitch line. 
 
 The involute is often considered to be a 
 rolled curve, because it can be formed by 
 a tracing point in a straight line that rolls on 
 its base line; but, although that is the fact, it 
 is a special feature and has nothing to do 
 with the rolled curve theory. The rolled 
 curve theory requires that the odontoid shall 
 be forme d by a roller that rolls on the pitch 
 line only . 
 
. THE; CYCLOIDAIv SYSTEM. 
 
 76. THE CYCLOIDAL SYSTEM. 
 
 If the curve known as the cycloid is 
 chosen as the determining rack odontoid, 
 (31), the resulting tooth system will be 
 cycloidal. 
 
 It is commonly called the " epicycloidal " 
 system, because the faces of its teeth are 
 epicycloids, but, as the flanks are hypocy- 
 cloids, it seems as if the name "epihypo- 
 cycloidal " would be still more clumsy and 
 accurate. 
 
 There is no more need of two different 
 kinds of tooth curves for gears of the same 
 
 pitch than there is need of two different 
 kinds of threads for standard screws, or 
 of two different kinds of coins of the same 
 value, and the cycloidal tooth would never 
 be missed if it was dropped altogether. But 
 it was first in the field, is simple in theory, is 
 easily drawn, has the recommendation of 
 many well-meaning teachers, and holds its 
 position by means of "human inertia," or 
 the natural reluctance of the average human 
 mind to adopt a change, particularly a 
 change for the better. 
 
 77. THE CYCLOIBAL TOOTH. 
 
 The cycloid is the curve A that 
 is traced by the point p in the circle 
 C that is rolled on the straight pitch 
 line p I, Fig. 54. The normal at 
 the point p is the line p q to the 
 point of tangency of the rolling 
 circle and the pitch line. 
 
 The line of action is the circle I a, 
 of the same size as the roller C. 
 
 As no tangent arc can be drawn to 
 the line of action from the pitch 
 point as a center, no terminal 
 point (18) exists. As there is no 
 point upon the line of centers from 
 which a circle can be drawn tangent 
 to the line of action, there will be no 
 cusps, (16) except on the pitch line. 
 
 The cycloidal tooth can be drawn 
 by the general method of (24), but 
 there are several easier methods 
 which will be described. There 
 are numerous empirical rules and 
 short cuts to save labor and spoil 
 the tooth, which will not be de- 
 scribed. 
 
 When the pitch line is of twice the diame- 
 ter of the line of action, the flank of the 
 tooth is a straight line. If the pitch line is 
 less than twice as large as the line of action, 
 the flank of the tooth will be under-curved, 
 
 as shown by Fig. 55, and it is customary to 
 avoid the resulting weak tooth by limiting 
 the line of action to a diameter not greatei 
 than half that of the smallest gear to be 
 used. 
 
Cycloid al Secondary Action. 
 
 78. SECONDARY ACTION. 
 
 The secondary line of action (21) is 
 a circle, Fig. 56, differing from the 
 pitch circle by the diameter of the 
 primary line of action, either inside or 
 outside of it. 
 
 When the internal secondary line of 
 action of an internal pitch line coin- 
 cides with the external secondary line 
 of action of its pinion _ there will be 
 secondary contact between the gears, 
 the face of the gear working with the 
 face of the pinion at a point of contact 
 upon the combined secondaries. Fig. 
 57 shows this for the cycloidal tooth, 
 the two faces working together at the 
 point a. As both secondaries are cir- 
 cles they must coincide, and the sec- 
 ondary action will be continuous. 
 
 When the teeth are also in contact at 
 b on the primary line OL action, there 
 will be double contact. 
 
 Undcrcurved flanks 
 
 Fig. 55. 
 
 Secondary 
 
 Fig 56. 
 
40 
 
 Cycloidal Interference. 
 
 79. INTERNAL INTERFERENCE. 
 
 If the secondary lines of action do not 
 come together the teeth will not touch each 
 other at all, but if that of the gear is smaller 
 than that of the pinion the teeth will cross 
 each other and interfere. The line c. Fig. 
 57, is the face of the gear tooth, and the line 
 d is the face of the pinion tooth having a 
 primary line of action equal to the difference 
 between the pitch lines. The secondary line 
 of each gear coincides with the pitch line of 
 the other, and the faces interfere with each 
 other the amount shown by the shaded 
 space. 
 
 The only remedy for internal interference 
 is to reduce the diameter of the primary line 
 of action to half the difference between the 
 diameters of the pitch lines, or else to leave 
 off one of the faces of the teeth. 
 
 The discovery of the law of internal cycloid- 
 al interference is due to A. K. Mansfield, 
 who published it in the ' ' Journal of the 
 Franklin Institute" for January, 1877. It 
 was afterwards re-discovered by Professor 
 MacCord, and most thoroughly applied and 
 illustrated in his " Kinematics." 
 
 When interference is avoided by omitting 
 one of the faces of the teeth the primary line 
 of action may be enlarged, but it must not 
 then be larger than the difference between 
 the pitch diameters. 
 
 Fig. 58 shows on the right the action 
 when the face of the gear is omitted, and on 
 the left the action when the face of the pin- 
 ion is left off. The teeth will just clear each 
 other, each one touching the other at a single 
 point a in its pitch line. 
 
 As the contact at a is not a point of practi- 
 cal action, care must be taken that the arc of 
 action at the primary line of action is as 
 great as the circular pitch, for otherwise, as 
 in the figure, the gears will not be in continu- 
 ous primary action. 
 
 The rule for internal interference, simply 
 stated, is that the diameters of the pitch lines 
 must differ by the sum of the diameters of 
 the lines of action if the teeth have both 
 faces and flanks, and by the diameter of the 
 acting line of action if the face of either gear 
 is omitted. For the standard interchangeable 
 system the gears must differ by twelve teeth 
 
 Fig 58 
 
 if both teeth have faces, and by six teeth if 
 one face is omitted. 
 
 Fig. 62 shows the secondary contact in the 
 case of a standard internal gear of twenty- 
 four teeth working with a pinion of twelve 
 teeth, and it is to be noticed that the teeth 
 nearly coincide between the two points of 
 contact. Where there is secondary contact 
 the teeth practically bear on a considerable 
 line instead of at a point. 
 
Cycloidal Odontograph. 
 
 41 
 
 80. THE STANDARD TOOTH. 
 
 The standard tooth (42), selected for the 
 
 cycloidal system, is by common consent the 
 one having a line of action of half the diame- 
 ter of a gear of twelve teeth, so that that 
 gear has radial flanks. 
 
 The standard adopted by manufacturers of 
 cycloidal gear cutters is that having radial 
 flanks on the gear of fifteen teeth, but it is 
 not and should not be in use for other pur- 
 
 poses. If any change is made, it should be 
 made in the other direction, to make the set 
 take in gears of ten teeth. 
 
 It must be borne in mind that the standard 
 adopted does not limit the set to the stated 
 minimum number of teeth, but that it sim- 
 ply requires that smaller gears shall have 
 weak under-curved teeth. 
 
 81. THE ROLLED CURVE METHOD. 
 
 It happens in this case, and in this case 
 only, that the rolled curve method, which 
 theoretically applies to all odontoids, can 
 be actually put into practical use, for the 
 generating roller is here the circle, the sim- 
 plest possible curve. 
 
 As in Fig. 59, roll a circle of the .diameter 
 of the circle of action upon the outside of the 
 pitch line for the faces, and upon the inside 
 for the flanks, and a fixed point in it will 
 trace the curve. 
 
 The method can be used by actually con- 
 structing pitch and rolling circles, but the 
 same result can be reached more easily and 
 quite as accurately by drawing several cir- 
 cles, and then stepping from the pitch point 
 along the pitch line, and back on the circles 
 to the desired point. If the length of the 
 
 Construction by rolling 
 
 Fig. 59. 
 
 step is not more than one-tenth of the diam- 
 eter of the circle, the error will not be over 
 one-thousandth of an inch for each step. 
 
 This method is the best one to adopt, ex- 
 cept for the standard tooth. 
 
 82. THE THREE POINT ODONTOGRAPH. 
 
 It is a simple matter to draw the tooth 
 curve by means of rolling circles, but such 
 a method requires skill on the part of the 
 draftsman. It is, moreover, nothing but a 
 method for finding points in the curve for 
 which approximate circular arcs are then 
 determined. 
 
 The "three point" odontograph is sim- 
 ply a record of the positions of the centers 
 of the circles which approximate the most 
 closely to the whole curve of the standard 
 tooth. The positions of two .points, a at the 
 center of the face or of the flank, Fig. 60, 
 and b at the addendum point or root point 
 of the curve, were carefully computed, and 
 then the position of the center C of the 
 circle which passes through these two 
 
 points and the pitch point 0, was calcu- 
 lated. The circle that passes through these 
 three points is assumed to be as accurately 
 approximate to the true curve as any pos- 
 sible circular arc can be. 
 
 The odontograph gives the radius " rad." 
 of the circular arc, and the distance "dis." 
 of the circle of centers from the pitch line, 
 for the tooth of a given pitch, and their 
 values for other pitches are easily found by 
 simple multiplication or division. 
 
 The advantages of this method lie in the 
 facts that the desired radius and distance 
 are given directly, without the labor of find- 
 ing them, and that as they are computed 
 they are free from errors of manipulation. 
 In point of time required, the advantage is 
 
\ 
 
 42 
 
 Cycloidal Odontograph. 
 
 WeofJW* 
 
 with the odontograph in the ratio of 
 ten to one. 
 
 The greatest error of the odonto 
 graphic arc, shown greatly exaggerated 
 by the dotted lines, is at the point c on 
 the face, and it is greater on a twelve- 
 toothed pinion than on any larger gear. 
 For a twelve-toothed pinion of three- 
 inch circular pitch, a large tooth, the 
 actual amount of the maximum error is 
 less than one one-hundredth of an inch, 
 and its average for eight equidistant 
 points on the face is about four-thousandths I that stated will be due to manipulation, and 
 of an inch. Any error that is greater than ! not to the method. 
 
 To apply the odontograph to any particu 
 lar case, tirst draw the pitch, addendum 
 
 83. USING THE ODONTOGRAPH. 
 
 distance "dis." inside of it. Take the face 
 radius "rad."on the dividers, and draw in 
 
 root, and clearance lines, and space the pitch 
 line, Figs. 60 and 61. 
 
 Then draw the line of flank centers at 
 the tabular distance "dis." outside of the 
 pitch line, and the line of face centers at the 
 
 all the face curves from centers on the line 
 of face centers; then take the flank radius 
 "rad."and draw all the flank curves from 
 centers on the line of flank centers. 
 
 THREE POINT ODONTOGRAPH. 
 STANDARD CYCLOIDAL TEETH. 
 
 INTERCHANGEABLE SERIES. 
 From a Pinion pf Ten Teeth to a Rack. 
 
 
 For One 
 
 /*** For One Inch 
 
 
 
 DIAMETRAL PITCH. 
 
 CIRCULAR PITCH. 
 
 
 NUMBER OF 
 TEETH 
 
 For any other pitch divide by 
 that pitch. 
 
 For any other pitch multiply by 
 that pitch. 
 
 IN THE GEAR. 
 
 
 
 
 
 Faces. 
 
 Flanks. 
 
 Faces. 
 
 Flanks. 
 
 Exact. 
 
 Intervals. 
 
 Rad. 
 
 Dis. 
 
 Rad. 
 
 Dis. 
 
 Rad. 
 
 I is. 
 
 Rad. 
 
 Dis. 
 
 10 
 
 10 
 
 1.99 
 
 .02 
 
 8.00 
 
 ^ 
 
 .62 
 
 .01 
 
 2.55 
 
 1.27 
 
 11 
 
 11 
 
 2.00 
 
 .04 
 
 11.05 
 
 6.50 
 
 .63 
 
 .01 
 
 -3.34 
 
 207 
 
 12 
 
 12 201 
 
 06 
 
 oo 
 
 ' 00 
 
 .64 
 
 .02 
 
 00 
 
 oo : 
 
 
 1314 
 
 2.04 
 
 .07 
 
 15.10 9.43 
 
 .65 
 
 .02 
 
 4.80 
 
 3.00 
 
 15Vi 
 
 1516 
 
 2.10 
 
 .09 
 
 7.86 
 
 ' 3 46 
 
 .67 
 
 .03 
 
 2.50 
 
 1 10 
 
 ITji 
 
 17-18 
 
 2.14 
 
 .11 
 
 6.13 
 
 2.20 
 
 .68 
 
 .04 
 
 1.95 
 
 .70 
 
 20 
 
 19-21 
 
 2.20 
 
 .13 
 
 5.12 / 
 
 1.57 
 
 .70 
 
 .04 
 
 .63 
 
 .50 
 
 23 
 
 22-24 
 
 2.26 
 
 .15 
 
 4.5Q V 
 
 1.13 
 
 .72 
 
 .05 
 
 .43 
 
 .36 
 
 27 
 
 25-29 
 
 2.33 
 
 .16 
 
 4.10 
 
 .96 
 
 .74 
 
 .05 
 
 30 
 
 .29 
 
 83 
 
 30-36 
 
 2.40 
 
 .19 
 
 3.80 
 
 .72 
 
 .76 
 
 .06 
 
 . .20 
 
 .23 
 
 42 
 
 37-48 
 
 2.48 
 
 .22 
 
 3.52 
 
 .63 
 
 .79 
 
 .07 
 
 .12 
 
 .20 
 
 58 
 
 4972 
 
 2.60 
 
 .25 
 
 3.33 
 
 .54 
 
 .83 
 
 .08 
 
 .06 
 
 , .17 
 
 97 
 
 73144 
 
 2.83 
 
 .28 
 
 3.14 
 
 .44 
 
 .90 
 
 .09 
 
 1.00 
 
 .14 
 
 290 
 
 145-300 
 
 2.92 
 
 .31 
 
 3.00 
 
 .38 
 
 .93 
 
 .10 
 
 .95 
 
 .12 
 
 00 
 
 Rack 
 
 2.96 
 
 .84 
 
 2.96 
 
 .34 
 
 .94 
 
 .11 
 
 .94 
 
 .11 
 
Cyclo idal Odo n tograph . 
 
 The table gives the distances and radii if 
 the pitch is either exactly one diametral or 
 one inch circular, and for any other pitch 
 multiply or divide as directed in the table. 
 
 Fig. 61 shows the process applied to a 
 practical case, with the distances given in 
 figures. 
 
 Fig. 62 shows the c'ame process applied to 
 an internal gear of twenty-four teeth work- 
 ing with a pinion of twelve teeth. It illus- 
 trates secondary action and double contact. 
 It also shows the actual divergence of the 
 Willis odontographic arp from the true 
 curve. 
 
 Odontographic example 
 Fig. 61. 
 
 Internal teeth 
 
 Fig. 62. 
 
44 
 
 Willis Odontograph. 
 
 84. THE WILLIS ODONTOGRAPH. 
 
 This is the oldest and best known 
 of all the odontographs, but it is 
 inferior to several others since pro- 
 posed, not only in ease of operation, 
 but in accuracy of result. 
 
 To apply it, find the pitch point* 
 a and a' half a tooth from the pitch 
 point 0, Fig. 63, draw the radii a c 
 and a' c', lay off the angles cab and 
 c' a' b', both 75, and Jay off the 
 distances a b and a' b' that are given 
 by table. 
 
 The centers b and b' thus found are 
 the centers of circular arcs that are 
 tangent to the tooth curves at d 
 and d'. The dividers are set to the 
 radius b or b' to draw the curves. 
 
 The Willis arc touches the true curve only 
 at the pitch point 0, and its variation else- 
 where is small, but noticeable. On the face 
 of the tooth of a twelve-toothed pinion of 
 three inch circular pitch, its error at the ad- 
 dendum point is four-hundred ths of an inch, 
 and it will average three times that of the 
 three point method (82). The error is shown 
 by Fig. 62. 
 
 The greatest error of the method is due to 
 manipulation. The angle is usually laid off 
 by a card, and the center measured in by a 
 scale on the card. The circle of centers is 
 
 TJie Willis odontograph 
 
 Fig. 63. 
 
 then drawn through the center, and unless 
 great care is used the chances of error are 
 great. 
 
 180 
 The angle 90 c ab = JFs= , and the 
 
 - sin. W, in which 
 
 distance a b = 5 - 
 
 27T t 
 
 is the number of teeth in the gear of the 
 same set which has radial flanks, usually 
 12 ; c is the circular pitch, and t is the num- 
 ber of teeth in the gear being drawn. The 
 positive sign is used for the face radius, and 
 the negative for the flank radius. 
 
 85. KLEIN'S CO-ORDINATE ODONTOGRAPH. 
 
 This is a method of finding the positions 
 of several points on the tooth curve by 
 means of their co-ordinates referred to axes 
 through the pitch point. Any point on the 
 curve is found by laying off a certain dis- 
 tance on the radius Y, Fig. 64, and then 
 a certain distance at right angles to it, the 
 distances being given by a table for a certain 
 standard tooth. 
 
 As many points as required are found by 
 this method, and then the curve is drawn in 
 by curved rulers, or by finding the approxi- 
 mating circular arc. 
 
 This odontograph is to be found in Klein's 
 Elements of Machine Design. 
 
 Coordinate odontograpli 
 
 Fig. 64. 
 
Obliquity of Action. 
 
 86. THE TEMPLET ODONTOGRAPH. 
 
 Prof. Robinson's templet odontograph is 
 an instrument, not a method. It is a piece 
 of sheet metal, Fig. 65, having two edges 
 shaped to logarithmic spirals. It is laid 
 upon the drawing, according to directions 
 given in an accompanying pamphlet, and 
 used as a ruler to guide the pen. It can be 
 fastened to a radius bar, and swung on the 
 center of the gear, to draw all the teeth. 
 See Van Nostrand's Science Series, No. 24, 
 for the theory of the instrument in detail. 
 
 The templet odontograph 
 
 Fig. 65. 
 
 87. OBLIQUITY OF THE ACTION. 
 
 When the point of contact between two 
 teeth is at the pitch point 0, Fig. 66, the 
 pressure between the teeth is at right angles 
 to the line of centers, but, as the point of con- 
 tact recedes from the line, the direction of 
 the pressure varies by an angle of obliquity 
 which increases from zero until the point K, 
 at the intersection of the addendum circle 
 with the line of action, is reached. 
 
 The angle K = K W, of the maximum 
 obliquity, can be found by solving the trian- 
 gle C c K, and for the standard set we have, 
 
 cos. 2 K = ^ ^, 
 3 n-\- 18 
 
 in which n is the number of teeth in the 
 gear. 
 
 For the smallest gear of the set, the one 
 having twelve teeth, K is 20 15', and for the 
 rack it is 24 5', so that it will always be be- 
 tween those two limits for external gears, 
 and greater for internal gears. 
 
 The friction between two gear teeth in- 
 creases with the angle of obliquity, but not 
 
 w 
 
 Obliquity 
 
 Fig. 66. 
 
 in direct proportion. With the involute 
 tooth the work done while going over a cer- 
 tain arc from the line of centers is propor- 
 tional to the square of the arc, and for 
 cycloidal teeth the increase with the arc is 
 still more rapid. Therefore it is the maxi- 
 mum obliquity of the action that principally 
 determines the injurious effects of friction. 
 
 THE CUTTER LIMIT. 
 
 When the number of teeth in the gear is 
 less than that in the gear having teeth with 
 radial flanks, the flanks will be under-curved, 
 and when too much so they cannot be cut 
 with a rotary cutter. The teeth of Fig. 55 
 could not be cut with a rotary cutter beyond 
 the points where the tangents to the two 
 sides are parallel. 
 
 The limit is reached when the last point 
 
 that is cut by the rotary cutter is also the 
 last point that is touched by the tooth of the 
 rack in action with it, not allowing for in- 
 ternal gears. 
 
 The diameter of the gear when this limit 
 is reached is found by the formula, 
 
 -" rt T C 
 
46 
 
 Limiting Cycloidal Teeth. 
 
 in which D is the diameter of the gear, d is 
 the diameter of the circle of action, c is the 
 circular pitch, and a is the addendum 
 
 For the common addendum of unity 
 divided by the diametral pitch this may be 
 put in the shape, ^ 
 
 n = s i 
 
 in which * is the number of teeth in the 
 radial flanked gear, and n is the number in 
 the required cutter limit. 
 
 For the common series, where 8 = 12, we 
 have n = 8.26; and for the cutter standard of 
 * = 15, we have n = 10.80, so that cutters 
 could easily be made to cut gears with less 
 than s teeth. 
 
 When the rolling circle for 
 the faces is of half the diam- 
 eter of the pitch line of the 
 mating gear, the flanks of 
 both gears will be straight 
 radial lines, as in Fig. 67. 
 
 Such gears are fitted to each 
 other in pairs, and are not 
 interchangeable with other 
 sizes. Their teeth are more 
 easily made than those of 
 standard gears. The maxi- 
 mum obliquity is less, but 
 the strength of the teeth is 
 also less than usual. There 
 is no reason for making such 
 teeth in preference to the 
 
 ). RADIAL FLANKED TEETH. 
 
 standard, al- 
 
 though, for that reason probably, they are 
 used to a considerable extent. It would be 
 
 Radial flanks 
 
 Fig. 67. 
 
 difficult to devise a form of tooth so whimsi- 
 cal that it would find no one to adopt and 
 use it. J 
 
 90. THE LIMITING NUMBERS OP TEETH. 
 
 When the number of teeth in a driving 
 gear is small, the point p, Fig. 68, of its 
 pointed tooth may go out of action by leav- 
 ing the line of action g before a certain 
 definite arc of recess r has been passed over, 
 and the problem is to find the smallest num- 
 ber of teeth in the following gear that will 
 just allow the given recess. 
 
 This question, which is not a particularly 
 important one, is discussed at length, and 
 applied to both bevel and spur gears, in either 
 external or internal contact, in an article in 
 the "Journal of the Franklin Institute" for 
 Feb., 1888, and we will here consider only 
 the case of the common spur gear. 
 
 The recess r is given as a times the cir- 
 cular pitch, and the thickness a r of the 
 tooth is given as b times the same. The 
 diameter of the circle of action is q times 
 
 Limiting ttftfth 
 
 Fig. 68. 
 
 that of the pitch line of the following gear. 
 The number of teeth in the driving gear is d, 
 and the number in the following gear is /. 
 
The Pin Tooth. 
 
 47 
 
 M is an auxiliary angle equal to -, and 
 W is an angle / \. 
 
 Then the required number / can be found 
 by a process of trial and error with the 
 formula, 
 
 sin. (M -f W) 
 sin. W 
 
 -Tf- M 
 
 For an example, let the recess be of the 
 pitch, the tooth equal to the space, and the 
 flanks of the follower to be radial. Let the 
 problem be to find a follower for a driver of 
 seven teeth. This gives a = , b = $,' q = i, 
 d = 7, and the formula becomes 
 
 gin. 
 
 in. l^L + 25 43' \ 
 \ J i 
 
 sin. 25 43' 
 
 If we put / at random, at 20, we shall 
 get, +.134 = 0. Next, trying/ =10, we 
 get, .132 = 0, and the opposite signs show 
 that / is between 20 and 10. Trying 12 the 
 result is positive, and for 11 it is negative, 
 showing that 12 is the required value of /. 
 That is, 7 teeth will not drive less than 12 
 teeth with radial flanks, unless it is allowed 
 an arc of recess greater than f of the pitch. 
 
 For another example, test MacCord's value 
 of 382 as the least driver for a follower of 
 10 teeth, when recess equals the pitch and 
 the follower has radial flanks. Trying d 
 382, the error is negative ; for 383 it is also 
 negative, but for 384 it is positive, and there- 
 fore the latter is the true number. 
 
 Extensive and sufficiently accurate tables of 
 limiting values are given by MacCord in his 
 "Kinematics." 
 
 5. 
 
 1PITST TOOTH SYSTEM. 
 
 91. THE PIN GEAR TOOTH. 
 
 The theory of the pin gear tooth is en- 
 tirely beyond the reach of the " rolled curve" 
 method of treatment, and, therefore, writers 
 who have adopted that method have had to 
 depend more on special methods adapted to 
 it alone than on general principles. The re- 
 sult is that its properties are often given in- 
 correctly, or with an obscurity and complica- 
 tion that is bewildering to the student. 
 Although the tooth is one of the oldest in 
 use, its theory is so difficult that its defect 
 was not discovered until within a very few 
 years, by MacCord, about 1880, and it was 
 
 not until it was examined by means of its 
 normals that a remedy for that defect was 
 discovered. 
 
 By treating the curve on the general prin- 
 ciples here adopted, as a special form of the 
 segmental tooth, it can be studied with ease, 
 and its peculiarities developed in a complete 
 and satisfactory manner. The method, in 
 general terms, is to find the conjugate tooth 
 curve of the gear, for the given circular tooth 
 curve of the pinion, and it presents no new 
 features or difficulties. 
 
 92. APPROXIMATE FORM OF PIN TOOTH CURVE. 
 
 Considered roughly, but accurate enough 
 for teeth of small size, the form of the 
 gear tooth b, Fig. 69, is a simple parallel 
 to the epicycloid E, formed by the center e 
 of the pin, and is to be drawn tangent to 
 
 any convenient number of circles having 
 centers on the epicycloid. 
 
 The action is practically all on one side of 
 the line of centers, the face of the gear tooth 
 working with the part of the pin that is 
 
The Pin Tooth. 
 
 inside of its pitch line. It is, therefore, all 
 approaching action when the pin drives and 
 all receding action when the gear drives, and 
 it is best to avoid the increased friction of 
 the approaching action by always putting the 
 pins on the follower. 
 
 Zantern wheel 
 
 Fig. 70. 
 
 Pin gearing 
 
 Fig. 69. 
 
 93. ROLLER TEETH. 
 
 The pin gear is particularly valuable when 
 the pins can be made in the form of rollers, 
 Fig. 70, for then the minimum of friction is 
 reached. The roller runs freely on a fixed 
 stud, or on bearings at each end, and can be 
 easily lubricated. 
 
 The friction between the tooth and pin, 
 otherwise a sliding friction at a line bearing, 
 is, with the roller pin, a slight rolling fric- 
 tion, and the sliding friction is confined to 
 
 the surface between the roller and its bear- 
 ings. 
 
 When the roller pin is used there can be 
 no increased friction of approach, and the 
 pin wheel can drive as well as follow. 
 
 For very light machinery, such as clock 
 work, there is no form of tooth that is su- 
 perior to the roller pin tooth, and, with the 
 improvement to be explained, there is no 
 better form for any purpose. 
 
 94. CUTTING THE PIN TOOTH. 
 
 The pin gear tooth can be very easily and 
 accurately shaped by mounting a revolving 
 milling cutter M, Fig. 71, of the size of the 
 pin, upon a wheel A, and causing it to roll 
 with a wheel B, carrying the gear blank Q. 
 The mill will shape the teeth to the correct 
 form. 
 
 Pin gear cutter 
 
 Fig. 71. 
 
 95. PARTICULAR FORMS OF PIN GEARS. 
 
 When the pins are supported between 
 two plates, as in Fig. 70, the wheel is called 
 a "lantern" wheel, and is the most common 
 
 form of clock pinion. The pins are some- 
 times called "staves," and are sometimes 
 known as "leaves." 
 
Defect of Pin Tooth. 
 
 4<J 
 
 When the diameter of the pin is 
 zero, Fig. 72, it being merely a 
 point, the correct tooth curve will 
 be a simple epicycloid. 
 
 When the pin gear is a rack, Fig. 
 73, the tooth bears on the pin only 
 at a single point on the pitch line, 
 and the action is therefore very de- 
 fective unless the roller form of pin 
 is used. This form is more properly 
 a particular case of the Involute tooth, 
 for the shape of the pin is immaterial if it does 
 not interfere with the gear tooth. The circle 
 with center on a straight line is not an 
 odontoid at all, for, although it coincides as 
 a whole and for a single instant with a cir- 
 cular space in the gear, it has no proper and 
 continuous tooth action. 
 
 The gears of Fig. 74, sometimes classed 
 with pin gearing, are not pin gears at all. 
 An epicycloidal face working with a radial 
 flank is a very common combination. 
 
 When the diameter of the pin wheel is half 
 that of the internal gear with which it works, 
 we have the combination of Fig. 75. The 
 pins may run in blocks fitted to the straight 
 slots. 
 
 Point gears 
 
 Fig. 72. 
 
 fin rack 
 
 Fig. 73. 
 
 Rot pin gears 
 
 Fig. 74. 
 
 Radial pin teeth 
 
 Fig. 75. 
 
 ). CORRECT FORM AND DEFECT OF PIN TEETH. 
 
 Although the pin tooth is apparently of a 
 very simple form, a close examination will 
 show that it is really quite complicated, and 
 that its practical action is incomplete and de- 
 fective. There is a cusp (16), and conse- 
 quent failure in the action, that is of small 
 importance when the teeth are small, but 
 which is troublesome when they are large. 
 This defect need not be considered when 
 pinions for clock work are in view, but if 
 pin wheels are to be used for large machinery 
 and heavy power it is important. 
 
 If the pin a, Fig. 76, is examined as an 
 odontoid, it will be seen lhat it is a true 
 odontoid only within the line TeT that is 
 tangent to the pitch line at the center of the 
 pin, for all normals, as pe, from points out- 
 side of that line, intersect the pitch line at 
 the center. 
 
 Drawing the normals, which are radii of 
 the pin, we can easily construct the line of 
 
 action and the conjugate tooth curve. The 
 line of action, commencing at the pitch point 
 0, Fig. 77, is there tangent to the line eOm, 
 which passes through the center e of the pin, 
 curves toward Oh, the tangent to the pitch 
 line at the pitch point, and touches it at the 
 point h, at the distance Oh, equal to the ra- 
 dius of the pin. From the point h it follows 
 the circle hJh' to the point h' , thence return- 
 ing to the pitch point and forming the loop 
 OKL. 
 
 From the center c of the gear, Fig. 78, we 
 can always draw a tangent arc FN to the 
 line of action at the point F, and therefore 
 there will always be a cusp at .ZV on the 
 tooth curve. The tooth curve must end at 
 the cusp, and, to avoid interference, the pin 
 must be cut off at the arc W, drawn through 
 the point F, from the center C. 
 
 The whole pin is generally used, and 
 when it is a roller it must be whole, and 
 
Improved Pin Tooth, 
 
 then interference can be avoided only by 
 cutting away the tooth curve until it will al- 
 low it to pass. 
 
 The complete tooth curve has a first 
 branch NOM, Fig. 78, which is the only 
 part that can be used, an in operative second 
 branch from the first cusp N to the second 
 cusp Q on the arc EQ, and thence an inopera- 
 tive circle ORQ'. 
 
 JLine of action 
 
 Fig. 77. 
 
 'Correct action 
 
 Fig. 78. 
 
 97. AN IMPROVED PIN TOOTH. 
 
 The cause of the broken action of the pin 
 tooth is the cusp, which is always present 
 when the center of the pin is on the pitch 
 line, and it can be avoided by placing the 
 center back, as in Fig. 79, to such a distance 
 inside the pitch line that the cusp does not 
 occur. 
 
 When the center of the pin is inside the 
 pilch line, the whole circle of the pin is a true 
 odontoid, and the distance en of the center 
 from the pitch line can be so chosen that 
 the cusp is not formed. 
 
 This distance does not appear to be sub- 
 ject to any simply stated rule, but in the 
 single case of the pin rack it is determined by 
 the formula: 
 
 2 d* 
 27 D ' 
 
 x 
 
 in which x is the required distance en, D is 
 the diameter of the gear, and d is the diame- 
 ter of the pin. 
 
 If the angle CeO, Fig. 79, is not less than 
 a right angle, there will be no cusp on the 
 
 Fig. ?9. 
 
 90 
 
 Corrected pin year 
 
 gear tooth if the diameter of the gear is 
 greater than that of the pin. 
 
6. TWISTED, SPIRAL,, AND \VORN1 
 
 5. STEPPED GEARS. 
 
 When two or more gears, Fig. 80, of the 
 same pitch diameter, are placed in contact en 
 the same shaft, they will evidently act as in- 
 dependently of each other as if they were 
 some distance apart, while they appear to act 
 together as a single gear with irregular teeth. 
 They are known as " Hooke's Gears." 
 
 It matters not how many different kinds or 
 numbers of teeth the several gears may have, 
 or in what order they are arranged, if those 
 that work together on opposite shafts are 
 matched. They may be given an irregular 
 arrangement, as in Fig. 80 ; a spiral arrange- 
 ment, as in Fig. 81 ; a double spiral, or "her- 
 ring-bone " arrangement, as in Fig. 82 ; a cir- 
 cular arrangement, as in Fig. 83, or other- 
 wise at will. 
 
 Stepped f/ear 
 
 Spiral arrangement 
 
 Fig. 81. 
 
 Double spiral arrangement 
 
 Fig. 82. 
 
 Circular arrangement 
 
 Fig. 83. 
 
 99. TWISTED TEETH. 
 
 The thickness of the component gears has 
 nothing to do with the theoretical action of 
 the stepped gear as a whole, and therefore 
 we can have them as thin as required. If the 
 thickness is infinitesimal the component 
 character of the gear is not apparent, and it 
 is known as a twisted gear, Fig. 84. 
 
 When the teeth are twisted there -may al- 
 ways be one or more points of contact at the 
 line of centers, where the theoretical fric- 
 tion is nothing, and therefore they are par- 
 ticularly well suited for rough cast teeth. 
 Furthermore, if the teeth are badly shaped 
 
 Tivisted arrangement 
 
 Fig. 84. 
 
52 
 
 Twisted Teeth. 
 
 the twisted arrangement tends to distribute 
 the errors so that they are not as noticeable. 
 The oblique action of twisted teeth tends 
 to produce a longitudinal motion of the 
 gears upon their shafts, which must be 
 guarded against. This end thrust may be j 
 avoided by so forming the twist that there ! 
 
 are aiways two oblique bearings between the 
 teeth, acting in opposite directions, as in the 
 herring-bone arrangement. 
 
 The twisted form of tooth is seldom found 
 in practice, except in the form of spiral and 
 double spiral teeth, for the difficulty of form- 
 ing other twists is great. 
 
 100. EDGE TEETH. 
 
 If the twist of the twisted tooth is such 
 that some part of the twist at the pitch cylin- 
 der is always upon the line of centers, the 
 gears will always be in action whether 
 there are full teeth or not, and they will 
 work with theoretical accuracy if they are 
 reduced to edges in the pitch cylinder, as in 
 Fig. 85. 
 
 The friction of the edge tooth is theoret- 
 ically nothing, as there is no sliding of the 
 teeth on each other. There is but one point 
 of contact, and that is always upon the line 
 of centers; but if any power is carried the 
 pressure will soon destroy the single point of 
 contact. 
 
 Fig. 85. 
 
 If the edges are thick the action will be 
 stronger, but there will still be but one point 
 of contact. 
 
 101. INVOLUTE TWISTED TEETH. 
 
 When the form of the tooth is the invo- 1 point contact. The straight involute tooth 
 
 lute, and the twist is such that some part of 
 it on the pitch cylinder always crosses the 
 line of centers, the teeth will remain in con- 
 tact, when the parallel axes are separated, 
 until their points are separated, although the 
 contact may sometimes be very short or even 
 
 will fail as soon as the arc of contact is less 
 than the tooth arc. 
 
 Twisted involute teeth are therefore partic- 
 ularly valuable for gears for driving rolls, or 
 for other purposes where the shaft distance 
 is variable. 
 
 102. FORMATION OF THE TWISTED TOOTH. 
 
 When the twist is a uniform spiral there 
 are convenient methods for shaping the 
 tooth, but the twisted tooth in general can be 
 formed only by the processes of (27), (28) and 
 (29), and then only when the twist is not 
 very irregular. 
 
 The principle of the linear planing opera- 
 
 tion of (29) is the same as for the straight 
 tooth, but the blank must be rotated accord- 
 ing to the form of the twist adopted, while 
 the tool is cutting. The twisting motions 
 are independent of the feeding motion, and 
 are repeated at every stroke. 
 
 103. SPIRAL GEARS. 
 
 The spiral gear is that particular form of 
 the twisted gear which has uniformly 
 twisted teeth, and it is, therefore, a particu- 
 
 lar form of the common spur gear. It has 
 such peculiar properties that it is often 
 classed by itself as a separate form of tooth. 
 
Spiral Act! 072. 
 
 The normal spiral section is that section of 
 the teeth of the spiral gear that is made by a 
 spiral surface, called a helix, that is at right 
 angles with the teeth. It is the equivalent, 
 for spiral teeth, of the normal section of the 
 spur gear that is made by a plane, or of the 
 normal section of the bevel gear that is made 
 by a sphere. As with spur and bevel gears, 
 the action of the teeth on> each other should 
 be studied upon this normal surface. As the 
 helix cannot be ^represented upon a plane 
 figure it must be imagined, and as it is ob- 
 scure it requires close attention. 
 
 Any two spiral teeth will work together, 
 provided their normal spiral sections are con- 
 jugate (24), and, as the shape of the normal 
 spiral section is independent of the angle of 
 
 the spiral, two spiral gears will work to- 
 gether, approximately, on shafts that are 
 askew. This will be seen more clearly if the 
 spiral section is imagined to be a flexible 
 sheet-metal toothed helix, which can be coiled 
 about the shaft of the gear, for it can evi- 
 dently be coiled close or loose without affect- 
 ing the shape of its teeth. If coiled close, 
 with a short lead, it runs nearly at right 
 angles to the shaft, and the gear approxi- 
 mates to the spur gear, while if the lead is 
 long the gear approximates to the screw. 
 
 As the diameter of the spiral gear increases, 
 the teeth straighten, and when the diameter 
 is infinite and it is a rack, they are straight 
 and in no way different from those of a com- 
 mon rack. 
 
 104. THEOKY OF SPIRAL TOOTH ACTION. 
 
 The Willis theory of the action of spiral 
 teeth is the one generally accepted, but it is 
 not correct. It assumes that the action be- 
 tween the gears is upon a section by a plane 
 through the axis of the gear and the common 
 normal to the two axes, and that the section 
 of the two gears made by the plane act to- 
 gether like a rack and gear. 
 
 When the axes are at right angles, and the 
 spiral angle is great, this theory is apparently 
 correct, the error being practically imper- 
 ceptible, but, as the axes become more nearly 
 parallel, the error is more apparent, until, 
 when they are parallel, the error is plain 
 enough. Willis applied his theory to worms 
 and worm gears, on axes at right angles, and 
 evidently did not consider the spiral gear in 
 general. 
 
 The action between spiral teeth is not upon 
 the axial section, and it is not that of a rack 
 and gear, but when there is any action at all 
 it is upon the normal spiral section. See the 
 AMERICAN MACHINIST for May 19th, 1888. 
 
 When the axes are parallel the normal 
 spiral sections, as well as the sections made 
 by a plane normal to the axes, are conjugate, 
 and therefore the action is correct and along 
 a line of action. The action is also continu- 
 ous when the axes intersect and the gears are 
 bevel gears. 
 
 When, however, the axes are askew, the 
 normal spiral sections are not necessarily con- 
 jugate, for they coincide only on one line, the 
 common normal to the two axes. Therefore, 
 there is no continuous tooth contact, except 
 in one particular case, the teeth being in con- 
 tact only for an instant as they pass the 
 normal. 
 
 The special case for which spiral teeth on 
 askew axes have continuous tooth contact, is 
 that case of the involute tooth when the base 
 cylinders are tangent and the gears become 
 spiraloidal skew bevel gears. See (175) and 
 (176). In that particular case the teeth have a 
 sliding conjugate action on each other. As 
 the spiraloidal gear is fully described in its 
 place, it will not be further considered here. 
 
 This theory is corroborated by experiment- 
 al gears made for the Brown & Sharpe Man- 
 ufacturing Company, for whom Mr. O. J. 
 Beale, to whom the theory of the spiral 
 gear is much indebted, made a pair of theo- 
 retically perfect spiral gears, exactly alike, 
 with a spiral angle of 45, working on shafts 
 at right angles, and of such a large size that 
 the action of the teeth could be plainly ob- 
 served. See Figs. 88 and 90. 
 
 Beale's gears cannot be made to run to- 
 gether properly at any shaft distance, but 
 if their ends are brought to the common 
 
54 
 
 Spiral Gearing. 
 
 normal, and their base cylinders are in con- 
 tact, they are skew bevel gears and show the 
 action required by Olivier's theory. 
 
 But, although the action of spiral gear 
 teeth is intermittent, and their contact is the- 
 oretically perfect at one instant only, when 
 
 they are passing the common normal, they 
 are very nearly in contact all the time and 
 the action is practically perfect. Spiral teeth 
 of ordinary sizes work together with a re- 
 markably smooth action. 
 
 105. FORMATION OF THE SPIRAL TOOTH. 
 
 As the spiral rack has an ordinary straight 
 tooth, we can conveniently derive the spiral 
 tooth in general from it by a method that is a 
 form of the molding method of (27) for spur 
 gears. 
 
 If a plane is moved in any direction upon a 
 cylinder it will move it, as if by friction, 
 with a speed that depends upon the direction 
 of the motion. If we imagine the same re- 
 sulting motion between the plane and the 
 pitch cylinder, and assume that the plane is 
 provided with hard and straight teeth run- 
 ning in any direction, it will mold the plastic 
 substance of the cylinder and form spiral 
 teeth upon it. All spiral teeth formed by the 
 same rack will have normal spiral sections 
 that are approximately conjugate to each 
 other, and they will work together inter- 
 changeably. 
 
 This process may be put into practical 
 shape by a modification of the process of (28) 
 for spur gears, by substituting a planing 
 tooth for the molding rack tooth. The tooth 
 has the shape of the normal section of the 
 rack, and, as it is reciprocated at an angle 
 with the axis of the gear blank being shaped, 
 both the tool and the gear blank receive the 
 motion of the plane and pitch cylinder. The 
 cutting face of the tool is normal to the direc- 
 tion of its motion, which motion is tangent 
 to the direction of the tooth spiral. 
 
 The linear process of (29) may be used, the 
 plane of Fig. 20 representing, approximately, 
 the normal spiral section of the gear. Thus, 
 if the planing tool or the equivalent milling 
 cutter receives a motion as if in a plane roll- 
 ing upon the base cylinder, the involute tooth 
 will be produced. 
 
 The spiral tooth may be formed by the 
 linear planing process of (29), directly ap- 
 plied on the principle that the spiral tooth is 
 a twisted spur tooth. The planing tool re- 
 ceives a planing motion in the direction of 
 the axis of the gear blank, and both tool and 
 blank receive the feeding rolling motion that 
 would produce the spur tooth of the section 
 that is normal to the axis. In addition, the 
 blank receives a motion of rotation while the 
 tool moves, that is repeated for every troke 
 of the tool. The cutting edge of the tool is 
 set normal to the axis of the gear. 
 
 The spiral tooth may also be formed by a 
 tool that is formed to the true shape of some 
 section of the tooth, preferably its normal sec- 
 tion, and which is guided in the tooth spiral. 
 This is the process used to shape a worm, the 
 tool being guided by a screw-cutting lathe. 
 
 The process generally used to mill the 
 teeth of the spiral gear is the equivalent of 
 the operation last described. The milling 
 cutter is shaped to the normal section of 
 the tooth space, and is guided in the tooth 
 spiral by a special feeding device that ro- 
 tates the blank while the cutter is working 
 in it. 
 
 Of these processes the planing process of 
 (28) is the best, as it produces the tooth with 
 theoretical perfection, and because all gears 
 formed with the same tool are conjugate and 
 interchangeable. But the screw-cutting and 
 milling processes are most in use, for the 
 reason that they are more expeditious and 
 better adapted to the common machine tools, 
 and it is therefore necessary to study the 
 shape of the normal section of the tooth with 
 1 some care. 
 
Spiral Gearing. 
 
 oo 
 
 108. THE NORMAL PITCH. 
 
 The real pitch of the spiral gear is meas- 
 ured on a section that is normal to its axis, 
 and, as in the case of the spur gear, it is 
 found by dividing the number of teeth by 
 the pitch diameter, but the shape of the tooth 
 must be regulated by the normal pitch, or 
 pitch of its normal section. 
 
 The normal pitch is found by dividing the 
 
 real pitch by the cosine of the angle made by 
 the tooth spiral with the axis of the gear. 
 Thus, if the pitch is 8, and the angle is 45, 
 the normal pitch is 8, divided by .707, or 
 11.3. 
 
 The normal circular pitch is found by mul- 
 tiplying the real circular pitch by the cosine 
 of the spiral angle. 
 
 107. THE ADDENDUM. 
 
 The addendum of the spiral gear should 
 not be determined by its real pitch, but by 
 its normal pitch, for it is then usually possi- 
 ble to mill the tooth with a milling cutter 
 that is made for a standard spur gear. A 
 gear of 8 pitch and 45 angle should have an 
 1 
 
 addendum of 
 
 11.3 
 
 = .089". 
 
 If the addendum is determined by the true 
 pitch when the angle is considerable, the 
 tooth will be long and thin. Fig. 86 shows 
 the normal pitch section of a rack to run 
 with a pinion of 45 angle, while Fig. 87 
 shows the true pitch of the same rack. Fig. 
 88 also shows the true pitch of the pinion, 
 and, although the tooth appears to be stunted, 
 it is really of the standard shape. 
 
 Spiral Rack, 
 Fig. 87. 
 
 Beetle's Experimental Gears. 
 Fig. 88. 
 
 108. THE AXIAL PITCH. 
 
 The section of the spiral gear by a plane 
 through the axis is that of a rack, and the 
 axial pitch, or pitch of the rack, is found by 
 dividing the true pitch by the tangent of the 
 
 spiral angle. Thus, if the angle is 45, the 
 axial pitch is the same as the true pitch, but 
 the axial pitch of a 70 spiral tooth is but .364 
 of the true pitch. 
 
 109. SHAPING THE TOOL. 
 
 When the spiral gear is cut in a milling 
 machine, or turned in a lathe, it is necessary 
 to give the tool the shape of the normal sec- 
 tion of the tooth to be cut, and this is most 
 readily accomplished by shaping it for the 
 spur gear that most nearly coincides with that 
 normal section. 
 
 The number of teeth in the gear that is 
 osculatory to the normal spiral, and therefore 
 most nearly coincides with it, is found by 
 dividing the actual number of teeth in the 
 gear by the third power of the cosine of the 
 spiral anple. 
 
 For example, if we are to cut a gear of 4" 
 
56 
 
 Spiral Gearing. 
 
 diameter, 6 pitch, and 24 teeth, at a spiral 
 angle of 45, the cutter should be shaped to 
 
 24 24 
 
 cut a spur gear of = = 69 teeth 
 
 of -=^ = 8.5 pitch. If the gear has 28 teeth 
 
 of 4 pitch, and an angle of 10, the equiva- 
 lent spur gear has 29 teeth of 4.08 pitch, as 
 the gear varies but little from a spur gear. 
 If the gear is of 5 pitch, and 15 teeth, with 
 an angle of 80, the equivalent spur gear has 
 2,830 teeth of 28.7 pitch, and in general, 
 when the gear has a great angle it is a 
 
 worm, the section is practically that of a rack. 
 Care must be taken, when the gear is a 
 screw, and is turned in the lathe, that the 
 tool should be set with its cutting edge nor- 
 mal to the thread of the screw, if it is shaped 
 by the above rule. If the tool is set in the 
 axial section of the screw, and it generally is, 
 it should be shaped to the axial section of the 
 worm, and have the axial pitch and adden- 
 dum. But when the lead of the thread of 
 the screw is small compared with its diam- 
 eter the difference between the normal and 
 axial sections is not noticeable. 
 
 110. VELOCITY RATIO OP SPIRAL GEARS. 
 
 The spiral gear does not follow the well- 
 known rule of spur gears, that the velocities 
 in revolutions in a given time are inversely 
 proportional to the pitch diameters, but re- 
 quires that ratio to be multiplied by the ratio 
 of the cosines of the spiral angles. 
 
 In the formula 
 
 v I) cos. A 
 
 ~V ~~ d cos. a 
 
 D and d are the diameters of the gears, A 
 and a are their spiral angles, and V and 
 are their velocities in revolutions. 
 
 If the angles are equal, the velocity ratio 
 is the same as for spur gears of the same 
 diameters. Fig. 88 shows a pair of gears B 
 and G that are of the same size and have the 
 same angle in opposite directions, requiring 
 the shafts to be parallel. See also Fig. 89. 
 The pair of gears A and B are exactly alike, 
 with equal angles in the same direction, re- 
 quiring the shafts to be at an angle equal to 
 
 Fig. SO 
 Spiral Spur Gears 
 
 Equa 
 Gears 
 
 Fig. 00 
 
 twice the spiral angle. See also Fig. 90. 
 The statement that like spiral gears will not 
 run together is founded on the Willis theory 
 of spiral gear contact, and is wrong. 
 
 111. SPIRAL WORM AND GEAR. 
 
 When the shafts are at right angles, and 
 the angle on one is so great that it is a screw, 
 the combination is known as a worm gear 
 and worm, Figs. 91 and 92, and is much used 
 for obtaining slow and powerful motions. 
 It is also too much used for wasting power 
 and wearing itself out, for its friction is very 
 great and consumes from one-quarter to 
 two-thirds of thd power received . 
 
 When the screw has a single thread, the 
 
 velocity ratio is simply the number of teeth 
 in the gear, and if th re are two or three 
 threads it must be modified accordingly. 
 
 The spiral worm is adjustable in its gear 
 both laterally and longitudinally, so that it 
 will change its position as required by wear 
 in the shaft bearings. 
 
 It is an excellent substitute for the hobbed 
 worm and gear, and in most cases will serve 
 practical purposes quite as well. 
 
Worm Gearing. 
 
 57 
 
 Spiral Worm Gear and Worm, 
 Fig. 91. 
 
 Fig. 92 
 
 Worm Gears 
 
 112. EFFICIENCY OF SPIRAL AND WORM GEARING. 
 
 Unless the shafts are parallel the teeth of a 
 pair of spiral gears are moving in different 
 directions, and therefore they cannot pass 
 each other without sliding on each other an 
 amount that increases rapidly with the angle 
 of divergence of the directions of motion, 
 that is, the shaft angle. 
 
 This sliding action creates friction and 
 tends to wear the teeth, and to a very much 
 greater extent than is generally supposed. 
 The friction is so great, in fact, that such 
 gears, particularly worm gears, should be 
 used only for conveying light powers. They 
 are extensively used, or rather misused, for 
 driving elevators, and are even found in mill- 
 ing machines, gear cutters, planers, and 
 similar places, in evident ignorance that they 
 waste from a quarter to two-thirds of the 
 power received. 
 
 The most extensive experiments on the 
 efficiency of spiral and worm gears ever made 
 were made by Wm. Sellers & Co., and they 
 may be found described in great detail in a 
 paper by Wilfred Lewis in the Transactions 
 of the American Society of Mechanical En- 
 gineers, vol. vii. Space will not permit ex- 
 tensive quotations from this valuable paper, 
 but the general result of the experiments is 
 
 shown by the diagram, Fig. 93. The diagram 
 shows that a common cast-iron spur gear and 
 pinion on parallel shafts have an efficiency of 
 from ninety to ninety-nine per cent., accord- 
 ing to the speed at which they are working ; 
 that a spiral pinion of 45, angle working in 
 a spur gear, with shafts at 45, has an effi- 
 ciency of from 81 to 97 per cent. ; that 
 the efficiency decreases as the angle of the 
 shafts increases, until, for a worm of a spiral 
 angle of 5, at a shaft angle of 85, it goes 
 as low as 34, and does not rise higher than 77 
 per cent. This includes the waste of power 
 at the shaft bearings as well as that at the 
 teeth of the gears. The efficiency is lowest 
 for slow speeds, and rises with the speed. 
 The diagram may be relied upon to give its 
 true value, under ordinary conditions, within 
 five per cent. 
 
 The same experiments developed the fact 
 that the velocity of the sliding motion of the 
 cast-iron teeth on each other should not be 
 over two hundred feet per minute in contin- 
 uous service, to avoid cutting of the surfaces. 
 It may be assumed that the efficiency will be 
 higher when the worm is of steel, particu- 
 larly when the gear is of bronze. 
 
 Diagram, Fig. 94, shows the result of simi- 
 
58 
 
 Worm Gearing. 
 
 Velocity at Pitch Xine in feet per minute, 
 
 o ,,^,,22 2 S g S 
 
 -feq 
 
 
 
 -S;., P. 20" 
 
 
 
 
 <M co r*o-oi-aoaioc*oo 
 
 Sellers' Experiments 
 
 Fig. 93. 
 
 lar experiments by Prof. Thurston, with a 
 worm of 6" diameter and one inch circular 
 pitch running in a gear of 16" diameter, both 
 cast-iron. 
 
 It is to be observed that it is the shaft angle, 
 and not the angle of the spiral, that deter- 
 mines the efficiency. A pair of spiral gears 
 on parallel shafts are practically as efficient 
 as gears with straight teeth. 
 
 The great friction of worm gearing is of 
 advantage for one purpose, and for one only, 
 to secure safety and prevent undesired mo- 
 tion of the gears. The worm of Fig. 97 will 
 easily move the gear, but the gear must be 
 moved with great force to start the worm. 
 When the angle of the worm is as small as 
 the "angle of repose" for the metals in 
 contact, it is impossible for the gear to drive 
 the worm. This may be an excuse for the use 
 of the worm gear in elevators, but it would 
 seem that the safety of the cage should de- 
 
 Revolutions of Driver per minute 
 50 100 150 200 250 300 350 400 
 
 Yale & Totvue Experiments 
 
 Fig. 94. 
 
 pend on devices attached to the cage itself, 
 rather than to the hoisting machinery or 
 other distant part. 
 
 Unless the friction of the gears must be 
 depended upon for safety, the worm gear 
 should be used only for purposes of adjust- 
 ment, or when speed must be greatly reduced 
 or power increased within a small compass, 
 and not for conveying power. 
 
Worm Gearing. 
 
 113. THRUST OF SPIRAL TEETH. 
 
 The oblique action of the teeth of 
 spiral gears on each other tends to throw 
 the gears bodily in the direction of their 
 axes, and this tendency creates a thrust 
 that must be opposed by thrust bear- 
 ings. The end pressure on the shaft of 
 a worm is greater than that exerted on 
 the teeth of the worm gear it is driving. 
 
 When the shafts are parallel the 
 thrust may be completely avoided by 
 the use of double spiral or "herring- 
 bone " teeth, Fig. 82 or 83, which act in 
 opposite directions, and neutralize each 
 other. 
 
 When the shafts are at right angles 
 the thrust may be neutralized by op- 
 posing a second gear in the manner 
 shown in section by Fig. 95. The two 
 worms with opposite spirals run in two spiral 
 worm gears that also work with each other, 
 and, as the pressure on one gear is opposite 
 that on the other, there is no thrust on the 
 shaft. When this combination is made with 
 worm gears having concave teeth, the teeth 
 can bear only at their ends. 
 
 Arrangement to avoid thrust 
 
 Fig. 95. 
 
 When the thrust cannot be avoided it 
 
 should be taken by a roller bearing, rather 
 
 j than by the common collar bearing. The 
 
 I diagram, Fig. 94, shows the superior efficiency 
 
 of the roller bearing as compared with the 
 
 collar bearing, the gain being from ten to 
 
 twenty per cent. 
 
 114. THE HOBBED OR CONCAVE WORM GEAR. 
 
 If a spiral gear is made of steel, provided 
 with cutting edges by making slots across its 
 teeth, and hardened, it will be a practical 
 cutting tool called a spiral milling cutter or 
 hob. Fig. 96 shows a spiral milling cutter, 
 having a great spiral angle, and therefore 
 called a worm. 
 
 If this cutting spiral gear is mounted in 
 connection with an uncut blank so that both 
 are rotated with the proper speeds, and the 
 shafts of the two gears are gradually brought 
 together while they are revolving, the edge 
 of the blank will be formed with concave 
 teeth that curve upwards about the sides of 
 the cutting gear. If the hob is then replaced 
 with a spiral gear that is a duplicate of it, ex- 
 cept that it has no cutting teeth, we shall 
 have the familiar worm and worm gear of 
 Fig. 97. 
 
 The principle of the concave gear applies 
 to any pair of spiral gears, on shafts at any 
 
 Concave Worm Gear and Worm. 
 Fig. 97. 
 
Worm Gearing. 
 
 angle, but in practice it is confined to the 
 worm and gear on shafts at right angles. 
 
 The nature of the contact between the 
 worm and the concaved worm gear has not 
 yet been definitely determined, but there is 
 no reason to suppose that it is different from 
 that between plain spiral teeth, a point con- 
 tact on the normal spiral, but it is probably 
 continuous. It is certain, however, that the 
 contact is considerably closer, more nearly 
 resembling surface contact, and being sur- 
 face contact when the diameter of the gear is 
 infinite. 
 
 The worm is adjustable in the concaved 
 teeth of the gear in the direction of its axis, 
 
 A Hob. 
 
 and will change its position as required by 
 the wear of the thrust bearing. It is not ad- 
 justable laterally. 
 
 115. HOBBING THE WORM GEAR. 
 
 When the hob is provided it is a simple 
 matter to cut the gear. The gear is generally 
 provided with the desired number of notches 
 in its edge, that are deep enough to receive 
 the points of the teeth of the hob, and 
 the hob will then pull it around as it 
 revolves. 
 
 It is a too common practice to make the 
 hob do its own nicking, for, if it is forced 
 into the face of the gear as it revolves, it will 
 pull it around by catching its last teeth in 
 the nicks made by the first. 
 
 If luck is good these nicks will run into 
 each other, and the gear will be cut with 
 teeth that appear to be correct, but, as the 
 outside diameter js greater than the pitch 
 
 diameter, there will be one, two, or three 
 teeth too many. The teeth of the finished 
 gear are therefore smaller than those of the 
 worm by an amount that is ruinous if the gear 
 is small, although it is not noticeable when 
 the diameter is large. If there are 12 teeth 
 where there should be but 10, each tooth will 
 be too small by two-twelfths of itself; but if 
 there are 102 teeth where there should be but 
 100, each tooth is too small by but two- 
 hundredths of itself. This handy makeshift 
 process will do very well on large ge^rs, but 
 not on small ones, unless the worm to run in 
 the gear is made to fit the tooth, with a tooth 
 that is smaller, and lead that is shorter than 
 that of the hob. 
 
 116. INVOLUTE WORM TEETH. 
 
 Worms are generally cut in the lathe, and ! 
 as a straight-sided tooth is most easily 
 formed, the involute tooth is generally 
 adopted. 
 
 Strictly, the form of the tool should be 
 that of the normal section of the thread, and 
 it should always be set in the lathe with its 
 cutting face at right angles to the thread. | 
 
 But custom and convenience allow the tooth 
 to have S'raight sides, and to be set with its 
 face parallel with the axis of the worm, and 
 the real difference is not generally notice- 
 able. 
 
 The standard tool has its sides inclined at 
 an angln of 30, and has a length and a width 
 dependent upon the pitch. 
 
 117. INTERFERENCE OF INVOLUTE WORM TEETH. 
 
 There is one difficulty that is seldom recog- 1 pected, and that is interference. The teeth 
 nized,' but which must be carefully guarded of worm gears will interfere with each other 
 against if properly running gears are ex- ! when the conditions are right for interference. 
 
Worm 
 
 Gearing'. 
 
 just as spur involute tee.th will interfere, as 
 shown by Fig. 36. Fig. 98 shows the gear 
 that would be formed by the usual process. 
 The difficulty can be remedied by rounding 
 
 over the tops of the teeth of the hob and 
 worm, as described in (55). 
 
 It is also a simple matter to avoid the inter- 
 ference by enlarging the outside diameter of 
 
 Interfering Worm, 
 Fig. 98. 
 
 Interference Avoided. 
 Fig. 99. 
 
 \ 
 
 Involute worm anil gear 
 twenty-one or more teeth 
 
 Fig. 100. 
 
Worm Gearing. 
 
 the worm gear. If, as shown by Fig. 99, the 
 tooth has but a short flank, or none at all, 
 and the addendum of the gear is about twice 
 that by the usual rule, the action will be con- 
 fined to the face of the gear and the flank of 
 the worm, and there can be no interference. 
 By adopting an obliquity greater than 15, 
 interference can be avoided without changing 
 the addendum. 
 
 This method has the advantage that the 
 same straight-sided worm and hob can be 
 used for small gears as for large ones, and 
 the disadvantage that the action is confined 
 to the approach and subject to greater fric- 
 tion (48). 
 
 When the standard 30 tool is used, all 
 gears of 26 teeth, or smaller, should be made 
 in this way, but the correction is not strictly 
 necessary for gears of more than 20 teeth, 
 unless particularly nice work is required. 
 
 Fig. 100 shows the proper construction of a 
 gear of 21 or more teeth, and Fig. 101 shows 
 that of a gear of less than 21 teeth. In the 
 former case, the teeth of the worm should be 
 limited by the limit line II, but the interfer- 
 ence for 21 to 25 teeth is not noticeable. 
 
 Draw worm teeth straight 
 Draw gear teeth by points (57) 
 
 Involute worm 
 for twenty^ or 
 
 118. CLEARANCE OF WORM TEETH. 
 
 There is another practical point that is sel- 
 dom recognized, and that is that worm teeth 
 should have clearance (40), for there is no 
 reason for clearing spur teeth that will not 
 apply quite as well to any other kind. 
 
 The clearance is easily obtained by making 
 the tooth of the hob a little longer than that 
 of the worm, as shown by the tooth a of Fig. 
 
 '100. For the same reason the hob should 
 have no clearance at the bottom of its thread, 
 so that the tops of the gear teeth will be 
 formed of the proper length. The custom 
 of making the hob and worm of exactly the 
 same diameters will apply only when the 
 worm "bottoms" in the gear and the gear 
 bottoms in the worm. 
 
 119. CIRCULAR PITCH WORM TEETH. 
 
 The old and clumsy circular pitch system \ were the same as those of common spur gears 
 is in universal use for worm teeth, for the | and racks on the circular pitch system. The 
 
 reason that worms are generally made in the 
 lathe, and lathes are never provided with the 
 proper change gears for cutting diametral 
 pitches. The error is so firmly rooted that it 
 is useless to attempt to dislodge it. 
 
 It is therefore necessary to figure the diam- 
 eters of worm gears as if their throat sections 
 
 table of diameters (35) will be of great assist- 
 ance. 
 
 One great objection to the use of the circu- 
 lar pitch system for spur gears does not ap- 
 ply to worm gears, that the center distance 
 between the shafts will always be an incon- 
 venient fraction, unless the pitch is as incon- 
 
Diametral Worm Gearing. 
 
 63 
 
 venient. The worm can be made of any 
 diameter, and can therefore be made to suit 
 the pitch diameter of the gear and the center 
 distance at the same time. 
 
 The sides of the tool for circular pitches 
 should come together at an angle of thirty 
 degrees, and the width of the point, as well 
 as the depth to be cut in the worm or in the 
 hob, should be taken from the following 
 table. The diameter of the hob should be 
 greater than that of the worm by the "in- 
 crease" given. 
 
 Make the tool with the proper width at the 
 point to thread the worm, and then, after 
 making the worm, grind off half the " in- 
 crease" from the length of the tool, and use 
 it to thread the hob. 
 
 TABLE FOR CIRCULAR PITCH WORM TOOLS. 
 
 Circular ]5itch 
 
 2 
 
 13 
 
 lUi 
 
 \\t 
 
 1^6 
 
 Point of hob tool 
 Point of worm tool. . . 
 Depth of cut in worm 
 or hob 
 Increase 
 
 .644 
 .620 
 
 1.416 
 .I6b 
 
 .564 
 .542 
 
 1.240 
 .146 
 
 .483 
 .466 
 
 1.062 
 1.249 
 
 .402 
 .388 
 
 .886 
 .104 
 
 .362 
 .349 
 
 .797 
 .094 
 
 Circular pitch 
 Point of hob tool 
 
 1 
 
 .322 
 
 % 
 .282 
 
 K 
 
 .241 
 
 % 
 .201 
 
 H 
 161 
 
 Point of worm tool. . 
 Depth of cut in worm 
 or hob 
 
 .310 
 708 
 
 .271 
 620 
 
 .233 
 531 
 
 .194 
 443 
 
 .155 
 354 
 
 Increase 
 
 .Ob3 
 
 .073 
 
 .062 
 
 .052 
 
 ^042 
 
 Circular pitch. 
 
 j. 
 
 % 
 
 JL 
 
 YA. 
 
 JL 
 
 Point of hob tool 
 Point of worm tool . . 
 Depth of cut in worm 
 or hob 
 
 .141 
 .135 
 
 310 
 
 .121 
 .116 
 
 265 
 
 100 
 
 .097 
 
 222 
 
 .080 
 .078 
 
 177 
 
 .060 
 .058 
 
 133 
 
 Increase 
 
 .036 
 
 .031 
 
 .026 
 
 .0^1 
 
 .016 
 
 120. DIAMETRAL PITCH WORM TEETH. 
 
 If the proper change gears are provided, it 
 is as easy to cut diametral pitch worm teeth 
 as any. The proper gears can always be 
 easily calculated by the rule that the screw 
 gear is to the stiyl gear as twenty-two times the 
 pitch of the lead screw of the lathe is to seven 
 times the diametral pitch of the worm to be cut. 
 
 For example, it is required to cut a worm 
 of twelve diametral pitch, on a lathe having 
 a leading screw cut six to the inch. We have 
 screw gear _ 22 X 6 _ 11 
 stud gear 7 X 12 ~ 7~' 
 and any change gears in the proportion of 11 
 and 7 will answer the purpose with an error 
 1 
 
 of 
 
 of an inch to the thread of the worm. 
 
 10,000 
 
 If 22 and 7 give inconvenient numbers of 
 teeth, the numbers 69 and 22 can be used 
 with sufficient accuracy, and 47 and 15, or 
 even 25 and 8 may do in some cases. 
 
 To save calculation and study, the table of 
 change gears for diametral pitches is provided, 
 and it will give the proportion of screw gear 
 to'slud gear to be used for all ordinary cases. 
 
 The pair on ihe left will give the proper 
 pitch within less than a thousandth of an 
 inch, and that on the riglit will serve with an 
 error always less than a hundredth of an inch, 
 and sometimes less than two or three thou- 
 sandths of an inch. 
 
 Having the change gears, figure the pitch 
 
 diameter of the gear as if the throat section 
 is a spur gear on the diametral pitch system. 
 The sides of the tool should come together 
 at an angle of thirty degrees, and the width 
 of the point of the tool, as well as the depth 
 to be cut in the worm or in the hob, should be 
 taken from the following table. The diame- 
 ter of the hob should be greater than that of 
 the worm by the "increase" given. 
 
 TABLE FOR DIAMETRAL PITCH WORM TOOLS. 
 
 Diametral pitch. 
 
 Point of hob tool 
 Point of worm tool 
 Depth of cut in worm or 
 hob 
 
 1 
 
 1.035 
 .968 
 
 2.125 
 
 2 
 
 .517 
 .484 
 
 1.063 
 
 3 
 
 .345 
 .323 
 
 .708 
 
 4 
 
 .258 
 .242 
 
 532 
 
 Increase 
 
 .250 
 
 .125 
 
 .083 
 
 .063 
 
 Diametral pitch 
 
 Pointof hob tool 
 Point of worm tool . . 
 Depth of cut in worm or 
 hob * 
 
 5 
 
 .207 
 .194 
 
 .425 
 
 6 
 
 .173 
 .162 
 
 .354 
 
 7 
 
 .148 
 .138 
 
 304 
 
 8 
 
 .129 
 1-A 
 
 .266 
 
 Increase .... 
 
 .050 
 
 .042 
 
 .036 
 
 .032 
 
 
 
 
 
 
 Diametral pitch 
 
 Point of hob tool 
 Point of worm tool 
 Depth of cut in worm or 
 hub 
 
 10 
 
 .104 
 .097 
 
 213 
 
 12 
 
 .086 
 .081 
 
 .177 
 
 14 
 
 .074 
 .069 
 
 .152 
 
 16 
 
 .065 
 .060 
 
 .138 
 
 Increase 
 
 .025 
 
 .021 
 
 .018 
 
 .016 
 
 Make the tool with the proper width at 
 the point to thread the worm; and then, 
 after making the worm, grind off half the 
 "increase " from the length of the tool, and 
 use it to thread the hob. 
 
64 
 
 Diametral Worm Gearing. 
 
 
 3 
 
 
 4 
 
 . 
 
 5 
 
 3 
 
 
 
 
 
 0) 
 JD 
 
 6 
 
 O 
 
 
 JC 
 
 7 
 
 a 
 
 8 
 
 "as 
 
 
 0) 
 ctf 
 
 10 
 
 5 
 
 12 
 
 14 
 16 
 
 Pitch of Leading: Screw. 
 
 4567 
 
 10 
 
 44 23 
 2l' 11 
 
 22 
 7 ' 
 
 88 46 
 2l' 11 
 
 110 
 21 
 
 44 69 
 7 'll 
 
 21 
 3 ' 
 
 176 92 
 21 ' 11 
 
 220 
 21 ' 
 
 11 
 
 7 ' 
 
 33 
 
 14' 
 
 22 
 
 7 ' 
 
 55 
 H' 
 
 33 
 
 7 ' 
 
 11 
 2 ' 
 
 44 
 
 7 ' 
 
 55 
 
 7 ' 
 
 44 5 
 35' 4 
 
 66 15 
 35* 8 
 
 88 5 
 ,35' 2 
 
 22 25 
 
 7 ' 8 
 
 182 15 
 35 ' 4 
 
 22 35 
 H' ~S 
 
 176 5 
 35 ' 1 
 
 44 25 
 
 7 ' 4 
 
 22 
 
 11 30 
 
 44 40 
 
 55 50 
 
 22 60 
 
 11 70 
 
 88 80 
 
 110 100 
 
 21- 
 
 7 19 
 
 21 19 
 
 21 19 
 
 7 19 
 
 3 19 
 
 21 19 
 
 21 19 
 
 44 9 
 
 66 27 
 
 88 9 
 
 110 9 
 
 132 27 
 
 22 63 
 
 176 18 
 
 220 9 
 
 49 10 
 
 49 20 
 
 49 5 
 
 49 4 
 
 49 10 
 
 7 20 
 
 49 5 
 
 49 2 
 
 11 4 
 
 33 6 
 
 11 8 
 
 55 2 
 
 33 12 
 
 11 14 
 
 22 16 
 
 55 4 
 
 14 5 
 
 28 5 
 
 7 5 
 
 28 1 
 
 14 5 
 
 4 5 
 
 7 5 
 
 14 1 
 
 22 5 
 
 33 15 
 
 44 5 
 
 11 25 
 
 66 15 
 
 11 35 
 
 88 5 
 
 22 25 
 
 35 8 
 
 35 16 
 
 35 4 
 
 7 16 
 
 35 8 
 
 5 16 
 
 35 4 
 
 7 8 
 
 11 10 
 
 11 15 
 
 22 20 
 
 55 25 
 
 11 30 
 
 11 35 
 
 44 40 
 
 55 50 
 
 21 19 
 
 14 19 
 
 21 19 
 
 42 19 
 
 7 19 
 
 6 19 
 
 21 19 
 
 21 19 
 
 22 4 
 
 33 2 
 
 44 8 
 
 55 10 
 
 66 4 
 
 11 14 
 
 88 16 
 
 110 20 
 
 49 9 
 
 49' 3 
 
 49 9 
 
 49 9 
 
 49 3 
 
 7 9 
 
 49 9 
 
 .49 9 
 
 11 2 
 
 33 3 
 
 22 4 
 
 55 1 
 
 33 6 
 
 77 7 
 
 11 8 
 
 55 2 
 
 28 5 
 
 56 5 
 
 28 5 
 
 56 1 
 
 28 5 
 
 56 5 
 
 7 5 
 
 28 1 
 
 Exact numbers on the left. Approximate on the right. 
 TABLE OP CHANGE GEARS FOR DIAMETRAL PITCH WORMS. 
 
 121. WIDTH OF WORM GEAR FACE. 
 
 The bearing between the tooth of the ' The length of the worm need be no more 
 
 worm and that of the gear is near the center 
 of the gear, and it is quite small (104). It is, 
 therefore, useless to make the gear with a 
 wide face. If the face is half the diameter 
 of the worm it will have all the bearing that 
 can be obtained, and any extra width will 
 simply add to the weight and cost of the gear. 
 
 than three times the circular pitch, for there 
 are seldom more than two teeth in contact at 
 once. If, however, the worm is made lorg, 
 it can be shifted when it becomes worn, so 
 as to bring fresh teeth into working position. 
 This provision is wise, for the reason that the 
 worm is always worn more than the gear. 
 
 122. THE HINDLEY WORM AND 
 
 If the cutting hob and the worm is shaped 
 by the tool a, and the process indicated by 
 Fig. 102, the resulting pair of gears is known 
 
 It is commonly but erroneously stated 
 that this worm fits and fills its gear on the 
 axial section, the section that is made by a 
 
 as the Hindley worm and gear. The worm j plane through the axis of the worm and 
 is often called the "hour-glass" worm. I normal to the axis of the gear. It has even 
 
Hindi ev Worm Gearing'. 
 
 (55 
 
 been stated that the contact is between sur- 
 faces, the worm tilling the whole gear tooth. 
 
 The real contact is not yet certain, but it is 
 certain that it is not a surface contact. It is 
 also certain that it is on the normal and not 
 on the axial section, and that the Hindley 
 worm hob will not cut a tooth that will till 
 any section of it. The contact may be linear, 
 along some line of no great length, but it is 
 probably a point contact on the normal sec- 
 tion. The order of the contact is certainly 
 very close, resembling that of two surfaces. 
 
 The worm is limited in length, for the 
 sides of the teeth cannot slant inward from 
 the normal to the axis. The end tooth m in 
 Fig. 102 cannot be used, for it will destroy 
 the teeth of the gear as it is fed towards 
 this axis in the operation of hobbing. 
 
 It has the one great defect that it is not 
 adjustable in any direction, and, therefore, 
 cannot change its position when the shaft 
 
 The Hindley 
 Worm Gear 
 
 bearings wear, unless it is itself worn the 
 same amount. It is doubtful if this form of 
 gearing has any advantage over the plain 
 spiral gearing, except when new and in per- 
 fect adjustment. 
 
 123. THE PIN WORM AND GEAR. 
 
 If the hob and the worm are shaped by Fig. 102, the gearing produced will have 
 the pin-shaped revolving milling tool b of ! linear bearing between the teeth. 
 
 The action will be the same as between a 
 series of pin teeth like the milling tool, each 
 pin being in the axial section of the worm, 
 but having a linear bearing on the normal 
 section of its teeth. 
 
 This form of gearing, which is a 
 modification of the Hindley form, 
 may take the shape of pin gearing, 
 the teeth being round pins like the 
 milling tool. If the pins are mounted 
 on studs, so as to revolve, a roller pin worm 
 gear will be produced. 
 
 Fig. 103 shows a form of roller pin 
 gearing in which the pins have been en- 
 
 2'in worm- year. 
 
 Fig. 103. 
 
 larged. 
 
 124. THE WHIT WORTH HOBBING MACHINE. 
 
 When the amount of work to be done will 
 warrant the use of a special machine, the 
 hobbing machine of Sir Joseph Whit worth 
 may be used. It was invented in 1835, and 
 has not been materially improved since then, 
 
 although there are numerous patents relating 
 to it. The worm gear to be hobbed is fixed 
 upon the same spindle with a master worm- 
 wheel. A driving worm runs in the master 
 wheel, and it is connected by a train of gear- 
 
66 
 
 Hobbing Machines. 
 
 ing with a hob that is so mounted on a 
 carriage that it caii be fed towards the gear 
 blank. The hob is slowly forced into the 
 blank, while both are revolving with the 
 proper speeds, and the gear is cut without 
 the assistance of previously made hicks. See 
 British patent 6,850 of 1835. 
 
 loc 
 
 Spiral and Spur Gear. 
 
 125. THE CONJUGATOR. 
 
 This is a machine for cutting spur or spiral 
 gears by means of a hob, and its principle is 
 an extension of that of the Whitworth worm 
 gear hobbing machine. 
 
 If, when the hob in the Whitworth ma- 
 
 Conjugator. Elevation 
 
 Fig.104. 
 
 flan 
 
 Tig. 105. 
 
 chine has reached the full depth of the tooth, 
 it receives a new .motion in the direction 
 of the tangent to its pitch spiral, it will 
 continue the tooth to the edge of the gear, 
 and form the plain spiral gear of Fig. 91 . 
 
 Fig. 104 is an elevation of the machine, and 
 Fig. 105 is a plan. The hob h is mounted 
 upon an arbor that is connected by a train of 
 gearing with the spindle * that carries the 
 blank gear g to be cut, so that the hob and 
 blank revolve together with any definite 
 proportionate speed. The hob is carried 
 upon a carriage that is fed on a frame /. 
 The hob swivels upon the carriage, so that 
 the tangent to its pitch spiral can be set 
 parallel with the direction of the feed, and 
 the frame swivels so that the tooth can be 
 cut at any angle with the gear spindle. 
 
 As the blank and the hob are revolving, 
 the latter is fed into the former, and it will 
 cut a perfect tooth in the direction that the 
 frame is set at. As the frame can be set in 
 any direction, the machine will cut the com- 
 mon straight tooth, as shown by Fig. 106. 
 All gears cut by the same cutter will run 
 together interchangeably, and if two spiral 
 gears are cut at the same angle in opposite 
 directions they will run together on parallel 
 shafts. See U. S. patent number 405,030, 
 June llth, 1889. 
 
7. IRREGUIvAR AND ELLIPTIC GEARS. 
 
 126. NON-CIRCULAR PITCH LINES. 
 
 The consideration of pitch lines that are 
 not circular, and of the teeth that are fitted for 
 them, is an interesting but not particularly 
 important branch of odontics. Such pitch 
 
 lines are largely used for producing variations 
 of speed and power, but have no other prac- 
 tical applications. 
 
 127. THE IRREGULAR PITCH LINE. 
 
 The most general case is that of two indefi- 
 nite irregular curves rolling together, Fig. 
 107, the only condition being that they 
 shall be so shaped that they will roll together 
 continuously. 
 
 As the practical importance of the free 
 pitch line is very small, we shall not ex- 
 amine it in detail. 
 
 Irregular pitch lines 
 
 Fig. 107. 
 
 128. PITCH LINES ON FIXED CENTERS. 
 
 When we attach the condition that the two 
 pitch lines shall revolve in rolling contact on 
 fixed centers, we have a definite problem of 
 more interest and importance than that of the 
 free pitch line. 
 
 If, as in Fig. 108, we have a pitch line A 
 revolving upon a fixed center a, we can con- 
 struct a pitch line B that will roll with it, 
 and revolve on the given fixed center b, by 
 the following process. 
 
 From any pitch point 0, step off equal arcs 
 Oc, cc, cc ; draw circular arcs cd from the 
 center a; draw circular arcs dn from the 
 center b; step off the same equal arcs Oe, 
 ee, ee, then Oeee will be the required mat- 
 ing pitch line. 
 
 These curves will always be in rolling con- 
 tact at a point on the line of centers ab, the 
 pitch point and the angle of the curves with 
 the line of centers continually changing. 
 
 The velocity ratio of the curves will be 
 
 Fixed centers 
 
 Fig. 108. 
 
 variable, and always equal to the inverse pro- 
 portion of any two mating radiants, ac and be. 
 
68 
 
 Multilobes. 
 
 129. CLOSED PITCH LINES. 
 
 When one of the curves of Fig. 108 is a 
 closed curve, the other will in general not 
 be closed, but by trying different centers, a 
 curve can be found that will be closed. 
 
 If the closed curve a,, Fig. 109, is taken, 
 the mating curve A l will be closed when the 
 center is chosen at a certain point B l , that can 
 be found by repeated trials. 
 
 The mating closed curves thus constructed 
 will seldom be alike, but will always have 
 points of similarity. A salient point q on one 
 will be paired with a reversed point or notch 
 on the other, and lobes on one will be repre- 
 sented by depressions on the other. Half a 
 revolution of one of the curves, from any 
 position, will turn the other through half a 
 revolution. 
 
 Set of Multilobes 
 
 Fig. 109. 
 
 1 30. M ULTILOBES. 
 
 If, after finding the center B lt Fig. 109, for 
 the closed mating curve, other centers are 
 tried, second, third, and succeeding centers, 
 B,, BZ, # 4 , will be found, about which the 
 mating curves will also be closed. 
 
 These closed curves, called multilobes, will 
 be each divided into like lobes, the second 
 curve, or bilobe, into two lobes ; the third, or 
 trilobe, into three lobes, and so on. 
 
 If the center is placed at infinity, the rack 
 lobe A oo will be formed. 
 
 If the center be taken negatively, on the 
 same .side as the original center b lf at Z> 3 , 
 #3, 6 4 , etc., negative multilobes a, a 3 , 4 , 
 etc., will be formed about the original curve 
 a,. 
 
 All these multilobes, positive and negative, 
 will roll together collectively about their fixed 
 centers, in rolling contact at a common and 
 shifting pitch point 0. 
 
 Any two, of the same sign, will roll in inter- 
 nal contact, and any two of opposite signs 
 
 Train of tnultilobcs 
 
 Fig. 110. 
 
 will roll in external contact, so that they can 
 be formed in train, Fig. 110. 
 
 When it so happens, as it does with the 
 ellipse revolving on its focus, or the logarith- 
 mic spiral revolving on its pole, is taken, that 
 the first derived pair of curves, or unilobes, 
 are exactly alike, all the multilobes will be 
 alike ; the positive trilobe like the negative 
 trilobe, and so on, so that any two curves of 
 such a set will work together in either inter- 
 nal or external contact, Fig. 111. 
 
Conic Pitch Lines. 
 
 69 
 
 131. CONIC SECTION PITCH LINES. 
 
 If two like conic sections are mounted upon 
 their foci, they will roll together. 
 
 Their free foci will revolve at a fixed dis- 
 tance from each other, and may be connected 
 by a link. The line of the free foci will in- 
 tersect the line of the fixed foci at the point 
 of contact of the pitch lines. 
 
 Fig. 112 shows a pair of ellipses, Fig. 113 
 a pair of parabolas, and Fig. 114 a pair of 
 hyperbolas. 
 
 The elliptic pitch line is the only one 
 known that will revolve with its equal, and 
 make a practical and complete revolution. 
 
 fixed 
 
 Elliptic tnultilobea 
 
 Fig. 111. 
 
 Ellipticpitch lines 
 
 Fig. 112. 
 
 Parabolic pitch lines 
 
 Fig. 113. 
 
 Hyperbolic pitch Hues 
 
 > Of 
 
 IIVEE3ITY, 
 
70 
 
 Logarithmic Pitch Lines. 
 
 132. THE LOGARITHMIC SPIRAL. 
 
 If the radiants a, b, c, d, e, Fig. 115, make 
 equal angles with each other, and each one 
 is equal to the adjacent one multiplied by a 
 constant number, their extremities will deter- 
 mine a logarithmic spiral. 
 
 If the first radiant a is given, with the con- 
 stant multiplier ??, the second radiant will be 
 na, the third will be n* a, the fourth will be 
 n 9 a, and so OD. 
 
 If the first and last radiants, a and e, are 
 given, and there are p equal angles between 
 them, the constant is 
 
 so that it is a simple matter to construct a 
 logarithmic spiral to connect any two given 
 radiants at any given angle with each other. 
 
 The curve possesses the singular property 
 that all tangents, A or E, make the same 
 angle with the radiants at their points of con- 
 tact. The curves are always inclined to the 
 line of centers at the constant angle. 
 
 The curve continually approaches the 
 center M, or "pole," making an infinite 
 number of turns about it, but ntver reach- 
 ing it. 
 
 It also has the entirely useless property that 
 the pole will trace an involute of the base 
 circle if it is rolled upon the pitch circle (75). 
 
 It possesses the property, not possessed by 
 any other curve, that it will roll with an 
 equal mate on fixed centers that can be varied 
 in position. The curve H will roll with the 
 curve C, whether its pole is at N, or at 8, or 
 at F. 
 
 Fig. 116 shows a pair of logarithmic spi- 
 rals in internal contact. 
 
 Logarithmic pitch lines 
 
 Fig. 115. 
 
 Internal logarithmic 
 pitch lines 
 
 Fig. 1W. 
 
 133. COMPOSITE PITCH LINES. 
 
 Instead of drawing a curve at random, 
 and finding the mate to run with it, Fig. 
 108, the complete pitch line may be built 
 up of a number of curves, of which the 
 properties are known. 
 
 Thus, Fig. 117 shows composite gears, 
 consisting of circular parts A and a, and an 
 elliptic trilobe B, working with an elliptic 
 
 bilobe b. Fig. 118 shows a combination of a 
 pair of logarithmic spiral arcs A and a, a 
 pair of elliptic bilobal arcs B and b, a pair 
 of logarithmic spiral arcs D and d, and a 
 pair of elliptic quadrilobal arcs E and e. 
 An endless variety of combinations can be 
 made in this way. 
 It is not necessary that the component 
 
Composite Pitch Lines. 
 
 71 
 
 curves be tangent, if they succeed each other 
 continuously. Fig. 119 shows a pair of equal 
 logarithmic spirals with a break at ab, the 
 action at b commencing just as it ends at a. 
 Care should be taken to avoid salient 
 
 points, breaks, and interruptions of the con- 
 tinuity of the curve, for there must be 
 defective tooth action at such points. The 
 cuives should run smoothly into each other 
 with gradual changes of curvature. 
 
 Composite pitch lines 
 Fig. 
 
 Composite pitch lines 
 
 Fig. 
 
 Broken pitch lines 
 
 Fig. 119. 
 
 134. TEETH OF NON-CIRCULAR PITCH LINES. 
 
 The action of the teeth of non-circular 
 pitch lines does not, at first sight, appear to 
 follow the laws pertaining to circular lines, 
 but there is really very little difference. 
 
 If we consider the two pitch lines to be 
 free, and to be so moved while they roll 
 together that the pitch point 0, Fig. 107, is 
 fixed, and so that the fixed line c C is always 
 at right angles to both curves at their com- 
 mon point 0, the laws of the tooth action will 
 be almost precisely the same as laid down 
 for the circular pitch line. Fig. 107 may be 
 easily applied to (24) as illustrated by Fig. 15. 
 
 When the centers are fixed, the same tooth 
 
 action takes place, but the line of action and 
 the pitch point continually change their 
 positions. 
 
 The teeth of non-circular pitch lines can 
 therefore be formed either by conjugating a 
 given odontoid, as in (24), or by the roiled 
 curve theory of (32). 
 
 By all means the most practicable method, 
 when the circumstances will permit, is to 
 make up the curve by joining approximating 
 circular arcs, and to provide each circular 
 arc with teeth in the ordinary way. See this 
 process as applied to the elliptic pitch line at 
 Figs. 129 -and 130. 
 
 135. TEETH AT SALIENT POINTS AND BREAKS. 
 
 When there is a salient point, or other inter- 
 ruption of the continuity of the action, as at q, 
 Fig. 109, or at Mm, Fig. 118, there must be 
 
 an interruption in the arrangement of the 
 normals of any tooth curve, and a consequent 
 failure of the tooth action. 
 
1-2 
 
 Elliptic Pitch Lines. 
 
 Fig. 120 shows a cycloidal tooth curve Jf, 
 at a corner or salient point S, between two 
 circular pitch arcs. There is a circular arc 
 A on the odontoid made while the describing 
 circle is turning about the point 8, and that 
 arc can have no continuous tooth action. 
 Therefore the tooth action will fail, unless 
 the next tooth curve .2V springs from the 
 salient point. 
 
 If a tooth springs from the salient point, 
 the tooth action will be correct, but mechani- 
 cally imperfect, as the arc of action of two 
 teeth cannot lap over each other to allow for 
 practical defects. And then, as two tooth 
 curves cannot spring from the same pitch 
 point in opposite directions, such gears can 
 run in but one direction, and are not reversi- 
 ble. 
 
 When there is a break, as at ab, Fig. 119, 
 the teeth must be so cut off that they will 
 
 The salient point 
 
 Fig. 12O. 
 
 separate at a just as they engage at b, for there 
 is a sudden change in the velocity ratio. 
 Such combinations are practicable, but in 
 every way undesirable. 
 
 136. THE ELLIPTIC GEAR. 
 
 The principal, and almost the only use of 
 the irregular gear, is to produce a variation 
 of speed between certain given limits, with- 
 out conditions as to the variations of speed 
 and details of the motion between the limits. 
 When that is the only object, the elliptic 
 pitch line is the only one that is required, and 
 it is chosen because it is the only known con- 
 tinuous closed curve that will work in roll- 
 ing'contact with an equal mate, and because 
 it is, next to the circle, the simplest known 
 curve. Of the elliptic multilobes, the uni- 
 lobe, or simple ellipse, revolving on one of 
 its foci as a center, is the only one used to 
 any appreciable extent, and therefore is the 
 only one that requires examination in detail. 
 
 The use of the elliptic gear is practically 
 confined to producing a simple variation of 
 speed between known limits, and to produc- 
 ing a "quick return motion" for planers, 
 shapers, slotters, and similar cutting tools, as 
 well as for pumps, shears, punches, shingle 
 machines, and others where the work is done 
 mostly during one-half of the stroke of a 
 reciprocating piece. The work of a planer 
 
 tool or of the plunger of a single acting 
 pump, is all done during the motion of the 
 tool or of the plunger in one direction, and 
 the only object on the return is to get the 
 piece ready for the next useful operation in 
 the quickest possible time. 
 
 For an example, the bobbin of a spinning 
 machine is to be wound in a conical form, 
 the thread being fed to it through a moving 
 guide, and the necessary variable motion of 
 the guide, fast at the point of the cone, and 
 slow at its base, is best given to it by a pair 
 of elliptic gears. For another example, the 
 motion of the platen of a printing press 
 should be rapid when the press is open, and 
 slow and powerful when the impression is 
 being taken, and the object can be reached 
 best by a pair of elliptic gears operating the 
 platen. 
 
 The practical uses of the elliptic gear are 
 endless, and it would be in greater use and 
 favor, if it were not for the fact that its pro- 
 duction, by the means ordinarily in use for 
 that purpose, is as difficult and costly as the 
 resulting gear is unsatisfactory. 
 
Elliptic Pitch Lines. 
 
 73 
 
 137. THE 
 
 To thoroughly understand the construc- 
 tion and operation of the ellipse, it is neces- 
 sary to learn but a few of its many proper- 
 ties. 
 
 The mechanical definition of the ellipse is 
 that it is one of the " conic sections." If the 
 cone, Fig. 121, is cut by a plane G at right 
 angles with its axis, the outline of the section 
 will be a circle; if the plane Ecuts the cone at 
 an angle, the section will be an ellipse; if the 
 plane P is parallel with the side of the cone, 
 the section is a parabola, and if the plane H 
 is at such an angle that it cuts both nappes of 
 the cone, the section is a hyperbola. All 
 these curves will roll together when fixed on 
 centers at certain points called foci, but the 
 ellipse, and its special case, the circle, are the 
 only ones that are capable of continuous mo- 
 tion. 
 
 In the ellipse, Fig. 122, the point C is the 
 center, the longest diameter, AA ', is the 
 major axis, the shortest diameter, BB' , is the 
 minor axis; A and A are the major apices, 
 and B and B' are the minor apices. 
 
 If an arc be drawn from the minor apex, 
 with a radius equal to the major semi-axis, it 
 will cut the major axis at points F and F, 
 called the foci, and one focus must be chosen 
 as the center, about which the curve is to re- 
 volve if used as the pitch line of a gear. 
 
 It is a property of the curve that the sum 
 of the distances, PF and PF', from any 
 point to the foci is equal to the major axis, 
 AA', and this feature is used as a means of 
 constructing the curve by points. Draw any 
 arc at random from one focus with radius 
 FP. Draw an arc from the other focus with 
 a radius equal to A A FP, and it will cut 
 the first arc at a point of the ellipse. When 
 the point P is near either major apex, the 
 arcs intersect at such a sharp angle that the 
 method is nearly useless. 
 
 Another, and much the best known method 
 for constructing the ellipse by points, is to draw 
 any radial line L, and also circular arcs W 
 and V, from the center through the apices. 
 From the intersections, w and v, of the radial 
 line and the circles, draw lines parallel to the 
 axes, and they will intersect, always at right 
 angles, at a point u on the curve. This 
 
 ELLIPSE. 
 
 Conic sections 
 
 Fiy. 121. 
 
 The ellipse 
 
 Fig. 122. 
 
 method is very accurate, and has no failing 
 position. 
 
 Another valuable property of the ellipse is 
 that if the line pab be so drawn that the dis- 
 tance pa is equal to BC, and pb to AC, the 
 point p will be upon the curve if the points a 
 and b are upon the axes. 
 
 The curvature of the ellipse is an important 
 feature in connection with its use as a gear 
 pitch line. It is sharpest at the major axis 
 A, and flattest at the minor apex B, else- 
 where varying between the two limits. 
 
 The radius of curvature at either apex, 
 that is, the radius of tne circle that most 
 nearly coincides with the curve, is found by 
 drawing the lines Bk and Ak at right angles 
 
74 
 
 Elliptic Pitch Lines. 
 
 with the chord AB. The distance Ch is the 
 radius of curvature at the major apex A, and 
 the distance Ck is the radius at the minor 
 apex B. 
 
 The normal PN to the curve at any point 
 P bisects the angle FPF' between the focal 
 lines, and the tangent PT is at right angles 
 to the normal. 
 
 138. ELLEPTOGRAPHS. 
 
 There are a multitude of elliptographs, or 
 instruments for drawing the ellipse, but only 
 two of them are of practical application in 
 this connection. 
 
 The simplest known elliptograph consists 
 of a couple of pins, a thread, a pencil, and a 
 stock of patience. The pins are inserted at 
 the foci, as in Fig. 123, and the curve is 
 drawn by moving the pencil with a uniform 
 strain against the string. After a number of 
 trials, depending in number on the skill of the 
 draftsman, the curve may be induced to pass 
 through the desired points. The best result 
 will be obtained by the use of a well waxed 
 thread running in a groove near the point 
 of a hard pencil. The pencil should be long, 
 .and held by the end, so that the strain on the 
 string will be uniform, for the elasticity of 
 the string is the greatest source of 
 error. This " gardener's ellipse " 
 will generally be accurate enough 
 for a tulip patch, but should not 
 be relied upon for mechanical pur- 
 poses, unless one or more points 
 between the apices are tested and 
 found to be correct. If the two 
 pins and the pencil are circular, 
 and of the same diameter, ~the ac- 
 curacy of the ellipse is independent 
 ' of their diameter. 
 
 The best elliptograph is the 
 "trammel," Fig. 124, which takes 
 a variety of shapes, but which in 
 its simplest condition consists of 
 a cross, with two grooves at right 
 angles, and a bar D with two pins 
 a and b, and a tracing point P 
 placed in line. The distance Pb 
 being set to the major semi-axis, and 
 the distance Pa to the minor semi- 
 axis, the point P will trace the ellipse if the 
 pins are confined to move in the grooves. 
 lf carefully made, the instrument works 
 
 with great precision, is easily handled and 
 set, and, if the curve drawn is not very 
 flat, it may be inked. The cheap wooden 
 
 Fig. 123. 
 
 Gardener's ellipse. 
 
 The trammel 
 
 Fig. 124=. 
 
 trammel should not be tolerated, for the 
 string and two pins cost less and are more 
 reliable. 
 
Elliptic Pitch Lines. 
 
 75 
 
 139. APPROXIMATE CIRCULAR ARCS. 
 
 If a well-made trammel is not at hand, the 
 best plan is to draw the ellipse with a string, 
 through several constructed points, and then 
 to ink it by finding centers for approximate 
 arcs, as shown by Fig. 125. An arc from a 
 center m on the major axis, will coincide 
 very well with the curve near the major 
 apex, a similar arc n from a center on the 
 minor axis will serve near the minor apex, 
 and a third center q can be found for an arc 
 to join the first two. More than three cen- 
 ters will seldom be required, and when the 
 ellipse is not very flat the two centers on the 
 axes will be sufficient. 
 
 . 125. 
 
 The elliptic involute 
 
 140. FOUR CENTER ELLIPSE. 
 
 When the ratio of the axes is not less than 
 eight to ten, as is generally the case, a prac- 
 tically perfect ellipse may be drawn from 
 four centers by the following method. 
 
 Draw the line CL, Fig. 126, parallel to 
 A'B, and construct the point u on the ellipse 
 by the method of (137). Find a point a on 
 the major axis, from which an arc from A 
 will pass through u, and it will be the major 
 center. It may be found by trial, or by 
 drawing um at right angles to uA, and 
 bisecting Am in a. 
 
 Through a draw ac at right angles to AB, 
 and its intersection with the minor axis will 
 be the minor center b. Lay off Co,' and CV 
 equal to Ca and Cb, and draw be', b'c", and 
 b'c'". 
 
 From the centers a draw the arcs cAc'", 
 and e'A'c", and from the centers b draw the 
 arcs cBc' and c"B'c'". 
 
 Fig. 126. 
 
 al 
 
 JS' 
 
 Four center method 
 
 Lines that are parallel to the pitch line, 
 such as the addendum, root, clearance, and 
 base lines, are to. be drawn from the same 
 centers. 
 
 141. ROLLING ELLIPSES. 
 
 When two equal ellipses, Fig. 127, are 
 arranged to revolve on their foci as centers, 
 with a center distance equal to the major 
 axis, they will roll together perfectly, and 
 be fitted to act as the pitch lines of gear 
 wheels. 
 
 of the arrow d, it will drive the follower 
 F by direct contact of the pitch ellipses, but 
 when turning in the other direction with 
 respect to the follower, as it must during 
 half of its revolution, it has no direct 
 driving action, and the follower must be 
 
 When the driver D turns in the direction ! kept in contact by some other force. 
 
Spacing tJic Ellipse. 
 
 As the two ellipses roll together, the free 
 foci F% and F will always move at a con- 
 stant distance apart, equal to the distance 
 between the fixed foci, and therefore they 
 may be connected by the link L. 
 
 The center line of the link will always 
 cross the fixed center line at the point of con- j 
 tact of the ellipses, and the tangent T at that | 
 point will pass through the intersection of the 
 axes. 
 
 142. SPACING THE ELLIPSE. 
 
 As the ellipses roll together it is essential | If the ellipse is drawn by means of the 
 that the axes come in line, and therefore, if trammel, Fig. 124, it can be accurately spaced 
 the teeth of one gear are fixed at random, by means of a graduated index circle /, hav- 
 those of the other must be fixed to corre- j ing. a diameter equal to the sum of the diam- 
 spond. If this requirement is satisfied, it j eters of the ellipse, for then the center line of 
 
 makes no 
 placed. 
 
 difference where the teeth are 
 
 It is, however, very desirable that the two 
 
 the bar will pass over an arc on the ellipse 
 that at the apices is exactly equal to half the 
 arc passed over at the same time on the circle, 
 
 gears shall be exactly alike, so that they 
 can be cut at one operation while mounted 
 together on an arbor through their focus 
 holes, and to do this, it is necessary to start 
 the teeth at different points, according to 
 whether their number is odd or even. 
 
 If the number of teeth is even, one tooth 
 must spring from the major axis, as shown 
 by Fig. 128. 
 
 If the number of teeth is odd, the major 
 axis must bisect a tooth and a space, as shown 
 by Fig. 129. In this case, if one of the 1 and that is elsewhere very nearly in the same 
 
 Fig. 128. 
 
 gears can be turned over, or.if its other focus 
 hole can be used as a center, it may have a 
 tooth springing from the major axis. 
 
 The simplest method of spacing the ellipse 
 is to step about it with the dividers. If the 
 curve is flat, the dividers should be set to less 
 than a whole tooth, for equal chords will not 
 measure equal arcs of the curve. 
 
 But this stepping method, although it is 
 sufficient and convenient for drafting pur- 
 poses, is wholly unfit for mechanical pur- 
 poses, and therefore we must have a method 
 that is not dependent on personal skill. 
 
 proportion. 
 
 This method is not mathematically exact, 
 but its accuracy is very far within the re- 
 quirements of practice. The space on the 
 quarter, at Q, will be greater than anywhere 
 else, but the maximum error will in general 
 be very minute. 
 
 For an example, take an extreme practical 
 case, a gear with axes eight and ten inches 
 long, and with seventy-two teeth The max- 
 imum error, the difference between the long- 
 est and shortest tooth arcs, will be not over 
 one five-hundredth of an inch. In the mor< 
 
Involute Elliptic Gears. 
 
 77 
 
 common practical case of a gear of nine and 
 ten inches axes, and seventy-two teeth, the 
 maximum error is about one two-thousandth 
 of an inch. In both these cases, the differ- 
 ence between the tooth arc at the major apex 
 and that at the minor apex is too small to be 
 
 readily calculated, but will be about one 
 twenty-thousandth of an inch. In all cases 
 likely to be met with in practice, the inevit- 
 able mechanical errors are greater than the 
 theoretical errors of the method, and it is 
 serviceable on ellipses as flat as three to one. 
 
 Involute elliptic teeth. 
 
 Fig. 129. 
 
 143. INVOLUTE ELLIPTIC TEETH. 
 
 As in the case of the circular gear, the 
 best form of tooth for the elliptic gear is the 
 involute, and for the same reasons. 
 
 The base line of the involute tooth is any 
 ellipse BE, Fig. 125, which is drawn from 
 the same foci as the pitch ellipse ; the limit 
 point i is the point of tangency of a tangent 
 from the pitch point 0, and the addendum 
 line a I of the mating gear must not pass 
 beyond that point. The method of laying 
 out the tooth and drafting it is so exactly 
 like the process for the circular gear that 
 the description need not be repeated. 
 
 The centers of involute elliptic gears can 
 be adjusted without affecting the perfection 
 of the motion transmitted, but, as the focal 
 
 distance remains fixed, the ratio of the axes 
 will be altered. 
 
 The work of drawing the teeth can be 
 much abbreviated by the process illustrated 
 by Fig. 129. Find the centers for approxi- 
 mate circular arcs, preferably by the method 
 of (140), and then consider the gear as made 
 up of four circular toothed segments. It is 
 then necessary to construct but two tooth 
 curves, one for the major and one for the 
 minor segment, and the flanks will be radii 
 of the circular arcs. 
 
 The line of action, la, Fig. 125, is not a 
 straight line, and it is not the same for all 
 the teeth. It is not fixed when the pitch 
 point and the line of centers is fixed (134). 
 
Cycloidal Elliptic Gears. 
 
 Cycloldal elliptic teeth. 
 
 Fig. 130. 
 
 144. CYCLOIDAL ELLIPTIC TEETH. 
 
 The cycloidal tooth is drawn, exactly as 
 upon a circular pitch line, by a tracing point 
 in a circle that is rolled on both sides of the 
 pitch line. The line of action is not a circle, 
 and it is not the same curve for all the teeth. 
 
 That the flanks shall not be under-curved, 
 the diameter of the rolling circle should not 
 be greater than the radius of curvature at 
 
 the tooth being drawn, and when, as usual, 
 the same roller is used for all the teeth, its 
 diameter should not be greater than the 
 radius of curvature at the major apex, the 
 distance Ch of Fig. 122. 
 
 Fig. 130 shows a cycloidal gear drawn as 
 four circular segments, by the methods of 
 (140) and (83). 
 
 145. IRREGULAR TEETH. 
 
 It is most convenient to draw all the teeth 
 alike, with the same rolling circle, or from 
 the same base line, and also to uniformly 
 space the pitch line, but such uniformity is 
 not essential. 
 
 The only requirement is that each tooth 
 curve shall be conjugate to the tooth curve 
 that it works with, and if that condition is 
 satisfied the teeth may be of all sorts and 
 sizes. 
 
 146. FAILURE IN THE TOOTH ACTION. 
 
 When the major axes are in line the action 
 of the teeth on each other is nearly direct, 
 but when the minor axes are in line the action 
 
 is more oblique, as shown by Fig. 127. The 
 teeth tend to jam together when the driver 
 is pushing the follower, and to pull apart 
 
The Link. 
 
 79 
 
 when the follower is being pulled, and when 
 the ellipse is very flat this tendency is so 
 great that the teeth fail to act serviceably. 
 
 At first glance it might appear that this 
 difficulty in the tooth action of very eccentric 
 gears might be overcome by making the teeth 
 radial to the focus, as shown by Fig. 131, 
 but examination will show that but little can 
 be gained in that way. 
 
 The teeth on the gear G were obtained by 
 the method of (28) from the assumed tooth 
 on the gear c, and the effect of the defective 
 shape of one side of the assumed tooth was to 
 cut away the conjugate curve of the derived 
 tooth. 
 
 Such teeth would not work as well as the 
 ordinary form, and their construction would 
 be very difficult. 
 
 Uttrfial teeth 
 
 Fig. 131. 
 
 147. THE LINK. 
 
 When the teeth of the elliptic gear fail to 
 properly engage, on account of the obliquity 
 of the action, the difficulty can be entirely 
 overcome by connecting the free foci by a 
 link (141), as shown by Fig. 127. 
 
 This link works to the best advantage 
 when the teeth are working at the worst, and 
 when it fails to act, as it passes the centers, 
 the teeth are working at their best. There- 
 
 fore gears that are connected by a link need 
 teeth only at the major apices. 
 
 When the tooth action is imperfect by rea- 
 son of its obliquity, and the link is not avail- 
 able or desirable, the difficulty can be over- 
 come by using three or more gears in a train, 
 as shown by Fig. 137, for then the same re- 
 sult can be obtained by the use of gears that 
 are much more nearly circular. 
 
 148. VARIABLE SPEED AND P.OWER. 
 
 If the shaft c, Fig. 132, turns uniformly, 
 the slowest speed of the shaft C will occur 
 when the gears are in the position of the 
 figure, and the proportion between the two 
 speeds will be the proportion between the 
 distances cO and CO. The greatest speed 
 of the driven shaft will occur when the shafts 
 have turned through a half revolution from 
 the position of the figure, and the relative 
 speed will be the same, reversed. 
 
 Fig. 132. 
 
 The ratio of speed, the ratio of the greatest variation of the axes to produce a decided 
 
 speed to the slowest speed, is the square of 
 the ratio between the speed of the driving 
 shaft and the greatest or the least speed of the 
 driven shaft, so that it requires but a slight 
 
 variation of the speed. 
 
 The following table will give the propor- 
 tion of minor to major axes that will give 
 any desired ratio of speeds. 
 
80 
 
 Elliptic Quick Return Motion. 
 
 Ratio of Speeds. 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7... 
 
 Ratio of Axes. 
 
 985 
 
 . .962 
 
 .907 
 .892 
 .878 
 .868 
 .854 
 .844 
 .8M 
 .824 
 .817 
 .807 
 .800 
 
 The power is always inversely proportional 
 to the speed. If the variable shaft is running 
 twice as fast as the uniform shaft, it will ex- 
 ert but one-half the force. 
 
 When the gears are arranged in a train, as 
 in Fig. 137, the speed ratio for the second, 
 third, and following gears will be in the pro- 
 portion of the first, second, third and follow- 
 ing powers of the first ratio. 
 
 Thus, the ratio for a pair of gears with 
 axes in the proportion of .952 to 1 being 4 
 for the second gear, will be 16 for the third 
 gear, 64 for the fourth gear, and so on. 
 
 The use of gears of troublesome eccentric- 
 ity can be avoided by this means. A train 
 of three gears of .952 axes, Fig. 137, is 
 equivalent to a single pair of very flat gears 
 with .800 axes, Fig. 138, and, in general, 
 three gears that are nearly circular are equiva- 
 lent to a single very flat pair. 
 
 149. ELLIPTIC QUICK RETURN MOTION. 
 
 If the gears are arranged with respect to 
 the piece to be reciprocated, in the manner 
 shown by Fig. 133, the time of the cutting 
 stroke will be to the time of the return stroke, 
 as the angle PEK is to the angle PEF, 
 where J^and F are the foci of the ellipse. 
 
 The following table will show the ratio of 
 axes that must be adopted to produce a re- 
 quired ratio of stroke to return. 
 
 Quick Return. Ratio of Axes. 
 
 J2tol 964 
 
 3 tol 910 
 
 4tol 861 
 
 5 tot 817 
 
 6 tol 778 
 
 To determine the ellipse that will give a 
 required quick return, we lay off the angles 
 PEK and PEF in the given proportion, 
 and then find by trial a point P such that the 
 length PE plus the length of the perpendicu- 
 lar PF is equal to the known center distance 
 Ee. F will be the other focus of the re- 
 quired ellipse. 
 
 When the driving gear has turned through 
 the angle P'EF, from the position of the 
 figure at the middle of the return, the varia- 
 ble gear will have turned through the angle 
 P"eO = P'FO, and we can study the action 
 of the tool by drawing equi-distant radii 
 about E, and finding the corresponding radii 
 about F. 
 
 Quick return 
 
 Fig. 133. 
 
 Fig. 134 shows the arrangement of the 
 radii (P'F = F'e of Fig. 133) in the case of a 
 four to one quick return, and it is seen, by 
 the parallel lines, that the motion of the tool 
 is very uniform, coming quickly to its maxi- 
 mum speed, and holding a quite uniform 
 speed until near the end of the stroke. Fig. 
 135 shows that the same motion derived from 
 a simple crank is not as uniform. 
 
 When the gears are arranged in a train, 
 Fig. 137, the quick return ratios can be de- 
 termined by the construction shown by Fig. 
 136. Draw Fc at right angles to A A', and 
 draw cEd through the other focus. The 
 quick return ratio of the second gear will be 
 the ratio of the angles a a and b a . Draw 
 dFe, and the ratio for the third gear will be 
 
Elliptic Trains. 
 
 81 
 
 Quick return crank 
 
 . 134. 
 
 Ordinary crunk 
 
 Fig. 135. 
 
 that of the angles 8 and &. Draw eEf, 
 and 4 and Z> 4 will give the ratio for the 
 fourth gear. And so on, in the same man- 
 ner, as far as desired, the ratio being greatly 
 increased by each gear that is added to the 
 train. 
 
 If carefully performed, the graphical pro- 
 cess is quite accurate. The case of axes in 
 the proportion of .98 to 1 gave a quick re- 
 turn of 1.6 for the second gear, and 2.8 for 
 the third gear, while their true computed 
 values are 1.66 and 2.74. 
 
 The chart will solve quick return 
 train questions involving gears not 
 flatter than .80, as accurately as need 
 be. For example, the ratio of axes 
 of .95 will give a quick return of 
 2.25 for the second gear, 4.85 for 
 the third gear, 9.80 for the fourth ' 
 gear, and 19.70 for the fifth gear. Again, 
 the proportion of axes to give a quick return 
 of 5 for the third gear is .948. 
 
 Quick return train 
 
 Fig. 130. 
 
 Elliptic train Fig. 137 
 
 Fig. 138. 
 
Elliptic Gear Cutting Machine. 
 
 Elliptic Quick Return Chart 
 
 ,81 .82 .83 .84 .85 
 
 .87, *88 ..89 .90 .91 .92 .S 
 Proportion of Axes 
 
 .94 .95 .96 .97 .98 .99 1.00 
 
 150. THE ELLIPTIC GEAR CUTTING 
 MACHINE. 
 
 The conditions of the described 
 -operation of drawing the ellipse by 
 means of the trammel (138) may be 
 reversed, the bar being held still 
 while the paper and the cross are 
 revolved, and it is evident that the 
 xesult will be the same ellipse on the 
 paper as if the bar is revolved as 
 described. 
 
 By thus reversing the process of 
 describing the ellipse, and by adopt- 
 ing the improved spacing device of 
 (142), we can construct a machine 
 for accurately cutting the teeth in 
 an elliptic gear, the main features 
 of which, omitting various unessen- 
 tial details, are shown by Figs. 139 
 and 140. 
 
 The blank to be cut is fastened 
 upon a trammel stand, which cor- 
 responds to the paper in the graphi- 
 cal process, and revolves upon the 
 fixed base. The adjustable trammel 
 pins a and b are fixed in a slot in 
 the bed, and they fit and slide in the 
 slots M and N in the under surface 
 of the stand. The cutter which 
 corresponds to the tracing point is 
 fixed with the pitch center of its 
 
 Plan 
 Fig. 139. 
 
 Cutter 
 
 Elevation 
 Fig. 14O. 
 
Elliptic Bevel Gear. 
 
 83 
 
 tooth curve directly over the point P in the 
 line of the pins. The index plate has a 
 diameter equal to the sum of the axes of the 
 ellipse, and it is held by an index pin p, 
 which slides in the slot, and is always'in the 
 line of the pins. 
 
 Thus arranged, the machine will always 
 cut its tooth in the true ellipse, and the teeth 
 will be accurately spaced. 
 
 The direction of the tooth will be sub- 
 stantially at right angles to the pitch line, 
 and a simple arrangement can be applied to 
 make it exactly so. An index plate of a 
 fixed diameter may be used for all sizes of 
 
 gears, if the index pin is carried by an arm 
 which swings about the center of the gear, 
 and has an adjustable pin that slides in the 
 slot. 
 
 The tops of the teeth are trued by a cutter 
 having a square .edge, and the line of the 
 tops will be substantially parallel to the pitch 
 line. 
 
 The blank is held by an arbor through its 
 focus hole, and the arbor is held by a slide, 
 which slides in a chuck upon the stand, so 
 that the focus can be accurately set in the 
 major axis at the proper distance from the 
 center. 
 
 151. CHOICE OF CUTTERS. 
 
 Theoretically, the teeth are of different 
 shapes, as they are in different positions upon 
 the ellipse, and, therefore, each space should 
 be cut with a cutter that is shaped for that 
 particular space. But as this is impracticable, 
 it is necessary to choose the cutter that will 
 serve the best on the average. 
 
 Strictly, the cutter should be the one that 
 is fitted to cut a spur gear having a pitch 
 radius equal to the radius of curvature of 
 
 the ellipse at the major apex, but as that 
 cutter will be much too rounding for the 
 minor apex, it is better to choose the one 
 that is fitted for the medium radius of cur- 
 vature. 
 
 The two radii of curvature are the dis- 
 tances Oh and Ck, Fig. 122, and the cutter 
 should be chosen for the radius half way 
 between the two, approximately half the 
 sum of the two. 
 
 152. THE ELLIPTIC BEVEL GEAR. 
 
 An ellipse may be drawn on the surface of 
 a sphere by means of a string and two pins, 
 according to the method of (138), and a 
 pair of such spherical ellipses will roll on 
 each other while fixed on their foci, their 
 free foci moving at a constant distance 
 apart. 
 
 Therefore we can have elliptic bevel 
 gears that are very similar to elliptic spur 
 gears, as shown by Fig. 141. The two gears 
 revolve on radial shafts through their foci, 
 and the link connects radial shafts through 
 the free foci. The velocity ratio is the ratio 
 of the perpendiculars a b and a c. The 
 elliptic bevel gear is the invention of Pro- 
 fessor MacCord. 
 
 The spherical ellipse cannot be drawn by 
 
 the trammel method of (138), and therefore 
 the method of spacing of (142), as well as 
 
 Elliptic bevel gears 
 
 Fig. 141. 
 
 the gear cutting machine of (150), does not 
 apply. 
 
84 
 
 Elliptic Calculations. 
 
 153. MATHEMATICAL TREATMENT. 
 
 If the major semi-axis is a, and the minor 
 semi- axis is b, the equation of the curve from 
 the origin at G is 
 
 a* y* -f&z=. a 8 b*, 
 the major axis being the axis of X. 
 
 The distance GF from the center to the 
 focus will be 
 c = 
 
 _&2 = a ^ i _ 
 
 in which TZ, is the ratio of axes = . 
 
 a 
 
 The radius of curvature at the major apex 
 
 is , and that at the minor apex is . 
 a b 
 
 There is no practicable formula for the recti- 
 fication of the curve, as the length is express- 
 ible only by a series. 
 
 The special spacing method of (142) is true 
 only at the instant of passing either apex, 
 for the tracing point describes half the arc 
 described by the line of the bar on the index 
 circle only when the bar is at right angles 
 with the curve. The error will be at its 
 maximum when the bar is at the maximum 
 angle with the normal, which is at about an 
 angle of forty-five degrees with the major 
 axis. The difference between an ordinary 
 tooth space at the major apex, and that 
 at the minor apex, is very minute. A very 
 careful calculation of the length of the chord 
 of a gear of seventy-two teeth, and eight and 
 ten inch axes, gave a chord of .41433" at the 
 major apex, a chord of .41495" at 45 for the 
 maximum, and a chord of .41441" at the 
 minor apex. The difference between the 
 chords at the apices is .00008", but as the cur- 
 
 vature at the major apex is greater than at the 
 minor apex, the difference between the arcs 
 would be less, perhaps not over .00004". 
 The ratio of speeds (148), is 
 
 /JL+jv^-^lY 
 \ i - vi K* i 
 
 The ratio of quick return being given as 
 qr, the value of n is 
 
 n = y 2 V rf 8 -f d* 3d 8 , 
 in which <Z = ton. /- 18 X 
 
 When the gears are in a train, there seems 
 to be no simple method for computing the 
 ratio of axes to produce a given quick 
 return, but, when the ratio is given, the 
 quick return for each gear can be computed 
 best by trial and error with the formula 
 sin. (M N) 
 
 sin. M+sin. N " 
 in which M is any known angle b, Fig. 136, 
 and -ZV is the angle b for the next following 
 gear in the train. Thus, assuming n = .98, 
 and MI = 90, we find N^ = 67 28'. Then 
 putting J/ 8 = 67 28', we find JT, = 48 5'. 
 Knowing the angles, we compute the 
 quick return ratio from 
 
 ' 
 
 which, for n = .98 gives qr for two gears 
 equal to 1.66, and for three gears equal to 
 2.74. The graphical process of Fig. 136 
 should first be employed to fix the angles 
 approximately. 
 
8. THE BEVEL GEAR. 
 
 154. THE BEVEL GEAR. 
 
 The theory of the bevel gear cannot be 
 properly represented, and can be studied 
 only with the greatest difficulty, upon a 
 plane surface. It is essentially spherical in 
 nature, and should be shown upon a spheri- 
 cal surface, as in Figs. 143 and 144. 
 
 This is best done upon a spherometer, 
 which is simply a painted sphere fitted in a 
 ring. The sphere rests upon a support, so 
 that the ring coincides with a great circle 
 upon it, and the ring is graduated to 360. 
 A very roughly made wooden sphere and 
 plain ring will be found to answer the gen- 
 
 eral purpose very well, and should be pro- 
 vided if the study of the bevel gear is 
 seriously intended. If painted, ink marks 
 can be scrubbed off, and pencil marks re- 
 moved with a rubber. 
 
 The mathematical treatment is unapproach- 
 able without a knowledge of the common 
 principles of spherical trigonometry. 
 
 A wide, interesting, and difficult field of 
 study is offered, but space will permit but a 
 brief examination of the more prominent 
 and practical points. A careful examination 
 would require ten times the available space. 
 
 155. THE GENERAL THEORY. 
 
 When thus represented upon the spherical 
 surface, the theory of the bevel gear is so 
 similar to that of the spur gear, as repre- 
 sented upon a plane surface, that any de- 
 tailed description would be mostly a repeti- 
 tion of what has already been stated. 
 
 All straight lines of the spur theory are 
 represented by great circles, the crown gear 
 being the rack among bevel gears, and all 
 distances are measured in degrees. 
 
 Irregular pitch lines and multilobes are 
 managed substantially as for spur gearing. 
 The elliptic bevel gear has been described in 
 connection with elliptic spur gears (152). 
 
 The tooth surfaces of the bevel gear arc 
 generally formed by drawing straight lines 
 from the spherical outline to the center of the 
 sphere, as in Figs. 143 and 144, the pitch lines 
 and tooth outlines being the bases of cones 
 with a common apex. 
 
 When limited in width, as is usually the 
 case, it is by a sphere concentric with the 
 outside sphere, so that a spherical shell is 
 formed. 
 
 These concentric spherical shells can be 
 moved on their axes to form twisted and 
 spiral teeth, Fig. 142, precisely as described 
 for spur gears (99). 
 
 The molding process of (27) will apply per- 
 fectly, but it has but one practical applica- 
 tion. 
 
 Fig. 142. 
 
 Twisted bevel gear. 
 
 The planing process of (28) will fail, for 
 practical purposes, except for one particular 
 form of tooth, because the shape of the cut- 
 ting tool cannot in the general case be 
 
Involute Bevel Gears. 
 
 changed in form as it approaches the apex, 
 and therefore the tooth will not be conical. 
 
 The planing process of (29) will apply per- 
 fectly, the strokes of the tool being radial, 
 and on this method we must depend for the 
 accurate construction of all forms of bevel 
 gear teeth except the octoid and the pin 
 tooth. 
 
 As the diameter of the sphere is increased, 
 the radii become more nearly parallel, until, 
 when the diameter is infinite, they are paral- 
 
 lel. Therefore the spur gear is a particular 
 case of the bevel gear, and all formulae and 
 processes that apply to the bevel gear will 
 apply to the spur gear if the diameter of the 
 sphere is made infinite. The most scientific 
 method of study would be to develop the 
 theory of the bevel gear, and from that pro- 
 ceed to that of the spur gear, but such a 
 method would be difficult to clearly carry 
 out, and is best abandoned for the more con- 
 fined process here adopted. 
 
 156. PARTICULAR FORMS OF BEVEL TEETH. 
 
 As in the case of spur gearing, there can 
 be an infinite number of tooth curves for 
 bevel gearing (31), each form having its own 
 line of action, but as there are only four 
 forms that are available for practical use by 
 means of simple processes of construction, 
 our attention will be confined to them. 
 
 These four particular forms are, first, the 
 
 involute tooth, having a great circle line of 
 action; second, the cycloidal tooth, having a 
 circular line of action; third, the octoid 
 tooth, having a plane crown tooth, and a 
 "figure eight" line of action; and, fourth, 
 the pin tooth, for which one gear of a pair 
 has teeth in the form of round pins. 
 
 157. THE INVOLUTE BEVEL TOOTH. 
 
 The spherical involute must be studied as 
 a whole if its form is to be clearly seen. 
 
 Its definition is that it is the tooth curve 
 having a great circle for a line of action. In 
 Fig. 143 the great circle line of action la ex- 
 tends around the sphere at an angle with the 
 crown pitch line pi, and it is tangent to 
 two base lines bl and bl' ' , that are paral- 
 lel with the crown line. 
 
 The most convenient method of draw- 
 ing the tooth curve is by rolling the line 
 of action on the base line, while a point 
 in it describes the curve on the surface 
 of the sphere. The equivalent graphical 
 process is to step along the base line 
 and any two tangent great circles, from 
 any point on the curve to any desired 
 point. 
 
 It will take the form shown by the 
 dotted lines ; rising at right angles to the 
 base line, it curves until the crown 
 line is reached, there reversing its curva- 
 ture and bending the other way until it 
 meets the other base line. At the base 
 line it has a cusp, and rises from it to 
 repeat the same course indefinitely. 
 
 Fig. 143 shows a crown gear or rack. The 
 pitch line is the great circle pi. The line 
 of centers cOCis a great circle at right angles 
 with the crown line pi. The line of action 
 is the great circle la set at a given angle of 
 obliquity with the crown line. The base 
 
 Fit/. 143. 
 
 The involtite Tooth 
 
Bilgram Bevel Gears. 
 
 87 
 
 circles are the small circles bl and bl'. The 
 spherical involutes have the same property of 
 adjustability as have the spur involutes, the 
 
 motion being confined to the sphere, and there- 
 fore the gears are adjustable as to their shaft 
 angle, the apex remaining common to both. 
 
 158. THE CYCLOIDAL BEVEL TOOTH. 
 
 The definition of the cycloidal tooth is I flank formed by a roller of half the angular 
 that it is that form which has a circular line diameter of the gear being nearly but not 
 
 of action. 
 
 The rolled curve method of treatment (32) 
 applies, and is the best means of studying 
 the curve. 
 
 There is no gear with radial flanks, the 
 
 exactly a plane. 
 
 The theory differs so little from that of 
 the spur gear, that but little of interest can 
 be found, and the curve will not be consid- 
 ered further. 
 
 159. THE OCTOID BEVEL TOOTH. 
 
 The definition of this tooth system is that 
 it is the conjugate system derived from the 
 crown gear having great circle odontoids. 
 
 In Fig. 144 the crown gear has plane 
 teeth cutting the sphere in great circles, 
 mOn, while a pinion would have convex 
 tooth curves conjugate to the great cir- 
 cles of the crown tooth. 
 
 The line of action, from which the 
 tooth derives its name, is the peculiar 
 ' ' figure eight " curve la, which is at right 
 angles to the tooth curve at the crown 
 line pi, and tangent to the polar circles 
 JS and &, to which the great circle crown 
 odontoids are also tangent. 
 
 This tooth owes its existence to the 
 fact that it is the only known tooth, and 
 probably the only possible tooth, that 
 can be practically formed by the mold- 
 ing planing process of (28).* The cutting 
 edge of the tool being straight, no 
 change is required while it is in motion, 
 except in its position, and that is accom- 
 plished by giving it a motion in such 
 a direction that its corner moves in the radial 
 line of the corner of the bottom of the tooth 
 space. 
 
 The octoid tooth, together with an ingeni- 
 
 * Since this statement was made, another bevel 
 practically constructed by the process of (28). 
 
 ous machine for planing it, was invented by 
 Hugo Bilgram, but it has always been mis- 
 taken for the very similar true involute tooth. 
 
 The Octoid Tooth 
 
 Fig. 144. 
 
 Bilgram's machine is described in the 
 AMERICAN MACHINIST for May 9th, 1885, 
 and in the Journal of the Franklin Institute 
 for August, 1886. 
 
 tooth, the " planoid " tooth, has been invented and 
 
 160. THE PIN BEVEL TOOTH. 
 
 If the tooth of one gear of a pair is a coni- 
 cal pin, Fig. 145, with apex at the center of 
 the sphere, that of the other will be conju- 
 
 gate to it, and the combination deserves 
 notice because it is one of the few forms that 
 are easily constructed. It may be said that 
 
88 
 
 TredgolcTs Method. 
 
 Tig. 145. 
 
 its practical construction is simpler and easier 
 than that of any other form of bevel gear 
 tooth except the skew pin tooth of (180). 
 
 The tooth is 
 preferably, but 
 not necessarily, 
 of the conical 
 form, for other 
 forms of circu- 
 lar pins would 
 serve the theo- 
 retical p u r - 
 pose. 
 Its theory is, 
 
 Pin bevel gears. 
 
 in the main, the same as that of the spur pin 
 
 tooth. It has the same troublesome cusp, 
 which can be avoided in the same way, b^r 
 setting the center of ihe pin back from the 
 pitch line. 
 
 It is the only known form of tooth that 
 can be formed in a practical manner by the 
 molding process of (27). If the cutting tool 
 is a conical mill, it will form the conjugate 
 tooth while the two pitch wheels are rolled 
 together. 
 
 The pins may be mounted on bearings at 
 their ends, ' forming roller teeth. They 
 would be weak, but would run with the 
 least possible friction, all the rubbing friction, 
 being confined to the bearings. 
 
 161. TREDQOLD'S APPROXIMATION. 
 
 The construction of the true bevel gear 
 tooth curve upon the true spherical surface is 
 impracticable with the-means in ordinary use, 
 and the true method of computation by means 
 of spherical trigonometry is equally unfitted 
 for common use. But, by adopting Tred- 
 gold's approximate method the difficulties can 
 be overcome. 
 
 By this method the tooth curves are drawn, 
 not on the true spherical surface, but, as in 
 Fig. 146, on cones A and B drawn tangent 
 to the sphere at the pitch lines of the gears. 
 The cones are then rolled out on a plane sur- 
 face, and the gear teeth drawn upon them 
 precisely as for spur gears of the same pitch 
 diameter. 
 
 Practically correct tooth curves could thus 
 be drawn on the spherical surface by cutting 
 the teeth to shape, and bending them down 
 to scribe around them, but in practice the 
 back rims of the gears are shaped to the tan- 
 gent cones so that the teeth lie directly upon 
 the conical surface. 
 
 This method is called approximate, but its 
 real error would be difficult to determine, 
 and is certainly not as great as the inevitable 
 errors of workmanship of any graphical pro- 
 cess. The tooth outline drawn by it upon 
 the spherical surface may be considerably 
 different from that which would be drawn 
 directly upon it, but it does not follow that 
 it is therefore incorrect. The only require- 
 ment is that the engaging curves shall be 
 
 Fig 
 
 Tredffold's method. 
 
 conjugate odontoids, and it is a matter of 
 very small consequence whether or not the 
 curve on the sphere is the same kind of curve 
 as that upon the cone. If the true plane in- 
 volute curve is drawn upon the developed 
 cone, the corresponding curve on the sphere 
 will not be an exact spherical involute, but its 
 divergence from some true odontoidal shape 
 must be minute, even when the teeth are very 
 large indeed. In ordinary cases it cannot be 
 sufficient to affect materially the constancy 
 of the velocity ratio. What is sometimes 
 given as its error is mostly the "difference in 
 shape" between the plane and the spherical 
 teeth. 
 
Drafting Bevel Gears. 
 
 89 
 
 162. DRAFTING THE BEVEL GEAR. 
 
 The practical application of Tredgold's 
 method is illustrated by Fig. 147. 
 
 Draw the axes CA and CB at the given 
 shaft angle ACE. Lay off the given pitch 
 radii a and b, and draw the lines c and d in- 
 tersecting at the pitch point 0. Dra the 
 center line OC, and lay off the face Of. 
 
 The pitch diameters are ON and OM, 
 and NCO and MCO are the pitch cones. 
 
 Draw the back rim line 02) at right an- 
 gles with the center line, lay off the addenda 
 Oe and Og, and the clearance gh. Draw the 
 front rim line parallel to the back rim line. 
 
 The center angle is X, the face increment 
 I is F, and W is the face angle. The cutting 
 I decrement is J, and T is the cutting angle. 
 ! Twice the distance mn is the diameter incre- 
 ! ment, and em is the outside diameter. 
 
 The pitch radius of the Tredgold back 
 cone is OD, and the figure shows the con- 
 struction of the gear teeth on this cone 
 developed. The teeth are represented as 
 drawn upon the figure, but it is better to use 
 a separate sheet. The odontograph should 
 be used, calculating the number of teeth in 
 the full circle of the developed cone. 
 
 Drafting the bevel gear* 
 
 163. THE BEVEL GEAR CHART. 
 
 The drafting of the bevel gear blanks by 
 means of the method of (162) is simple, but 
 the method requires drafting instruments, 
 not always at hand, as well as the ability to 
 use them accurately. The drawing must be 
 carefully made, to give correct results, par- 
 ticularly when the gears are small. After 
 the drawing is made the various angles and 
 
 diameters must be taken off for use at the 
 lathe, and that is by no means a simple mat- 
 ter. 
 
 So great are the practical difficulties that 
 any one who has a knowledge of simple arith- 
 metic will find it not only easier, but more 
 accurate to use the chart and method by 
 means of the following rules. 
 
90 
 
 THE BEVEL GEAR CHART. 
 
 Shafts at 90 
 I'roportion. 
 
 Center 
 Angle. 
 
 Angle Incr. 
 Div. by Teeth 
 
 p S 
 
 Shafts at 00 
 Proportion. 
 
 Center 
 Angle. 
 
 Angle Incr. 
 Div. by Teeth 
 
 II 
 
 .10 110 
 
 5.72 
 
 11 
 
 2.00 
 
 10.00 
 
 101 
 
 84.28 
 
 114 
 
 .20 
 
 .11 19 
 
 6.33 
 
 13 
 
 2.00 
 
 9.00 
 
 91 
 
 83.67 
 
 114 
 
 .22 
 
 .13 1 8 | 7.12 
 
 14 
 
 1.99 
 
 8.00 
 
 81 
 
 82.88 
 
 113 
 
 .25 
 
 .14 
 .17 
 
 1 7 
 
 8.13 
 
 16 
 
 1.98 
 
 7.00 
 
 71 
 
 81.87 
 
 113 
 
 .28 
 
 1 6 
 
 9.47 
 
 19 
 
 1.97 
 
 6.00 
 
 61 
 
 80.53 
 
 113 
 
 .33 
 
 .20 1 5 | 11.32 
 
 23 
 
 1.96 
 
 5.00 
 
 51 
 
 78.68 
 
 112 
 
 .89 
 
 .22 2 9 
 
 12.53 
 
 25 
 
 1.95 
 
 4.50 
 
 92 
 
 77.47 
 
 111 
 
 .43 
 
 .25 1 4 
 
 14.03 
 
 28 
 
 1.94 
 
 4.00 
 
 41 
 
 75 97 
 
 111 
 
 .29 2 7 
 
 15.95 32 
 
 1 92 
 
 3.50 
 
 72 
 
 74.05 
 
 110 
 
 .55 
 
 .30 310 
 
 16.70 
 
 33 
 
 1.92 
 
 3.33 
 
 103 
 
 73.30 
 
 109 
 
 .58 
 
 .33 1 3 
 
 18.44 
 
 36 
 
 1.90 
 
 3.00 
 
 31 
 
 71.57 
 
 109 
 
 .63 
 
 .38 3 8 
 
 20.55 
 
 40 1.87 
 
 2 67 
 
 83 
 
 69.45 
 
 107 
 
 .70 
 
 .40 25 
 
 21.80 
 
 43 1.86 
 
 2.50 
 2.33 
 
 52 
 
 68.20 
 
 106 
 
 .74 
 
 .43 
 
 3 7 
 
 23.20 
 
 45 1.84 
 
 73 
 
 66.80 
 
 105 
 
 .79 
 
 .44 
 
 4 9 
 
 23.97 
 
 46 1.83 
 
 2.25 
 
 94 
 
 66.03 
 
 104 
 
 .81 
 
 .50 
 
 1 2 
 
 26.57 
 
 51 1.79 
 
 2.00 21 
 
 63.43 
 
 103 
 
 .89 
 
 .56 
 
 5 9 
 
 29.05 
 
 56 1.74 
 
 1.80 
 
 95 
 
 60.95 
 
 101 
 
 .97 
 
 .57 
 
 4 7 29.75 
 
 57 1.74 
 
 1.75 
 
 74 
 
 60.25 
 
 99 
 
 .99 
 
 .60 
 
 3_ 5 30.97 
 
 59 1 72 
 
 1.67 
 
 53 
 
 59.03 
 
 98 
 
 1.03 
 
 .63 
 
 5 8 32.00 
 
 61 1.69 
 
 1.60 
 
 85 
 
 58.00 
 
 97 
 
 1.06 
 
 .67 
 
 2 3 33.68 
 
 64 1.66 
 
 1.50 
 
 32 
 
 56.32 
 
 95 | 1.11 
 
 .70 
 
 710 34.99 
 
 66 1.64 
 
 1.43 
 
 107 
 
 55.00 
 
 94 
 
 1.15 
 
 .71 
 
 5 7 35.53 
 
 67 1.63 
 
 1.40 
 
 75 
 
 54.47 
 
 93 
 
 1.16 
 
 .75 
 
 3_ 4 36.87 
 
 69 
 
 1.60 
 
 1.33 
 
 43 
 
 53.13 
 
 92 
 
 1.20 
 
 .78 
 
 7 9 37.87 
 
 70 
 
 1.58 
 
 1.29 
 
 97 
 
 52.13 
 
 91 
 
 1 22 
 
 .80 
 
 4 5 38.67 
 
 72 
 
 1.56 
 
 1.25 
 
 54 
 
 51.33 
 
 90 
 
 1.25 
 
 .83 
 
 5 6 39.80 
 
 73 
 
 1.54 
 
 1.20 
 
 65 
 
 50.20 
 
 88 
 
 1.28 
 
 .86 
 
 6 7 40.60 
 
 75 
 
 1.52 
 
 1.17 
 
 76 
 
 49.40 
 
 87 
 
 .88 
 
 7 8 41.18 
 
 76 
 
 1.50 
 
 1.14 
 
 87 
 
 48.82 
 
 86 
 
 1.32 
 
 .89 
 
 8 9 41.63 
 
 76 
 
 1.49 
 
 1.13 
 
 98 
 
 48.37 
 
 86 
 
 1.33 
 
 .90 
 
 910 
 
 41.98 
 
 77 
 
 1.49 
 
 1.11 
 
 109 
 
 48.02 
 
 85 
 
 1.34 
 
 1.00 
 
 1 1 
 
 45.00 
 
 81 
 
 1.41 
 
 1.00 
 
 11 
 
 45.00 
 
 81 
 
 1.41 
 
Specimen Chart Calculations. 
 
 91 
 
 FIG. 148. 
 SAMPLE COMPUTATION. 
 
 SHAFTS AT 
 A RIGHT ANGLE. 
 
 Pitch = 3 Prop. = 7 5 
 
 Shaft ang. 90j 
 
 Teeth = 42) 93 (2.22 = 
 84 .37 + 
 
 face incr. | 
 * 
 
 90 2.59 = 
 
 84 
 
 60 
 
 cut deer. 
 
 Center angles = 54.47 
 _j_ incr 2 22 
 
 35.53 
 2 22 
 
 
 
 Face angles 56.69 
 
 37.75 
 
 Center angles = 54.47 
 deer 2.59 
 
 35.53 
 2.59 
 
 
 
 Cut angles 51.88 
 
 32.1)4 
 
 
 
 Pitch =3)1.16 
 
 3) 1.63 
 
 Diam. incr. = .39 
 -\- p. diams. = 14. 
 
 .54 
 10. 
 
 o. diams. = 14.39 
 
 10.54 
 
 FIG. 149. 
 SAMPLE COMPUTATION. 
 
 SHAFTS AT 
 ANY ANGLE. 
 
 Pitch = 5 Prc 
 
 >p. = X 
 
 Shaft ang. 52.8 
 
 Teeth = 20) 
 
 66 (3.30 = 
 
 .55 + 
 
 3.85 = 
 
 face incr. 
 i 
 
 cut deer. 
 
 Center angles 
 
 ' 
 
 = 35.80 
 3.30 
 
 17.00 
 3.30 
 
 
 
 
 Face angles 
 
 = 39.10 
 
 20.30 
 
 Center angles 
 deer .... 
 
 = 35.80 
 ... 3.85 
 
 17.00 
 3.85 
 
 
 
 
 Cut angles 
 
 = 31.95 
 
 13.15 
 
 Pitch = 5) 
 
 1.66 
 
 5) 1.91 
 
 Diam. incr 
 -j- p. diams 
 
 .33 
 
 4. 
 
 .38 
 
 2. 
 
 o. diams. . . . 
 
 4.33 
 
 2.38 
 
92 
 
 Chords of Angles. 
 
 164. SHAFTS AT BIGHT ANGLES. 
 
 1st. Divide the pitch diameter by that of 
 the other gear of the pair, or else the number 
 of teeth by that of the other gear, to get the 
 proportion. Enter the table by means of the 
 proportion. All numbers for that pair will 
 be found on the same horizontal line in the 
 two columns. 
 
 2d. The center angles are given directly 
 by the table at the proper proportion. 
 
 3d. Divide the tabular angle increment 
 by the number of teeth in the gear, to get 
 the angle increment. This need be done for 
 but one gear of a pair, as the increment 
 is the same for both. 
 
 4th. Add the angle increment to the cen- 
 ter angle, to get the face angle. 
 
 5th. Increase the angle increment by one- 
 sixth of itself, to get the cutting decrement, 
 and subtract this decrement from the center 
 angle, to get the cutting angle. 
 
 6th. Divide the tabular diameter incre- 
 ment by the diametral pitch, to get the 
 diameter increment, and add that to the pitch 
 diameter, to get the outside diameter. 
 
 Fig. 148 is a sample computation for shafts 
 at right angles. 
 
 165. SHAFTS NOT 
 
 The table cannot be entered by means of 
 the proportion, and the numbers for the two 
 gears of the pair will not be found on the same 
 horizontal line, and it will be necessary to 
 determine the center angles. 
 
 As in Fig. 147, draw the axes, at the given 
 shaft angle, and find the center angles, by the 
 method described in (162). 
 
 Then enter the table, for each gear by 
 itself, by means of the center angles, and 
 proceed as for shafts at right angles. The 
 angle increment and decrement is the same 
 for both gears of a pair. 
 
 Fig. 149 is a sample computation applied 
 
 AT BIGHT ANGLES. 
 
 to the case of Fig. 147, the center angles be- 
 ing found by means of the table of chords. 
 If preferred, the center angles can be found 
 by means of the formula, 
 
 sin. 8 
 
 tan. C = 
 
 
 in which C is the center angle of the gear, P 
 is the proportion found by dividing the num- 
 ber of the teeth in the gear by the number in 
 the other gear, and 8 is the shaft angle. 
 Having found one center angle, subtract it 
 from the shaft angle to get the other center 
 angle. 
 
 166. THE TABLE OF CHOBD8 AT SIX INCHES. 
 
 When the lathesman is provided with a 
 graduated compound rest which feeds the 
 tool at any angle, nothing but the computa- 
 tion is required; but when there is nothing 
 but the common square feed, the faces must 
 be scraped with a broad tool. A templet for 
 guiding the work can easily be made by 
 means of the table of chords at six inches. 
 
 To lay out a given angle, draw an arc 
 with a radius of six inches, draw a chord of 
 the length given by the table for the angle, 
 
 and then draw the sides oc and ob of the 
 angle boc, Fig. 150. 
 
 For tenths of a degree use the small tables. 
 The chord of 37.5 is 3.81 + .05 = 3.86 
 inches. 
 
 Fig. 151 shows the manner of using the 
 angle templet at the lathe. 
 
 This table of chords is very convenient for 
 many purposes not connected with gearing, 
 and it is more accurate than the common 
 horn or paper protractor. 
 
 167. BILGBAM'S CHABT. 
 
 A graphical method for determining the 
 angle and diameter increments, the invention 
 of Hugo Bilgram, is described in the AMEBI- 
 CAN MACHINIST for November 10, 1883. It 
 
 determines the required values by the inter- 
 sections of lines and circles, and requires no 
 computation. 
 
Chords of Angles. 
 
 Chord of * angle 
 
 Fig. 150. 
 
 Using the templet 
 
 TABLE OF CHORDS OF ANGLES, 
 
 AT RADIUS OF SIX INCHES. 
 
 Degrees. 
 
 Chord. 
 
 Tenths. 
 
 Degrees. 
 
 Chord. 
 
 Tenths. 
 
 Degrees. 
 
 Chord. 
 
 Tenths. 
 
 1 
 2 
 3 
 4 
 5 
 
 .10 
 .20 
 .31 
 .42 
 .52 
 
 
 31 
 32 
 33 
 34 
 35 
 
 3.20 
 3.31 
 3.41 
 3.51 
 3.61 
 
 
 61 
 62 
 63 
 64 
 65 
 
 6.10 
 6.19 
 6.28 
 6.36 
 6.45 
 
 
 6 
 
 7 
 8 
 9 
 10 
 
 .62 
 .73 
 .84 
 .94 
 1.04 
 
 
 36 
 37 
 38 
 39 
 40 
 
 3.71 
 3.81 
 3.91 
 4.01 
 4.10 
 
 
 66 
 67 
 68 
 69 
 70 
 
 6.54 
 6.62 
 6.71 
 6.80 
 6.89 
 
 
 11 
 12 
 13 
 14 
 15 
 
 .15 
 .26 
 .36 
 .46 
 .57 
 
 .1 .01 
 .2 .02 
 .3 .03 
 
 41 
 42 
 43 
 44 
 45 
 
 4 20 
 4.30 
 4.40 
 4.50 
 4.60 
 
 .1 .01 
 .2 .02 
 .3 .03 
 
 71 
 
 72 
 73 
 74 
 75 
 
 6.97 
 7.06 
 7.14 
 7.22 
 7.31 
 
 .1 .01 
 .2 .02 
 3 02 
 
 16 
 17 
 
 18 
 19 
 20 
 
 .67' 
 
 .77 
 .87 
 98 
 2.08 
 
 .4 .04 
 .5 .05 
 .6 .06 
 .7 .07 
 .8 .08 
 
 46 
 47 
 
 48 
 49 
 50 
 
 4.69 
 4.79 
 4.88 
 4.98 
 5.08 
 
 .4 .04 
 .5 .05 
 .6 .05 
 .7 .06 
 
 .8 .07 
 
 76 
 77 
 78 
 79 
 80 
 
 7.39 
 7.47 
 7.55 
 7.63 
 7.71 
 
 .4 .03 
 .5 .04 
 .6 .05 
 .7 .06 
 .8 .06 
 
 21 
 22 
 23 
 24 
 25 
 
 2.18 
 2.29 
 2.39 
 2.49 
 2.59 
 
 .9 .09 
 
 51 
 52 
 53 
 54 
 55 
 
 5.17 
 5.26 
 5.35 
 5.45 
 5 54 
 
 .9 .08 
 
 81 
 
 82 
 83 
 84 
 85 
 
 7.79 
 7.87 
 7.95 
 8.03 
 8.11 
 
 .9 .07 
 
 26 
 27 
 
 28 
 29 
 30 
 
 2.70 
 2.80 
 2.90 
 3.00 
 3.10 
 
 
 56 
 
 57 
 58 
 59 
 60 
 
 5.63 
 5.72 
 
 5.82 
 5.91 
 600 
 
 
 86 
 
 87 
 88 
 89 
 90 
 
 8.18 
 8.26 
 8.34 
 8.41 
 8.48 
 
 
Templet Planer. 
 
 168. ROTARY CUT BEVEL TEETH. 
 
 The most common method of forming the ! ehape of the tooth changes, while that of the 
 teeth of the bevel gear is by cutting them cutter is invariable. Therefore the result 
 
 from the solid blank by the use of the com- 
 mon rotary cutter. 
 
 The cutter should be shaped to cut the 
 tooth of the correct shape at the large end, and 
 the small end must be shaped either by an- 
 other cut with a different cutter, or with a file. 
 
 It is impossible to cut the tooth correctly 
 at both ends, for the simple reason that the 
 
 must always be an approximation depending 
 upon the personal skill and experience of the 
 workman. It is a too common practice to 
 make the teeth fit at the large ends, and to 
 increase the depth of the tooth toward the 
 point, so that the teeth will pass without 
 filing, but such teeth can be in working con- 
 tact only at the large ends. 
 
 ). THE TEMPLET GEAR PLANER. 
 
 The most common method of planing the 
 teeth of bevel gears is by means of devices 
 adapted to guide the tool by a templet that 
 has previously been shaped, as nearly as 
 may be, to the true curve. The arm that 
 carries the tool is hung by a universal joint 
 at the apex of the gear, so that all of its 
 strokes are radial, and a finger placed in the 
 line of the stroke of the cutting point of the 
 tool is held against the templet. There are 
 many different arrangements for the purpose, 
 but they are all founded on the same princi- 
 ples, and differ only as to details. 
 
 The invention of the templet gear planer is 
 commonly credited to George H. Corliss, 
 who patented it in 1849, and was the first to 
 use it in this country. But it was patented 
 in France, by Glavet, in 1829, and may be 
 even older. 
 
 It is largely used for planing the teeth of 
 heavy mill gearing, but has not been, and 
 cannot be, profitably applied to common 
 small gear work. Its product is, in any case, 
 superior to the rough cast tooth, but its accu- 
 racy is dependent on that of the templet, and 
 is therefore dependent on personal skill. 
 
9. THE SKEW BEVEL OEAR. 
 
 170. THE SKEW BEVEL GEAR. 
 
 When a pair of shafts are not parallel, and 
 do not intersect, they are said to be askew 
 with each other, and they may be connected 
 by a pair of skew bevel gears, having 
 straight teeth, which bear on each other 
 along a straight line. Such gears are to be 
 carefully distinguished from spiral gears, 
 used for the same purpose but having spiral 
 teeth bearing on each other at a single point 
 only. 
 
 We will endeavor to describe the skew 
 bevel gear so that its general nature can be 
 understood, but it is impossible to do so in 
 simple language. It is the most difficult ob- 
 ject connected with the subject. The theory 
 cannot even be considered as yet settled, for 
 writers upon theoretical mechanism do not 
 agree upon it, and there are points yet in 
 controversy. 
 
 In the theory of the bevel gear the surface 
 of reference is the spherical surface upon 
 which the tooth outlines are drawn, and upon 
 
 which the laws of their action may be 
 studied, for spheres of reference of two sep- 
 arate gears may be made to coincide so that 
 the lines upon one will come in contact with 
 those upon the other. For the spur gear, the 
 spheres become planes and the process is the 
 same. But for the skew bevel gear there is 
 no analogous process, for it is impossible to 
 imagine a surface of such a nature that it can 
 be made to coincide with a similar surface 
 when both are attached to revolving askew 
 shafts. There are spiral surfaces which will 
 approximately coincide, and are analogous to 
 the Tredgold tangent cones of bevel gears 
 (161), but any tooth action developed upon 
 such approximate surf aces must, of necessity, 
 be not only approximate, but also very diffi- 
 cult to define and formulate. 
 
 Of all the skew tooth surfaces that have 
 been proposed, there is but one, the Olivier 
 involute spiraloid, that can be proved to be 
 theoretically correct. 
 
 171. THE HYPOID. 
 
 The pitch surface of the ?kew bevel gear 
 is the surface known as the " hyperboloid of 
 revolution," and it is so intimately connected 
 with the subject that it must be thor- 
 oughly understood before going further. 
 The clumsy name may be abbreviated to 
 "hypoid." 
 
 If a line D, Figs. 152 and 153, called 
 a generatrix, is attached to a revolving 
 shaft A, so that it revolves with it, it 
 will develop or ' ' sweep up " the hypoid 
 H in the space surrounding the shaft. 
 A section of the surface by any plane 
 normal to the axis is a circle. The com- 
 mon normal to the generatrix and the 
 axis is the gorge radius G, and circular 
 section through that line is the gorge 
 circle. A section by a plane D, Fig. 152, 
 parallel to the axis, at the gorge distance 
 from the axis, will be the pair of straight 
 
 lines d and d ', Fig. 153, either one of which 
 is an element of the surface, and will form it 
 if used as a generatrix. A section by any 
 
 sections. 
 
 Fig. 152. 
 
 Hyperbolic sections. 
 
 Fig. 153. 
 
 other plane parallel to the axis will be a 
 hyperbola, to which the elements d and d' 
 
Rolling Hypoids. 
 
 are assymptotes, or lines which the curves 
 continually approach, but reach only at in- 
 finity. Fig. 153 shows at Q the hyperbolas 
 cut by the plane Q of Fig. 152, and at R 
 those cut by the plane R. The principal 
 hyperbola H is the only one with which 
 we are concerned . 
 
 The hypoid is best studied as projected 
 upon a plane parallel to the axis, as in Fig. 
 154, in which A is the projection of the axis, 
 d is that of the generatrix, dGA is the skew 
 angle, and //is the principal hyperbola. 
 
 When the skew angle and the gorge radius 
 are given, the hyperbola is easily constructed 
 by points. Any line ab is drawn normal to 
 the axis and the gorge distance be= Gg is laid 
 off from b, the distance ab is made equal to 
 ec, and a is then a point on the curve. The 
 curve is to be drawn through several points 
 thus constructed. 
 
 The hyperbola. 
 
 Fig. 154. 
 
 To draw a tangent to the curve at any 
 point a, draw a line am parallel to the 
 assymptote d, lay off mn equal to Gm, and 
 draw the tangent an. 
 
 172. THE PITCH HYPOIDS. 
 
 The utility of the hypoid as the pitch sur- 
 face of the skew gear depends upon the pe- 
 culiar property that any number of such 
 surfaces will roll together, and drive each 
 other by frictional contact with velocity ratios 
 in the proportions of the sines of their skew 
 angles, if their gorge radii are in the propor- 
 tions of the tangents of their skew angles. 
 
 It is required to construct a pair of rolling 
 hypoids that will transmit a given velocity 
 ratio between two shafts that are set at a 
 given angle with each other. In Fig. 155, 
 A and .Bare the given axes, and AGB the 
 given shaft angle. The directrix D is to be 
 so drawn that the sines of the skew angles 
 AGD and BGD are in the proportion of the 
 given velocity ratio, and this is best done by 
 drawing lines parallel to the axes, at distances 
 from G that are in the given ratio, and drawing 
 the directrix through their intersection D. 
 
 In the figure the axes are situated one over 
 the other at a distance GH called the gorge 
 distance, and the directrix D is situated be- 
 tween them so as to pass through the gorge 
 line and divide the gorge distance into gorge 
 radii, #TFand HW, which are in proportion 
 to the tangents of the skew angles. This is 
 
 Pitch hypoids. 
 
 Fig. 155. 
 
 best done by drawing cd normal to GD in 
 any convenient position, laying off the gorge 
 distance ce at any convenient angle with cd, 
 and drawing de and gf parallel to it; cf will 
 be the goige radius GW for the axis GA, 
 
Rolling Hypoids. 
 
 97 
 
 and fe will be the gorge radius HW for the 
 axis OS. 
 
 Then, if the directrix, thus situated, is at- 
 tached first to one shaft and then to the 
 oiher, and used as a generatrix, it will sweep 
 up a pair of pitch hypoids that will be in 
 contact at the directrix, and which will roll 
 on each other. 
 
 They will not only roll on each other in 
 contact at the directrix, but they will also 
 have a sliding motion on each other along 
 that line, the two motions combining to form 
 a resulting motion that must be seen to be 
 understood. It is this sliding motion that 
 makes all the difficulty in the construction of 
 the teeth, for they must be so constructed 
 as to allow it. It is also the cause of the 
 great inefficiency of such teeth in action, for 
 any possible form must have a lateral sliding 
 motion, with the consequent friction and 
 destruction. 
 
 If we draw two diameters mn and m'n' 
 through the same point C on the directrix, 
 they will be the diameters of circles that will 
 touch each other while revolving, and may 
 
 be called pitch circles. If they are thin, and 
 provided with teeth in the given velocity 
 ratio, they will drive each other with a con- 
 tact that is approximately correct, and if 
 there are several pairs of such thin gears set 
 so far apart that they do not interfere with 
 each other, they will serve light practical 
 purposes fairly well. 
 
 If a face distance CE is laid off on the 
 directrix and another pair of pitch circles 
 constructed, the frustra of the hypoids in- 
 cluded between the circles may be called 
 pitch frustra, and they will roll together in, 
 contact at the directrix. 
 
 It is to be noticed that the pitch diameters 
 thus determined are not, as in spur and bevel 
 gearing, in the inverse proportion of the 
 velocity ratio of the axes, and therefore if 
 one diameter of a pair of skew gears to have 
 a given velocity ratio is given, the other must 
 be constructed. When the skew angles are 
 equal, the pitch diameters are equal, but 
 otherwise the proportion cannot be expressed 
 in simple terms, and must be determined by 
 making the drawing. 
 
 173. THE LOCUS OF AXES. 
 
 The rolling hypoids may be examined 
 from another and most interesting point of 
 view. In Fig. 156 the gorge line G is nor- 
 mal, and the directrix D is parallel to the 
 plane of the figure. The plane P is normal 
 to the directrix, and below is a front view of 
 it. On the plane P draw any straight line L 
 through the directrix. From any two points 
 a and b on this line draw lines A and B 
 normal to the gorge line G, and they will 
 be axes of pitch hypoids that will roll on 
 each other in contact at the directrix. 
 
 Axes drawn from all points of the line L 
 will form a continuous surface called a "hy- 
 perbolic paraboloid," which will be the locus 
 of all the axes of a set of hypoids that will 
 mil together collectively in contact at the 
 directrix. 
 
 The locus of axes. 
 
 Fif/. 156. 
 
98 
 
 Olivier Skew Bevel Gears. 
 
 174. CYCLOID AL TEETH FOR SKEW GEARS. 
 
 As any number of hypoids, on axes in the 
 same locus of axes, will roll together in 
 either external or internal contact at the 
 directrix, it might be supposed that a tooth 
 similar to the cycloidal tooth for bevel and 
 spur gears might be formed by an element in 
 an auxiliary hypoid X, Fig. 156, which rolls 
 inside of one and outside of the other pitch 
 bypoid. 
 
 This is such a plausible supposition that it 
 long passed for the truth, not only with its 
 inventor, the celebrated Professor Willis, but 
 with many other prominent writers, until 
 shown by MacCord to be wrong. It serves 
 to illustrate the confusion in which the whole 
 subject has been and now is. 
 
 The tooth surfaces which Willis supposed 
 to be tangent at the generating element of 
 the auxiliary hypoid really intersect at that 
 line, and Fig. 157 shows a pair of such in- 
 tersecting teeth. The curves of the figure 
 were drawn by an instrument made for the 
 
 Cycloidal tooth Curves 
 Fig. 157. 
 
 purpose, and are, therefore, a better proof of 
 the intersection of the surfaces than solid 
 teeth would be. 
 
 The cycloidal tooth is examined at consid- 
 erable length, and the instrumental proof of 
 its failure is given in the AMERICAN MACHIN- 
 IST for September 5th, 1889. 
 
 175. INVOLUTE TEETH FOR SKEW BEVEL GEARS. 
 
 Herrmann's form of the Olivier spirrloidal 
 tooth is constructed with the directrix of 
 (172) as a generatrix, as follows : 
 
 Suppose that cylinders are constructed 
 upon the gorge circles of a pair of pitch 
 hypoids, Fig. 158, and suppose a plane K to 
 be placed between them. This plane will be 
 tangent to both cylinders, and will contain 
 the directrix, and if moved will move the 
 ^cylinders as if by friction. Then imagine 
 ithe plane to move in a direction normal to 
 the directrix, and it will carry that directrix 
 with it as a generatrix always parallel to its 
 first position. It will sweep up the spiraloid 
 tooth surfaces 8^ and S a imperfectly shown 
 by the figure, or by Fig. 159, and they will 
 be correct tooth surfaces always in tangent 
 contact. 
 
 Fig. 159 shows #, full involute tooth sur- 
 face or " spiraloid," and Fig. 160 is a full 
 Olivier skew bevel gear. 
 
 The particular involute skew tooth above 
 described is not the only possible form, but 
 it has the least possible sliding action, and is, 
 therefore, the best. 
 
 Involute tooth action 
 Fig. 158. 
 
 If the plane K has a generatrix line at any 
 angles with the axes of the gears, and is 
 moved in a direction at right angles with that 
 line, correct tooth surfaces will be swept up. 
 In fact, any two spiraloids on any two cylin- 
 ders will work correctly with each other, and 
 therefore any two spiraloidal gears of the 
 same normal pitch will work correctly to- 
 gether. 
 
Beetle's Gears. 
 
 Olivier Involute skew Bevel Gear 
 Fig. 160. 
 
 176. HERRMANN'S LAW. 
 
 Herrmann gives a law, and claims it to be 
 universal, to the effect that the skew bevel 
 tooth must be swept up by a straight line 
 generatrix that is always parallel to the direc- 
 trix. He mentions the Olivier tooth, and 
 claims that it cannot be correct, evidently not 
 understanding that Olivier's theory clearly 
 includes the form he himself proposes. His 
 form of tooth, claimed to be the only possi- 
 ble form, is really only the best form of the 
 Olivier tooth. 
 
 We will not undertake to state wherein 
 Herrmann's law is incorrect, but that it is 
 wrong is clearly shown by the most con- 
 
 vincing of all proofs, the reduction to prac- 
 tice. Beale, for the Brown & Sharpe Mfg. 
 Co., has made working Olivier gears on a 
 large scale, which are directly contrary to 
 Herrmann's law, but which work perfectly, 
 and demonstrate the truth of Olivier's theory 
 in a way that admits of no question. 
 
 Indeed, the closest possible scrutiny of 
 Olivier's theory, without the aid of Beale's 
 experimental work, fails to detect a flaw in 
 it. Herrmann's condemnation of it is not 
 based on direct consideration, but simply on 
 the fact that it does not agree with his own 
 law. 
 
 177. BEALE'S SKEW BEVEL GEARS. 
 
 . Beale's gears are the same as Olivier's gears 
 in general theory, but the improvement in 
 practical form and application is so great that 
 they may be considered a distinct invention. 
 
 Fig. 161 is a section through one axis, and 
 at right angles to the other axis of a pair of 
 Beale gears. Both surfaces of the teeth are 
 true Olivier spiraloids of Fig. 159, and the 
 gears will run in either direction. When 
 corrected for interference they are reversible, 
 like spur or bevel gears. The gorge cylin- 
 ders are tangent to each other, and are so cut 
 away inside as to allow the teeth of the ma- 
 ting gear to pass. 
 
 The Olivier theory requires the teeth to 
 
 ! vanish at the gorge, as shown by the single 
 j full tooth of Fig. 160, in order to pass, while 
 i the Beale gear is cylindrical in form as a 
 I whole, and passes the full tooth at the gorge, 
 with action over its whole surface. The 
 ; difference is practically very great. 
 
 When in action a pair of uncorrected Beale 
 ! gears must be placed as shown by Figs. 161 
 to 163, and Fig. 169, with one end of each at 
 I the gorge, and they will not run together if 
 placed at random. If either gear extends 
 beyond the gorge line there is an interfer- 
 ence between the involute spiraloids which 
 is the same in kind as that between the in- 
 volute curves of common spur gear teeth. 
 
100 
 
 Beetle's Gears. 
 
 In comparison, the Beale gear is taken so 
 near the gorge that it is practical and service- 
 able, having large teeth and small obliquity. 
 
 Fig. 161. 
 
 Section of Beale's Gears 
 
 Each gear can drive in but one direction, 
 depending upon the position of tbe gear and 
 the direction of the spiral, and if turned 
 backwards the action is intermit- 
 tent and practically useless. The 
 gears must be placed as in Fig. 
 162 for right-hand spirals, and as 
 in Fig. 163 for left-hand spirals, 
 and the direction of the rotation 
 is shown by the arrow D, when 
 the gear bearing the arrow is the 
 driver. 
 
 But, if the direction is to be re- 
 versed, the gears can be arranged gorge c y u. 
 as in Fig. 162a, or as in Fig. 163a. 
 This resetting is the same in effect 
 as turning the gears half around, 
 except that opposite sides of the 
 teeth are in contact in the two 
 positions of the same gears. 
 
 If, however, the interfering parts 
 of the tooth surface are removed, 
 the gears will run together per- 
 fectly and in either direction when 
 put together at random as in Fig. 
 168. 
 
 In the cases shown by the figures, 
 the spirals make the angles of forty- 
 five degrees with the shafts, con- 
 trary to Herrmann's law, but the 
 action will be smoother, and the 
 sliding of the teeth on each other 
 will be less, if Herrmann's angles 
 are adopted. These angles are the 
 same as those made by the conical 
 face of common bevel gears of the 
 same proportion with the axes, and 
 the best angles for the two-to-one 
 proportion of figures are those of 
 the line X of Fig. 162, making the 
 angles 26 34' and 63 26' with the 
 axes. 
 
 The Olivier gear of Fig. 160 is 
 perfect in theoretical action, but the 
 teeth must be taken so far from the 
 gorge that the obliquity of the ac- 
 tion is excessive, and the arc of 
 action is so limited that the teeth* 
 
 must be small. The sliding and wedging The working length of each gear is as 
 action is so great that the gears are practically determined by the line L of Fig. 161, and 
 useless. ! the whole surface of the tooth within that 
 
 1 Fig. 162a. 
 
 
 Beale Skew Bevel Gears. 
 
 Right Hand Spiral at 45 
 
 i Fiy. 163a. 
 
 Left Hand Spiral\at 45 
 
Approximate Skew Teeth. 
 
 101 
 
 limit will be swept over "by the line of 
 contact. It the length of each gear is equal 
 to the radius of the other gear it will always 
 be long enough. 
 
 The action between two gears will be at 
 the straight, equidistant, parallel lines a a, 
 Figs. 161 and 162, in the plane of action 
 tangent to both gorge cylinders. 
 
 The shafts of a pair of skew bevel gears 
 should be as near together as possible, just 
 far enough apart to allow the shafts to T ass, 
 so as to avoid the excessive sliding action. 
 In that case both Beale and Olivier gears 
 are practically useless, the former on ac 
 count of the small size of the teeth, and the 
 latter on account of the great obliquity of 
 the action. 
 
 The common bevel gear becomes the spur 
 gear when the shaft angle becomes zero, but 
 the analogous transformation of the skew 
 
 bevel gear into a bevel gear by reducing the 
 gorge distance to zero is not possible. 
 
 The skew bevel gear becomes a spur gear 
 if we imagine the axes to be brought parallel 
 by removing the gorge to an infinite distance, 
 for the spiraloids on the gorge cylinders then 
 become involute surfaces on base cylinders. 
 But, and it is a curious circumstance, when 
 the shafts are brought parallel by imagining 
 the shaft angles to become zero without 
 changing the position of the gorge, the gorge 
 cylinders become tangent and the gears do 
 not become spur gears. 
 
 Involute skew bevel gears do not appear 
 to have any possible adjustment correspond- 
 ing to the adjustment of the shaft distance 
 of involute spur gears, or of the shaft angle 
 of involute bevel gears, (56) and (157). 
 
 Beale's gears are fully described in the 
 AMERICAN MACHINIST of Aug. 28th, 1890. 
 
 178. TWISTED SKEW TEETH. 
 
 As no two surfaces of reference attached 
 to a pair of revolving askew shafts can be 
 made to coincide with each other, like the 
 planes of spur gears or the spheres of bevel 
 gears, the twisted or spiral tooth is impossi- 
 ble, for such a tooth would not permit the 
 required sliding action. 
 
 But, if a line is drawn upon one pitch 
 hypoid of a pair, a corresponding line may 
 be drawn upon the other, as if the given 
 
 line could leave an impression. Therefore 
 a tooth having edge contact (100) may be 
 constructed, provided the twist is such that 
 one pair of lines always crosses the directrix. 
 These teeth are purely imaginary, but if the 
 edges are thick they will have an action upon 
 each other, at a single point of contact, that 
 is closely approximate to the theoretical 
 action, and they will serve the general pur- 
 pose, if the power carried is inconsiderable. 
 
 179. APPROXIMATE SKEW TEETH. 
 
 As the true involute skew tooth is diffi- 
 cult to construct, and in many cases is of 
 small practical utility, and all other proposed 
 forms are incorrect, it follows that we must 
 depend for practical purposes mostly upon 
 some approximation, provided it is not pos- 
 sible to avoid the skew gear altogether. 
 
 The blanks can be constructed by a definite 
 process Construct the frustra of the pitch 
 hypoids by the method of (172) and Fig. 155. 
 Consider the end sections mn and pq as ends 
 of a frustrum of a pitch cone, and on this 
 
 pitch cone construct the blank gear exactly 
 as for a common bevel gear. 
 
 Having constructed the blanks, the general 
 direction of the tooth is to be marked upon 
 them. Mount each blank as in Fig. 155, 
 with its axis parallel with a plane surface Z. 
 Set a surface gauge with its point at the line 
 of the directrix W, and with it mark the po- 
 sition of the directrix on the pitch line at 
 each end of the blank. 
 
 The tooth must then be cut so that its 
 direction follows the directrix, and it is to be 
 
102 
 
 Substitute Skew Trains, 
 
 noticed that it is not only askew with the 
 axis, but that the tooth outline twists. The 
 appearance of the tooth on either rim, as 
 well as upon any section between the two 
 rims, is the same as upon a common bevel 
 gear, symmetrical, and not canted to one 
 side, as is sometimes taught. 
 
 The approximate tooth is very similar to 
 the twisted bevel tooth, see (155) and (99), 
 with the twist following a straight line set 
 askew with the axis, and as the line of the 
 twist is not parallel with the conical face, 
 that face should be as short as possible. 
 
 The process of cutting is not capable of 
 description, for it depends upon personal 
 skill and judgment. The workman must 
 imagine that he sees the twisted cut in the 
 body of the blank, and then must persuade 
 
 the cutter to follow it. Gear cutting ma- 
 chines are seldom so made that the cutter 
 can be turned while it feeds, and theretore 
 i it must be set to a medium path, and reset 
 | two or three times to get the desired form. 
 The beginner will fail the first time, and 
 there may be several failures. The best pos- 
 sible result can be bettered with a file, after 
 running the cut gears together to find where 
 they interfere. 
 
 In the hands of a skillful workman, a pass- 
 able approximation can be reached, and if the 
 axes are very near together compared with the 
 diameters of the gears, the teeth are small, and 
 the face is short, the result is satisfactory. 
 In fact, when the conditions are favorable, 
 this approximate tooth is more serviceable 
 than the true tooth. 
 
 Substitute train. 
 
 Fig. 165. 
 
 180. SUBSTITUTES FOR THE SKEW BEVEL 
 GEAR. 
 
 When there is a chance to introduce an 
 intermediate shaft, the skew bevel gear can 
 be avoided, and it is not only better, but 
 cheaper to avoid the objectionable gear at 
 the cost of the extra mechanism. 
 
 Fig. 164 shows how to place an inter- 
 mediate shaft and gears, when the shafts are 
 so far apart that the shortest or gorge 
 distance can be used. Fig. 165 shows how 
 the skew shafts can be connected by one 
 pair of bevel gears and one pair of spur 
 gears, and that is the best device for general 
 purposes. 
 
 Substitute train. 
 Fig. 164. 
 
Skew Pin Gears. 
 
 105 
 
 181. SKEW PIN GEARING. 
 
 Fig. 166 shows a pair of skew pin gears 
 commonly called face gears. They will run 
 together with a uniform velocity ratio if 
 they are exactly alike and at right angles 
 with shafts at a distance apart equal to the 
 diameter of the pins. 
 
 If the gears are not alike or not at right 
 angles, the teeth on one may be straight 
 pins, but those on the other must be shaped 
 to correspond. 
 
 Such gears are objectionable because they 
 have but a single point of contact for each 
 pair of teeth, at which they slide on each 
 other with great friction. 
 
 Face gearing in its various forms is thor- 
 oughly examined in MacCord's Kinematics. 
 At the present day they are not in use, and 
 do not deserve much study. 
 
 Skew pin Ttevel gears. 
 
 Fig. 206. 
 
 Beale gears corrected for interference. 
 Fig. 168. 
 
 Formation of Beale gear. 
 Fig. 167. 
 
 UncorrecUd 
 Beale gears. 
 
 Fig. 169. 
 
INDEX. 
 
 SECTION. 
 
 Addendum 39 
 
 Arc of Action 25 
 
 Axoids 7 
 
 Backlash 41 
 
 Base Circle 16, 54 
 
 Beale's Gears 177 
 
 Beale's Treatise 4 
 
 Begin at the Beginning 3 
 
 Bevel Gears 6,7,8, 152, 154 to 169 
 
 Box's Mill Gearing 4, 51 
 
 Brown & Sharpe Co.'s Treatise 4 
 
 Cast Gearing, Friction 49 
 
 Chart for Bevel Gears 163 
 
 Chordal Protractor 166 
 
 Clearance 40 
 
 Complete Tooth 22 
 
 Conic Pitch Lines 131 
 
 Conjugator 125 
 
 Consecutive Action 12 
 
 Construction by Points 23, 57 
 
 Cusp 16, 17, 54 
 
 Cutter Limit 88 
 
 Cutter Series 45 
 
 Cycloidal System 31, 76 to 90, 158, 174 
 
 Dedendum 39 
 
 Demonstrations Avoided 2 
 
 Disputed Points 9 
 
 Double Secondary Action 21 to 78 
 
 Double Terminal Action 18 
 
 Edge Teeth 100 
 
 Efficiency in Transmission 49, 73, 112 
 
 Elliptic Gears 136 to 153 
 
 Elliptic Gear Cutting Machine 150 
 
 Elliptic Pitch Lines 131 
 
 Elliptographs 138 
 
 Epicycloidal Teeth 76 
 
 Equidistant Series 45 
 
 Extent of the Subject 2 
 
 Fillet 28, 44 
 
 Friction 26 
 
 Friction of Approach , 48, 49 
 
 Gear Cutting Machines 27, 28, 29, 
 
 94, 102, 105, 122, 125, 150, 159, 168, 169 
 
 Gear Teeth, Theory 1 to 34 
 
 " Spur 35 to 52 
 
 " Involute 53 to 75 
 
 Cycloidal 76 to 90 
 
 Pin 91 to 97 
 
 Spiral 98 to 110 
 
 Worm Ill to 125 
 
 " Irregular 126 to 135 
 
 Elliptic 136 to 153 
 
 Bevel 154 to 169 
 
 SkewBevel. 170 to 181 
 
 SECTION. 
 
 Gears, Beale's 177 
 
 Bilgram's 159 
 
 Composite 133 
 
 Elliptic Bevel 152 
 
 Herrmann's 175 
 
 Hindley 122 
 
 Hooke's 98 
 
 Hyperbolic 181 
 
 Lantern t>3 
 
 Mortise 47 
 
 Parabolic 181 
 
 Pin Bevel 1(50 
 
 " Skew Pin 1S1 
 
 Stepped JH 
 
 Herrmann's Erroneous Law 17(1 
 
 Herrmann's Treatise 4 
 
 Hindley Worm Gear 122 
 
 Robbing Machines 124, 125 
 
 Hobbing Worm Gears 114, 115 
 
 Hooke's Gears 98 
 
 Horse Power of Gears 51, 52 
 
 Hunting Cog 46 
 
 Hyperbolic Pitch Lines 131 
 
 Hyperboloid of Revolution 7, 171 
 
 Hypoid 171 
 
 Integrater, Odontoidal 34 
 
 Interchangeable Odontoids 14, 22 
 
 Interchangeable Rack Tooth 22 
 
 Interference 1 6, 55 
 
 Interference, Internal 79 
 
 Interference, Worm 117 
 
 Internal Contact 15 
 
 Internal Double Action 21 
 
 Internal Gears 64 
 
 Internal Friction 49 
 
 Involute System 31, 53 to 75, 157, 175 
 
 Irregular Pitch Lines 126 to 128, 133 
 
 Kinematics 
 
 Klein's Treatise 
 
 Klein's Odontograph 
 
 85 
 
 Law of Tooth Contact 11, 12 
 
 Limiting Numbers of Teeth 66 to 72, 90 
 
 Limit Line 16 
 
 Line of Action 13 
 
 Literature 4 
 
 Logarithmic Pitch Lines 132 
 
 Logarithmic Spiral 32, 75, 132 
 
 MacCord's Treatise 4 
 
 Molding Construction 27 
 
 Mortise Gear 47 
 
 Multilobes 130 
 
 Natur Tooth Action 20 
 
 Normals 11 
 
 Normal Surfaces 8 
 
 
SECTION. 
 
 Obliquity of Action 26, 74r, 87 
 
 Octoid Teeth 31, 159 
 
 Odontics 6 
 
 Odontographs 43, 59, 62, 82, 83 
 
 Odontoid and Line of Action 33 
 
 Odontoids 12 to 34 
 
 Olivier's Spiraloidal Teeth 175 
 
 Parabolic Pitch Lines 131 
 
 Parabolic System 31 
 
 Pin Tooth System 91 to 97, 160, 181 
 
 Pitch, Circular 35, 119 
 
 Pitch Cylinders 10 
 
 Pitch Diameters, Table 35 
 
 Pitch, Diametral 36, 120 
 
 Pitch Lines 11 
 
 Pitch Point 11 
 
 Pitch Surfaces 7 
 
 Pitch Table 37 
 
 Planing Construction 28, 29 
 
 Quick Return Motion 149 
 
 Radial Flank Teeth 89 
 
 Rankine's Treatise 4 
 
 Rack Originator 30, 31 
 
 Retrograde Action 18 
 
 Reuleaux's Treatise 4 
 
 Robinson's Odontograph 86 
 
 Rolled Curve Theory 32, 75, 81, 91 
 
 Roller Teeth 93 
 
 SECTION. 
 
 Secondary Action 21, 78 
 
 Segmental System 31 
 
 Sellers' Experiments 49, 112 
 
 Skew Bevel Gears 6, 7, 8, 170 to 181 
 
 Smallest Pitch Circle 17 
 
 Speed of Point of Action 19 
 
 Spiral Gears 98 to 103 
 
 Spur Gears 6, 7, 8, 35 to 153 
 
 Stahl & Wood's Treatise 4 
 
 Standard Teeth 42, 58, 80 
 
 Stepped Teeth 98 
 
 Strength of Teeth 49 
 
 Systems of Teeth 31 
 
 Templets 38 
 
 Terminal Point 18 
 
 Twisted Teeth .99, 101, 102, 15ft 
 
 Unsymmetrical Teeth 22 
 
 Variable Speed 148 
 
 Willis' Odontograph 84 
 
 Willis' Treatise 4 
 
 Wooden Teeth 47 
 
 Worm Gears ..Ill, 113, 114, 116 
 
 Yale & Townes Experiments 112 
 
 MICHIGAN BRICK AND TILE MACHINE CO. 
 
 MORENCI, MICH., Nov. 24th, 1891. 
 
 GEORGE B. GRANT. Dear Sir : Two years ago you sent me one of your books on Teeth of 
 Gears, and I have replaced all of the gears in our brick machinery with new ones from your in- 
 volute odontograph table. I find that we now have the finest cast gears in the world. I do not 
 understand why pattern makers don't catch on to your book. It is a sight to see the gear pat- 
 terns that are made by some men who are called good pattern makers. 
 
 O. S. STURTEVANT, 
 
 Pattern Maker for M. B. & T. M. Co. 
 
ADVERTISEMENT 
 
 IT'S NO USE TRYING TO GET 
 
 ELLIPTIC GEARS 
 
 OF ANY ONE BUT 
 
 - \ 
 
 CEO. B. GRANT, 
 
 LEXINGTON, MASS. 
 PHILADELPHIA, PA. 
 
-n] 
 O 
 ./*/ 
 
 SHi/^/ 
 
 UNIVERSITY OF CALIFORNIA LIBRARY 
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