THE LIBRARY 
 
 OF 
 
 THE UNIVERSITY 
 
 OF CALIFORNIA 
 
 SANTA BARBARA 
 
 COLLEGE 
 
 PRESENTED BY 
 
 William E. Roberts

 
 THE 
 
 TEETH OF SPUR WHEELS; 
 
 THEIR CORRECT FORMATION 
 
 IN 
 
 THEORY A^D PRACTICE. 
 
 BY PROF. C. W. MAcCORD. 
 
 HARTFORD, CONN. : 
 
 PUBLISHED BY THE PRATT & WHITNEY COMPANY, 
 RS ov MACHINISTS' TOOLS, GUN AND SBWING MACHINE MACHINERY, &c., &C.
 
 PREFACE. 
 
 As a preliminary to the description of the machines for the accurate 
 formation of cutters for spur wheels, given in the second part of this 
 treatise, it seemed appropriate to explain the manner of laying out the 
 teeth. And in the belief that it may be acceptable to many who are 
 interested in the subject, we have endeavored to give, in as simple, clear 
 and brief a manner as possible, the reasons and the proof of every step in 
 the construction. The method of drawing rolled curves by means of 
 tangent arcs, which we believe to be the most accurate and expeditious 
 known, may be new to some; and this, as well as Prof. Rankine's elegant 
 graphic processes relating to circular arcs, will be found applicable to 
 many other purposes, and exceedingly useful. To which we may add, 
 finally, the hint that the same is true of the principles and the methods 
 made use of in the demonstrations. 
 
 C. W. MAC-CORD. 
 STEVENS INSTITUTE OF TECHNOLOGY, 
 Hoboken, N. J., Jan. 28, 1881.
 
 \ VS (^5 UNIVERSITY OF CALIFORNIA 
 
 SANTA BARBARA L^LLESE LIBRARY 
 
 73256 
 
 PART I. 
 
 THE TEETH OF WHEELS. 
 
 GENERAL PRINCIPLES. 
 
 The proper action of many pieces of mechanism depends so largely 
 upon that of spur wheels, that any means of effecting a radical improve- 
 ment in the making of such wheels cannot but be of interest and im- 
 portance. 
 
 There was a time when the teeth of wheels were made in rude hap- 
 hazard ways, of almost any shapes that would permit them to engage, 
 with a mistaken idea that they would wear themselves into correct forms. 
 The machine was expected not only to do its own proper work, but partly 
 to finish itself; small wonder, then, that it failed to do either of these 
 things well. Naturally, these crude methods gave place to better ones. The 
 mechanician perceived the necessity of greater care in making the teeth 
 of proper form ; the mathematician soon became interested in the problem 
 of determining what forms were proper, and the results of their combined 
 efforts, leave little to be desired in relation to the latter. 
 
 And as little would seem to be left in regard to the former after the 
 introduction of the gear-cutting engine, by which, if the milling cutter be 
 of the correct outline, all the teeth of a wheel are made with the utmost 
 regularity and precision. But on closer consideration, it will be seen that 
 something is yet lacking in reference to the formation of the cutter itself.
 
 It is one thing to know what its outline should be, but quite another thing 
 to make it so. 
 
 The process most extensively employed involves, 1st, the laying out of 
 the required curve ; 2nd, the filing of a template to that exact form, and, 
 3rd, the turning of the cutter to fit the template. 
 
 In some cases a specially-contrived apparatus has been used for me- 
 chanically tracing the curve by continuous motion, but, until recently, the 
 two remaining steps have been executed by hand, which makes the per- 
 fectly accurate formation of a cutter, especially if it be a small one, vexy 
 difficult, and its exact duplication still more so. 
 
 The time has now come when all this ought to be changed. No one 
 who considers for a moment the vast numbers of accurate machines em- 
 ployed in the various industrial arts, and of others equally accurate, em- 
 ployed in making them, can fail to perceive the advantages over the 
 system above described, of one in which the template is not merely lined 
 out, but cut out to the true form, and the contour of the milling cutter, be 
 it large or small, is made to correspond to that of the template, by mechan- 
 ism nearly automatic. Of such a system, and of the means by which 
 these results are effected, we propose to give a detailed description. Before 
 entering upon this, however, we shall briefly explain the principles upon 
 which the correct forms of the teeth depend, and the method of laying 
 out epicycloidal teeth in outside gear. 
 
 GRAPHIC REPRESENTATION OF MOTION. 
 
 The motion of a point at any instant may be represented in magnitude 
 and direction by a right line. 
 
 It is true that the path of the point may be a curve of any kind ; but 
 at any given instant it can occupy but one position, and its direction "Will
 
 be that of the tangent to the path at that point. The length of that tan- 
 gent may, evidently, be made to indicate the velocity; therefore the motion 
 is fully represented. 
 
 COMPOSITION AND RESOLUTION OF MOTION. 
 
 The composition (or finding the resultant) of two motions, is effected 
 as shown in Fig. 1. Suppose the point A, to receive simultaneously two 
 
 Fig. 1. 
 
 impulses, the motions due to which are represented in velocity and direc- 
 tion by AB, AC. Draw BD parallel to AC, and CD parallel to AB : then 
 AD, the diagonal of the parallelogram thus formed, will represent the re- 
 sultant motion in both direction and velocity. That is to say, the point A 
 will go to D, in the same time in which it would have reached either B or 
 C, had it received but one of the impulses. 
 
 Evidently, if one component and the resultant be known, the other 
 component may be found in a similar manner. If, for instance, we know 
 that AD is the resultant of two components, one of which is AC, draw 
 CD; then the other component must have the direction AB parallel to CD, 
 and its magnitude is found by drawing DB parallel to AC. 
 
 The resolution of motion is the converse of composition : thus, it is evi- 
 dent that the motion AD in Fig. 1 may be separated into the components
 
 from which it was derived, by drawing the parallels DC, AB, in one direc- 
 tion, and DB, AC, in the other; by which the original parallelogram is 
 reconstructed. 
 
 But again, AD may be the diagonal of a great number of other 
 parallelograms : from which we see that a given motion may be resolved 
 into two components, having any directions we please to assign. 
 
 ANGULAR VELOCITY AND VELOCITY RATIO. 
 
 This term angular velocity is applied only to circular motion, like that 
 of a wheel revolving on its axis. Every point in the revolving body turns 
 through the same angle in the same time, whatever be its distance from 
 the axis: thus in Fig. 2, the point A goes to D, in the same time that it 
 
 Fig. 2. 
 
 takes the point B to reach E. Clearly the arcs AD, BE, represent the 
 linear velocities of the moving points; but AD is as many times greater 
 than BE, as AC is greater than BC. If then we divide AD by
 
 BE by BC, either quotient may be taken as the measure of the angular 
 motion represented by A CD; or, in general, we say that 
 
 linear velocity 
 
 angular velocity = -. 
 
 radius 
 
 If we consider the motion of a revolving point at a single instant 
 only, its direction is that of the tangent to its circular path. In represent- 
 ing it, then, we set off the linear velocity perpendicular to the radius 
 through the point, as AM, BN. Drawing CM, it will be observed that the 
 angle ACJIfis not the same as A CD. Nor should it be, since the former 
 represents what is happening at a given instant, the latter a motion con- 
 tinuing through a period of definite duration. 
 
 The velocity ratio of two revolving bodies, at any instant, is simply the 
 quotient obtained by dividing the angular velocity of one by that of the 
 other, at the given instant. If this quotient be the same at every instant, 
 the velocity ratio is said to be constant; as in the case of two pulleys 
 driven by a belt which does not slip if one be half the size of the other, 
 it will always turn twice as fast, whether the actual speed be uniform 
 or not. 
 
 DETERMINATION OF VELOCITY RATIO. 
 
 In Fig. 3, C and D are fixed centers, about which turn the two curved 
 levers, CH, DK, which touch each other at A. Through this point draw 
 TV 1 the common tangent, and -<?WVthe common normal, of the two curves, 
 and also AC and AD, the radii of contact. If CH turn in the direction 
 indicated by the arrow, DK will be driven in the opposite direction ; we 
 now wish to find the angular velocity ratio of the two motions.
 
 8 
 
 The point A, of the driver CH, must move in a direction perpendicular 
 to CA y and we will suppose its linear velocity at the instant to be repre- 
 sented by AB, which may be resolved into the components AP, AM; the 
 first is the effective component of rotation, the latter being the sliding 
 component. 
 
 Fig. 3. 
 
 The point A, of the follower D K, must go in a direction perpendicular 
 to AD, and its velocity AL must be such as to have the same normal com- 
 ponent AP, the tangential or sliding component being AR. 
 
 We perceive, then, that when one revolving piece drives another by 
 contact, the motion is transmitted in the direction of the common normal, 
 which is therefore called the line of action. 
 
 Now draw CE and DF perpendicular to NN, CD the line of centrrs,
 
 cutting NN at 7, also CB, DL. We shall thus have three pairs of similar 
 triangles, viz. : 
 
 CAE is similar to ABP, 
 DAF " " ALP, 
 CIE " " DIP. 
 
 Let the angular velocities of the driver and the follower respectively 
 
 lin. vel. 
 be represented by v,d ' ; then since ang. vel. , 
 
 radius 
 we shall have 
 
 AB AP 
 
 CA' CE'' 
 
 and 
 
 AL AP 
 
 qj __j, 
 
 DA DF 
 
 whence 
 
 v DF DI 
 
 ' v' CE Cf 
 
 That is to say: The angular velocities are inversely proportional to the 
 perpendiculars let fall from the centers of motion upon the line of action. 
 
 Or otherwise : The angular velocities are inversely proportional to the 
 segments into which the line of centers is cut by the line of action. 
 
 CONDITION OF A CONSTANT VELOCITY RATIO. 
 
 If it be required that the velocity shall remain constant, the second of 
 the values, above deduced, indicates a condition which the curves must
 
 10 
 
 satisfy, viz. : they must be of such forms that their common normal shall 
 always cut the line of centers at the same point. For the line of centers is of 
 fixed length, whence the segments into which it is cut must always be the 
 same, in order to maintain a constant ratio. 
 
 Now the outlines of the teeth of spur wheels must be curves which 
 O//7/ transmit rotation with a constant velocity ratio, and it will readily be 
 seen that parts of the curved levers in the immediate neighborhood of the 
 point of contact A, Fig. 3, might be of the forms proper for teeth of wfctrels 
 whose centers are Cand D. In the figure, the sliding components of ^he 
 motions AS, AL, are respectively AM, AR; these lie in the same direc- 
 tion, but are not equal ; so that at this instant, the levers are sliding upon 
 each other with the velocity RM, equal to their difference. There would 
 be no sliding if these were of the same length as well as in the same direc- 
 tion ; but since the normal component AP, is the same for each motion, 
 this can only happen when the resultants AJB, AL, also coincide. And 
 these being perpendicular to the radii of contact, cannot coincide unless 
 CA, AD, lie in one right line, which must, of course, be CD. That is to 
 say, there will be more or less sliding, except at the instant when the point of 
 contact is on the line of centers. 
 
 NATURE OF ROLLING CONTACT. 
 
 The finding of curves which will satisfy the above condition, and also 
 the best means of drawing them, depend upon a feature of perfect rolling 
 contact, best seen by a study of that which is not perfect. The polygon, 
 in Fig. 4, rolls along the fixed right line with a hobbling motion ; the point 
 A is at rest, and the whole figure turns about it as a center until B comes 
 into LM at D, then about 13, and so on ; the perimeter of the polygon 
 measuring itself off upon the line. If the number of sides be increased,
 
 11 
 
 LAD M 
 
 Fig. 4 
 
 the hobbling will be diminished, and if the number become inconceivable, 
 it will become imperceptible. The broken outline then becomes the 
 dotted curve, tangent to the line, and the change from one center of rota- 
 tion to another goes on continuously. But this does not alter the facts, 
 that at any instant the point of contact is at rest, and that every point in 
 the figure is at the instant turning about that point of contact as a fixed 
 center. 
 
 DRAWING OF ROLLED CURVES. 
 
 In pig. 5, AA is a curved ruler fixed to the drawing board, and BB is 
 a free one rolling along it. Let a pencil be fixed to and carried by the 
 latter, either in the contact edge, as at Z>, or at any distance from it, as at 
 E. At the instant the rulers are in contact at P, the motion of D 
 is in the direction DF, perpendicular to DP, the contact, radius. DF, 
 then, is tangent to the path of D, traced as the ruler BB rolls: but it is also 
 tangent to the circular arc whose center is D and radius PD, consequently 
 the path of D is also tangent to that arc. Let the arcs PC, Po of BB,
 
 12 
 
 be equal to the arcs Pd ', Po' of AA, then cD will be contact radius when c 
 reaches d ' , and oD when o reaches o' . If, then, we describe, with these 
 radii, circular arcs about c 1 and o' , the curve traced by D will be tangent to 
 those arcs ; and that traced by E will be tangent to arcs about the same 
 centers with cE and oE as radii. 
 
 Curves thus described, by points carried by one line which rolls upon 
 another, are called rolled curves or epitrochoids ; and the drawing of a series 
 of tangent arcs as above explained is the readiest and most reliable method 
 of laying them out. The line which carries the tracing point is called the 
 describing line, and the one in contact with which it rolls is called the base 
 liffe; either of these may be straight, or both may be curved.
 
 13 
 
 RECTIFICATION OF CIRCULAR ARCS. 
 
 For our present purpose we have to do only with the rolling of a circle, 
 either upon its tangent or upon another circle, and shall have frequent 
 occasion to set off upon a right line a length equal to that of a given arc, 
 or upon a given circle an arc equal in length to a given right line. 
 
 Since the circumference is 3.1416 times the diameter, these operations 
 can be performed arithmetically ; but the following graphic process will be 
 found equally accurate and much more expeditious. 
 
 a 
 
 F E 
 
 Eig* ti 
 
 I. In Fig. 6, let AE be tangent at A to the given arc AB. Draw 
 BA, produce it, making AG=^AD=y 2 chord AB. With center G and 
 radius GB, describe an arc cutting AE in F. Then AF=a.rc AB (very 
 nearly). 
 
 II. In Fig. 7, let the given line AB, be tangent at A, to the given 
 circle. Make AD~}{ AB ; with center D, and radius JDB=i/ AB, 
 describe an arc cutting the given circle in E. Then arc AE=AB (very 
 nearly). 
 
 NOTE. The arc thus rectified or found should not measure over 60. 
 If the given arc or line exceed this limit, it should be bisected.
 
 14 
 
 The particular rolled curves to be used are : 
 
 I. The Cycloid, Fig. 8. Traced by a point in the circumference of a 
 
 /i 
 
 E 
 
 circle rolling upon its tangent. Find Aa', the length of a convenient 
 fraction Aa, of the circumference; step this off the required number of 
 times, making ^^"^semi-circumference. Divide both into the -same
 
 15 
 
 number of equal parts, draw chords from P to the points of division on 
 the circle, with which, as radii, strike arcs about the corresponding points 
 on AE; the cycloid is tangent to all these arcs. 
 
 To find points on the curve. When aC becomes contact radius, it has 
 the position of a'R, perpendicular to AE. The angle a CP remaining un- 
 changed, make a'jRL equal to it: then RL is the generating radius, and L 
 a point on the cycloid. Also a'L is the normal, and a perpendicular to it 
 is tangent to the curve at L. 
 
 Fig. 9 
 
 Conversely. Let O be any point on the curve ; about this as a center,
 
 16 
 
 describe an arc with radius equal to CP, cutting CD, the path of the 
 center, in S. Then OS is the generating radius; Stf, perpendicular to 
 AE, is the contact radius, and b' O is normal to the cycloid. 
 
 II. The Epicycloid, Fig. 9. The describing circle rolls on the outside 
 of another, whose center is G. Draw the common tangent at A, set off on 
 this the length of Aa (any convenient fraction of semi-circumference AP), 
 and find the arc of the base circle equal to that length. Step this off as 
 above, making ^.fi 1 semi-circumference AP. The curve is drawfTby 
 tangent arcs in the same manner as the cycloid. The path of the center 
 of the describing circle, is, in this case, another circle, whose center is G, 
 and the contact radii a'JR, b'S, are prolongations of the radii Ga' , Gb' , of 
 the base circle, which slightly modifies the processes of finding the point 
 of the curve corresponding to a given point of contact and the converse. 
 Fig. 10 
 
 A 
 
 III. The Hypocycloid, Fig. 10. Traced by a point in the circum- 
 ference of a circle rolling inside another. Construction in all respects the
 
 17 
 
 same as in the case of the epicycloid, and the diagrams being lettered to 
 correspond throughout, no further explanation is needed. 
 
 In all three of these curves, if the rolling continue beyond , a new 
 branch EF springs up, which is, of course, perfectly symmetrical with EL. 
 It is to be particularly noted that these branches are tangent to ED, and to 
 each other, at E. These parts near E are the ones which require the 
 greatest care in their construction, as they only are employed in the forma- 
 tion of teeth.
 
 18 
 
 LAYING OUT THE TEETH THE PITCH CIRCLE AND 
 CIRCULAR PITCH. 
 
 If the line of centers of a pair of spur wheels be divided into two 
 parts which are to each other in the same ratio as the numbers of thejjeeth, 
 the circles of which these parts are the radii are called the pitch circles. 
 And the first step in laying out a pair of wheels is to determine the fadii 
 and draw these circles. Suppose, for example, that the distance CD, 
 between centers, in Fig. 11, is given, and it is required to make two wheels 
 
 1 
 
 Fig. 11. 
 
 whose angular velocities shall be as 2 : 1. Divide CD into three equal 
 parts, of which AD is one, then AC will measure two, and the tangent 
 circles shown are the pitch circles. Evidently they can move in perfect 
 rolling contact about their fixed centers; the linear motion AB is the 
 same, whether we regard the point A, as belonging to one circle or the 
 other. But one will not drive the other without the possibility of slipping, 
 which would cause the velocity ratio to vary ; hence the necessity of teth.
 
 19 
 
 The next step is to divide each pitch circle into as many equal parts 
 as its wheel is to have teeth. We may give the smaller wheel any number 
 we please, but the larger one must have twice as many in this instance. 
 The pitch of the teeth is the length of the circular arc obtained by this 
 subdivision. Since the larger circumference is twice the smaller, but is 
 divided into twice as many parts, the pitch arc is the same in both wheels. 
 Each of these arcs must contain a tooth and a space ; hence we may say 
 that the pitch is the distance between the centers or the corresponding 
 edges of rwo adjacent teeth, measured on the pitch circle, not in a right line. 
 This is sometimes Jcalled thejfC/>r///<zr Pitch, in distinction from what is 
 known as the Diametral Pitch, of which hereafter. 
 
 GENERATION OF THE TOOTH OUTLINE. 
 
 In Fig. 12, let Cand D be the centers" of the pitch circles LM, RN.
 
 Tangent to these at A, is a smaller circle whose center is O. Suppose all 
 the centers to be fixed, then the three circles can move in rolling contact, 
 with equal linear velocities. Set off from A the three equal arcs, AB, AE, 
 AP. Suppose a marking point originally fixed at A, in the circumference 
 of the small circle ; then while this travels to P, it must trace, with refer- 
 ence to N, the curve BP, and with reference to LM, the curve EP. Now 
 the relative motions of the circles are precisely the same as though the 
 small circle had rolled upon the outside of RN, and upon the insidtrof 
 LM, regarding these as fixed base lines ; the curves are, therefore, an epi- 
 cycloid and a hypocycloid respectively; and AP is their common normal. 
 
 If the tracing point go on to P', the arcs AP', AE', AB' , being 
 equal, the resulting curves B'P', E'P', are clearly but extensions of the 
 first pair, and AP' , is their common normal. 
 
 We perceive, then, that the curves thus simultaneously generated are 
 tangent to each other at some point throughout the generation ; that the 
 point of tangency is always in the describing circle; and that the com- 
 mon normal always passes through the fixed point A, upon the line of 
 centers. 
 
 Consequently these curves are correct outlines for parts, at least, of 
 teeth; if the curved lever CE' turn, as shown by the dotted arrow, it will 
 drive the other before it, the point of contact following the arc P'PA, 
 until E' and B' meet at A; and as the common normal always cuts the 
 line of centers at the same point, the velocity ratio will be constant. 
 
 FACE AND FLANK. 
 
 The epicycloid B' P', which lies without its pitch circle, is called the 
 face; and the hypocycloid E'P', which lies within its pitch circle, is called 
 the flank. Usually, each tooth has both ; but wheels can be made; and
 
 21 
 
 sometimes used to great advantage, in which one of a pair has faces only, 
 the other only flanks: we will consider this case first. 
 
 SIZE OF THE TOOTH. 
 
 This depends upon the pitch, for the pitch-arc must contain a tooth 
 and a space, which might be exactly equal, were perfection in workmanship 
 possible. Practically, the space must be a little wider than the tooth; the 
 difference is called back-lash, and should be made as small as practicable. 
 In drawing we may disregard it, and make the thickness of the tooth just 
 half the pitch. 
 
 ARC AND ANGLE OF ACTION. 
 
 The angle through which a wheel turns, while one of its teeth is in 
 contact with a tooth of another wheel, is called the angle of action, and the 
 arc by which it is measured is the arc of action. This latter must evidently 
 be at least equal to the pitch-arc, in order that each tooth may continue in 
 gear until the next one begins to act, and it should be considerably greater. 
 
 A PAIR OF WHEELS LIMITING CASE. 
 
 In Fig. 13, the pitch and describing circle being drawn as in Fig. 12, 
 let AB, AE, be the pitch arcs, and AP an equal arc on the describing 
 circle. Then the face for the tooth of RN, can not be less than BP, since, 
 if made, as shown, of exactly that height, contact is ending at P, at the 
 very instant the next tooth begins to act at A. Bisect AB in H, which 
 gives the thickness of the tooth ; and draw through H, a reversed face sim- 
 ilar to JBP. The conditions are purposely so chosen that this second face
 
 Fly. 13 
 
 passes through P. The case is, therefore, a barely possible one ; the tooth 
 is pointed, and just high enough to continue in gear until the next one 
 begins to act. We found that the face must be of the height BP, in order 
 to secure this arc of action ; drawing PD, which cuts the pitch circle in G, 
 we find in this case that BG is just half the thickness of the tooth. Had 
 it been greater, GH must have been less, so that the face through H t would 
 not have passed through P, but between P and G, and the case would 
 have been impracticable ; it would then have been necessary to reduce the 
 pitch and give both wheels more teeth. But if BG had been less than half 
 the thickness of the tooth, we could either make the tooth higher, or give 
 it some thickness at the top, as in Fig. 14.
 
 o Is 
 
 The acting flank is EP '; but in order to let the teeth of the other 
 wheel pass, the hypocycloid is extended to /, making the depth of the 
 space a little greater than PG; the difference is called clearance, and a 
 similar provision is made in the other wheel by cutting in radially, as 
 shown* at A, H, JB, a little below the pitch circle. The tooth of LM, is 
 completed by bisecting the pitch-arc AE, at F, and drawing the curves 
 AK', FJ, etc., similar to El. 
 
 A PRACTICAL CASE. 
 
 Limiting cases like the preceding are to be avoided in practice. A 
 pointed tooth is bad, as being weak and liable to wear at the top, and
 
 24 
 
 even if it be not pointed, the angle of action should be greater, as other- 
 wise, the least wear at the top reduces the face below the requisite height, 
 which affects the velocity ratio. A reasonable case is shown in Fig. 14 ; 
 the arc of action is \y 2 times the pitch, and drawing the radial line PS, we 
 find BG much less than y 2 BH, thus giving the tooth a thickness PK, at 
 the top. 
 
 APPROACHING AND RECEDING ACTION. 
 
 In Figs. 13 and 14, the action takes place wholly on one side of the 
 line of centers. If filVbe the driver (the direction being as shown by the 
 arrows) the action begins at A and ends at P, the point of contact contin- 
 ually receding from the line of centers ; in which case AJ3, AE, are called 
 arcs of recess, or of receding action. If LM drive (in the opposite direc- 
 tion) the action begins at Pending at A; the point of contact is always 
 approaching the line of centers, and AB, AE, are then called arcs of ap- 
 proach, or of approaching action. 
 
 It has been found by experience that the friction is greater and more 
 injurious in the latter case than in the former ; hence when such wheels are 
 used, the one with faces only should always drive. But even then, there is 
 one drawback, which will be seen by reference to Fig. 12. The longer the 
 arc of action, the longer the face of the tooth, and the greater the 
 obliquity of the line of action, that is, its inclination to the common tangent 
 TT, of the pitch circles. The pressure as well as the motion is transmitted 
 in this line, and the greater its inclination to TT, the greater will be the 
 component of pressure in the line of centers, tending to cause friction in 
 the bearings. 
 
 Consequently such wheels are better suited for use in light mechanism 
 where the teeth can be made small and numerous, and smoothness of action 
 is important, than for the transmission of heavy pressure.
 
 25 
 
 TEETH WITH BOTH FACES AND FLANKS. 
 
 By giving faces and flanks to the teeth of each wheel of a pair, we 
 can secure a given angle of action with shorter faces, consequently with 
 less sliding and less obliquity of action. Also, the action will take place 
 partly before, and partly after, the point of contact reaches the line of 
 centers. If a wheel is both to drive and to follow, the arcs of approach 
 
 . 15 
 
 R 
 
 and recess may be made equal; but if one of a pair is always the driver, 
 it may be desirable to make the arc of recess the greater, in order to reduce 
 the amount of the more injurious friction. 
 
 The construction is shown in Fig. 15 ; all that relates to the face BP
 
 for RN, and the flank EP for LM, is precisely the same as in Fig. 13, and 
 the lettering being so far made to correspond, no further explanation is 
 needed. To 'complete the teeth, another describing circle is used, on the 
 opposite side of the pitch circles, which generates the face OF tor LM, and 
 flank OK for RN. If we assume the arc of action on that side of CD, as 
 AFvr AK, the possibility of securing it with a given number of teeth is 
 at once ascertained by making the arc AO equal to Af, and drawing OC, 
 cutting LM in /.- if FI be less than half the thickness of the tooth- as 
 required by the pitch, or equal to it, the construction is possible, the tooth 
 in the latter event being pointed ; if greater it is impracticable. If it be 
 found feasible, we have only to draw the epicycloid OF, which joined to 
 EP completes the outline of the tooth for LM, and the hypocycloid OK, 
 joining it to BP, which finishes the outline of the tooth of RN. That is 
 to say, these are the whole of the acting outlines ; the flanks must be "ex- 
 tended to a greater depth in order to give clearance, as already explained. 
 
 The operation will be readily seen ; as in the diagram the acting side 
 of a tooth of each wheel is drawn in two positions. Supposing RJV to 
 drive, the action begins at O, the driver's flank pushing the face of the 
 follower, and the point of contact moving in the arc OA, until K and F 
 meet at A. 
 
 The face of the driver then urges the follower's flank, the point of 
 contact now traveling in the arc OP, and at P the action ends. 
 
 We see, then, that the angle of approach depends upon the length of 
 the follower's face, and the angle of recess upon that of the driver's face ; 
 and if these lengths be assumed or given, those angles are readily found, 
 as for instance, had the length FO been assigned, it is only necessary to 
 strike an arc about C with radius CO, which, cutting the describing circle 
 in O, gives OA the length of the arc of approach, which is then to be set 
 off on ZJ/and RN, as AF, AK.
 
 27 
 
 A Practicable Example. 
 
 The diagram Fig. 15, is drawn without regard to practical propor- 
 tions, in order to make the construction clear; but in Fig. 16 we have 
 
 I) 
 
 shown a feasible case ; the cut is half size, ' and the conditions are as 
 follows : 
 
 Distance between centers, 27 inches. Wheels to have 63 and 45 teeth, 
 
 the smaller to drive. 
 
 Angle of action to be 2f times the pitch. 
 Angle of recess to be one-third greater than the angle of approach.
 
 We have, then, 63:45::7:5, 7+5=12, tf=2J, 2Jx7=15f, 2Jx5=ll#, for 
 the radii of the pitch circles. And 2|=V, which is to be divided into parts 
 in the ratio of 3:4; whence, 3+4=7, Y-=-7=f, f X3=li times the pitch 
 
 =angle of approach, 
 X4=1 times the pitch 
 =angle of recess. 
 
 INTERCHANGEABLE WHEELS. 
 
 Inasmuch as the face and the flank which act in contact are generated 
 by the same describing circle, it makes no difference whether the diameter 
 of the one which traces the other face and flank be the same or not, in 
 laying out a single pair of wheels; and in Fig. 15 the describing circles are 
 of different diameters. But for the very reason just stated, it is clear that 
 if we wish to make a number of wheels, any one of which will gear with 
 any other one, we must use the same describing circle for all the faces and 
 all the flanks. 
 
 SIZE OF THE DESCRIBING CIRCLE. 
 
 In making such a set of wheels, the question at once arises, how large 
 shall the describing circle be? This depends upon a property of the 
 hypocycloid, illustrated by Figs. 17 and 18. In Fig. 17, the describing 
 circle is half the size of the base circle, and the line traced by the point A 
 is merely the diameter AD. For after rolling till the point of contact is 
 B (taken at pleasure), the describing circle cuts AD in P, its center mean- 
 time going from E to F. Draw BFC, and PF; then the angle A CB is 
 half the angle PFB, but the radius A C is twice the radius BF, therefore 
 the arcs AB, PJB, are equal. In Fig. 18, the point P will trace the curve
 
 Fig. 18 
 
 AS, by rolling in one direction, if we regard it as in the circumference of 
 the smaller circle ; but if the same point be carried by the larger describ- 
 ing circle, it will trace the same curve by rolling in the other direction. 
 If, then, in any case the describing circle be half the size of the pitch circle, 
 the flanks will be radial, as in Fig. 19 ; if it be less, they will spread out 
 Fly. 19 Fig. 20 Fig. 21
 
 30 
 
 toward the root of the tooth, giving a stronger form, as in Fig. 20 ; but if 
 greater, the flanks will curve in toward each other, as in Fig. 21, whereby 
 the teeth become weaker and difficult to make. 
 
 From this, the safe practical deduction is, that the describing circle for 
 a set of wheels should not be more than half the diameter of the smallest 
 wheel ; and in laying out a single pair, two describing circles, each of I 
 the diameter of its pitch circle, give good practical forms to the teeth^ for 
 general purposes. 
 
 Still, the face is shorter, and the obliquity of action less, for a given 
 arc of action, the larger the describing circle ; so that for very delicate 
 mechanism, it is possible that the gain from these causes would sometimes 
 render it advisable to use teeth of the form shown in Fig. 21. They may 
 be much strengthened by the use of large fillets at the junction of the^side 
 and bottom of the space, which is quite admissible, since the acting depth 
 of the flank is comparatively small, as has been shown. 
 
 RACK AND WHEEL. 
 
 A rack is simply an infinitely large wheel. The curvature of a circle 
 diminishes as the radius increases, and disappears when the radius becomes 
 infinite ; so that the pitch line of a rack is only a straight tangent to the 
 pitch circle of the wheel with which it works, and the line of centers become 
 a perpendicular to this pitch line, through the center of the wheel. 
 
 The rack will travel through a distance equal to the circumference of 
 the pitch circle of the wheel during one revolution of the latter, whatever 
 the number of teeth, and in the same proportion for any fraction of a revo- 
 lution. The pitch of the rack teeth, therefore, is found by rectifying the 
 pitch arc of the wheel, whatever that may be, and setting off that length 
 upon the pitch line.
 
 31 
 
 The construction is shown in Fig. 22. We have here shown the two 
 describing circles as of the same size, and it is clear that if the same circle 
 be used to generate the faces and flanks of a -set of wheels, any one of 
 them will gear with the rack if the pitch be also the same. 
 
 Fig. 22 
 
 Evidently both faces and flanks of the rack teeth are cycloids, being 
 generated by the rolling of a circle upon the pitch line. If the length BP 
 of the face, be assumed or given, a line parallel to RN cute the generating 
 circle in P, thus determining AP, to which AB must be made equal, and 
 fixing the part of the action which will take place on the right of CD. Or 
 if AB be assigned, we make AP equal to it, thus ascertaining the necessary 
 length of face. In either case, PS is now to be drawn perpendicular to
 
 32 
 
 the pitch line, which it cuts at G, and as in the preceding constructions 
 BG cannot be greater, and should be less, than half the thickness of the 
 tooth as determined by the pitch. The part of the action, which will take 
 place on the left of CD, depends upon the length of the face of the wheel 
 tooth, and is ascertained as in the cases previously explained. 
 
 ARBITRARY PROPORTIONS. 
 
 It is not necessary in all cases, indeed probably not in the majority of 
 cases, to pay particular attention to the relative amounts of the approach- 
 ing and receding action. It is a very common practice to make the 
 whole height of the tooth a certain fraction of the pitch ; the part which 
 projects outside the pitch circle being made a little less than that within, 
 by which the clearance is provided for. Thus, in Fig. 23, the whole height 
 
 Fig. 23. 
 
 /, is | of the pitch ; and the part h is to d as 11 : 13. 
 tions which have been extensively used are as follows : 
 /= pitch ; h : d :: 4 : 5. 
 " h : d :: 3 : 4. 
 
 Two other propor-
 
 33 
 
 Teeth proportioned according to either of these rules will work satis- 
 factorily for most purposes, if there be at least 12 teeth upon the smallest 
 wheel. But if occasion arises, as it may, for the use of a pinion with only 
 six or eight "leaves," (as the teeth of very small wheels are sometimes 
 called,) it will be found advantageous, if not absolutely necessary, to make 
 the faces of those leaves longer. As for the angle of action, and the pro- 
 portion of the angle of approach to that of recess, both these things will, 
 of course, vary according to the numbers of teeth, whichever of these 
 systems be adopted. It may be added, that in connection with these rules, 
 instructions are frequently given as to the amount of back-lash, which is 
 also, according to them, a definite fraction of the pitch. This is, no doubt, 
 very proper in reference to wheels cast from patterns, but there certainly 
 does not seem to be any reason for it if the teeth are to be cut ; for whatever 
 the pitch, it is practically sufficient that the backs of the teeth should barely 
 clear each other when the fronts are in driving contact. 
 
 DIAMETRAL PITCH. 
 
 In designing spur gearing it is necessary to find the circular pitch, not 
 only because, as we have seen, it is used in the graphic construction, but 
 because the strength of the tooth depends upon its thickness. Were there 
 nothing to the contrary, it would be most convenient to express the pitch 
 in whole numbers or manageable fractions, as 2 inch pitch, f inch pitch, 
 and so on. But as the circumference is 3.1416 times the diameter, awk- 
 ward decimals will often appear in the values of the diameters of the pitch 
 circles, if this plan be adhered to. Now, if the tooth be strong enough, it 
 matters not if it be a little stronger ; and it is practically much more im- 
 portant to have the diameter a whole number, or a convenient fraction, 
 than that the circular pitch should be either the one or the other. This is
 
 34 
 
 accomplished by the use of what is called the diametral pitch ; which is 
 simply the quotient found by dividing the diameter of the pitch circle, 
 instead of the circumference, into as many equal parts as the wheel has 
 teeth. 
 
 Whence, Diametral Pitch= Diameter and circular Pitch=Di- 
 No. of Teeth. 
 
 ametral Pitch X 3.1416. 
 
 In the use of this system, convenient values of the diametral pitch are 
 selected, each being a fraction with unity for its numerator and an integer 
 for its denominator, as 1, *, i, J-, T V, TJ, etc. 
 
 The denominators of these fractions only are commonly used in giv- 
 ing the diametral pitch; thus, an "8-pitch wheel" is one which has eight 
 teeth for each inch of diameter, or whose diametral pitch is \" . -This is, 
 
 in fact, merely inverting the fraction, and giving the value of ; 
 
 Diameter 
 
 thus, let a wheel of 16 inches diameter have 80 teeth; then, $$=\" = 
 diametral pitch, but | = 5, and we call it a "5-pitch" wheel. By this 
 system the calculations as to diameter and number of teeth are made very 
 simple ; as, for example : 
 
 Required, the diameter of a 4-pitch wheel with 37 teeth : 
 
 V = 9i diameter. 
 
 How many teeth of 16-pitch on a wheel of 3| diameter ? 
 3| X 16 = 62 = No. of teeth. 
 
 The tooth may be made to project outside the pitch circle, a definite 
 fraction of the diametral pitch, as in the case of the circular pitch; and 
 thus the size of the blank may be readily ascertained. If this projection be
 
 35 
 
 made, for instance, 1 J times the diametral pitch, the face of the tooth will 
 be nearly as long as that found by the first of the arbitrary rules above 
 given ; and the diameter of the blank is determined by simply adding to 
 that of the pitch circle, 2} times the diametral pitch.
 
 PART II. 
 
 THE MANUFACTURE OF ACCURATE GEAR CUTTERST* 
 
 In cutting a spur wheel it is essential that the contour of the milling 
 cutter conform precisely to that of the space between two teeth. The 
 obvious disadvantages of turning the cutter by hand to fit a template filed 
 out, if not laid out, by hand, have already been alluded to ; also the fact 
 that they have been in part avoided, by the use of mechanical mearis^for 
 describing the required curves on the template. 
 
 This is, however, but one step, and that of the shortest; for by the 
 methods previously explained, the epicycloidal curves can be drawn on 
 metal with comparative speed and extreme accuracy. There yet remain the 
 laborious processes of making the template, and making the cutter fit it 
 when made. When these things are done by hand, exact duplication of 
 templates, cutters or wheels, is a matter of impossibility, to all practical 
 intents and purposes. 
 
 But this becomes, on the contrary, a matter of ease and perfect cer- 
 tainty, when the work is done by the two machines of which we give 
 illustrations. They were recently introduced by The Pratt and Whitney 
 Company, in whose shops they may be seen in successful operation ; and 
 they are well worthy of study, not only on account of the ingenuity and 
 beauty of their movements, but as models of skillful planning and rare 
 examples of the practical embodiment of correct theory.
 
 , : 
 
 o 
 
 a 
 
 w 
 
 w 
 
 71 
 bi)
 
 THE EPICYCLOIDAL MILLING ENGINE. 
 
 The object of this machine is to form the template subsequently used 
 as a guide in shaping the cutter. Its general appearance is shown in Fig. 
 1 ; the operating parts are more distinctly shown in Fig. 2, which is a 
 view taken nearly from above ; the action is illustrated in Figs. 3, 4, 5, 
 
 6, 7, 8. 
 
 In Fig. 3, A,A, is a portion of a flat ring, fixed to the framing ; this 
 represents a pitch circle. B, is a disc, representing the describing circle ;
 
 40 
 
 this turns freely upon a tubular stud E, fixed in the carrier C, which turns 
 about a pivot D, fixed to the frame at the center of A. By means of the 
 clamped socket, capable of sliding upon the rod, the position of D may 
 be adjusted to suit the radius of A. Thus as C moves, the disc can roll 
 upon the edge of A, and is compelled to do so by the flexible steel ribbon 
 shown by the heavy line, which is wrapped round and secured to both 
 pieces, due allowance for its thickness being made in adjusting their radii. 
 
 E' is a second tubular stud fixed in the carrier, at the same distance from 
 the pitch circle as the other, but on the opposite side ; the centers of~the 
 two studs lying on a right line through D. Upon these two studs turn the
 
 41 
 
 two worm wheels, F, f, shown in Fig. 4 ; these are in a plane above A 
 and B, so that the axis of the worm G, is vertically over the common 
 tangent of the pitch and describing circles; the relative positions of these 
 and other parts will be most clearly seen by a study of the vertical section, 
 Fig. 8. The worm G, is supported in bearings secured to the carrier C, 
 and is driven by another small worm turned by the pulley /, as seen in 
 Fig. 2 ; the driving cord, passing through suitable guiding pulleys, is kept 
 at uniform tension by a weight, however C moves ; this is shown in Figs. 
 1 and 2. 
 
 Upon the same studs, in a plane still higher than the worm-wheels, 
 turn the two discs H, H' , Figs. 5, 6, 7. The diameters of these are equal, 
 and precisely the same as those of the describing circles which they 
 represent, with due allowance, again, for the thickness of a steel ribbon, 
 by which these, also, are connected. It will be understood that each of 
 these discs is secured to the worm-wheel below it, and the outer one of 
 these to the disc B, so that as the worm G turns, H and H' are rotated in 
 opposite directions, the motion of H being identical with that of B ; this 
 last is a rolling one upon the edge of A, the carrier Cwith all its attached 
 mechanism moving around D at the same time. Ultimately, then, the 
 motions of H, H' , are those of two equal describing circles rolling in ex- 
 ternal and internal contact with a fixed pitch circle. 
 
 In the edge of each disc a semi-circular recess is formed, into which is 
 accurately fitted a cylinder /, provided with flanges, between which the 
 discs fit so as to prevent end play. This cylinder is perforated for the 
 passage of the steel ribbon, the sides of the opening, as shown in Fig. 5, 
 having the same curvature as the rims of the discs. Thus when these 
 recesses are opposite each other, as in-Fig. 6, the cylinder J fills them both, 
 and the tendency of the steel ribbon is to carry it along with H when C
 
 42 
 
 moves to one side of this position, as in Fig. 7, and along with H' when C 
 moves to the other side, as in Fig. 5. 
 
 This action is made positively certain by means of the hooks K K' , 
 which catch into recesses formed in the upper flange of J, as seen in Fig. v 
 6. The spindles, with which these hooks turn, extend through the hollow 
 studs, and the coiled springs attached, to their lower ends, as seen in Fig. 8, 
 urge the hooks in the directions of their points ; their motions being 
 limited by stops o, o', fixed, not in the discs H, H' , but in projectig*-ct>llars 
 on the upper ends of the tubular studs. The action will be readily traced 
 by comparing Fig. 6 with Fig. 7 ; as C goes to the left, the hook K' is left 
 
 behind, but the other one, K, cannot escape from its engagement with the 
 flange of J; which, accordingly, is carried along with H by the combined 
 action of the hook and the steel ribbon. 
 
 On the top of the upper flange of /, is secured a bracket, carrying the 
 bearing of a vertical spindle Z, whose center line is a prolongatioivof that
 
 43 
 
 of J itself. This spindle is driven by the spur wheel N, keyed on its 
 upper end, through a flexible train of gearing seen in Fig. 2 : at its lower 
 end it carries a small milling cutter M, which shapes the edge of the tem- 
 plate T, Fig. 7, firmly clamped to the framing. 
 
 When the machine is in operation, a heavy weight seen in Fig. 1, acts 
 to move C about the pivot D, being attached to the carrier by a cord guided 
 by suitably arranged pulleys ; this keeps the cutter Mup to its work, while 
 the spindle L is independently driven, and the duty left for the worm G to 
 perform, is merely that of controlling the motions of the cutter by the 
 means above described, and regulating their speed. 
 
 The center line of the cutter is thus automatically compelled to travel 
 in the path RS, Fig. 7, composed of an epicycloid and a hypocycloid if A A 
 be a segment of a circle as here shown ; or of two cycloids, if A A be a 
 straight bar. The radius of the cutter being constant, the edge of the tem- 
 plate Tis cut to an outline also composed of two curves ; since the radius 
 M is small, this outline closely resembles RS, but particular attention is 
 called to the fact that it is not identical with it, nor yet composed of truly 
 epicycloidal curves of any generation whatever : the result of which will be 
 subsequently explained. 
 
 NUMBER AND SIZES OF TEMPLATES. 
 
 With a given pitch, every additional tooth increases the diameter of 
 the wheel, and changes the form of the epicycloid j so that it would ap- 
 pear necessary to have as many different cutters, as there are wheels to be 
 made, of any one pitch. 
 
 But the proportional increment, and the actual change of form, due to 
 the addition of one tooth, becomes less as the wheel becomes larger ; and 
 the alteration in the outline soon becomes imperceptible. Going still far-
 
 44 
 
 ther, we can presently add more teeth without producing a sensible varia- 
 tion in the contour. That is to say, several wheels can be cut with the 
 same cutter, without introducing a perceptible error. It is obvions that 
 this variation in the form, is least near the pitch circle, which is the only 
 part of the epicycloid made use of; and Prof. Willis many years ago 
 deduced theoretically, what has since been abundantly proved by practice, 
 that instead of an infinite number of cutters, 24 are sufficient of one pitch, 
 for making all wheels, from one with 12 teeth up to a rack. 
 
 Accordingly, in using the epicycloidal milling engine, for forming the 
 template, segments of pitch circles are provided of the following diameters 
 (in inches) : 
 
 12, 16, 20, 27, 43, 100, 
 
 13, 17, 21, 30, 50, 150, 
 
 14, 18, 23, 34, 60, 300, 
 
 15, 19, 25, 38, 75, oo.
 
 45 
 
 The diameter of the discs which act as describing circles, is 7 inches, 
 and that of the milling cutter which shapes the edge of the template, is I 
 of an inch. 
 
 Now if we make a set of 1 -pitch wheels with the diameters above 
 given, the smallest will have twelve teeth, and the one with fifteen teeth 
 will have radial flanks. The curves will be the same whatever the pitch ; 
 but as shown in Fig. 9, the blank should be adjusted in the epicycloidal 
 engine, so that its lower edge shall be iVh of an inch (the radius of the 
 cutter Af) above the bottom of the space ; also its relation to the side of 
 the proposed tooth should be as here shown. As previously explained, 
 the depth of the space depends upon the pitch. In the system 
 adopted by The Pratt & Whitney Company, the whole height of the tooth 
 is 2^ times the diametral pitch, the projection outside the pitch circle being 
 just equal to the pitch, so that diameter of blank = diameter of pitch circle 
 + 2 X diametral pitch. 
 
 We have now to show how, from a single set of what may be called 
 1-pitch templates, complete sets of cutters of the true epicycloidal contour 
 may be made of the same or any less pitch. 
 
 THE PANTAGRAPHIC ENGINE FOR FORMING CUTTERS. 
 
 In Fig. 9, the edge TT, is shaped by the cutter M, whose center 
 travels in the path J?S, therefore these two lines are at a constant normal 
 distance from each other. Let a roller P, of any reasonable diameter, be 
 run along TT, its center will trace the line UV, which is at a constant 
 normal distance from TT, and therefore from RS. Let the normal distance 
 between 7Fand fiSbe the radius of another milling cutter N, having the 
 same axis as the roller P, and carried by it, but in a different plane, as 
 shown in the side view ; then whatever ^Vcuts will have RS for its contour, 
 if it lie upon the same side of the cutter as the template.
 
 46 
 
 Now if TT be a 1-pitch template as above mentioned, it is clear that 
 ^Vwill correctly shape a cutting edge of a gear cutter for a 1-pitch wheel. 
 The same figure, reduced to half size, would correctly represent the for- 
 mation of a cutter for a 2-pitch wheel of the same number of teeth; if to 
 quarter size, that of a cutter for a 4-pitch wheel, and so on. 
 
 But since the actual size and curvature of the contour thus determined, 
 depend upon the dimensions and motion of the cutter N, it will be seen 
 that the same result will practically be accomplished, if these, onlyj-be re- 
 duced ; the size of the template, the diameter and the path of the roller 
 remaining unchanged. 
 
 Fig 10 
 
 The nature of the means by which this is effected in the Pantagraphic 
 Engine, is illustrated in Fig. 10. The milling cutter N, is driven by 
 a flexible train acting upon the wheel O; its spindle is carried -by the 
 bracket JB, which can slide from right to. left upon the piece A, and this,
 
 47 
 
 again, is free to slide in the framed. These two motions are in horizontal 
 planes, and perpendicular to each other. 
 
 The upper end of the long lever PC, is formed into a ball, working 
 in a socket which is fixed to B. Over the cylindrical upper part of this 
 lever slides an accurately fitted sleeve D, partly spherical externally, and 
 working in a socket which can be clamped at any height on the frame f. 
 The lower end P, of this lever being accurately turned, corresponds to the 
 roller P in Fig. 9, and is moved along the edge of the template T, which 
 is fastened in the frame in an invariable position. 
 
 By clamping D at various heights, the ratio of the lever arms PD, 
 DC, may be varied at will, and the axis of N made to travel in a path 
 similar to that of the axis of P, but as many times smaller as we choose ; 
 and the diameter of TV is made less than that of P in the same proportion. 
 
 The template being on the left of the roller, the cutter to be shaped is 
 placed on the right of N, as shown in the plan view at Z, because the lever 
 reverses the movement. 
 
 This arrangement is not mathematically perfect, by reason of the 
 angular vibration of the lever. This is, however, very small, owing to 
 the length of the lever; it might have been compensated for by the intro- 
 duction of another universal joint, which would practically have introduced 
 an error greater than the one to be obviated, and it has, with good judg- 
 ment, been omitted. 
 
 The gear cutter is turned nearly to the required form, the notches are 
 cut in it, and the duty of the pantagraphic engine is merely to give the 
 finishing touch to each cutting edge, and give it the correct outline. It is 
 obvious that this machine is in no way connected with, or dependent upon, 
 the epicycloidal engine; but by the use of proper templates it will make 
 cutters for any desired form of tooth; and by its aid exact duplicates may 
 be made in any numbers with the greatest facility. Its general appearance 
 is shown in Fig. 11. It will be noted that the universal joints are not
 
 Fig. 11. Pantagraphic Engine for forming Cutters.
 
 49 
 
 actually of the ball and socket kind, which suggests the explanation, that 
 in Figs. 3-10 inclusive, we have made no attempt to give precise details 
 or proportions, but only to make as clear as we are able to, the principles 
 and mode of action of these remarkably ingenious machines, as well as of 
 the system adopted in using them. 
 
 THEORETICAL DEFECTS OF THE SYSTEM. 
 
 It forms no part of our plan to represent as perfect that which is not 
 so, and there are one or two facts, which at first thought might seem 
 serious objections to the adoption of the epicycloidal system. These are; 
 
 1. It is physically impossible to mill out a concave cycloid,*by any 
 means whatever, because at the pitch line its radius of curvature is zero, 
 and a milling cutter must have a sensible diameter.
 
 50 
 
 2. It is impossible to mill out even a convex cycloid or epicycloid, by 
 the means and in the manner above described. 
 
 This is on account of a hitherto unnoticed peculiarity of the curve at 
 a constant normal distance from the cycloid. In order to show this clearly, 
 we have, in Fig. 12, enormously exaggerated the radius CD, of the milling 
 cutter (M of Figs. 7 and 8). The outer curve HL, evidently, could be 
 milled out by the cutter, whose center travels in the cycloid CA; it re- 
 sembles the cycloid somewhat in form, and presents no remarkable Teatures. 
 But the inner one is quite different ; it starts at D, and at first goes down, 
 inside the circle whose radius is CD, forms a cusp at E, then begins to rise, 
 crossing this circle at G, and the base line at f. It will be seen, then, that 
 if the center of the cutter travel in the cycloid A C, its edge will cut away 
 the part GED, leaving the template of the form OGI. Now if a roller of 
 the same radius CD, be rolled along this edge, its center will travel in the 
 cycloid from A, to the point P, where a normal from G, cuts it ; then the 
 roller will turn upon G as a fulcrum, and its center will travel from P to 
 C, in a circular arc whose radius to GP= CD. 
 
 That is to say, even a roller of the same size as the original milling 
 cutter, will not retrace completely the cycloidal path in which the cutter 
 traveled. 
 
 Now in making a rack template, the cutter, after reaching C, travels 
 in the reversed cycloid CR, its left-hand edge, therefore, milling out a 
 curve DK, similar to HL. This curve lies wholly outside the circle DI, 
 and therefore cuts OG at a point between ^and G, but very near to G. 
 This point of intersection is marked S in Fig. 13, where the actual form of 
 the template OSK is shown. The roller which is run along this template, 
 is larger, as has been explained, than the milling cutter. When the point 
 of contact reaches S (which is so near to G that they practically coincide), 
 this roller cannot now swing about S through an angle so great as PGC of 
 Fig. 12 ; because at the root D, the radius of curvature of DK is only
 
 51 
 
 equal to that of the cutter, and G and 6" are so near the root that the curva- 
 ture of SK, near the latter point, is greater than that of the roller. Con- 
 sequently there must be some point U in the path of the center of the 
 roller, such, that when the center reaches it, the circumference will pass 
 through S, and be also tangent to SK. Let T be the point of tangency ; 
 draw SC7 and TU, cutting the cycloidal path AR in JSf and K Then, UY 
 being the radius of the new milling cutter (corresponding to N oi Fig. 9), 
 it is clear that in the outline of the gear cutter shaped by it, the circular 
 arc .ATKwill be substituted for the true cycloid. 
 
 THE SYSTEM PRACTICALLY PERFECT. 
 
 The above defects undeniably exist ; now, what do they amount to ? 
 The diagrams, Figs. 12 and 13, are drawn purposely with these sources of 
 error greatly exaggerated, in order to make their nature apparent and their 
 
 FIG. 14. SET OF WHEELS AND RACK. 
 
 existence sensible. The diameters used in practice, as previously stated, 
 are: describing circle, 7 inches; cutter for shaping template, \ of an inch; 
 roller used against edge of template, H inches; cutter for shaping a I -pitch 
 gear cutter, 1 inch. 
 
 l7NivERSt*r? OP CALIFORNIA 
 
 SANTA BARBARA COLLEGE L
 
 52 
 
 With these data the writer has found that the total length of the arc 
 XY of Fig. 13, which appears instead of the cycloid in the outline of a 
 cutter for a 1-pitch rack, is less than 0.0175 inch; the real deviation from 
 the true form, obviously, must be much less than that. It need hardly be 
 stated that the effect upon the velocity ratio of an error so minute, and in 
 that part of the contour, is so extremely small as to defy detection. And 
 the best proof of the practical perfection of this system of making epicy- 
 cloidal teeth is found in the smoothness and precision with which the-wheels 
 run ; a set of them is shown in gear in Fig. 14, the rack gearing as 
 accurately with the largest as with the smallest. To which is to be added, 
 finally, that objection taken, on whatever grounds, to the epicycloidal form 
 of tooth, has no bearing upon the method above described of producing 
 duplicate cutters for teeth of any form, which the pantagraphic engine will 
 make with the same facility and exactness, if furnished with the proper 
 templates.
 
 Table of Cutters for Teeth of Gear Wheels, 
 
 MADE BY 
 
 THE PRATT & WHITNEY CO., HARTFORD, CONN., U. S. A 
 
 All Gears of the same pitch cut with our Cutters are perfectly interchangeable. 
 
 Diameter of 
 Cutters. 
 
 Diametral 
 Pitch. 
 
 Price of Cutters. 
 
 Size of Hole 
 in Cutters. 
 
 SET OF 24 CUTTERS. 
 
 For each pitch coarser than 10. 
 
 inches. 
 
 11 
 
 $25 00 
 
 1} inches. 
 
 No. 1 cuts 12 T 
 
 
 9 
 
 20 00 
 
 a u 
 
 No. 2 " 13 
 
 " 
 
 2J 
 
 18 00 
 
 " " 
 
 No. 3 " 14 
 
 
 3 
 
 15 00 
 
 " " 
 
 No. 4 " 15 
 
 
 3 
 
 12 00 
 
 1 
 
 No. 5 " 16 
 
 
 4 
 
 9 00 
 
 u 
 
 No. 6 ' " 17 
 
 
 5 
 
 7 00 
 
 it 
 
 No. 7 " 18 
 
 
 
 6 
 
 6 00 
 
 (( U 
 
 No. 8 " 19 
 
 " 
 
 7 
 
 5 00 
 
 (I U 
 
 No. 9 " 20 
 
 
 
 8 
 
 4 50 
 
 I ;; 
 
 No. 10 " 21 to 22 
 
 " 
 
 9 
 
 4 00 
 
 
 No. 11 " 23 " 24 
 
 " 
 
 10 
 
 3 50 
 
 " 
 
 No. 12 " 25 " 26 
 
 " 
 
 12 
 
 3 50 
 
 ( 
 
 No. 13 " ' 27 " 29 
 
 r " 
 
 14 
 
 3 50 
 
 
 
 No. 14 " 30 " 33 
 
 r ;; . 
 
 16 
 
 3 00 
 
 ti 
 
 No. 15 " 34 " 37 
 
 
 18 
 
 3 00 
 
 u 
 
 No. 16 " 38 " 42 
 
 i 
 
 20 
 
 3 00 
 
 " " 
 
 No. 17 " 43 " 49 
 
 1 ;; 
 
 22 
 
 3 00 
 
 u 
 
 No. 18 " 50 " 59 
 
 
 24 
 
 3 00 
 
 " " 
 
 No. 19 " 60 " 75 
 
 " 
 
 26 
 
 3 00 
 
 
 
 No. 20 " 76 " 99 
 
 " 
 
 28 
 
 3 00 
 
 " " 
 
 No. 21 " 100 " 149 
 
 tt 
 
 30 
 
 3 00 
 
 " " 
 
 No. 22 " 150 " 299 
 
 
 
 32 
 
 3 00 
 
 " " 
 
 No. 23 " 300 Rack. 
 
 
 
 
 
 No. 24 " Rack. 
 
 The Cutters are made for diametrical pitches. By diametrical pitch is meant, the 
 number of teeth per inch in the diameter of the gear at pitch line. Two pitches should 
 always be added to this diameter in preparing a gear for cutting. For example : a gear of 
 80 teeth, 8 to the inch, diametrical pitch, would be 10 inches on pitch circle, but the gear 
 should be turned 10 2-8 (or J). The teeth should always be cut two pitches deep beside 
 clearance. 
 
 The Cutters are made for a clearance of 1-16 of the depth of the tooth : example : 8 
 to the inch has a clearance of 1-64; therefore the tooth should be cut two pitches (1-4) 
 and 1-64 deep. The gears must be set to run with this clearance to give the best results. 
 
 In ordering bevel gear cutters, give the diameter of gear at outside pitch line, and 
 number of teeth, also the width of face. For the present all cutters are made to order.
 
 THE PRATT -& WHITNEY COMPANY, 
 
 HARTFORD, CONN., 
 
 MANUFACTURERS OF 
 
 js/n^oHiisrisTS' TOOLS 
 
 for general use, comprising a large variety of Lathes, Planers, Drilling, Milling, Boring, 
 
 Screw making, Bolt cutting, Die sinking, Grinding, Polishing, Shaping, Tapping 
 
 and Marking Machines, Planer and Milling Machine Vises, Planer, 
 
 Milling Machine and Bench Centers, Cam cutting Machines for 
 
 various purposes, Power Shears, a variety of Power 
 
 and Foot Presses, Iron Cranes for shops and 
 
 other purposes, Lathe Chucks, etc. 
 
 MACHINES FOR GUN AND SEWING MACHINE MAKERS AND FOR 
 ALL USES IN METAL WORKING. 
 
 Having furnished several plants complete for the manufacture of Guns, Pistols, Sewing 
 
 Machines, etc., we particularly solicit such business, and where only drawings or 
 
 models are furnished, are prepared to complete such tools and machines 
 
 as may be required, and to send competent men to superintend 
 
 their erection, and to run them if required. 
 
 A^ND DIES 
 
 Special attention given to this branch, and in form of thread, mathematical exactness 
 
 and workmanship, are unsurpassed, and have become the " standard " 
 
 in many leading workshops. 
 
 Forging Machinery, 
 
 Consisting of DROP HAMMERS (a specialty) in six sizes, of best and most modern con- 
 
 struction; TRIP HAMMERS, TRIMMING PRESSES, SHEARS, etc., 
 
 FORGES and DROP HAMMER DIES. 
 
 CUTTERS FOR GEAR WHEELS, 
 
 made by new process and entirely by machinery, which secures absolute correctness as to 
 form and interchangeability of any given pitch. 
 
 SPECIAL MACHINERY (to order). 
 
 Are prepared to perfect plans and models for same. Our facilities for the above, and for 
 
 the different branches of Forging, Finishing, Casting, etc., are unsurpassed. 
 IVIany of the above Tools and Machines are kept in stock. 
 
 Illustrated Catalogue and Price Lists furnished on application. 
 
 THE PRATT & WHITNEY CO., Hartford, Conn.
 
 UNIVERSITY OF CALIFORNIA 
 
 Santa Barbara College Library 
 Santa Barbara, California 
 
 Return to desk from which borrowed. 
 This book is DUE on the last date stamped below. 
 
 LD 21-10m-10,'ol 
 (8066s4)476
 
 000 590 362 
 
 TJ 
 
 136 
 
 M3