REESE LIBRARY 
 
 OF THE 
 
 UNIVERSITY OF CALIFORNIA. 
 
 ,:;...,,,.::. v .,,^9o-: r f 
 
 ^Accession No. .92342 Class A': 
 
WORKS OF J. H. CROMWELL 
 
 PUBLISHED BY 
 
 JOHN WILEY & SONS. 
 
 A Treatise on Toothed Gearing:. 
 
 1-2010, cloth, $1.50 
 
 A Treatise on Belts and Pulleys. 
 
 12010, cloth, $1.50. 
 
A TREATISE 
 
 ON 
 
 TOOTHED GEARING. 
 
 Containing Complete Enstructions 
 
 FOR DESIGNING, DRAWING, AND CONSTRUCTING 
 
 SPUR WHEELS, BEVEL WHEELS, LANTERN 
 
 GEAR, SCREW GEAR, WORMS, ETC., 
 
 THE PROPER FORMATION OF TOOTH-PROFILES. 
 
 FOR THE USE OF 
 
 MACHINISTS, PATTERN-MAKERS, DRAUGHTSMEN, 
 DESIGNERS, SCIENTIFIC SCHOOLS, ETC. 
 
 BY 
 
 J. HOWARD CROMWELL, Pn.B. 
 
 FOURTH EDITION. 
 SECOND THOUSAND. 
 
 NEW YORK: 
 
 JOHN WILEY AND SONS, 
 
 43-45 EAST NINETEENTH STREET. 
 1901. 
 

 COPYRIGHT, 1883, 
 BY J. HOWARD CROMWELL. 
 
PREFACE. 
 
 IN presenting to the mechanical public this little work, I am 
 fully aware that I am treading upon well-worn ground, and that 
 I have devoted time and labor to a subject which is well-nigh 
 "old as the hills," and likewise, to many, as familiar. It may 
 also seem to some, who have read more extensively than I have 
 upon the subject of toothed gearing, that this book contains 
 nothing new, or original with its author : had such been my 
 belief, the book would never have been written, much less 
 published. 
 
 In my experience as a mechanical engineer I have sought 
 often and earnestly, but always in vain, for a terse, compact, 
 yet complete and comprehensive work on the subject of toothed 
 gearing. Compelled, therefore, by necessity to gain the requi- 
 site knowledge from many works, and also from some failures 
 on my own part, and believing, that, in the crowded field of 
 technical literature, room yet remained for such a publication. 
 I decided to write a book on toothed gearing, which should 
 contain all that I had dug out from so many sources, and ns 
 much more as my experience and originality had taught me, 
 yet being concise, terse, and simple enough to suit even " the 
 wayfaring man, though a fool." Such were the somewhat 
 
 iii 
 
 92342 
 
IV PREFACE. 
 
 exalted intentions of the author in writing this book : whether or 
 not the reality equals the anticipation, is for the reader to judge. 
 
 Notwithstanding the apparent tendency to lay aside the old 
 and simple " rules of thumb " for the surer and better methods, 
 involving, to a certain extent, a knowledge of algebra and geom- 
 etry, there are still many mechanics who continue to look with 
 extreme distrust upon any thing in the shape of a book, because 
 " books are generally too deep and too theoretical." For this 
 reason I have given throughout the following pages simple rules, 
 as well as formulas, for performing each and every operation 
 necessary in designing and laying out the various kinds of gears. 
 
 He who possesses the requisite knowledge of algebra and 
 geometry for which any man will be the better off may 
 make use of the formulas in designing the gears he may have 
 to construct ; while he whose knowledge of mathematics goes 
 not beyond the simple rules of arithmetic may obtain precisely 
 the same results, and do in every way as good work, by using the 
 corresponding rules. Throughout the book I have used a uni- 
 form system of notation in order to avoid confusing or burden- 
 ing the memory of the reader, and the numerous examples will 
 serve to illustrate sufficiently the application of the various rules 
 and formulas. In all cases where the contrary is not stated, 
 forces and weights are taken in pounds, and dimensions in 
 inches. I have also carefully avoided any use of the metric 
 system ; because I believe the good old English inch, foot, and 
 pound to be accurate enough for the proper construction of 
 any machine, engine, or thing which can be made by the use 
 of the metric system. In fact, American and English machin- 
 ery being the best in the world, I see no reason to doubt the 
 efficacy of the English system of weights and measures, from 
 a machinal point of view at least. In writing upon a subject 
 
PREFACE. V 
 
 so old, and upon which so much has been written from time 
 to time, it is impossible that I should not, to a certain extent, 
 have copied the thoughts of others, even though in many cases 
 they are also honestly my own. I deem it best, therefore, to say 
 that I have taken the liberty of referring to and quoting such 
 standard writers as Reuleaux, Camus, Unwin, Haswell, and 
 others, but never, I believe, without giving them due credit. 
 In writing the paragraph on " Special Applications of the Prin- 
 ciples of Toothed Gearing," I have been greatly assisted by 
 referring to Mr. Henry T. Brown's valuable little book entitled 
 " 507 Mechanical Movements," without which the work of col- 
 lecting the various contrivances explained in this paragraph 
 would have been indeed laborious. I trust, that, while much 
 that is printed in this book may be found in other works on 
 the subject, it also contains much that cannot be found else- 
 where, and that my earnest desire to make it a simple, compre- 
 hensive, and convenient companion in the shop and scientific 
 school, may be in some measure, if not fully, realized. 
 
 J. H. C 
 
 NEW YORK, Feb. i, 1884. 
 
TABLE OF CONTENTS. 
 
 SECTION I. 
 
 PAGE. 
 Introduction. Fundamental Principles. The First Gear-Wheel. 
 
 First Transformation i 
 
 SECTION II. 
 
 Proper Form of Tooth- Profiles. The Epicycloid and Hypocycloid. 
 Conditions necessary for Minimum Friction. Conditions 
 necessary for Uniform Velocity. Proper Size of Generating 
 Circle. The Involute 9 
 
 SECTION III. 
 
 Comparison. Advantages and Disadvantages of Cycloidal and 
 Involute Teeth. Experiments with Involute Teeth. The In- 
 volute a Limiting Case of the Epicycloid 
 
 SECTION IV. 
 
 Practical Methods for laying out Teeth, Exact and Approximate. 
 Epicycloidal Faces and Hypocycloidal Flanks. Involute 
 Teeth. Straight Flanks 28 
 
 SECTION V. 
 
 Rack. Internal Gears. Methods for laying out their Teeth .. 37 
 
 vii 
 
viii CONTENTS. 
 
 SECTION VI. 
 
 PAGE. 
 Special Forms. External and Internal Lantern Gears. Mixed 
 
 Gears. Gear at Two Points ....... 42 
 
 SECTION VII. 
 
 Bevel Gears. Pitch Cones. Supplementary Cones. Method for 
 laying out the Teeth. Internal Bevels. The Disk or Plane 
 Wheel ........ .... 49 
 
 SECTION VIII. 
 
 Screw Gears. Angles of the Teeth and Shafts. Screw Gear and 
 Spur Pinion. Screw Rack and Pinion. Method for laying 
 out the Teeth. Worm and Wheel. Worm and Rack. In- 
 ternal Worm Wheel ......... 54 
 
 SECTION IX. 
 
 Hyperbolic Gears. Calculations. Examples. Teeth of Hyper- 
 
 bolic Gears ........... 65 
 
 SECTION X. 
 
 Relations between Diameter, Circumference, Pitch, Number of 
 Teeth, etc. Diametral Pitch. Methods for stepping off the 
 Pitch on the Pitch Circle ........ 72 
 
 SECTION XI. 
 Ratios. Velocity. Revolution. Power. Examples ... 78 
 
 SECTION XII. 
 
 Line of Contact. Arcs of Approach and Recess. Arc of Con- 
 tact . .......... 
 
. CONTENTS. IX 
 
 SECTION XIII. 
 
 PAGF.. 
 Strength of Teeth. Rules and Formulas for determining the Pitch 
 
 and Other Tooth Dimensions. Tables for determining the 
 Pitch. Examples. Table for converting Decimals into Frac- 
 tions. High Speed Gears 89 
 
 SECTION XIV. 
 
 Strength of Arms. Rectangular, Circular, Elliptical, and Flanged 
 Cross-Sections. Number of Arms. Rim, Nave, Shafts, etc. 
 Tables for determining Diameters of Steel and Wrought- 
 Iron Shafts. Approximate Weight of Gear-Wheels . . 107 
 
 SECTION XV. 
 Recapitulation of Formulas and Rules, with Uniform Notation . 139 
 
 SECTION XVI. 
 
 Complete Design of Spur Wheel, Bevel Wheels, Screw Gears, 
 Worm and Wheel, Internal Gears, Lantern Gears, and Gear 
 Train, with Full Working Drawings 151 
 
 SECTION XVII. 
 
 Special Applications of the Principles of Toothed Gearing. 
 Devices for producing Variable Motion. Rectangular Gears. 
 Triangular Gears. Elliptical Gears. Scroll Gears, etc. . 197 
 
 APPENDIX. 
 
 Relative Values of Circumferential and Diametral Pitches. Ex- 
 planation of the Process of cutting Gear-Teeth. Diametral 
 Rules and Formulas -^ 3 
 
TOOTHED GEARING. 
 
 I. Introduction. Fundamental Principles. 
 
 IN the Science of Machinery, a science of vast conse- 
 quence to the world, and vital to the wealth and power 
 of any nation, there is, perhaps, no more important 
 branch than the transmission of power and motion by 
 means of toothed gearing ; for in toothed gearing we 
 have practically the only means of the all-necessary 
 transmission. Having been known for thousands of 
 years, and in practical use for centuries, in reviewing 
 this subject we should naturally look for many succes- 
 sive alterations and improvements, even in fundamental 
 principle ; but no such result will be found by the most 
 diligent research. Contrary to the natural and seem- 
 ingly inevitable course of mechanical contrivances, in 
 principle toothed gearing stands as an exception to the 
 well-nigh universally accepted theory of "small begin- 
 ning and gradual development." Improvement in this 
 branch of machinal science has been slow and retarded ; 
 and strangely discordant with the general belief that 
 first principles are always erroneous, or at least faulty 
 ones, is the fact that the fundamental principle of 
 toothed gearing, as it may be expressed to-day, is pre- 
 
TOOTHED GEARING. 
 
 cisely what it was ten centuries ago. The slow-moving 
 centuries which have witnessed the successive changes 
 in water-motors from the simple undershot wheel, 
 driven in mid-stream by the impulsive force of the 
 river's current, first to the overshot and Poncelet, then 
 to the turbine and water-engine of the nineteenth cen- 
 tury, each involving a different, and, in its turn, an 
 improved, principle can tell of no such advance in 
 the essential principle of toothed gearing. Throughout 
 the years which have changed the steam-engine from 
 an atmospheric-pressure engine to a high-pressure ex- 
 pansion steam-motor ; throughout the years which have 
 produced the locomotive-engine, the ocean steamer, the 
 telegraph, the electric light, the gas-engine, and the 
 telephone, with all their successive alterations in prin- 
 
 Fig. I 
 
 /* 
 
 & 
 
 B 
 
 A ( 
 
 ciple and theory, the science of toothed gearing almost 
 alone has been able to attest, that in one case at least, 
 if no more, first principles have been sound and per- 
 fect, so perfect as to stand the test of years without 
 change or improvement. This principle, most simple, 
 although the underlying principle of the whole theory 
 and study of toothed gearing, may be succinctly ex- 
 
TOOTHED GEARING. 3 
 
 pressed as follows : If two cross-shaped pieces be placed 
 as in Fig. i, the arms of A being somewhat shorter than 
 those of B, and the pieces being allowed only the motion 
 of rotation about their fixed axes, or centres, then, if a 
 continuous rotary motion in the direction indicated by 
 the arrow be given to the piece B, a similarly contin- 
 uous rotary motion in the opposite direction will be 
 given to the piece A. For the arm a, in contact with 
 the arm a 1 , will act as a lever upon it, forcing it down- 
 ward, and at the same time bringing the arms b and b' 
 into such relative positions, that a similar action will 
 take place between them. Thus successively each arm 
 of the piece B will act upon the corresponding arm of 
 the piece A, and a continuous rotary motion will be 
 transmitted from the piece B to the piece A. Simple 
 
 Fig.2 
 
 and crude as our sketch may appear, and however 
 childish and primary our statement of this fundamental 
 principle may seem, a most complete analogy exists 
 between them and the most smoothly and accurately 
 running gears of the present day ; for each one of the 
 countless scores of accurately profiled teeth, working 
 so industriously and almost noiselessly in our machine- 
 
TOOTHED GEARING. 
 
 shops and factories, is but the projecting arm of our 
 cross-shaped pieces, modified in accordance with the 
 advance in machine manufacture, and shaped to suit 
 the increased demand for accuracy of transmission. 
 Since, doubtless, the first gear-wheels were similar to 
 those represented in our figure, let us examine a little 
 more minutely their action and the conditions neces- 
 sary for such action. Let us suppose each wheel to 
 consist of three long, slender pieces, or arms, crossed 
 and fixed in such a manner that their ends divide the 
 circumscribing circles into six equal arcs ; that is, they 
 form the diagonals of a regular hexagon (Fig. 2). The 
 arrows indicate the directions in which the wheels 
 revolve. Now, in order that the rotary motion be 
 continuous, it is obvious that contact between the arms 
 
 d and d' must not cease until contact is begun between 
 the following pair of arms, c and c: otherwise the 
 wheel o would move some distance without moving the 
 wheel </, and consequently the motion of the wheel o 
 would be intermittent. It is also necessary that the 
 arms of o f be somewhat shorter than those of o; for if 
 they were equal (Fig. 3), the arcs xqy and xcfy being 
 also equal, the arms a and a would come in contact at 
 
TOOTHED GEARING. 
 
 5 
 
 their ends, and rotation would be impossible, or, for a 
 greater separation, the arm /; would leave the arm // 
 before the arms a and a f had reached their proper posi- 
 tions, and the wheel o would move on indefinitely with- 
 out touching the wheel o 1 (Fig. 4). 
 
 Fig.4 
 
 A glance at Fig. 5 is sufficient to show the impos- 
 sibility of continuous transmission from o to o 1 when 
 the arms of o r are longer than those of o. Let r be the 
 (ength of each arm of the wheel o (Fig. 2), and / 
 
 the length of each arm of the wheel o'. Let us sup- 
 pose that contact between the arms c and c' begins at 
 the moment when contact between d and d r is just 
 about to cease. We have then the distance pR, be- 
 tween the two points of contact equal to /, because 
 
6 TOOTHED GEARING. 
 
 opR is an equilateral triangle. But we have seen that 
 r' must be less than r: consequently pR must be less 
 than r. The distance pR must obviously be greater 
 than the distance pS, else there would be no contact at 
 all between the arms c and c' . Since, now, the line/ W 
 is perpendicular to and bisects the arm c, we have 
 
 ==\/x;2 
 
 2 = y/f r 2 = .866r, 
 
 but pR = r' is greater than pS: hence the 'conditions 
 necessary for uniform transmission from the wheel o to 
 the wheel o' are, that r' be less than r and greater than 
 .866r. If there were a wheel of this sort given, to be 
 used as a driver, and we wished to construct a wheel 
 
 Fig. 6 
 
 which would gear continuously with it, we would pro- 
 ceed as follows: From the point /, with a radius less 
 than r and greater than .866r, say .gr, we describe an 
 arc cutting the arm c in some point R. Then, with 
 the same radius, we describe a circle passing through the 
 points R and/, and draw the diagonals of the regular 
 inscribed hexagon, of which pR is one side. The end 
 of the arm o'c (Fig. 6) comes in contact with the arm 
 oe at the point r, and slides along its surface until the 
 
TOOTHED GEARING. J 
 
 arms have assumed the positions o'g and of respective- 
 ly. Then the end of the arm oe (now in the position 
 of} comes in contact with the arm o'c (now o'g) at the 
 point f, and slides along its surface until the positions 
 ob and o'b are reached, after which contact between this 
 pair of arms ceases. That is, during each revolution 
 the end of the arm o'c rubs along the surface of the arm 
 oe for the distance cd, and the end of oe rubs along the 
 surface of o'c for the greater distance ab. The wearing- 
 surfaces being unequal in the two wheels, the wear will 
 be unequal, or, in other words, one wheel will wear out 
 before the other : thus the accuracy of transmission 
 will soon be destroyed, and the wheels rendered useless. 
 Such rude contrivances can, of course, be of no practical 
 use, and are given here, not as practical examples, but 
 because of their natural primitiveness, and because they 
 embody principles from which has been built up the 
 present complete theory of toothed gearing. Whether 
 or not these primitive gear-wheels were ever used for 
 actual transmission, is indeed uncertain ; and aside from 
 the natural conclusion that the science of toothed gear- 
 ing, like all other sciences, must have sprung from a 
 mere germinal conception, and that our simple crossed 
 pieces were most probably the first tangible form, the 
 evidence of their real existence is confined to a few 
 rough old drawings, such as those representing the 
 ancient Greek and Asiatic norias for hoisting water, in 
 which crossed pieces of wood precisely similar to our 
 Fig. i are delineated. Certain it is, however, that if 
 these crossed pieces were ever in actual use, time soon 
 effaced the crudeness of their construction, and obliter- 
 ated the faults which caused their inutility. The num- 
 
8 TOOTHED GEARING. 
 
 her of arms was greatly increased ; the arms themselves 
 changed into pegs, or teeth, projecting at regular inter- 
 vals from the circumferences of drums or wheels, and 
 formed with curved profiles, in order to distribute the 
 wear evenly over the whole surfaces of the teeth, and, 
 if possible, to diminish the friction between the teeth, 
 and so also the wear itself (Fig. 7). Since, now, the teeth 
 
 Fig.7 
 
 of the wheels A and B rub or slide against each other 
 when in contact, and thus produce friction and wear, 
 there must be some form of profile, straight or curved, 
 simple or compound, which, when given to the teeth, 
 will reduce the friction between them to a minimum, 
 some form which will be more advantageous for accurate 
 transmission and uniformity of motion than any other. 
 It is needless here to state of what vast importance is 
 this desired form of tooth-profile ; for the perfection of 
 almost every machine the most simple and compact, as 
 well as the most complicated and extensive depends, 
 to a very great degree, upon the action of its gear- 
 wheels, and consequently upon the formation of the 
 tooth-profiles of the gears. The proper formation of 
 
TOOTHED GEARING. 9 
 
 the tooth-profile must insure, in the words of another, 
 "a more equable performance of the work in hand, a 
 diminution of the moving-power wasted by friction, and 
 hence the accomplishment of more work with the same 
 amount of power, and a greater durability, and conse- 
 quently a less cost for repairs in the whole machine." 
 Recognizing, then, the fact that the subject with which 
 we are dealing is of more than ordinary importance, we 
 propose an investigation which aims to present, in as 
 clear and terse a manner as possible, the method of 
 reasoning by which the present development of tooth- 
 profiles has been attained, an investigation from 
 which have been purposely omitted all the more intri- 
 cate and tedious mathematical calculations pertaining 
 to the subject, which have been so laboriously worked 
 out by other writers and investigators. Far from think- 
 ing, or even wishing, to disparage the labors of men of 
 genius and ability who have devoted their time and 
 energies to the promotion of the purely mathematical 
 and theoretical part of the great study of toothed gear- 
 ing, on the contrary, believing their investigations and 
 calculations to be the foundation upon which have been 
 built the present more abstruse theories, their investi- 
 gations have been omitted, because they may be found 
 in almost any comprehensive work on the subject, and 
 because it is thought unnecessary to repeat them here. 
 
 II. Proper Form of Tooth- Profiles. 
 
 Let C and C r (Fig. 8) be two circles, in contact at 
 the point a. If the circle C be made to revolve in the 
 direction indicated by the arrow, the circle C f will be 
 made to revolve in an opposite direction by the friction 
 
10 
 
 TOOTHED GEARING. 
 
 Fig. 
 
 between the two circles, supposing, of course, the fric 
 tion to be great enough to overcome the resistance. 
 Suppose, now, it is required of the circle C' to perform 
 work, for example, to lift a weight 
 W by means of a string wound 
 around its axle. By varying the 
 pressure of the circle C upon C' at 
 the point a of contact, the friction 
 between the circles may be made 
 just sufficient for the lifting of the 
 weight : the friction between the 
 circles will then be the smallest 
 possible for the given amount of 
 work. Also, if the circle C is 
 driven by a constant and uniform 
 force, since the resistance and 
 motion are constant and' uniform, 
 the weight W will be lifted by a 
 constant and uniform force, or, in 
 other words, power and motion 
 will be uniformly transmitted. 
 We may therefore conclude, that in order that toothed 
 wheels may work together most uniformly, with the 
 least friction and wasted power, and with the greatest 
 durability, the tooth-profiles must be such that the 
 driving wheel shall cause the driven wheel to revolve 
 as if moved by simple contact. If the circle a roll, in 
 the direction indicated by the arrow, upon the circum- 
 ference of the circle B (Fig. 9), the point o of the circle 
 a will assume successively the positions o' t o", o'", etc., 
 the arc p'o being equal to the arc p'o', the arc p"o being 
 equal to the arc p"o", etc., and the position o of the 
 
TOOTHED GEARING. 
 
 T I 
 
 point o corresponding to the position b of the rolling 
 circle, etc. The point of contact, o, generates during 
 the rolling the curve o o'o"o'"o"" , obtained by drawing a 
 
 curve through the successive positions of the point o. 
 This curve, described by a point of the circumference 
 of circle which rolls upon the circumference of another 
 circle, is called an epicycloid. In the same manner, if 
 
 a circle a roll, in the direction of the arrow, within the 
 circumference of another circle B (Fig. 10), the point o 
 on the circumference of the rolling circle will generate 
 ; the arcs p'o', p"o", and p m o'" being 
 
 the curve o o'o"o'" 
 
12 TOOTHED GEARING. 
 
 respectively equal to the arcs p'o, p"o, and /"'<?, and the 
 positions 0', o", and o'" of the point o corresponding to 
 the positions b, c, and d of the rolling circle a. This 
 curve, described by a point on the circumference of a 
 circle which rolls ivithin the circumference of another 
 circle, is called a hypocycloid. In Fig. 9 the motion 
 of the circle a, relative to the motion of the circle B, is 
 precisely similar to the motion of the circle C, relative 
 to the motion of the circle C r (Fig. 8). For in Fig. 9 
 equal arcs of the rolling circle are developed, in equal 
 times, upon the circumference of the circle B ; and the 
 same is true of the circles C and C' (Fig. 8). Conse- 
 quently the motion of any point, as o, of the circle a, 
 with reference to the motion of the corresponding point 
 o of the circle B (Fig. 9), must be similar to the motion 
 of the point a of the circle C with reference to the 
 motion of the corresponding point a of the circle C' 
 (Fig. 8). But we have shown that the point o of the 
 circle a (Fig. 9) generates an epicycloid with reference 
 to the motion of the point o of the 
 circle B : hence, also, in Fig. 8, the 
 point a of C generates an epicycloid 
 with reference to the motion of the 
 point a of C '. For the same reasons, 
 the point a of the circle O, revolving 
 about its fixed centre, and thereby 
 causing the circle O f to revolve about 
 its centre (Fig. 1 1), generates, with reference to the 
 motion of the point a of the circle O f , a hypocycloid. 
 
 Let C and C (Fig. 12) be two teeth, contact between 
 which has just begun, C being the driving, and C' the 
 driven tooth. It is plain, from what has been said, 
 
TOOTHED GEARING. 13 
 
 that the motion of the point a of the tooth C', with 
 reference to the motion of the point a of the tooth 
 C, is similar to the path described by the point a of 
 a circle O', which rolls within the circumference of the 
 circle O. This path, as before explained, is a hypocy- 
 cloid ; and consequently, if we give to the portion ab 
 of the tooth C (called the flank of the tooth) a hypocy- 
 cloidal form, the profile ab' will slide along it with the 
 least possible friction. While the point a of the tooth 
 C r slides along the profile ab, the point a of the tooth C 
 
 also slides along the profile ab', and generates, with re- 
 spect to the motion of the point a of the tooth C', the 
 epicycloid ab', the path described by the point a of 
 a circle (9", which rolls upon the circumference of the 
 circle O. If, therefore, we give to the portion ab' of 
 the tooth C (called the face of the tooth) an epicycloidal 
 form, the profile ab will slide along it with the least 
 possible friction. Again : let the teeth of the wheels O 
 and (7 be in contact at the point / (Fig. 13), and suppose 
 (J to be the driver. The driving-force of the wheel C? 
 will be transmitted to the wheel O through the point /, 
 
TOOTHED GEARING. 
 
 and in the direction of AB, the common normal to the 
 surfaces in contact at the point /. From the centres 
 O' and O draw the lines O'A and OB, each perpen- 
 dicular to AB. Let F denote the driving-force of the 
 wheel O', or the force exerted by the circumference Y, 
 and F' the force exerted by the point A. From the 
 principles of the simple lever, we have the propor- 
 
 tion F\F' '.'. 
 
 Hence F' = 
 
 Since 
 
 or 
 
 F f = 
 
 O'A 
 
 the lines O'A and OB are 
 parallel, the perpendicular 
 AB will be tangent to two 
 circles drawn with and 
 O as centres and O'A and 
 
 PB as radii, and the force 
 ' of the point A will be 
 directly transmitted -to the 
 point B through the line 
 AB. Let P denote the 
 force transmitted to the 
 circumference X. As be- 
 fore, we shall have the 
 proportion P\F' \\OB\Oa, 
 
 From this we obtain 
 
 F:P::OaXO f A:O'aXOB. From the right - angled 
 triangles cOB and cO'A we may write the proportion 
 A (7: OB\: cO 1 \cO, which, multiplied by aOiaC?:: 
 aO : aa, gives OA X Oa : OBX O'a : : cO'X Oa : cOX Va, 
 and consequently we shall have F\P : \cCfxOa : 
 cOxO'a. But, for best results, all the force of the 
 wheel Cf must be transmitted to the wheel O ; also, in 
 
TOOTHED GEARING. 
 
 Fig. 14 
 
 order that the wheels may move as simple friction 
 wheels, the velocities at the circumferences must be 
 equal. Hence the forces F and P must be equal; and 
 we will consequently have cO'^aO cOXaO', which 
 can only be true when the points c and a coincide, and 
 form one point. We may conclude from this, that the 
 most advantageous form for the profiles of the teeth is 
 such that the common normal to the profiles at the 
 point of contact will pass through the point of inter- 
 section of the line of centres with the pitch circles X 
 and F. This point is 
 called the pitch point. 
 Suppose, now (Fig. 14), 
 the pitch circle O and 
 the rolling or generating 
 circle O' to be regular 
 polygons, having each 
 an infinite number of 
 sides. As the polygon 
 O' rolls in the direction 
 shown by the arrow, the 
 point A generates an 
 epicycloid ; and there is, for an instant, a rotation of the 
 polygon O' about the point C. The point A, for that 
 instant, describes an arc of a circle, the centre of which 
 is the point C, and the radius of which is the line CA. 
 But, since the radius of a circle is always normal to the 
 circumference at the point of their intersection, the line 
 CA is a normal to the epicycloid at the point A : it also 
 passes through the pitch point C. These two demon- 
 strations were, we believe, first given by M. Camus in 
 his " Cours de Mathematiques." By a similar course oi 
 
l6 WOTHED GEARING. 
 
 reasoning it may be proved that the normal CA' of the 
 hypocycloid BA' (generated by the point A' of the poly- 
 gon O", which rolls within the polygon O), at the point 
 A' passes through the pitch point C. If, now (Fig. 13), 
 we give to the face of the tooth of O' an epicycloids! 
 form, and to the flank of the tooth of a hypocycloichJ 
 form, the point of contact of the teeth will be the point 
 of contact of two infinitely small circle-arcs, the radii 
 of which are parallel, coincide to form the common 
 normal, and pass through the pitch point a. We may 
 now briefly sum up our arguments in order, and the 
 conclusions which must be drawn from them. We have 
 shown (Fig. 8), that, in order that the teeth of wheels 
 work most uniformly together and with the least detri- 
 mental friction possible, the action of the driving wheel 
 upon the driven wheel must be such that the wheels 
 shall move as if driven by simple contact. We have 
 also proved (Fig. 12) that this desired action takes 
 place between the teeth when the faces of the teeth 
 are given the epicycloidal and the flanks of the teeth the 
 hypocycloidal form. Further: we have proved (Fig. 
 13) that the condition necessary for uniform power and 
 velocity is that the common normal to the teeth in con- 
 tact, at the point of contact, shall pass through the 
 pitch point, and (Fig. 14) that this condition is fulfilled 
 by teeth having epicycloidal faces and hypocycloidal 
 flanks. From these demonstrations but one logical con- 
 clusion can be drawn, that teeth having epicycloidal 
 faces and hypocycloidal flanks fulfil all the conditions 
 required of gear-teeth, and that the desired form of 
 tooth-profile has been determined. Roomer, the cele- 
 brated Danish astronomer and inventor, is said to have 
 
TOOTHED GEARING. I/ 
 
 been the first to demonstrate the advantages of these 
 curves for tooth-profiles. But De la Hire, who is 
 credited with having first discovered, that, " if the pro- 
 files of the teeth of one wheel have an epicycloidal 
 form, the profiles of the teeth of its fellow will prop- 
 erly have the form of a hypocycloid the generating 
 circle of which has the same diameter as that of the 
 epicycloid forming the teeth of the first wheel," * - 
 Brewster, Young, Buchanan, and Reuleaux have been 
 the chief promoters of the application. 
 
 Our investigation has now given us the required 
 forms of tooth-profile ; but since these curves, like all 
 others, are susceptible of a considerable number of 
 variations, it remains to determine somewhat more 
 specifically the conditions upon which their applica- 
 bility to wheel-teeth depends. In the first place, then, 
 the amount of curvature, or amount of deviation, of 
 epicycloidal and hypocycloidal curves from the diameter 
 of the primitive or pitch circle, which passes through 
 the pitch point, depends upon the diameter of the gen- 
 erating circle and upon the diameter of the primitive 
 circle, or, in other words, upon the ratio of the diame- 
 ter of the generating circle to that of the primitive 
 circle. Thus, in Fig. 15, the epicycloids <?, b, and c 
 were generated by circles having diameters respectively 
 equal to \, f , and \ the diameter of the primitive circle 
 OO' ; and the hypocycloids a f , b', and c' had for generat- 
 ing circles respectively the same as the epicycloids. 
 If d denote the diameter of the generating circle, and 
 D that of the primitive circle, it is plain from the 
 
 * Mr. J. I. Hawkins's translation of Camus on the Teeth of Wheels. 
 
i8 
 
 TOOTHED GEARING. 
 
 figure, that, as the ratio jz becomes smaller, the 
 
 tion of the curve from the diametral line Bcf, passing 
 through the pitch point/, becomes greater. If the diam- 
 eter of the generating circle of a hypocycloid is equal 
 to one-half the diameter of the primitive circle, the 
 curve described will be a straight line coinciding with 
 the diameter of the primitive circle passing through 
 
 Fig. 15 
 
 a'.- 
 
 the starting position of the generating point. To 
 prove this, let C' (Fig. 16) be the generating circle, and 
 C the primitive circle. Let o be the starting position 
 of the generating point. Since the diameter of C f is 
 equal to one-half that of C, the circumference of C' 
 will be equal to one-half the circumference of C, one- 
 
TOOTHED GEARING. 
 
 half circumference of C r one-fourth circumference 
 of C, one-fourth circumference of C r = one-eighth cir- 
 cumference of C, etc. Then, when the circle C r rolls 
 sufficiently, the point A will fall upon the point A' ; 
 Ao one-fourth circumference of C' =. oA f =. one- 
 eighth circumference of C. The diameter AB will then 
 have the position A'o" ; the diameter oo" will have the 
 position A"o , at right angles 
 to A'o" ; and the point o will 
 have the position o' on the 
 diameter oE. Again : when 
 the circle C' rolls sufficiently, 
 the point o" will fall upon the d ] 
 point d ; arc oAo" = one-half 
 circumference of C' = arc 
 0./4V one-fourth circumfer- 
 ence of C. The diameter o"o 
 will then have the position 
 do", and the point o will have the position o", still on 
 the diameter oE. Thus it may be proved, that, for any 
 position of the generating circle C', the point o will 
 fall upon the diameter oE, and consequently that diam- 
 eter is the path of the point ; or the curve generated 
 by the point o will coincide with the diameter of the 
 primitive circle, which passes through the starting 
 position of the point. If, therefore, we use for the 
 generating circle of the tooth-profiles one which has 
 for a diameter one-half that of the primitive circle, the 
 flanks of the teeth will be simply radial straight lines, 
 as is sometimes the case in practice. 
 
 Fig. 17 shows forms of tooth-profile for different gen- 
 erating circles. Thus profile \a\ was generated by a 
 
2O 
 
 TOOTHED GEARING. 
 
 circle having one-half the diameter of the primitive cir- 
 cle O(J\ the generating circle of profile 20,2 had for a 
 diameter three-eighths that of the primitive circle ; and 
 the diameter of the generating circle of profile 3^3 was 
 one-quarter that of OO r . In profile 3^-3 the inclination 
 of the faces is so great, that there may be, by the princi- 
 ples of the inclined plane, a tendency to produce press- 
 ure upon the axles of the wheels ; 
 while profile \a\ is a weak form 
 for teeth, being narrowest at the 
 base \b, where it should be widest, 
 because this part of the tooth 
 bears the greatest strain when in 
 action. Profile lai is also a bad 
 form for wear, because the friction 
 between the face of one tooth of 
 this form and the flank of another 
 is much greater than it would be if the face and flank 
 were more nearly envelopes of each other. Therefore, 
 for greater strength, less friction and wear, and best 
 action between the teeth, we should take for the diam- 
 eter of our generating circle less than one-half and 
 greater than one-quarter of the diameter of the primi- 
 tive circle. Generating circles of one-third the diam- 
 eter of the primitive circle give very good results in 
 practice. 
 
 Let us investigate the subject of profiles further. 
 Let op p' . . . p vi be a string, wound around the circum- 
 ference of the circle C, and fastened at the point p vi 
 (Fig. 1 8). If, now, the string be unwound from the 
 point o, and held rigid as it unwinds, the end, or point 
 o t will assume successively the positions </, o" 9 </", etc.; 
 
TOOTHED GEARING. 
 
 21 
 
 the line po' being equal to the arc po, p'o" p'o, 
 p"o" r p"o> etc. The curve o o'o"o" r . . . o vf , gener- 
 ated by a point of a string as it unwinds from the cir- 
 cumference of a circle, is called an involute to the circle, 
 or an involute simply. Suppose (Fig. 19) the primi- 
 
 Fig.18 
 
 01V 
 
 tive circle to be a regular polygon, having an infinite 
 number of sides. As the string bao' unwinds, there 
 will be, for an instant, a revolution about the point a ; 
 and the point o' of the string will then generate 
 a circular arc having its centre in the point a, and 
 
 Fig. 19 
 
 a radius ao'. Therefore, as was shown in Fig. 14 for 
 the epicycloid, the involute also fulfils the condition 
 necessary for uniform power and velocity. For this 
 reason the involute curve has been, and still is, exten- 
 sively used for tooth-profiles, the curve forming the 
 
22 
 
 TOOTHED GEARING. 
 
 whole profile, cd (Fig. 20) ; or the teeth having involute 
 
 faces and radial straight flanks, 
 as in ab. We have now two 
 kinds of tooth-profile, cycloidal 
 ^ and involute ; each having, it is 
 presumable, its advantages and 
 its disadvantages in practice. 
 A comparison between the two 
 is therefore necessary. 
 
 III. Comparison. Advantages and Disadvantages of Cycloidal and 
 Involute Teeth. 
 
 Cycloidal teeth have a great advantage over involute 
 teeth, in that the number of teeth, for wheels of the 
 same diameter gearing together, may be reduced to 
 seven, without in any degree interfering with the 
 uniformity of action. Reuleaux gives the smallest 
 number of involute teeth necessary for proper action, 
 eleven. In cycloidal teeth the loss of power and wear 
 due to friction is not so great as in involute teeth ; 
 also the effect upon the action of the teeth by wear 
 is less in cycloidal than in involute teeth, because the 
 wear is evenly distributed in the former, and the teeth, 
 even when considerably worn, present more nearly the 
 original form of profile. Involute teeth, on the other 
 hand, have the advantage of being easier and cheaper 
 to construct than the compound profiles of cycloidal 
 teeth. They are also stronger for the same width on 
 the pitch circle. Again : the axles of wheels having 
 involute teeth may be moved slightly from or toward 
 each other without disturbing the proper action ; while 
 a very slight alteration of the distance between the 
 
TOOTHED GEARING. 2$ 
 
 axles of cycloiclal gears destroys the accuracy of motion. 
 Straight flanks are acknowledged by all to be poor 
 forms, both on account of their weakness, and loss of 
 work by friction. They should never be used except 
 for large wheels, where the distance of the centres from 
 the pitch circles renders them more nearly parallel, and 
 consequently stronger. The principal objection offered 
 to involute teeth is, that, especially in small wheels, 
 the great obliquity of the profiles tends to produce a 
 pressure upon the journals and bearings, as before 
 noticed. Considerable difference of opinion exists as 
 to the truth of this objection, and of late years actual 
 experiment seems to assert its falsity. The following 
 experiments were tried by Mr. John I. Hawkins, and 
 are taken from his English translation of that portion 
 of M. Camus's " Cours de Mathematiques " relating to 
 the teeth of wheels. Simi- Fi fl ,2i 
 
 lar experiments tried by the N 
 author of this book, with 
 wheels carefully sawed out 
 of black walnut, gave es- 
 sentially the same results. 
 The approach noticed by Mr. Hawkins in his Experi- 
 ment II., however, failed to appear in the experiments 
 of the author. Having constructed the sectors of two 
 wheels, each of two feet radius, and each containing 
 four teeth of the same curve as those shown in Fig. 21, 
 one of the sectors (No. i) was mounted on a fixed 
 axis, and the other on an axis so delicately hung, that a 
 force of even a few grains would cause the axis of the 
 latter to recede from that of the former in a direct line. 
 The following experiments were then made: 
 
24 TOOTHED GEARING. 
 
 EXPERIMENT I. 
 
 The teeth of both sectors being engaged their full 
 depth of an inch and a half, No. i was moved forwards 
 and backwards a great number of times, without exhib- 
 iting the least tendency to thrust No. 2 to a greater 
 distance, notwithstanding the tangent to the surfaces 
 of the teeth in contact formed an angle of nearly sixteen 
 degrees with the line of centres. The points of contact 
 of the teeth at the line of centres were three-quarters of 
 an inch from the ends of the teeth. 
 
 EXPERIMENT II. 
 
 The teeth were engaged an inch and a quarter deep : 
 consequently the ends of the teeth were a quarter of 
 an inch free from the bottoms of the spaces ; the 
 tangent of contact made an angle of nearly seventeen 
 degrees with the line of centres ; and the point of con- 
 tact at the line of centres was five-eighths of an inch 
 from the ends of the teeth. The sector No.- i, being 
 repeatedly moved forwards and backwards, sometimes 
 caused sector No. 2 to approach, but never to recede. 
 In Experiment I. the approach could not take place, 
 because the teeth were engaged their full depth. 
 
 EXPERIMENT III. 
 
 The teeth were engaged one inch deep, leaving half 
 an inch between the ends of the teeth and the bottoms 
 of the spaces. The angle of the tangent of contact 
 with the line of centres was eighteen degrees ; the 
 points of contact at the line of centres were half an 
 inch from the ends of the teeth. On the sector No. i 
 being moved frequently forwards and backwards, no 
 motion of the axle of No. 2 appeared, 
 
TOOTHED GEARING. 2$ 
 
 EXPERIMENT IV. 
 
 The teeth of the sectors were engaged three-quarters 
 of an inch deep : consequently the ends of the teeth 
 were three-quarters of an inch free from the bottoms 
 of the spaces ; the points of contact of the teeth at 
 the line of centres were three-eighths of an inch from the 
 ends of the teeth ; the angle of the tangent of contact 
 v,-ith the line of centres was nineteen degrees. The 
 axle of sector No. 2 neither approached nor receded 
 on numerous trials made by moving No. i. 
 
 EXPERIMENT V. 
 
 The teeth were engaged half an inch deep ; the point 
 of contact was a quarter of an inch from the ends of 
 the teeth at the line of centres ; the ends of the teeth 
 were one inch from the bottoms of the spaces ; the 
 tangent of contact formed an angle of full twenty 
 degrees with the line of centres. In a great number 
 of repetitions of this experiment, a slight receding of 
 sector No. 2 sometimes appeared. 
 
 EXPERIMENT VI. 
 
 The teeth were engaged a quarter of an inch : the 
 ends of the teeth, therefore, were one inch and a 
 quarter from the bottoms of the spaces ; and the points 
 of contact, one-eighth of an inch from the ends of the 
 teeth at the line of centres ; the angle of the tangent 
 of contact with the line of centres was rather more 
 .than twenty-one degrees. In this experiment, which 
 was repeated very frequently, a tendency to recede 
 appeared several times, but so slightly as to be of no 
 practical importance. The quiescent state of the axle 
 was much oftener manifest than the receding. 
 
26 TOOTHED GEARING. 
 
 " These experiments," says Mr. Hawkins, "tried 
 with the most scrupulous attention to every circum- 
 stance that might affect the results, elicit this important 
 fact, that the. teeth of wheels in which the tangent of 
 the surfaces in contact makes a less angle than twenty 
 degrees with the line of centres, possess no tendency 
 to cause a separation of their axes : consequently there 
 can be no strain thrown upon the bearings by such an 
 obliquity of the tooth." Such an obliquity as twenty 
 degrees must, unless counteracted by an opposite force, 
 tend to separate the axes ; and, as suggested by Mr. 
 Hawkins, this opposite force is most probably the fric- 
 tion between the teeth, which tends to drag the axes 
 together with as much force as that tending to separate 
 them. Of course the friction between teeth sawed out 
 of wood is greater than in metal teeth ; but Mr. Haw- 
 kins cites experiments tried by a Mr. Clement, with 
 metal wheels lying loosely upon a work-bench, in which 
 no tendency to separate the axes of the wheels could be 
 noticed. This very serious objection to involute teeth 
 having once been fairly removed, then the relative 
 value of the two kinds of profile must depend upon the 
 action between the teeth in each case, the amount of 
 friction and wasted power, and the relative expense and 
 difficulty of construction. The fact, that, in cycioidal 
 teeth, less power is lost in overcoming friction than in 
 involute teeth, seems to be well established, in theory 
 at least, if, perhaps, not so well in practice ; but whether 
 or not the gain in this respect is sufficient to compen- 
 sate for the additional expense of construction over the 
 involute system, is still a question which must be finally 
 settled by practice and actual experiment. In this 
 
TOOTHED GEARING. 27 
 
 practical age, the value of any one mechanism, com- 
 pared with that of another, is simply a comparison 
 between the relative amounts of work to be obtained 
 from them and the relative costs ; and that system of 
 tooth-profiles from which can be obtained "the most 
 work for the least money " must eventually gain the 
 supremacy. In Fig. 18, while generating the involute 
 curve, as fast as any portion of the string is unwound, 
 it is held rigid, and forms a straight line tangent to 
 the circle at the point of contact ; as, for instance, the 
 portion p iv o v is tangent to the circle C at the point p iv . 
 Since this portion is that which generates the curve, 
 and upon which alone a 
 the curve depends, we 
 may assume the whole 
 string to be rigid and 
 straight, and the re- 
 sult will be the same, a' 
 Let oa (Fig. 22) be a a . 
 straight line, which 
 rolls from right to 
 left upon the circumference of the circle C. When 
 the line oa has rolled sufficiently, the point b will fall 
 upon the point / (the arc op being equal to the line ob), 
 and the point o will take the position o' . When the 
 line has rolled sufficiently, the point b' will fall upon 
 the point /' (the arc p'o being equal to the line b'o\ 
 and the point o will then take the position o". When 
 the point b" falls upon the point /", the point o will 
 take the position o"\ etc., and the curve o o'o"o'" thus 
 generated will be an involute to the circle C. Thus we 
 have generated an involute by rolling a straight line 
 
28 7VOTHED GEARING. 
 
 upon the circumference of a circle. But a straight line 
 is the circumference of a circle the radius of which is 
 infinitely long; and the curve generated by a point of 
 a circle which rolls upon the circumference of another 
 circle is an epicycloid : consequently an involute curve 
 is simply an epicycloid the generating circle of which 
 has an infinitely long radius ; or, in other words, the 
 involute is but a limiting case of the epicycloid. Thus, 
 without coming to any actual decision as to the relative 
 mechanical value of these two curves, or rather two 
 different forms of the same curve, we have, neverthe- 
 less, the satisfaction of having verified our former con- 
 clusion, and may still assert that the cycloidal form of 
 tooth-profile fulfils all the conditions and requirements, 
 and is therefore the most useful and advantageous. 
 
 IV. Practical Methods for laying out Teeth, Exact and 
 Approximate. 
 
 Because of the difficulty with which exact epicycloidal 
 and hypocycloidal profiles are constructed, approximate 
 methods are very generally used ; and they are found to 
 answer the practical purpose very well. Any one of 
 the following approximate methods will give very good 
 results, and will, in ordinary cases, answer as well as the 
 more difficult and tedious exact methods, also given 
 here for use in special cases : 
 
 METHOD i (exact). Let O (Fig. 23) be the primitive 
 or pitch circle. Take the diameters of the rolling cir- 
 cles C and K, each equal to one-third the diameter of the 
 pitch circle. Strike the circles C', C" , C'" , etc., which 
 represent the different positions of the rolling circle (7, 
 and from the points of tangency, b, b', b"> etc., measure 
 
TOOTHED GEARING. 
 
 2 9 
 
 off the following arcs : bp' bp, b'p" = b'p, b"p r " b"p, 
 etc. The points /',/",/"', etc., thus found, are points 
 of the epicycloid which is to form the face of the tooth ; 
 and the curve pp r p" . . . p* v , drawn through them, is 
 the face-profile. For the hypocycloidal flank, after 
 having struck the circles /, O f> ', O'" y lay off the arcs 
 da' = dp, d'a" = d'p, etc., from the points of tangency 
 d, d', etc. The curve pa' a" a'", drawn through the 
 points a' t a", a'", thus found, is the hypocycloid which 
 
 Fig. 23 
 
 is to form the flank of the tooth. The other profile, 
 xyz t which is similar to the one just found, is found 
 by starting at the point y (py being the given width 
 of the tooth at the pitch circle), and rolling the gen- 
 erating circles in the opposite directions from those 
 just described. We have now but to limit the tooth 
 at the top and bottom by circle-arcs, AA and BB, and 
 the profile is complete. 
 
 METHOD 2 (exact}. In Fig. 24 O is the pitch circle, 
 
3O TOOTHED GEARING. 
 
 Cf and O" the rolling circles, and A the pitch point. 
 Divide the pitch circle and rolling circles into an equal 
 number of small parts, equal each to each, as shown in 
 the figure. Let the point 5 of O' correspond to point 5' 
 of O, the point e of O" correspond to the point e of O, 
 etc. From A and 5' as centres, with 5 5' and the chord 
 A$ respectively as radii, describe arcs intersecting in 
 the point c ; then from the centres A and 4', with the 
 radii 4 4' and ^4, describe arcs intersecting at <:', etc. 
 
 Fig.24 
 
 The points thus found are points of the epicycloid Ac'c. 
 Similarly, for the hypocycloid, from the points A and 
 /, with radii e'e and Ae, describe arcs intersecting at 
 the point /, and thus determine the curve Ap'p. In 
 these two methods, the closer together the positions 
 of the rolling circles and the points of division of the 
 pitch and rolling circles are taken, the more accurate 
 will be the curves. When either of these methods is 
 used, the work of laying out the teeth may be greatly 
 simplified by accurately working out one entire profile 
 
TOOTHED GEARING. 31 
 
 upon a smooth piece of wood, and cutting out this 
 profile for a template with which to trace the profiles 
 around the pitch circle. 
 
 METHOD 3 (approximate}. From the points i', 2', 3', 
 etc., a', b f , c f , etc. (Fig. 24), as centres, and with the 
 corresponding chords of the rolling circles as radii, 
 draw circle-arcs. Thus the radius for centre 5' is A$ t 
 for centre 3' is A 3, for centre e f is Ae, etc. The en- 
 velope of these arcs, or the curve which is tangent to 
 them, is very nearly the correct profile of the tooth. 
 
 Fig.25 
 
 METHOD 4 (approximate). In Fig. 25 let A A be the 
 pitch circle, and B and C the rolling circles. Let, also, / 
 be the pitch point, and te and tk the heights of the tooth 
 above and below the pitch circle. Take /;/ = fte, and 
 strike through ;/ the arc ;/;/ concentric with the pitch 
 circle. Step off on the pitch circle to tu, and from o 
 as a centre, with the chord n't for a radius, strike an arc 
 cutting ;/;/ in the point /. Draw po. The point / is a 
 point of the epicycloid, and po is the normal to the 
 curve at the point p. Find now, upon the line po l 
 
32 TOOTHED GEARING. 
 
 the centre for an arc passing through the points / and 
 /. In the figure, o' is this centre, and o'p is the radius 
 for the faces of the teeth. The centres for all the faces 
 are upon the circle aa drawn through o' t and concentric 
 with the pitch circle. Similarly, for the flanks, take 
 tm | tk, strike the arc mm' , step off tb = tin', and, 
 with the centre b and radius bt, strike an arc cutting 
 mm' in the point x. Draw xb, and find the centre b' 
 for an arc passing through / and x. The radius for the 
 flanks is b'x ; and the centres are all upon the circle dd y 
 drawn through tf, and concentric with the pitch circle. 
 METHOD 5 (approximate}. Let A (Fig. 26) be the 
 
 pitch circle, and a the pitch 
 point. Draw af tangent to 
 the pitch circle at the pitch 
 point, and make it equal to 
 0.57 the diameter of the roll- 
 ing circle, or \\ times the 
 circular pitch of the ' teeth. 
 Draw dfe parallel to the 
 diameter aO, make df = af, 
 and ef the diameter of the 
 rolling circle. Draw Od and 
 O Oep, and, taking ab = ac 
 
 \af, draw bp and gp r parallel each to af. The point / of 
 the intersection of Op and bp is the centre for the flank 
 ax. Make p'c = eg, and /' is the centre for the face ay. 
 As before, all the face centres are upon a circle drawn 
 through /' concentric with the pitch circle, and ali the 
 flank centres are upon a circle drawn through p. 
 
 METHOD 6 (approximate}. Let A (Fig. 27) be the 
 pitch circle, C and B the rolling circles, and a the pitch 
 
TOOTHED GEARING. 
 
 33 
 
 point. Draw a'c'b and cB'd through the centres of the 
 rolling circles, each making angles of 30 with the line 
 of centres. Draw the line 
 cabf through the points c 
 and b, and join a' and d 
 with the centre O. The 
 points g and f are the 
 centres for the face bx 
 and flank cy respectively. 
 These approximate meth- 
 ods are from Reuleaux's A' 
 " Const ructeur," and Un- 
 win's " Elements of Ma- 
 chine Design," and are as 
 accurate as any in use at 
 the present time. When 
 a set of wheels is to be 
 constructed so that any 
 wheel of the set will gear 
 with any other, the same 
 generating circles must be 
 taken for all the teeth of 
 the set. Sometimes the 
 generating circle is taken 
 with a diameter equal to 
 the radius of the smallest 
 wheel of the set. The 
 following are some of the 
 simpler and rougher meth- 
 ods of approximation in 
 use : they are convenient and easy, but give poor re- 
 sults, and should only be used in rough work. Fig. 28, 
 
34 
 
 TO OT PI ED GEARING. 
 
 draw ac, making 75 with the line of centres bd, and 
 make be equal to one-tenth the pitch of the teeth multi- 
 plied by the cube root of the number of teeth. Take 
 ab = ^bc : c is the centre for the face bi, and a the centre 
 for the flank bk. The following values of ba and be give 
 
 better results: ba - - -, and be = o. 1 2p\lN t in 
 
 2N 20 
 
 which / represents the pitch, and N the number of 
 teeth. In Fig. 29, oo is the pitch circle, and bb and aa 
 the circles which limit the teeth at top and bottom. 
 
 Fig. 29 
 
 The centres for both faces and flanks are taken upon 
 the pitch circle ; the flank centre for gk and mn being 
 in the centre of the tooth width at x, and the face 
 centre for cd and ef being* in the centre of the space 
 width at y. Still another rough rule is to take the 
 centres upon the pitch circle, and take the radius for 
 the faces equal to one and one-fourth times the pitch, 
 making the flanks radial straight lines. 
 
TOOTHED GEARING. 
 
 35 
 
 For laying out involute teeth, the exact method is as 
 follows : Fig. 30, O is the circle of the bottoms of 
 the teeth, and / the starting-point of the involute, or the 
 root of the tooth. Lay off the distances pp ', p'p", p"p f ", 
 etc., along the circle OO ; draw the tangents /Y, p"a", 
 etc. ; and step off p'a arc //, p"ct' = arc />, p'"a'" = 
 p'"p, etc. The curve a a" a'", etc., drawn through the 
 points thus found, is the true involute profile. In 
 the same manner, the profile cf is found, and the tooth 
 limited in height by the circle bb. Radial straight 
 flanks are often used in involute teeth ; but, for reasons 
 already given, they should never be used except for 
 large wheels, and even then only for rough work. 
 True involute profiles 
 may be easily traced by 
 means of a straight 
 spring arranged to hold 
 a pencil, or other mark- 
 er, at one end, and fas- 
 tened at the other end 
 to the circumference of 
 a wooden circle-segment of the same radius as the bot- 
 tom or root circle of the teeth of the wheel. Because of 
 the comparative ease with which true involute profiles 
 may be traced, approximate or circle arc methods are 
 not much in use. The following methods, however, 
 give very close approximations to the true curve, and 
 are, perhaps, more in use than any others. In Fig. 31 ei 
 is the working height of the tooth, i.e., the actual height 
 less the clearance between the end of the tooth of one 
 wheel and the bottom of the corresponding space of 
 the other wheel, and im is the actual height. Make 
 
TOOTHED GEARING. 
 
 ea = \ei, and draw ad tangent to the circle A ; make 
 pd \ad, and / is the centre for the profile bak. A 
 circle through /, concentric with the circle A, gives 
 the positions of the centres for all the profiles. The 
 
 part kc may be a straight line 
 tangent to bak at k, since the 
 profile which engages with 
 bak does not touch this part 
 at all. It is better, however, 
 to round this part, as in kf, 
 for greater strength and bet- 
 ter casting. Let O (Fig. 32) 
 be the pitch circle, c and d 
 the circles limiting the tooth 
 at top and bottom (top circle 
 and root circle], and a 
 the pitch point. Draw 
 the straight line ap 
 through the pitch point, 
 and making angles of 
 75 with the line of cen- 
 V tres ; draw fp through 
 the centre /, and per- 
 pendicular to ap; and p 
 is the centre for the 
 profile shown in the fig- 
 ure. For small teeth, 
 the centres are often 
 taken on the pitch circle, and the radius taken equal 
 to the pitch of the teeth. 
 
 Fig.32 
 
TOOTHED GEARING. 
 
 V. Rack. Internal Gears. 
 
 37 
 
 If, in a pair of gear-wheels, we assume the radius of 
 one of the pitch circles to be infinitely long, this pitch 
 circle becomes a straight line tangent to the other 
 pitch circle at the pitch point, and the wheel becomes 
 a rack. The rolling circles which generate the tooth- 
 profiles for the rack now roll along a straight line in- 
 stead of upon and within the circumference of a circle, 
 and consequently the faces and flanks of the teeth 
 are no longer epicycloids and hypocycloids, but both are 
 ordinary cycloids. Fig. 33 represents one of the exact 
 methods for tracing the teeth. OO is the pitch circle, 
 
 Fig. 33 
 
 and a the pitch point. The generating circles roll in 
 the directions indicated by the arrows, and the points 
 a, a", b r , etc., are found as in Fig. 23 ; the arc/V being 
 equal to p'a, p"a" p"a, rb" ra, etc. The approxi- 
 mate methods for cycloidal teeth, explained in the pre- 
 ceding paragraph, are applicable to the rack, Some of 
 these we give as examples, 
 
TOOTHED GEARING. 
 
 Fig. 34 
 
 METHOD 4 (approximate). Let A (Fig. 34) be the 
 pitch circle, B and C the rolling circles, and / the pitch 
 point. Let also // and tk be the heights of the tooth 
 
 above and below the 
 pitch circle. Take /;/ 
 = f//, and draw ;/;/, cut- 
 ting the rolling circle C 
 in the point ;/'. Step off 
 to arc /;/, and from 
 ;^7 o as a centre, with the 
 chord in' as a radius, 
 
 strike an arc cutting ;/;/ 
 in the point /. Draw 
 po, and on it find the 
 centre o' for an arc of a 
 circle passing through the points / and /. It is obvious 
 that the curvature of the flank will be the same as that 
 
 of the face. Therefore, to 
 find the flank centre, make 
 o"o"' = o m o', draw o"b par- 
 allel to the pitch circle A, 
 and make ab = xo': b is 
 A the flank centre. The 
 centres for all the flanks 
 will be on the line o"b, 
 and all the face centres will 
 be on the line </;;/, drawn 
 through the points o' and 
 b t and parallel to the pitch 
 circle A. 
 
 METHOD 5 (approximate). Fig. 53, A is the pitch cir- 
 cle, and a the pitch point. Take af=o.$? the diameter 
 
 V 
 
TOO THED GEA RING. 
 
 Fig.36 
 
 of the rolling circle, and through / draw dfe parallel to 
 
 the line of centres. Take ab ac =. \af, and draw bd 
 
 and p'g parallel to AA : d is the flank centre. Make 
 
 cp' = eg, and/' is the face centre. Method 6 of the pre- 
 
 ceding paragraph is greatly 
 
 simplified when applied to 
 
 the rack. The lines Odfand 
 
 Oga' (Fig. 27) become paral- 
 
 lel to the line of centres, 
 
 c'd and a'p' (Fig. 36), and 
 
 intersect the line cf in the 
 
 points d and /, where this 
 
 line meets the 3O-degree 
 
 lines,* giving these points 
 
 as centres for the profiles. 
 
 Hence this method, when 
 
 used for rack teeth, reduces 
 
 to the following : AA is the pitch circle, a the pitch 
 
 point, and o and o' the rolling circles. Through the 
 
 centres o and o draw the lines a'od and cfo'p', each 
 
 making angles of 30 with the line of centres. The 
 
 points d and p' ', in which these 
 
 lines meet the circumferences 
 
 of the rolling circles, are the 
 
 centres respectively for the 
 
 flank p'x, and face dx*. 
 
 When involute teeth are 
 used for a rack, the profiles 
 
 FFg.37 
 
 \_7 
 
 reduce to straight lines, making angles of 75 
 
 the pitch circle (Fig. 37). This may be very prettily 
 
 * This is true only when the rolling circles are equal. 
 
40 TOOTHED GEARING. 
 
 demonstrated by means of the approximate method of 
 Fig. 32 in the preceding paragraph, as follows : Let OO 
 (Fig. 38) be the pitch circle, a the pitch point, and ap 
 the 75-degree line. Since the centre of the pitch circle 
 is infinitely distant from the pitch point, the perpendic- 
 ular pf y which passes through this centre, will also be 
 infinitely distant from the pitch point. The radius ap 
 of the profile will therefore be infinitely great, and the 
 profile a straight line perpendicular to this radius, and 
 passing through the pitch point. But since the line ap 
 
 makes angles of 75 with 
 the line of centres, the per- 
 pendicular ab will make 
 angles of 75 with the 
 pitch circle, which is per- 
 " pendicular to the line of 
 centres. 
 
 In internal gears, the 
 curves forming the faces 
 and flanks of the teeth are 
 reversed as compared with 
 external gears ; that is, the 
 faces are hypocycloids, and the flanks epicycloids. The 
 exact method for constructing internal cycloidal teeth 
 is shown in Fig. 39. O is the pitch circle, a the pitch 
 point, c' and c the rolling circles, and o" and o' the top 
 and root circles. Find the profile bad by rolling the 
 circles, as in Fig. 23 ; find, in similar manner, the profile 
 gxf (ax being the given width of the tooth at the pitch 
 circle), and the tooth b'ad'fxg is complete. The ap- 
 proximate methods given for external cycloidal teeth 
 are applicable, without change or difference, to internal 
 
TOOTHED GEARING. 
 
 gears, remembering, however, that the faces are hypo- 
 cycloidal, and the flanks epicycloidal curves. The 
 
 Fig. 39 
 
 Fig.40 
 
 following, for example, is method 5, of the preceding 
 paragraph applied to internal gear-teeth. The centres 
 p and /' (Fig. 40), for the 
 profiles ax and ay respec- 
 tively, are found as before 
 explained (see Fig. 26), 
 the curves drawn, and the o 
 tooth limited by the top 
 and root circles O" and 
 
 a. 
 
 In generating involute 
 teeth for internal gears, 
 the primitive circle, upon 
 which the generating line 
 rolls, or from which the string unwinds, may be taken 
 the same as the top circle of the teeth with very good 
 
42 TOOTHED .GEARING. 
 
 results. Thus in Fig. 41, for the exact method, A is 
 the pitch circle, T and R the top and root circles. 
 Find the profile ca, as before explained. (See the 
 preceding section, Fig. 30.) In a similar manner find 
 
 the profile de, and the tooth 
 is complete. The approxi- 
 mate methods for external 
 involute teeth may be used 
 without change for internal 
 gears. Internal gears were 
 formerly quite extensively 
 used ; but of late years they 
 have come to be considered 
 as clumsy contrivances, and 
 are rarely used except in special mechanisms. 
 
 VI. Special Forms. Lantern-Gears. Mixed Gears. 
 
 This paragraph has been translated from the French 
 edition of Professor Reuleaux's valuable work, " Le 
 Constructeur." Straight lines are often used for the 
 profiles of the teeth of gear-wheels, the straight line 
 forming the flank of the tooth, and a curve the face. 
 But teeth obtained thus do not gear together with the 
 necessary exactness, and for this reason ought not to 
 be used in the construction of ordinary machinery. In 
 the teeth of clock-work gears, this kind of profile can 
 be advantageously used ; because it permits, at the same 
 time, of the easy cutting-out of the spaces with a file, 
 and of the use of a small number of teeth. If we take 
 the diameter of the generating circle greater than a 
 certain fraction of the radius of the corresponding prim- 
 itive circle, we obtain teeth which are still of a possible 
 
TO O THE r> GEA RJiVG. 
 
 43 
 
 execution, but which, in practice, are admissible only 
 for particular cases. If we ^ake, for the generating 
 circle, the pitch circle of one ot the wheels, we obtain, 
 for the profiles of the teeth of the wheel corresponding 
 to the pitch circle upon which it rolls, epicycloidal arcs, 
 while for the other wheel the profiles are reduced to 
 points. It is in this kind of profile that we include 
 Ian tern-gears. 
 
 External Lantern Gears (Fig. 42). From the pitch 
 
 FIg.42 
 
 point a describe a circle having a radius equal to \^ the 
 pitch. This gives the profile of the rung, or spindle, 
 corresponding to the point a. The face of the tooth 
 of the wheel R' is formed by a curve parallel to or 
 equidistant from the epicycloidal arc ab, generated by 
 the point a in the rolling of the circle R upon R' (the 
 arc tb the arc to). The envelope of circles described 
 
44 
 
 TOOTHED GEARING. 
 
 from different points of ah. with a radius equal to that 
 of the rung, gives the face profile cd: the flank di is a 
 circle quadrant. The arc of contact coincides with the 
 circled; its length al, of which the limit / is deter- 
 mined by the top circle k, ought to be greater than the 
 pitch, and hence at least i.i times the pitch. This last 
 value serves to determine the height g and the real 
 height g f of the face. 
 
 Internal Lantern-Gears (Fig. 43). The following 
 manner of proceeding is similar to the one just de- 
 
 Fig.43 
 
 scribed : The portion cd of the tooth-profile is found by 
 a curve parallel to the hypocycloidal arc ab, generated 
 by the point a in the rolling of the circle R within the 
 circle R f (the arc tb the arc to). The arc of contact 
 al ought to be taken at least equal to i.i times the 
 pitch. The flank ci is a radial straight line connected 
 with the rim of the wheel by a small circle arc. 
 
 In Fig. 44 the hollow wheel is the lantern : the face 
 
TOOTHED GEARING. 
 
 45 
 
 cd is parallel to the pericycloidal arc ab, generated by 
 the point a in the rolling of the circle R f upon R (the 
 
 Fig. 44 
 
 Plfl,45 
 
 arc tb = the arc to). The arc of contact al ought to 
 be at least i.i times the pitch : the flank ci is a radial 
 straight line connected 
 with the rim by a small 
 circle arc. 
 
 Fig. 45 represents a 
 particular case of Fig. 
 43. We have R = %R', 
 and consequently the 
 number of teeth in R 
 = J the number of teeth 
 in R' (N = N f ). In 
 this case, N = 2, and N' 
 4. The profile cd is 
 parallel to the straight 
 line ai y to which the hy- 
 pocycloid reduces (the 
 arc ab = the arc bi) : al 
 is the arc of contact. This arc is here necessarily greater 
 than the pitch : since, however, the straight form of the 
 
46 TOOTHED GEARING 
 
 flanks of the teeth of the wheel R' permits the sup- 
 pression of all play between the teeth, so that the same 
 rung gears at the same time with two opposite flanks, 
 the arc of contact may be considered equal to twice 
 al. Many writers regard this kind of gear as a special 
 mechanism, since in actual practice the rungs are mov- 
 able rollers provided with axles. If in Fig. 43 we 
 consider the radius R' as infinitely long, we obtain the 
 mechanism of the rack, in which the profiles of the 
 teeth upon the rack itself afe formed by curves parallel 
 to ordinary cycloids. If, again, in Fig. 44, we consider 
 the radius R' as infinitely long, we obtain a very simple 
 form of rack, which is very often used in preference to 
 the preceding. Upon the pinion the profiles of the 
 teeth are formed by curves parallel to an involute to 
 the pitch circle. Lantern-gears, in cases which require 
 a certain precision and not very frequent use, offer the 
 advantage that the rungs can be easily and exactly 
 described with a pair of compasses. Lantern-racks of 
 wrought iron are very useful in practice for apparatus 
 exposed to cold and wet ; such as for lifting gates, draw- 
 bridges, etc. 
 
 Gear at Two Points (Fig. 46). If we connect togeth- 
 er two gears at a single point, we obtain a new style 
 of gear, which allows us to adopt for one of the wheels 
 a very small number of teeth, and consequently a great 
 difference in the revolutions of the two wheels, even 
 though both wheels are quite small. In the figure the 
 two pitch circles are at the same time the generating 
 circles of the profiles of the teeth : ac is an epicycloidal 
 curve (generated by the rolling of R' upon R), which, 
 for the length of contact al, gears with the point a of 
 
TOOTHED GEARING. 
 
 47 
 
 the wheel R' ; ab is a second epicycloidal curve (gen- 
 erated by the rolling of R upon R f ), which, for the 
 length of contact all, gears with the point a of the 
 wheel R ; ai and ai' are the profiles for the flanks of 
 the teeth for the wheels R' and R. The small wheel 
 is used frequently for shrouded wheels. This kind 
 
 Fig.46 
 
 of gear is frequently met with in cranes and hoisting- 
 machines. 
 
 Mixed Gear (Fig. 47). - This kind of gear, which 
 is very convenient for the small pinions of hoisting- 
 machines, has the advantage of diminishing the space 
 
4 8 
 
 TOOTHED GEARING. 
 
 at the root of the tooth. This result is due to the use 
 of radial straight lines for the flanks of the teeth of the 
 small wheel. In order to obtain a sufficient duration of 
 engagement, it is convenient to use upon both wheels 
 the curves which form the faces of the teeth as far as 
 their points of intersection. In the figure, ac is an arc 
 of a cycloid, or involute, generated by the rolling of R/ 
 
 Fig.47 
 
 (which here, for a rack, is a straight line) upon R: ai' is 
 a radial straight line generated by the rolling of the 
 circle J^upon the inside of R (the radius of W= | that 
 of R). The gearing of the profile ac with the point a 
 takes place for the length of contact all. The cycloidal 
 arc ab, generated by the rolling of W upon R', gears 
 with the flank ai' for the length of contact ai. 
 
TOOTHED GEARING. 
 
 49 
 
 Fi ? .48 
 
 VII. Bevel Gears. 
 
 The different gears hitherto described are intended 
 to transmit power from one shaft to another parallel 
 shaft. If we wish to transmit from one shaft to another 
 which is not parallel, or which makes an oblique angle 
 with the first, we must make use of either bevel or 
 screw gears. A bevel or 
 conical gear differs from 
 a cylindrical or spur gear 
 in that its two pitch cir- 
 cles (at the two ends of 
 the teeth) are of differ- 
 ent diameters, and conse- 
 quently the ends of any 
 one tooth are of differ- 
 ent heights, widths, etc. 
 The pitch circles of a 
 pair of bevel wheels limit 
 frusta of cones, the api- 
 ces of which meet at the 
 point of intersection of 
 the axes of the wheels. 
 Thus, in Fig. 48, o is the 
 point of intersection of 
 the axes ox and oy ; a'b', 
 ab> a'c f , and ac are the 
 pitch circles; a'c'ca and a'b'ba, the "pitch frusta;" and 
 a' do and a'b'o, the "pitch cones." The axes may make 
 any angle with each other. It should, however, be 
 remarked that wheels such as are represented in (c) 
 are seldom used in practice, since the same angle may 
 be obtained with the wheels shown in (b). To lay out 
 
 \ 
 
5<D TOOTHED GEARING. 
 
 the pitch cones and frusta, we have, then, the following 
 simple rule : Draw a'c and a'b' (Fig. 48), making with 
 each other the required angle, and equal respectively to 
 the diameters of the larger pitch circles of the wheels. 
 Draw now the axes oy and ox perpendicular to their 
 respective pitch circle planes. The point of intersec- 
 tion o determines the cones ; and the given length a' a 
 or tfb, the frusta. The ends of bevel teeth lie upon the 
 
 Fig.49 
 
 surfaces of cones which are supplementary to the pitch 
 cones, and of which the top and root circles limit frusta. 
 In Fig. 49, a"cc'a'" and V'dd'b'" represent two teeth ; ab 
 and a'b' are the pitch circles ; a'b'o, the pitch cone ; and 
 a'b'ba, the pitch frustum. The root circles are a"b n 
 and a'"b"'\ and the top circles, cd and c'd'. These 
 circles limit the frusta cdb"a" and dd r b'"a'" of the sup- 
 plementary cones cdo" and c'd'd. The larger of the 
 
TOOTHED GEARING. 
 
 two pitch circles determines the size of the wheel and 
 the tooth dimensions. For instance, a fifteen-inch bevel 
 
 of one inch pitch means a bevel of which the larger 
 pitch circle is fifteen inches in diameter, and of which 
 
52 TOOTHED GEARING. 
 
 the pitch of the teeth upon this pitch circle is one 
 inch. 
 
 The teeth of bevels may be either cycloidal or invo- 
 lute. The latter are, however, more often used, because 
 they are much easier to construct. Fig. 50 shows a 
 convenient method for laying out the teeth of bevels : 
 abo is the pitch cone ; pn and gi are the top circles ; 
 and ef and kl, the root circles. Produce pc as far as 
 its intersection with the axis of the wheel, and from 
 this point o' as a centre, with o'a, o'p, and o'e as radii, 
 describe the circle-arcs q, /, and ;;/. These are the 
 virtual pitch, top, and root circles. Find by any of 
 the preceding methods the centres for the faces and 
 flanks, regarding a as the pitch point : r and s are the 
 centre circles for the faces and flanks respectively. The 
 teeth xyy"x" and it/t"wt are correct in size, and are 
 drawn to give the pattern-maker his dimensions. Now 
 project and describe the actual pitch, top, and root 
 circles q', /', and ;//', also the same circles for the small 
 end of the tooth (q" t /'' ', and ;;/'), and the centre circles 
 r', /, etc. Set off now x'x'" = xx", y'n = ak ', and z'z" 
 = yy", and find upon the centre circles the centres for 
 arcs passing through the points xf and y' (face), and y' 
 and z' (flank). The widths of the small end of the tooth 
 at the pitch, top, and root circles, are determined by the 
 lines o"y' t o"x' ', and 0'V, etc., and the faces and flanks 
 drawn as above. Draughtsmen sometimes find the 
 centres for the tooth-profiles upon the actual instead 
 of upon the virtual pitch circle, as we have done ; but 
 the height of the teeth upon the actual pitch circle is 
 less than the real height, as a glance at Fig. 50 will 
 show ; and consequently the widths upon the top and 
 
TOOTHED GEARING. 
 
 53 
 
 root circles are respectively too great and too small, 
 thus marring the correctness of the drawing. In Fig. 
 48 (a), the planes of the pitch circles of the two bevels 
 are at right angles with each other (the angle = 90). 
 If, now, we gradually increase this angle 6, the wheels 
 take the form of Fig. 48 (b) ; and finally, when the 
 planes become parallel (0 180), become external 
 cylindrical or spur 
 gears. If, on the 
 contrary, we gradu- 
 ally decrease the an- 
 gle 0, the bevels take 
 first the form of Fig. 
 48 (c), and finally, 
 when the planes be- 
 come coincident (6 
 = 0), become inter- 
 nal cylindrical gears. Between these latter two cases 
 (Fig. 48 (c) and internal cylindrical gears) we have two 
 interesting, if not altogether practicable, cases, the 
 
 Fig.52 
 
 internal bevel and the disk 
 wheel. A pair of bevels, 
 the internal bevel being 
 in section, is represented 
 in Fig. 51, and the "disk 
 wheel" in Fig. 52. The 
 internal bevel, because of 
 the difficulties in the way 
 of its construction, is 
 never used in practice. It may be constructed, however, 
 if desired, by the rules already given for internal cylin- 
 drical and bevel gears. The disk wheel is the least 
 
54 TOOTHED GEARING. 
 
 difficult of all bevels to construct, and is, although 
 seldom used, for this reason entitled to a place among 
 practical gears. The disk wheel and pinion possess 
 one peculiarity not found in any other bevel; viz., the 
 ratio of the radius of the pinion to that of the disk 
 wheel depends upon the angle included between the 
 axes of the wheels. If we let r and R be the radii of 
 the pinion and disk wheel, and tf the angle included 
 
 between the axes, we shall have the relation -^ = cos tf. 
 
 The supplementary cones upon which lie the ends of 
 the teeth become, for the disk wheel, right cylinders 
 having diameters equal to those of the pitch circles, 
 and, when cycloidal teeth are used, the profiles are or- 
 dinary cycloids, the disk wheel being regarded as (and 
 is sometimes called) the "bevel rack." In order that 
 a set of bevel gears shall gear together each to each, it 
 is necessary, not only that the pitch and kind of tooth 
 profile be the same, but that the slant height (b'o, Fig. 
 48) of the pitch cones be the same in all the wheels of 
 the set. Practice, however, allows a slight variation 
 from this rule ; and, according to Reuleaux, bevels will 
 work sufficiently well together if the difference in the 
 lengths of these slant heights does not exceed five per 
 cent. Such wheels are called bastard wheels, and are 
 quite commonly used in cases where there is no neces- 
 sity for very accurate gear. 
 
 VIII. Screw Gears. Worm and Wheel. 
 
 Screw gears are cylindrical gears, in which the teeth 
 are not parallel to the axes of the wheels, but make 
 oblique angles with them. All the lines of the teeth, 
 
TOOTHED GEAKING. 
 
 55 
 
 which in spur gears arc parallel to the axes of the 
 wheels, are in screw gears parts of helices drawn 
 around the pitch, top, and root circles. Let Fig. 53 
 represent two screw gears ; being the angle included 
 between the axes, and < and </>' the angles made by the 
 teeth with the "middle planes " of the wheels, as shown 
 in the figure. It is plain that the angle aob is equal to 
 the angle : conse- 
 quently we have from 
 the figure, < + <' + 
 = 1 80. This condi- 
 tion must be fulfilled, 
 else the wheels will 
 not gear properly to- 
 gether. Another ne- 
 cessary condition in 
 screw gears is, that ~~ 
 the pitches of two 
 gears which work to- 
 gether, taken normal 
 to the directions of 
 the teeth or the nor- 
 mal pitches, must be 
 equal. It is more 
 convenient to lay off the pitches on the pitch circles ; 
 that is, to lay off the circumferential pitches, instead 
 of the normal. In Fig. 54, ab represents the normal 
 pitch, and ae the circumferential. The angle aeb being 
 
 equal to <, we have ae = -^ , the circumferential 
 
 sin < 
 
 pitch equal the normal pitch divided by sin <. In order 
 that the wearing surfaces may be equal, the lengths of 
 
50 TOOTHED GEARING. 
 
 the teeth of a pair of screw gears should be equal. 
 The width of face depends upon the length of tooth 
 and the angle <. Thus, in Fig. 54, / = ec being the 
 length of the tooth, and I' = dc the width of face, we 
 have the angle ced =: angle </>, and consequently /' / 
 sin <. Suppose (Fig. 53), = 40, and <f> 60 : hence 
 <' + 60 + 40 = 1 80, </>' = 80. If / and / represent 
 
 Fig.54 
 
 Fig.55 
 
 the circumferential pitches, and // the common normal 
 pitch, we shall have, 
 
 sin 
 
 and 
 
 sin 60 0.866 
 
 sin $' sin 80 0.985 
 
 Also, for the widths of faces of the two wheels, we shall 
 have 
 
 /' = / sin </> = 0.866/, and I" = / sin <j>' 0.9857. 
 
TOOTHED GEARING. 
 
 $'/ 
 
 If we make 9 = 90, we will have an ordinary spur- 
 wheel gearing with a screw gear (Fig. 55). In this 
 *igure, < 90 and = 40 : hence <f>' =. 180 - 
 (90 + 40) = 50. We therefore have, for the circum- 
 ferential pitches and widths of faces, 
 
 = , p = 
 
 n 
 0.766' 
 
 sm 90 sm 50" 
 
 /' = / sin 90 = /, and /" = / sin 50 = 0.766^. 
 Let 6 90, that is, the 
 axes are at right angles 
 with each other (Fig. 56) : 
 consequently 
 
 <f> -f- 9' = 1 80 90 = 90. 
 
 The angles 9, 9', may be 
 
 equal or unequal : in the 
 
 figure they arc taken equal. 
 
 9 = 0' = 91 = 45 . 
 
 2 
 
 From this, 
 
 and I' = I" = I sin 45 ^ 0.7077.* In Fig. 57 the axes 
 are parallel, or = o : hence <f> -f- 9' = 180. This sig- 
 nifies that 9 and 9' are supplementary, 9 = 180 9'. 
 The inclinations of the teeth across the faces of the 
 wheels are in opposite directions. We have taken 9' = 
 
 60 : hence <A 120, and we have ^'= = - 
 
 -sm6o 0.866 
 
 * If we make the angle </> less than the angle of repose, which, for 
 cast-iron on cast-iron, is about 10, only the wheel /" can be the driver: 
 the wheel I' then restrains motion in the direction opposite to that in 
 which it is driven. 
 
TOOTHED GEARING. 
 
 Since the sin. of an angle equals the sin. of its supple 
 ment, / = / and /' = /" = / sin < = / sin $ = o.86f 
 
 Frg.57 
 
 Screw Rack and Pinion. If we make the radius of 
 one of a pair of screw gears infinitely long ( oo ), the 
 
 Fig.58 Fig.59 
 
 I I 
 
 wheel becomes a screw rack, and the pair constitutes a 
 screw rack and pinion, shown in Fig. 58. Let = 45, 
 
TOOTHED GEARING. 
 
 59 
 
 and </> = 
 ;/ 
 
 75: hence $ = 180 - (45 + 75) = 60, 
 ;/ ;/ ;/ 
 
 _ //___ 
 
 ~ sTnT' "~ 5^66' - ' sin * 
 
 = 0.9667, and 7" = / sin <' = O.866/. 
 
 Fig. 59 represents a spur rack gearing with a screw 
 
 pinion, 6 = 45, </> = 90 : hence <// = 
 
 45, / = 
 
 S1U tp 
 
 7 sin </>' = 0.7077. 
 
 - 
 = 7 ' and l " 
 
 Fig. 6 I 
 
 \ 
 
 \ 
 
 \ 
 
 We may also have a screw rack gearing with a spur 
 pinion, by making </>' = 90 (Fig. 60). Let 45, and 
 
 sin 9 
 = ;/, I' I sin </> = 0.7077, and 7" = 7 sin <' = 7. 
 
 If we make the radii of both wheels of a pair of 
 screw gears equal to infinity, the pair becomes two 
 screw racks gearing together (Fig. 61) ; and if we make 
 </> or <' = 90, we have a spur-rack gearing with a 
 screw rack (Fig. 62). 
 
6o 
 
 TOOTHD GEARING. 
 
 To draw the tooth profiles for a screw gear we pro- 
 ceed as follows : Having determined the angle < of the 
 teeth, and the length /, draw the horizontal line xy (Fig. 
 63). Draw db, making the angle < with xy, and make 
 it equal in length to /. Drop the lines dc and be per- 
 pendicular respectively to xy and dc. The line be is the 
 length of the tooth projected in the plane of the pitch 
 circle P. Strike, now, the pitch, top, and root circles, 
 P, /, and r, and make aU = be (a being the pitch point). 
 
 Fig. 62 
 
 Fig.63 
 
 Find the centres for faces 
 and flanks, as in spur gears, 
 and draw the profiles through a, b', and f. In con- 
 structing screw gears, it is advantageous to make the 
 angles of the teeth equal (< = <//). The circumferen- 
 tial pitches, and tooth dimensions in the planes of the 
 pitch circles, as also the face widths, will then be equal, 
 thus saving calculation and extra work. The friction 
 between the teeth is also more evenly distributed by 
 this means. 
 
 The motion between two well-constructed screw gears 
 
TOOTHED GEARING. 
 
 6l 
 
 Is very regular and uniform. They are therefore useful 
 in cases where uniformity of motion is requisite ; but, 
 owing to the friction between the teeth, these gears are 
 not very durable, and should be used for the transmis- 
 sion of small powers only, and at comparatively slow 
 motion. y 
 
 Worm and Wheel. The mechanism known as the 
 worm and wheel, or the worm and worm-wheel, is a 
 modification of screw 
 gears with axes at 
 right angles, the prin- 
 cipal object of which is 
 to obtain conveniently 
 a great difference in Fig. 64 
 the revolutions of two 
 shafts. The worm is an 
 endless screw, and the 
 worm-wheel a screw 
 gear (Fig. 64). It is 
 evident from the fig- 
 ure, that (the worm being the driver), at each revo- 
 lution of the worm, the wheel will be moved through 
 a distance equal to one tooth. Hence, if the wheel 
 has thirty teeth, the worm will make thirty revo- 
 lutions while the wheel makes one, or the worm-shaft 
 will revolve thirty times as fast as the wheel-shaft. 
 The common angle A of the teeth is usually taken 
 such that the worm will drive the wheel, while the 
 wheel will not drive the worm ; so that, if at any 
 time the driving-power is taken off, the gearing will 
 remain stationary. For this purpose, the angle A. may 
 be taken from 4^ to 9. If, however, the worm-wheel 
 
62 TOOTHED GEARING. 
 
 is to be the driver, X must be taken greater than 10. 
 The pitch radius R' of the worm may be from one to 
 two times the circumferential pitch.* The tooth-pro- 
 files of the worm and wheel may be either cycloidal or 
 involute ; and, in either case, those of the worm are 
 drawh as for a rack, and those of the wheel as for a 
 screw gear. 
 
 Involute profiles are particularly useful in worms, be- 
 cause the worm is, at best, difficult to construct, and the 
 straight 75 profiles of the involute rack very much fa^ 
 cilitate the construction. If we make the radius of the 
 worm-wheel infinitely long, the wheel becomes a screw 
 rack, and the mechanism becomes a worm and screw 
 rack (Fig. 65). We may also have a worm and internal 
 worm-wheel (Fig. 66), or an internal worm and worm- 
 wheel. In either of these cases the profiles are drawn 
 as explained in sections IV. and VIII. As in screw 
 gears, by placing the axes at oblique angles, we may 
 have a worm gearing with an ordinary spur wheel, a 
 
 Fig.,64a * If we develop in the straight line ac (Fig. 
 
 64 a) the circumference of the pitch circle, and 
 in the straight line ab the length of one revolu- 
 tion of the screw, we shall have be the pitch 
 = /, and ac = the circumference = 2irR' : hence 
 
 This condition must be fulfilled : hence, if we 
 
 make R'= 2/, -- f \, tan A = O.I59X = 0.0795, 
 A 
 
 A = 4 33'. If R' = p, ^ i, and tan 1 
 o. 1 59, A = 9 2'. Inversely, if 7i 1 2, 
 tan a = 0.213 = 0.159-,, -^=1.34, R' = $p. 
 
TOOTHED GEARING. 
 
 Fig. 65 
 
 spur rack, or an internal spur wheel. It must, however, 
 not be forgotten that the pitch of the spur gear must 
 be taken equal to the pitch of the worm multiplied by 
 cos A. In order to obtain more bearing surface between 
 the teeth of the worm and 
 those of the wheel, the bot- 
 toms of the spaces in the ^ 
 wheel are sometimes cast 
 in the form of circle-arcs, to 
 fit the threads or teeth of 
 the worm (the radius of curvature equals radius of 
 ends of worm-teeth plus the clearance), and the ends 
 of the wheel-teeth formed to fit the bottoms of the 
 spaces in the worm (radius of curvature equals radius of 
 bottoms of worm-spaces plus the clearance), as shown 
 
 Fig.66 
 
 Fig.67 
 
 in Fig. 67. The figure gives a section through the 
 centre of the wheel, showing two teeth entire and an 
 end view of the worm. As with plain screw gears, so 
 with the worm and wheel, the wear is excessive ; and, 
 for this reason, only comparatively small powers can be 
 advantageously transmitted by this mechanism. In 
 
6 4 
 
 TOOTHED GEARING. 
 
 cases, however, where the gears are not in motion 
 continuously, as in hoisting-machines, cranes, some 
 
 CO 
 
 machine tools, etc., worms may be used for the trans- 
 mission of considerable powers. 
 
TOOTHED GEARIXG. 65 
 
 IX. Hyperbolic Gears.* 
 
 Hyperbolic, or, more properly, hyperboloidal, gears 
 are intended to be fixed upon arbors, the axes of which 
 cross, without intersecting each other. Their primitive 
 surfaces (surfaces limited by the primitive or pitch cir- 
 cles) are hyperboloids of revolution, which touch along 
 a common generatrix. This generatrix may be deter- 
 mined as follows : 
 
 In Fig. 68, which is a projection made normally to 
 the shortest distance between the axes, let us divide 
 the angle of inclination of the axes into two other 
 angles, (3 and ft, in such a manner that the perpendicu- 
 lars AB and AC, drawn to some point A of the line of 
 division SA, shall be inversely proportional to the 
 numbers of revolutions of the wheels, i.e., directly pro- 
 portional to the diameters. SA is, then, the generatrix 
 of contact of the two hyperboloids. AB = R' and 
 AC = R' represent the projections of the radii of two 
 normal sections through the point A, and we have, 
 
 R' _sinff _n f _ N^ 
 57 " siiiT? ~~ ~ N" 
 
 n and ;/ being the numbers of revolutions, and N and 
 N' the numbers of teeth, of the wheels. The real radii, 
 R and R,, are still to be determined, as also are the 
 radii SD r and SE = r'. Between these last we 
 have the relation, 
 
 -+COS0 
 r _ tan p _ n 
 
 7 = tan/?""^ 
 
 f -h cos 
 
 * From Le Constructeur. 
 
66 TOOTHED GEARING. 
 
 That is, r and r' are in the same relation to each other 
 as are the two segments AF and AG, which are deter- 
 mined by the projections of the axes upon the right 
 line FG, drawn through the point A perpendicular to 
 the generatrix of contact. Representing by a the 
 shortest distance between the axes, we have, 
 
 i-h-cos<9 i+-'cos<9 
 
 r n , r n 
 
 and = 
 
 n' \n 
 
 a n . In V a n' A /n'\ 2 
 
 i -f 2 cos -f ( , ) i + 2 cos -H J 
 
 n 
 
 The radii R and R l are the hypothenuses of right- 
 angled triangles, of which the sides are respectively R f 
 and xy = r, Rf and x'y' = r', and consequently have 
 the values, 
 
 R = y^ 2 + >' 2 and R, = \A#/ 2 + r' 2 . 
 
 R' and R t ' are known from what precedes when we 
 have given the length SA = !. The angles ft and f? 
 are determined by the relation 
 
 -_ and tan/T= 
 
 n A ,. 
 
 . + cos v -f- cos 
 
 
 
 As in bevel gearing, the problem permits of two 
 solutions, according as the line SA is drawn withir 
 the angle 0, or within the supplementary angle BSC' 
 (Fig. 69). These two solutions differ from each other 
 in the direction of rotation of the driven arbor. One 
 of these solutions leads to an internal gear, as in bevel 
 gears ; but this, to our knowledge, has never been 
 actually constructed, and it cannot possibly have any 
 
TOO THED GEARING. 
 
 practical value. When the angle of inclination, 0, is 
 made equal to 90, we have, 
 
 and 
 
 '-,-* B -ffi 
 
 r \a I 
 
 a n 2 -f- ;/ 2 ' 
 
 n* -f / 
 
 It is easily seen, from what precedes, that hyperbolic 
 gears present a more limited number of solutions than 
 
 Fig.69 
 
 ordinary screw gears, with which, however, they pre- 
 sent many analogies. In the latter, for one value of 
 the angle of inclination of the axes, we can give an 
 arbitrary value to the angle of inclination of the teeth 
 of one of the wheels ; while in hyperbolic gears there is 
 only one pair of values admissible for the angles of 
 inclination. 
 
 The primitive surfaces of two hyperbolic gears are 
 
68 TOOTHED GEARING. 
 
 formed by corresponding zones of two hyperboloids of 
 revolution. When the distance (shortest) between the 
 axes is small, the zones comprising the circles of the 
 gorge, of which r and r (Fig. 68) are radii, cannot be 
 utilized as primitive surfaces, and we must have re- 
 course to zones somewhat removed from these circles. 
 These may ordinarily be replaced by simple frusta of 
 cones, and the construction thus rendered compara- 
 tively simple. The following examples will serve to 
 illustrate the preceding formulas and remarks : 
 
 Example I. = 40, - = -, a = 4". From the 
 
 R' ri R f i 
 
 formula 7 = we have -=- f =. - = 0.5 ; 
 
 RI n K. l 2 
 
 also we have 
 
 r 0.5 4- 00340 _ 1.266 
 
 7 =: 2 + cos 40 := ^66 = 4577 
 
 r i 4- 2 cos 40 _ 2.532 = 
 
 a i -f 2 x 2 cos 40 -f 4 8.064 
 
 r= 1.2559", r' = 2.744". 
 For the angles ft and fi' we have 
 
 o sin 40 0.6428 
 tan ft := 2 + cos 40 = ^66- = 
 
 or ^813 5', and jtf = 40 = 26 55'. For the 
 distance SA = / = 8" we have 
 
 R' = /sin 13 5' = 8 X 0.226368 = 1.81" 
 
 ^P/ = 8 X sin 26 55' = 8 x 0.452634 = 3.62". 
 
TOOTHED GEARING. 69 
 
 Finally, 
 
 R =Vi^ 
 and 
 
 R, = N/p^ 2 + ^74 2 - 4-54". 
 
 Example 2. 90, - (a value which will be 
 
 satisfied by the numbers of teeth TV^^ 36 and N' = 20), 
 and # = 0.8". From the preceding formulas we have 
 
 8 1 
 
 5 2 4- 9 2 106 
 
 and 
 
 / = o.i 86". 
 
 We have also tan /? 1.80, or /8 = 60 57', and 
 
 consequently /^ = 29 3'. For R = 2" we have the 
 formula 
 
 R' = \JR 2 r* = \2 2 - o.6i 2 = 1.90" 
 
 and 
 
 9 9 
 
 Also for R, we have 
 
 Rt = ViTo6 2 -f- oT89 2 = i. 08". 
 
70 TOOTHED GEARING. 
 
 Example^. = 90, = i. As before, tan ft 
 = '=i or/2=45, = V=i, orr = /. Also 
 
 R =. R It and the hyperboloids are congruent. 
 
 Example 4. In the particular case where the rela- 
 
 tion is numerically equal to cos 0, and the line of 
 
 division which determines the angle ft is situated within 
 the supplementary angle of in such a manner, that, tak- 
 
 ing into consideration the sign, we have = cos 0, 
 
 one of the primitive surfaces reduces to a cone, and the 
 other to a hyperboloidal plane. This hyperbolic plane 
 (or disk) wheel corresponds to the disk wheel in bevel 
 gears, and can be made to gear with an ordinary bevel 
 wheel. It offers, however, no practical advantage, since 
 the disk wheel interferes with the" prolongation of the 
 
 arbor of the bevel. For = 60, -- = cos 60, 
 
 11 2 
 
 T _ 
 
 we obtain the disk wheel, and have tan ft - y/3, ft = 30, 
 
 R=R', R, V^/ 2 + (i 2 \/4^ 2 + a 2 . If -were neg- 
 
 ative, and less than cos 0, we would obtain a hyperbolic 
 internal gear ; but gears of this kind are not at all 
 practical. 
 
 With hyperbolic gears we may obtain, as a limiting 
 case, the mechanism of a rack and pinion. The rack, 
 m this case, carries oblique teeth ; while the pinion is 
 
TOO THED GEA RING. 
 
 Fig.70 
 
 formed by the zone corresponding to the circle of the 
 gorge of a hyperboloid of revolution. But since the 
 construction of this pinion is much more difficult than 
 that of a screw gear, the effect of which is equivalent, 
 it results that the latter should be used in all cases 
 where this effect is to be produced. 
 
 Teeth of Hyperbolic Gears. If we wish to give to the 
 teeth of hyperbolic gears perfectly accurate forms, we 
 meet with very serious difficulties in the execution. 
 We may, however, content ourselves with approximate 
 forms. In this case, to determine the teeth of a hyper- 
 bolic gear, we begin by tracing the supplementary cone 
 of the hyperboloidal zone, which 
 is to be used as the primitive sur- 
 face. The apex H of this cone 
 (Fig. 70) is obtained by drawing 
 a perpendicular AH to the gen- 
 eratrix SA, parallel to the plane 
 of the figure. We then deter- 
 mine the profiles of the teeth 
 for the normal pitch p tl upon the 
 circle of the gorge as if it was 
 acted upon by a screw-wheel 
 having a diameter r, and an in- 
 clination of teeth 90 /?; then 
 we continue the profiles thus 
 obtained upon the conical surface HJL, taking care to 
 increase the dimensions parallel to the circle of division 
 in the proportion of / to p n (p being the circumfer- 
 ential pitch), and the lengths in the proportion of K to 
 r, K representing the length of the generatrix of the 
 supplementary cone. We repeat the same construction 
 
72 TOOTHED GEARING. 
 
 for the supplementary cone corresponding to the other 
 base of the zone, being careful to decrease the values 
 of / and K. Thus we obtain for each tooth two pro- 
 files, sufficiently exact, of which the corresponding 
 points must be joined by straight lines to form the 
 body of each tooth. 
 
 In certain cases a cone frustum may be substituted 
 for the hyperboloidal zone, upon the condition of prop- 
 erly determining the apex. To this effect, we revolve 
 the generatrix SA about the axis HS until the point A 
 becomes coincident with the point J : the projection 
 of the generatrix, in this position, determines by its 
 intersection with HS the desired apex of the cone. 
 
 X. Relations between Diameter, Circumference, Pitch, Number of 
 Teeth, etc. Diametral Pitch. Methods for stepping off the 
 Pitch. 
 
 The circumference of a circle is expressed by the 
 formula 
 
 C= irD, or C= 271-7? (i) 
 
 where C is the circumference, D the diameter, R the 
 radius, and TT the constant 3.14159. From these formu- 
 las we may write, 
 
 c c 
 
 /? = -, R=~ (2). 
 
 TT' 2?r 
 
 Thus, to find the circumference, multiply the diameter 
 by 3.14159, or the radius by 2 X 3.14159 = 6.28318. 
 Inversely, to find the diameter, divide the circumfer- 
 ence by 3.14159: to find the radius, divide the circum- 
 ference by 6.28318. The simple, old rule, which says, 
 
TOOTHED GEARING. 
 
 73 
 
 This 
 
 " To find the circumference of a circle, multiply the 
 diameter by 22, and divide by 7, to find the diameter, 
 multiply the circumference by 7, and divide by 22," 
 ordinarily answers the purpose well enough. The cir- 
 cumferential pitcJi or circular pitch (generally called 
 simply the pitch) of a gear of any kind is the distance 
 from the centre of one tooth to the centre of an adja- 
 cent tooth, measured on the pitch circle, or, what is the 
 same thing, the distance on the pitch circle, which 
 includes one tooth and one space, 
 distance, laid off a certain num- 
 ber of times around the pitch 
 circle, divides the pitch circle 
 into a certain number of equal 
 parts, each containing one tooth : 
 consequently the circumference 
 of the pitch circle divided by 
 the pitch will give the number 
 of teeth, and the pitch multi- 
 plied by the number of teeth 
 will give the circumference of the pitch 
 formula, N being the number of teeth, and / the pitch, 
 
 From formula (i) we may write , and, from the 
 
 6 wD 
 
 i N i N 
 
 third of formula (3), - = -^. Hence - = , or 
 
 N 7T 7T 
 
 _ = _=^ and - = / (4). 
 
74 TOOTHED GEARING, 
 
 This ratio of the constant quantity TT 3.14159 to the 
 circumferential pitch is called the diametral pitch, 
 because it is equal to the ratio of the number of teeth 
 to the diameter of the pitch circle. We represent this 
 diametral pitch by p d . The diametral pitch gives the 
 number of teeth in a gear wheel per unit (say inch) 
 of length of the pitch-circle diameter. To illustrate., 
 suppose we have a pitch circle of 10" diameter and i 
 circumferential pitch of 3. 141 59". From formula (i) the 
 circumference is C * X 10 = 31.41-59", and from for- 
 mula (3) the number of teeth is N =. =- ^ = 10. 
 
 P 3.HI59 
 
 Hence, from formula (4), p d = i ; that is, 
 
 there is one tooth in the gear for each inch of length 
 in the diameter of the pitch circle. In order to distin- 
 guish the diametral from the circumferential pitch, the 
 former is often designated as "pitch No. ." Diame- 
 tral pitch No. i = circumferential pitch of - =. 3.14159", 
 
 diametral pitch No. 2 = circumferential pitch of 
 
 = i.57079"> etc. 
 
 Since the circumference of a circle cannot be meas- 
 ured exactly (the quantity TT being irrational), it is often 
 tedious work to step off the circumferential pitch arounc 
 the pitch circle (especially in large gears), a great many 
 trials being necessary before the equal division of the 
 pitch circle is obtained. A formula, by the use of whicl: 
 this work is simplified, may be obtained as follows : Let 
 bed (Fig. 72) be a circle, be a circle chord. In the tri 
 angle abc we have, from trigonometry, the proportior 
 
TOOTHED GEARING. 
 sin angle bac \bc\\ sin angle bca : ab. 
 
 75 
 
 But ab = R, the radius of the circle, and be = I", the 
 circle chord. Calling the angle bac 6, we have, since 
 ac = ab = R, the relation, 
 
 - 
 
 2 
 
 angle bca = 
 
 Substituting these values in the above proportion, we 
 obtain 
 
 Hence 
 
 But 
 
 . /i8o - 
 
 sin I - 
 V 2 
 
 /i8o-0\ / 0\ 
 sin f J = sin I 90 j = cos 
 
 and, from trigonometry, sin = 2 sin J 6 cos J 0. 
 
 These values, substituted in the 
 last expression for I' 1 ', give 
 
 Fig.72 
 
 sn 
 
 cos 
 
 cos4<9 
 
 or 
 
 (5). 
 
 Suppose, now, the arc be to repre- 
 sent the pitch laid off on the pitch 
 circle of a gear. If we represent by N the number of 
 
 360 
 teeth in the gear, we shall have for 6 the value 6 = , 
 
76 TOOTHED GEARING. 
 
 1 80 
 
 and consequently \ = -^-. From this, by substitu- 
 
 tion in formula (5), we have for the length of the 
 chord be, 
 
 (6). 
 
 Rule. To find the length of the chord subtended 
 by the pitch arc, multiply the diameter of the pitch 
 circle by the sine of the angle obtained by dividing 
 1 80 by the number of teeth. 
 
 Example i. Suppose D = 24" and N= 80. Hence 
 
 T 0-.O 
 
 i o if^L 2 15',- sin 2 15' = 0.03926, and /" = 24 
 
 oO 
 
 X 0.03926 = 0.942". 
 
 Example 2. D = 39!" and the pitch =. p =. 4". 
 From formula (i) the circumference is C = trD = 124", 
 
 and from formula (3) N =. 331. Formula (6) 
 
 4 
 
 therefore gives I" 39^ sin /L 8o !\ 39^ sin 5 48' 23!" 
 
 V 31 / 
 
 = 392 X 0.1011683 = 3-99 6// - 
 
 Mr. W. C. Unwin, in " Elements of Machine Design," 
 gives the following : 
 
 " To lay off the Pitch on the Pitch Line. The follow- 
 ing construction is convenient when the wheel is so 
 large that it is impossible to find the exact pitch by 
 stepping round the pitch line. Let the circle (Fig. 73) 
 be the pitch line. At any point, a, draw the tangent ab. 
 Make ab equal to the pitch. Take ac equal to \ab. 
 With centre c and radius cb, draw the arc bd. Then 
 the arc ad is equal to ab, and is the pitch laid off on the 
 
TOOTHED GEARING. 77 
 
 pitch line. When the wheel has many teeth, the arc 
 ad sensibly coincides with its chord ; but, if it has few 
 teeth, there is an appreciable error in taking the chord 
 ad equal to the pitch." 
 
 Unfortunately neither of these rules gives exactly the 
 
 required distance; for, in the first case, the sin f _J 
 
 is usually a number containing six, eight, or even more 
 decimal places, and consequently the chord be will be 
 such a number, not capable of exact measurement with 
 the compasses ; and, in the second case, the pitch 
 (being the circumference F| 73 
 
 an irrational quantity di- j> c a 
 
 vided by the number of teeth) 
 cannot be exactly laid off on 
 the line ab. Such simple and 
 easily remembered rules, how- 
 ever, simplify in some degree 
 the work of the draughtsman and mechanic, and are 
 therefore worthy of our notice. An accurately con- 
 structed ""Tr-rule" (pi-rule), used in connection with the 
 preceding method, gives very close results. To con- 
 struct such a rule, have a four-inch circle turned, as 
 accurately as possible, out of wood or metal. Mark a 
 point anywhere upon the circumference, and starting 
 with this point tangent to a straight, true ruler about 
 14" long, roll the circle along (taking care not to slip or 
 slide) until the point is again tangent to the ruler. The 
 distance thus developed upon the ruler is equal to the 
 circumference of the 4" circle, equals 4?r. Divide 
 the developed length into four parts : each part is equal 
 to one TT (pi), and may be divided into halves, quarters, 
 
7 8 TOOTHED GEARING. 
 
 eighths, etc., or into tenths and hundredths. The total 
 distance now marked off is 477, and the divisions are 
 equal to TT, JTT, JTT, JTT, etc., or -^TT, and yJ-^TT. As an 
 example to illustrate the use of the 7r-rule, suppose 
 the diameter of a gear to be constructed is 10", and the 
 number of teeth 100. The circumference of the pitch 
 circle is IOTT, and the pitch is IOTT divided by 100, or 
 -^QTT. This, measured on the 7r-rule, and laid off on the 
 tangent line ab (Fig. 73), will give the arc ad (or chord 
 ad) as accurately as any method with which we are 
 acquainted. 
 
 XI. Ratios. Velocity. Revolution. Power. 
 The velocity ratio of two gear wheels is the velocity at 
 the circumference of one wheel divided by the velocity 
 at the circumference of the other, both velocities being 
 taken in terms of the same unit (generally feet per 
 second), or the ratio of the velocity at the circumference 
 of one to the velocity at the circumference of the other. 
 The velocity ratio of two toothed wheels which gear 
 together is always constant, and equal to unity ; that is, 
 the velocity at the circumference of one is equal to the 
 velocity at the circumference of the other.* To prove 
 this, let the circles of Fig. 74 represent the pitch circles 
 of a pair of gear wheels. Suppose R to be the driver, 
 
 * When two gear wheels are fixed upon the same shaft, their veloci- 
 ties are proportional to their diameters or radii. Thus, 
 let D and D' be the diameters of two such wheels. 
 The velocities at the circumferences of the wheels are 
 v = Cn, and v' C'n ; v, v', C, and C f being the velo- 
 cities and circumferences. Hence 
 
 i/ nzy* jy R r 
 
 - = - r _ =:gr = 
 
TOOTHED GEARING. 
 
 79 
 
 Fi 8 .74 
 
 and r the driven wheel. As the wheels revolve, it is 
 plain, that, as each tooth of R passes the imaginary line 
 AB, it carries with it a tooth of the wheel r. Thus 
 equal numbers of teeth of the two wheels pass the line 
 AB in equal times. But, since the pitches of the wheels 
 are equal, equal numbers of teeth must lie on equal 
 arcs of the two pitch circumferences : therefore, with- 
 out reference to the relative sizes of the wheels, equal 
 arcs of their pitch circumferences 
 pass the line AB in equal times, or, 
 in other words, the velocities at the 
 circumferences are equal. 
 
 The revolution ratio of two gear 
 wheels which gear together is the 
 greater number of revolutions di- \ \ 
 vided by the less, or the ratio of the \ 
 greater number of revolutions to 
 the less. For example, if one of a 
 pair of gear wheels makes 100 revo- 
 lutions per minute and the other 20, 
 the revolution ratio is ^V" = i> anc ^ 
 we say the wheels are geared 5 to i. 
 in Fig. 74 that equal numbers of teeth of the wheels R 
 and r pass the line AB in equal times. Let us suppose 
 the number of teeth (N) of the wheel R to be 100, and 
 that (N') of r to be 25. When 25 teeth of-.-/? have 
 passed the line AB, 25 teeth (all) of r have also passed 
 the line ; that is, R has made J of a revolution, and r has 
 made i entire revolution. When 50 teeth of R have 
 passed the line AB, 50 teeth of r have also passed the 
 line, or R has made of a revolution, and r has made 
 2 entire revolutions. Thus, when 100 teeth of R have 
 
 We have proved 
 
80 TOOTHED GEARING. 
 
 passed AB, or when R has made i entire revolution, r 
 has made *- = 4 entire revolutions. The revolution 
 ratio of the pair is therefore -*, the small wheel making 
 4 revolutions while the large wheel makes i. But the 
 ratio of the number of teeth of the small wheel (r) to 
 that of the large wheel (R) is -f^ = ^ : therefore it is 
 plain that tJie revolution ratio of a pair of toothed wheels 
 is inversely equal to the ratio of the numbers of teeth of 
 the wheels. Letting ;/, N, R, D, and C represent the 
 number of revolutions, number of teeth, radius, diameter, 
 and circumference respectively, of the smaller wheel, 
 and ;/, N', R', D f , and C the number of revolutions, 
 etc., of the larger wheel, we have, since the number of 
 teeth is directly proportional to the radius, diameter, or 
 circumference, 
 
 n N' R D C' 
 
 Rule. The number of revolutions of the smaller 
 wheel is to the number of revolutions of the larger 
 wheel as the number of teeth, radius, etc., of the 
 larger wheel are to the number of teeth, radius, etc., of 
 the smaller wheel. 
 
 Example i. Two bevel wheels are to gear together 
 so that the revolutions per minute are respectively 
 n = 1 60 and ;/ = 40. The diameter of the smaller 
 wheel is D = 8", and the pitch of the teeth, p = $ '. It 
 is required to find the diameter of the larger wheel (ZX) 
 and the numbers of teeth (N and N'} of each wheel. 
 
 We have here * = = i From formula (7), i = ^ 
 ;/ 40 i I 8 
 
 D' = 32". From formula (i), C = * X 8 = 25. i", and, 
 
TOOTHED GEARING. 8 1 
 
 from formula (3), N = -~- 50. From formula (7), 
 
 2 
 
 again, ^ , or N' = 200. 
 
 Example 2. A shop shaft makes 120 revolutions per 
 minute. From this shaft it is required to gear down 
 to 8 revolutions per minute. The diameter of the 
 wheel on the first shaft is 12" . Find other diameters 
 and the numbers of teeth of each wheel, supposing the 
 
 Fig. 75 
 
 pitch = i". The revolution ratio is 152=11. From 
 
 o I 
 
 formula (7), I* = , D' 1 80" = 1 5 feet. A wheel 
 
 of this size is out of the question : we therefore must 
 have recourse to a train of wheels such as is repre- 
 sented in Fig. 75. We may take the revolution ratio 
 between D and D' \ , and that between D" and D'" \ : 
 we then have f X f = ^ as the ratio between D and 
 
 D'". From formula (7), then, - t = 3 = . ZX = 36", 
 
 and n ~ = 1 = ^. Taking Z> /x = D = 12", we have 
 D"' = 6o". From formula (i), C= * X 12 = 37.7, and, 
 
82 TOOTHED GEARING. 
 
 from formula (3), JV= ^ZiZ =38. Hence, from formula 
 
 (7), N' = 1 14, N" = 38, and N"' =. 190.* 
 
 In a pair of gears in which N= 25 and IV' = 100 
 
 the revolution ratio is , = -. The same 
 
 ;/ TV 25 i 
 
 teeth are therefore in contact once in every revolution 
 of the larger wheel, or once in every 4 revolutions of 
 the smaller wheel. Contact taking place so frequently 
 between the same two teeth, if these teeth happen to 
 be rough and poor, the wear between them must be 
 greater than in any other part of the wheels. If, how- 
 ever, we make N= 26, the revolution ratio is Ye ' 3iJ> 
 practically the same as before, and the two poor teeth 
 are in contact only once in 13 revolutions of the larger, 
 or 50 revolutions of the smaller wheel. By means of 
 this "wear tooth " the wear of the wheels may be more 
 evenly distributed, and the durability of the wheels con- 
 siderably increased, without seriously interfering with 
 the revolution ratio of the wheels. 
 
 Power Ratio. The power or force of a gear wheel 
 is the force with which the circumference of the wheel 
 .turns : it is equal to that force, which, when applied 
 to the circumference in a direction contrary to that of 
 rotation, is just sufficient to stop the rotation of the 
 wheel. The power ratio or force ratio- of two gears 
 is the greater power divided by the less, or the ratio of 
 the greater power to the less. The powers of two 
 wheels which gear together are equal, the power of the 
 
 * The gears D' and Z>", being fixed upon the same shaft, of course 
 make the same number of revolutions per minute, regardless of diameters 
 or radii. 
 
TOOTHED GEARING. 
 
 Fig.76 
 
 driver being transmitted directly to the driven wheel : 
 in this case, therefore, the power ratio, as the velocity 
 ratio, is constant, and equal to unity. Let R and R' 
 (Fig. 76) represent the radii of a pair of gears, and ;' 
 the radius of a pulley which is fixed upon the axle of R, 
 and arranged to lift a weight W by means of a string 
 passing around its circumference. Let the power or 
 force of the driver R' be denoted by P. This force is 
 transmitted to R in the direction shown by the arrow. 
 We may regard the imaginary line ac as a simple lever, 
 the fulcrum of which is at 
 b, and the arms of which are 
 abr and bc=.R, The 
 force P acts upon the long 
 arm, and the force W upon 
 the short arm. By the prin- 
 ciples of the lever, the mo- 
 ments of the forces with 
 reference to the fulcrum 
 must be equal : hence we 
 have 
 
 W R 
 Wr = PR, or -p = - (8). 
 
 That is, the forces of the wheels R and r are inversely 
 proportional to their radii. Since the radii R and r are 
 directly proportional to the velocities of the circumfer- 
 ences, and the power and velocity of R are equal to the 
 power and velocity of R r , we may write, 
 
 W 
 
 PV 
 
 Wv 
 LL^L 
 
 v 
 
84 TOOTHED GEARING. 
 
 where V and v are the circumferential velocities of R 
 and r respectively. From this formula we may write 
 the following : 
 
 Ride. The relative powers of the wheels of a train 
 of gears are inversely proportional to the circumferen- 
 tial velocities of the wheels. To find the power of any 
 wheel of a train of gears when the power of the next 
 wheel is known, multiply the power of the latter by its 
 own velocity, and divide by the velocity of the former. 
 
 Example i. In a train of gears such as is repre- 
 sented in Fig. 76, the force of the driver is ^==50 
 pounds, the velocity of the driver is V=. 10 feet per 
 second ; that of the pulley, v = 5 feet per second. Re- 
 quired the weight W which can be lifted by the pulley. 
 The force of R is equal to that of the driver, since their 
 velocities are equal. By the rule, 
 
 . force of R' X velocity of K _ 
 
 force of R = - = : ^-~- = force of tf. 
 
 velocity of R 
 
 From formula (9), or the rule, 
 
 ___ PV 50 x 10 
 
 W = = = 100 pounds.* 
 
 5 
 
 Example 2. In the gear train represented in Fig. 
 77 the force of the driver R" is P= 500 pounds, 
 R = R' = 12", r r' = 5". It is required to find the 
 
 * The gain in power is obtained by a sacrifice of time ; for the wheel 
 y?, having twice the velocity and half the power of the pulley r, can lift 
 twice as far a weight equal to ^ W in the same time, or just as far a 
 weight equal to W in half the time. The work inherent in these two 
 wheels is therefore the same : r simply does double work in double time. 
 If, however, we have only 50 pounds of force at our disposal, we can 
 lift 100 pounds at one lift only by means of such a train, or a similar 
 mechanism. 
 
TOOTHED GEARING. 
 
 weight, W, which can be lifted by the pulley r, and the 
 distance per minute which W can be lifted, supposing 
 the wheel R' to make 15 revolutions per minute. The 
 power of R' is equal to that of the driver = P. From 
 formula (8), P' representing the power of r', 
 
 r>f pf n' 
 i J\ * 
 
 ~P^V 
 
 12 
 
 = , Jr= 1 200 pounds. 
 500 5 
 
 This power is transmitted directly to R : hence 
 W R W 
 
 P' 
 
 , W ' = 2880 pounds. 
 
 1 200 5 
 
 R' and r' make the same number of revolutions, being 
 
 Fig.77 
 
 on the same shaft. From formula (7), ' and n being 
 the numbers of revolutions of r r and R, we have 
 
 r 
 R 
 
 15 
 
 12 
 
 The circumferential velocity of r, which is the velocity 
 with which the weight W is lifted, is 
 
 2-rrrn 
 12 
 
 12 
 
 = 16.36 feet per minute. 
 
86 TOOTHED GEARING. 
 
 Example 3. Required an expression for the weight 
 which can be lifted by a train similar to that of Fig. 77, 
 containing any number of wheels. From Example 2, 
 
 P r R f , PR' A W R ,,, P'R 
 
 - rr = -r or JT = T- and -TV - or W= - . 
 P r r r' P' r r 
 
 Substituting in this expression the value of /", just 
 
 r> r> r>/ 
 
 written, we obtain W-=- -7. In the same manner, 
 
 rr 
 
 for any number of wheels, R, R ', R" , R"' y etc., repre- 
 senting the radii of the large wheels, and r y r', r", /", etc., 
 
 f .. , . 1J7 PRR'R"R'", etc. 
 
 those of the pinions, we obtain W 7 __ _____ 
 
 rrrr, etc. 
 
 , D Wrrr"r m etc. 
 Inversely, P = 
 
 Example 4. We have a shaft which drives a gear 
 with a force of 250 pounds : we wish with this power to 
 lift a weight of 1,500 pounds. Required the radii of the 
 wheels of the necessary train. We can see at a glance 
 that a simple train, such as Fig. 76, will not be 1 practi- 
 
 cable, for in this case = - ; and, if r= 6" 
 
 P r 250 I 
 
 (as small as is convenient), R r X 6 36", or the 
 diameter of our large gear will have to be 6 feet. This 
 is practically out of the question : we must therefore 
 use a train with 4 or more wheels. Let us try 4. 
 
 r) r> ir>/ 
 
 From Example 3 we have W= -- j. Taking r r' 
 
 36 36 250 
 
 = 216. We can now assume a value for R, and find 
 the corresponding value of R r . Say R=.\2" t then 
 
TOOTHED GEARING. 8/ 
 
 In the preceding examples no account has been taken 
 of the friction of the gear teeth and axles, since they 
 are given simply to illustrate the use of the rules and 
 formulas which precede them. The detrimental fric- 
 tion is, of course, very considerable, even in the best 
 wheels, and increases rapidly as we increase the num- 
 ber of wheels in a train : therefore the trains spoken 
 of in the examples, if actually made and used, would 
 accomplish considerably less than the examples give 
 them credit for. Were this not the case, we could, 
 with the slightest possible amount of power, by means 
 of a train containing a sufficient number of wheels, per- 
 form an infinitely great amount of work manifestly, 
 from a practical point of view at least, an absurdity. 
 
 XII. Line of Contact Arc of Contact. 
 
 In a pair of toothed wheels, each tooth of one wheel 
 is in contact, for a certain, definite length of time or 
 distance of revolution, with a tooth of the other wheel, 
 and there is always at least one pair of teeth in contact. 
 Whether or not the same two teeth come into contact 
 at each revolution depends, as we have already seen, 
 upon the relative numbers of teeth of the two wheels. 
 If, during the contact of a pair of teeth, a curve be 
 drawn through all the successive points of contact, this 
 curve will represent the entire contact of the teeth. 
 Such a curve is called the line of contact, and its length 
 represents the duration of the contact. The line of 
 contact may be found by drawing different positions of 
 two teeth while in contact, and drawing a curve through 
 the points of contact thus determined. This operation 
 is, however, often a difficult one, because the effect of 
 
88 
 
 TOOTHED GEARING. 
 
 the preceding pair of teeth upon the early contact of 
 the pair in question cannot easily be taken into consid- 
 eration, and this effect is very often too important to 
 be neglected. Reuleaux has pointed out the following 
 method for determining the line of contact : Let O and 
 O' (Fig. 78) be the centres of two toothed wheels which 
 gear together, OpO' the line of centres, and / the pitch 
 
 point. From different points along the profile apc f draw 
 normal lines intersecting the pitch circle in the points 
 b, b' y b" d r , etc., and from O as a centre strike circle- 
 arcs through the points a, a', a", c, etc. We have seen, 
 that, for uniform velocity ratio, it is necessary that the 
 common normal to two teeth in contact at the point of 
 contact shall pass through the pitch point. If, there- 
 fore, from the pitch point / as a centre, with radii equal 
 to ab, a'b ', a"b", dc, etc., we strike arcs intersecting the 
 
TOOTHED GEARING. S() 
 
 above-mentioned arcs, the points of intersection will be 
 points of contact of the teeth, and a curve drawn 
 through these points will be the line of contact. The 
 arcs Kp and K'p, taken on the pitch circles, and limited 
 by the top circles, are called the arcs of approacJi and re- 
 cess, according to the direction of rotation, and together 
 form the arc of contact. The length of the arc of con- 
 tact depends upon the diameters of the pitch circles of 
 the gears and the height of the teeth between the pitch 
 and top circles ; while the length and position of the line 
 of contact depend not only upon these dimensions, but 
 also upon the form of the profiles of the teeth and the 
 number of teeth in contact at one time. In ordinary 
 gearing, where the height of the teeth between the 
 pitch and top circles is the same for both wheels, the 
 arcs of approach and recess are equal, and, in wheels 
 having cycloidal profiles, the lengths of the line and 
 arc of contact are, according to Reuleaux, equal. The 
 length of the arc of contact must be at least equal to 
 the pitch of the teeth, else there would be less than 
 one pair of teeth in contact at one time : in ordinary 
 machine gearing this length varies from i to 2| times 
 the pitch. 
 
 XIII. Strength of Teeth. Rules for determining the Pitch, and 
 other Tooth Dimensions. 
 
 Before taking up the subject of strength of wheel 
 teeth, our notation for the calculations under this head 
 must be explained. The total height of the tooth, 
 i.e., the sum of the heights above and below the pitch 
 circle, we denote by // ( = // + //') ; the breadth of the 
 tooth on the pitch circle, by b ; and the face width of 
 
90 TOOTHED GEARING. 
 
 the tooth, by /(see Fig. 79). In calculating the strength 
 of a wheel tooth, the curved profile is disregarded, as is 
 also, in ordinary gearing, the influence of the velocity 
 of the wheel, and the tooth regarded as a simple beam 
 or semi-girder supported at one end, and having a 
 weight or force, W, acting at the other end (Fig. 80). 
 The width b is taken equal to the width of the tooth on 
 the pitch circle. The safe working-load for a beam 
 
 Fig.79 Fig.80 
 
 such as is represented in Fig. 80 is expressed by the 
 formula 
 
 W-^ 
 W 6k 
 
 in which W is the safe working-load, f the greatest 
 safe working-stress in pounds per square inch for the 
 material used, and the other quantities the same as in 
 Fig. 80. It is evident that the width b of the tooth 
 must be less than half the pitch, else the space would 
 not be wide enough to admit the tooth of the mate- 
 wheel ; and, in order that the tooth may be sufficiently 
 strong when it becomes worn, we take, at the sugges- 
 tion of Unwin, b Q.^p; p being the circumferential 
 pitch. Also we may take, as is now generally done, 
 
TOOTHED GEARING. 9 1 
 
 // = // + //' = o.4/ + o. ip = o.7/. These values, sub- 
 stituted in the above formula, give 
 
 In this expression W^is the actual load or strain on one 
 tooth. It is more convenient to use this formula in 
 terms of the total force, P, transmitted by the wheel. 
 Ordinarily, more than one pair of teeth are in gear at 
 once : therefore the whole force transmitted is not sus- 
 tained by one tooth. The number of teeth in gear at 
 once varies considerably in different wheels ; but we 
 may safely say that no tooth bears more than three- 
 fourths of the entire force transmitted. We have, then, 
 J'F !/>, and consequently 
 
 flx 0.1296^ 
 
 Reducing this equation, we obtain 
 > 0.041 14^/ 
 
 From this, by transposing, 
 
 p p 
 
 V _ 
 
 0.04 1 1 4/ / 
 
 or 
 
 This formula may be termed the general formula for 
 determining the pitch. It may be used for any ma- 
 
92 TOOTHED GEARING. 
 
 terial whatever by substituting for the quantity f its 
 proper value. From the formula, therefore, we may 
 write the following : 
 
 General Rule. To determine the pitch of a gear 
 of any material, divide the total force to be transmitted 
 by the greatest safe working-stress per square inch for 
 the material of the wheel, multiply the quotient thus 
 obtained by the ratio of the pitch to the face width,* 
 extract the square root of this product, and multiply 
 the result by 4.93. 
 
 The degree of safety necessary in calculations for 
 strength of gear teeth varies with the work to be done 
 by the. gear, in other words, with the amount of clanger 
 to be incurred. Thus the degree of safety necessary is 
 greater when the gear is to be subjected to sudden, 
 violent shocks than when no such shocks occur, be- 
 cause the danger of breakage or accident is greater. 
 This degree of safety we obtain conveniently by vary- 
 ing the value of the quantity f, taking small Values 
 when the danger is great, and vice versa. 
 
 For ordinary, good cast-iron we may take f= 4,000 
 pounds when there are sudden, violent shocks upon the 
 gear, f= 5,000 pounds when only moderate shocks 
 occur, and /= 10,000 pounds when there is little or 
 no shock. By substituting these values of f, in turn, 
 in formula (10), and reducing, we obtain the following 
 formula for determining the pitch of a cast-iron gear : 
 
 * Ordinarily this ratio is assumed. For example, we may assume 
 
 =j = - 
 sumed. 
 
 -, or y = -3, and determine the pitch for the particular value as- 
 
TOOTHED GEARING. 
 
 93 
 
 For violent shock, / = 0.078^ P x -. (a) 
 
 For moderate shock,/ = 0.07 
 
 For little or no shock, / = 0.05 P x 
 
 t 
 
 P 
 
 . To determine the pitch for a cast-iron gear, 
 multiply the total force to be transmitted by the ratio 
 of the pitch to the face width, extract the square root of 
 the product, and multiply the result by 0.078 for violent 
 shock, 0.07 for moderate shock, or 0.05 for little or no 
 shock. 
 
 In ordinary machine-gearing the face width is very 
 
 often taken equal to twice the pitch j/=2/, y -j; 
 
 because a greater relative face width does not, in the 
 same degree, add strength to the tooth, the principal 
 effect being to increase the stiffness of the tooth. If 
 
 we make y = i in each of the formulas (u), we obtain 
 
 p = 0.078^ X |, / = o.07\//' x J, and / = 0.05^ Px ?. 
 
 Reducing these, we have, for the three cases given 
 above, the formulas 
 
 / = 0.055^ 
 / = 0.05 ^P 
 p = 0.035^ 
 
 (o 
 
 (0 
 
 (12). 
 
 Rule. To determine the pitch for cast-iron gears 
 when the face width is equal to twice the pitch, multi- 
 
94 TOOTHED GEARING. 
 
 ply the square root of the total force to be transmitted 
 by 0.055 f r violent shock, 0.05 for moderate shock, or 
 0.035 f r little or no shock. 
 
 A horsc-poiver, as commonly used, is that force which 
 will lift a weight of 33,000 pounds one foot high in one 
 minute, 33,000 foot-pounds. If we let //represent the 
 horse-power, and v the velocity at the circumference of 
 the wheel in feet per second, we shall have the expres- 
 sion, 
 
 _ 550^ 
 
 607; v 
 
 This value of P, substituted in formulas (11), gives the 
 following convenient formulas for the pitch when the 
 horse-power and the velocity in feet per second, instead 
 of the force transmitted in pounds, are given : 
 
 For violent shock, p = 1 &Z\ ~~ 7 ( a ) 
 
 I j-f -ft 
 For moderate shock, / = i.64\/ -j (b) 
 
 For little or no shock, p i.i 7y - (c) 
 
 Rule. To determine the pitch from the horse- 
 power and velocity in feet per second, multiply the 
 ratio of the horse-power to the velocity by the ratio of 
 the pitch to the face width, extract the square root 
 of the product, and multiply the result by 1.83 for 
 violent shock, 1.64 for moderate shock, or 1.17 for 
 little or no shock. 
 
 By substituting the above value of P in formulas (12), 
 we obtain for the pitch, when the face width is not less 
 than twice the pitch, the formulas : 
 
TOOTHED GEARING. 
 
 95 
 
 For violent shock, p 
 
 ' Jl 
 
 For moderate shock, / = i.iyV/ (b) 
 
 For little or no shock, / = 0.8 2V/ (<r) 
 
 vv 
 
 (14) 
 
 Rule. To determine the pitch from the horse- 
 power and velocity in feet per second, when the face 
 width is equal to twice the pitch, multiply the square 
 root of the ratio of the horse-power to the velocity by 
 1.29 for violent shock, 1.17 for moderate shock, or 0.82 
 for little or no shock. 
 
 If we represent by ;/ the number of revolutions per 
 minute, and by D the diameter of the wheel in inches, 
 we may obtain, for the velocity in feet per second, the 
 value, 
 
 v = 
 
 12 X 60 
 
 = 0.0043 6-Z?. 
 
 This value, substituted for v in formulas (13), gives 
 For violent shock, 
 For moderate shock, / = 24.84X7 yr- j (b) 
 
 I TT L 
 
 For little or no shock, / = 1 7-7 2 \/ 77- f (0 
 
 Rule. To determine the pitch from the horse- 
 power, diameter, and number of revolutions, divide the 
 horse-power by the product of the diameter, in inches, 
 into the number of revolutions per minute, multiply 
 the quotient by the ratio of the pitch to the face width, 
 
96 TOOTHED GEARING. 
 
 extract the square root of the product, and multiply the 
 result by 27.71 for violent shock, 24.84 for moderate 
 shock, or 17.72 for little or no shock. 
 
 By substituting v = o.oo^6Dn for v in formulas (14), 
 the following formulas for the pitch, when the face 
 width is equal to twice the pitch, may be obtained : 
 
 For violent shock 
 
 / // 
 
 , / = I9.54y 2^ (a) 
 
 -=- 
 
 (*) 
 
 For moderate shock,/ = i7.72y 
 
 I~S" 
 For little or no shock,/ = 1 2.42V/ -=r- (t) 
 
 06). 
 
 Rule. To determine the pitch from the horse-power, 
 diameter, and number of revolutions, when the face 
 width is equal to twice the pitch, divide the horse- 
 power by the product of the diameter into the number 
 of revolutions, extract the square root of the quotient, 
 and multiply the result by 19.54 for violent shock; 17.72 
 for moderate shock, or 12.42 for little or no shock. 
 
 Example I. A cast-steel gear wheel is required 
 which will transmit a force of 100,000 pounds. The 
 gear is to be subjected to severe shock. Suppose a 
 specimen of the steel to be used has been tested, and 
 the breaking-weight found to be 140,000 pounds per 
 square inch. We may take, for the greatest safe work- 
 ing-stress per square inch, / = \ X 140000 = 23333 
 
 pounds, say, /=. 23000. If, now, we take ^ = -, for- 
 
 / 4 
 
 mula (10) gives, 
 
 = 4 . 93 V/^^X- = 4-93V/T^ = 4-93 X 1.042 = 5.137 
 ^V 23000 4 
 
 n. 
 
TOOTHED GEARING. 97 
 
 For the other dimensions we have 
 
 / = 4/ = 4 X 5.137 = 20.548 inches 
 
 h = Q.ip = 0.7 x 5.137 = 3.5959 inches 
 and 
 
 b = 0.46^ = 0.46 x 5.137 = 2.363 inches. 
 
 Taking these values to the nearest eighth, sixteenth, 
 etc., taking care to err only on the safe side, we have 
 / = 5 6i inches, / = 2O T 9 g inches, // = 3 if inches, and 
 b = 2 J inches. 
 
 Example 2. Required the tooth dimensions for a 
 wooden cog-wheel to transmit a force of 10,000 pounds, 
 moderate shock. Let us suppose the cogs to be of 
 oak, the breaking-strength of which is 15,000 pounds 
 per square inch. We may safely take 
 
 /= \ X 15000 = 2500 pounds. 
 If j = -, formula (10) gives 
 
 = 4-93^/ 
 
 IOOOO I . 
 
 X - = 4-93V2 = 4-93 X 1.414 = 6.971 m. 
 
 /= 2p = 13.942 inches 
 
 h = 0.7^ = 4.8797 inches 
 and 
 
 b = 0.46^ = 3.20666 inches. 
 
 In fractions the dimensions are, / = 6\\ inches, 
 /= i3|-J inches, h = 4^ inches, and d= 3^2 inches. 
 
 Example 3. With the data of Example 2, required 
 the tooth dimensions for a wheel to be subjected to 
 
98 TOOTHED GEARING. 
 
 little or no shock. In this case we may take 
 
 /= i x 15000 = 3750. 
 As before, formula (10) gives for the pitch, 
 
 P = 4 ' 9 X = 4 * 93V/I '333 = 4-93 X 1.154 = 5-689 
 
 k = o.7/ = 3.982" 
 b = 0.46^= 2.617". 
 
 Example 4. Required the tooth dimensions for a 
 cast-iron gear wheel which is to be worked by hand 
 and crank. Suppose a man can exert by means of the 
 crank a force of 300 pounds on the wheel. From 
 formula (12, c) we have, for little or no shock, 
 
 p = 0.035^300 = 0.035 x I 7-3 2 = 0-606" 
 /= 2 /= 1.212" 
 
 ^ = 0.7^ = 0.4242" 
 and 
 
 = 0.46^ = 0.37876". 
 
 If, for any reason, it is necessary to make 7=-, we 
 
 / 4 
 
 must use formula (u, c), which gives 
 
 p = 0.05^300 x J = 0.05 x 8.66 = 0.433". 
 
 Hence 
 
 /= 4 /= 1.732", etc. 
 
 Example 5. A cast-iron gear is to have a circumfer- 
 
TOOTHED GEARING. 99 
 
 ential velocity of 10 feet per second, and is to transmit 
 a force of 15-horse power, moderate shock. Required 
 
 the dimensions of the teeth when -,-=. . From for- 
 
 / 3i 
 mula (13, b), 
 
 p = i.64 x- = 1.64^0.4286 = 1.64x0.6546 = 1.0725 
 Hence 
 
 ", etc. 
 It j = -, formula (14, b) gives 
 
 p = ItX Vxo = '-'f^S * = l '^ x x 
 
 xo = 1-43*9" 
 
 and 
 
 /= 2 p = 2.8658", etc. 
 
 Example 6. The diameter of a cast-iron gear which 
 is to transmit 25-horse power, excessive shock, is 30". 
 
 Required the dimensions, taking "? = - and the number 
 of revolutions per minute ;/ = 80. From formula (i6,#), 
 
 = 19-54^0.0104 = 19.54 X 0.102 = 2" 
 
 / = 4 ",etc. 
 
 Example J. Required a cast-steel gear which will 
 safely transmit loo-horse power, excessive shock. The 
 diameter of the gear is to be 36", and the number of 
 revolutions per minute 20. From the expressions 
 
100 TOOTHED GEARING. 
 
 v 0.00436/7/2 and P = - we have, by combining, 
 
 v 
 
 cco/7 550 x 100 
 
 P = JJ . _ = - - = 17520 pounds. 
 
 0.00436 X 36 X 20 
 
 Taking f= 23,000 pounds, as in Example i, and 7 = -, 
 
 / 4 
 
 formula (10) gives 
 
 x 7 = 4.93^0.19 = 4-93 X 0.436 = 2.149" 
 23000 4 
 
 / = 4 / = 8.596", etc. 
 
 As has already been noticed, the ratio ^ is com- 
 monly taken equal to J ; that is, the face width equals 
 twice the circumferential pitch. Formulas (12), (14), 
 and (16), which have been obtained from formulas (11), 
 
 (13), and (15) by substituting for - the value |, are 
 
 therefore much the more often used. The following 
 tables, obtained from formulas (12), (14), and (16), will 
 be found convenient : 
 
TOOTHED GEARING. 
 
 101 
 
 TABLE I. 
 
 From formula (12, a, b, and c). 
 
 p 
 
 in inches. 
 
 7>for 
 
 No. 
 
 Little or no shock. 
 
 Moderate shock. 
 
 Violent shock. 
 
 r 
 
 816 
 
 4OO 
 
 331 
 
 , 
 
 l| 
 
 1,276 
 
 625 
 
 513 
 
 2 
 
 '* 
 
 1,837 
 
 900 
 
 743 
 
 3 
 
 If 
 
 2,500 
 
 1,225 
 
 1,012 
 
 4 
 
 2 
 
 3,265 
 
 1, 6OO 
 
 1,322 
 
 5 
 
 2* 
 
 4J33 
 
 2,025 
 
 1,670 
 
 6 
 
 2 
 
 5,102 
 
 2,500 
 
 2,066 
 
 7 
 
 2f 
 
 6,173 
 
 3,025 
 
 2,5OO 
 
 8 
 
 3 
 
 7,347 
 
 3,600 
 
 2,975 
 
 9 
 
 31 
 
 8,622 
 
 4,225 
 
 3,49i 
 
 10 
 
 3 
 
 10,000 
 
 4,900 
 
 4,050 
 
 n 
 
 sf 
 
 11,480 
 
 5,625 
 
 4,649 
 
 12 
 
 4 
 
 13,061 
 
 6,400 
 
 5,289 
 
 13 
 
 41 
 
 14,745 
 
 7,225 
 
 5,97i 
 
 14 
 
 4* 
 
 16,531 
 
 8,100 
 
 6,694 
 
 15 
 
 4f 
 
 18,418 
 
 9,025 
 
 7,459 
 
 16 
 
 "5 
 
 20,408 
 
 10,000 
 
 8,265 
 
 17 
 
 5i 
 
 22,500 
 
 11,025 
 
 9,111 
 
 18 
 
 5 
 
 24,694 
 
 12,100 
 
 10,000 
 
 19 
 
 Sf 
 
 26,990 
 
 13,225 
 
 10,930 
 
 20 
 
 6 
 
 29,388 
 
 14,400 
 
 11,900 
 
 21 
 
 6 
 
 34,490 
 
 16,900 
 
 13,967 
 
 22 
 
 7 
 
 40,000 
 
 19,600 
 
 16,198 
 
 23 
 
 7* 
 
 45,918 
 
 22,500 
 
 i8,595 
 
 24 
 
 8 
 
 52,245 
 
 25,600 
 
 21,157 
 
 25 
 
102 
 
 TOOTHED GEARIfiG. 
 
 TABLE II. 
 
 From formula (14, #, b, and c.) 
 
 p 
 
 in inches. 
 
 H , 
 
 - for 
 v 
 
 No. 
 
 Little or no shock. 
 
 Moderate shock. 
 
 Violent shock. 
 
 I 
 
 1.49 
 
 0-73 
 
 O.6O 
 
 I 
 
 '* 
 
 2.32 
 
 1. 14 
 
 0.94 
 
 2 
 
 * 
 
 3-35 
 
 1.6 4 
 
 1-35 
 
 3 
 
 I4 2 
 
 4-55 
 
 2.24 
 
 1.84 
 
 4 
 
 2 
 
 5-95 
 
 2.92 
 
 2.40 
 
 5 
 
 2* 
 
 7-53 
 
 3.69 
 
 3.04 
 
 6 
 
 2 
 
 9-30 
 
 4.56 
 
 3.76 
 
 7 
 
 2| 
 
 11.25 
 
 5.52 
 
 4-54 
 
 8 
 
 3 
 
 13-38 
 
 6-57 
 
 541 
 
 9 
 
 3* 
 
 15.71 
 
 7.72 
 
 6-35 
 
 10 
 
 3 
 
 18.22 
 
 8.95 
 
 7.36 
 
 ii 
 
 si 
 
 20.91 
 
 IO.27 
 
 8.45 
 
 ' 12 
 
 4 
 
 23.80 
 
 11.69 
 
 9.61 
 
 13 
 
 4* 
 
 26.86 
 
 I3-I9 
 
 10.85 
 
 H 
 
 4 
 
 30.11 
 
 14.79 
 
 12.11 
 
 *5 
 
 4f 
 
 33-56 
 
 16.48 
 
 I3.56 
 
 16 
 
 5 
 
 3M8 
 
 18.26 
 
 15.02 
 
 17 
 
 5* 
 
 40.99 
 
 20.13 
 
 16.56 
 
 18 
 
 5 
 
 44.99 
 
 22.09 
 
 18.18 
 
 19 
 
 5f 
 
 49.17 
 
 24.15 
 
 19.87 20 
 
 6 
 
 53-54 
 
 26.30 
 
 21.63 2I 
 
 6* 
 
 62.83 
 
 30.86 
 
 25-39 
 
 22 
 
 7 
 
 72.87 
 
 35.80 
 
 29.45 
 
 23 
 
 7 
 
 83.66 
 
 41.09 
 
 33-80 
 
 24 
 
 8 
 
 95.18 
 
 46.75 
 
 38.46 
 
 25 
 
TOOTHED GEARING. 
 
 TABLE III. 
 From formula (16, #, <5, and c). 
 
 p 
 
 in inches. 
 
 H 
 Ti~ for 
 
 z># 
 
 No. 
 
 Little or no shock. 
 
 Moderate shock. 
 
 Violent shock. 
 
 I 
 
 0.0065 
 
 0.0032 
 
 O.OO26 
 
 I 
 
 I* 
 
 O.OIOI 
 
 0.0050 
 
 O.004I 
 
 2 
 
 I* 
 
 0.0146 
 
 0.0072 
 
 O.OO59 
 
 3 
 
 If 
 
 0.0198 
 
 0.0098 
 
 O.OO8O 
 
 4 
 
 2 
 
 0.0259 
 
 0.0127 
 
 O.OIO5 
 
 5 
 
 2* 
 
 0.0328 
 
 0.0161 
 
 0.0133 
 
 6 
 
 2* 
 
 0.0405 
 
 0.0199 
 
 0.0164 
 
 7 
 
 2j 
 
 0.0490 
 
 O.024I 
 
 0.0198 
 
 8 
 
 3 
 
 0.0583 
 
 0.0287 
 
 0.0236 
 
 9 
 
 31 
 
 0.0685 
 
 0.0336 
 
 0.0277 
 
 10 
 
 3 
 
 0.0794 
 
 0.0390 
 
 0.0321 
 
 ii 
 
 3! 
 
 0.0912 
 
 0.0448 
 
 0.0368 
 
 12 
 
 4 
 
 0.1037 
 
 0.05IO 
 
 0.0419 
 
 13 
 
 4J 
 
 0.1171 
 
 0.0575 
 
 0.0473 
 
 H 
 
 4 
 
 0.1313 
 
 0.0645 
 
 0.0530 
 
 15 
 
 . 4f 
 
 0.1463 
 
 0.0719 
 
 0.0591 
 
 16 
 
 5 
 
 0.1621 
 
 0.0796 
 
 0.0655 
 
 17 
 
 Si 
 
 0.1787 
 
 0.0878 
 
 0.0722 
 
 18 
 
 5* 
 
 0.1961 
 
 0.0963 
 
 0.0792 
 
 19 
 
 si 
 
 0.2143 
 
 0.1053 
 
 0.0866 
 
 20 
 
 6 
 
 0.2334 
 
 O.II46 
 
 0.0943 
 
 21 
 
 ^ 
 
 0.2739 
 
 0.1346 
 
 O.II07 
 
 22 
 
 7 
 
 0.3177 
 
 0.1560 
 
 0.1283 
 
 23 
 
 7* 
 
 0.3647 
 
 0.1790 
 
 0.1473 
 
 24 
 
 8 
 
 0.4149 
 
 0.2038 
 
 0.1676 
 
 25 
 
IO4 TOOTHED GEARING. 
 
 Example I. Required the pitch of a cast-iron bevel 
 wheel which will transmit a force of 10,000 pounds, 
 moderate shock. In Table L, column for moderate 
 shock, line 17, we find P = 10,000 pounds. In the pitch 
 column, and directly opposite this value of P, we find 
 the required pitch, / = 5". Hence /= 2p 10", etc. 
 
 Example 2. The force transmitted by a cast-iron 
 gear under violent shock is 6,000 pounds. Required 
 the necessary pitch. Table L, column for violent shock, 
 line 14, gives P = 5,971 pounds ; and the corresponding 
 pitch is / 4j". Since this pitch corresponds to a 
 value of P slightly less than the required one, we may 
 take for our required pitch / = 4". 
 
 Example 3. The pitch of a cast-iron gear subjected 
 to little or no shock is 2^". Required the force in 
 pounds which can be safely transmitted by the gear. 
 In Table L, pitch column, line 6, we find/ = 2%'. The 
 value of P for little or no shock, corresponding to this 
 pitch, is 4,133 pounds. 
 
 Example 4. Required the pitch for a cast-iron gear 
 which will safely transmit 24-horse power, violent 
 shock, at a circumferential velocity of 8 feet per sec- 
 ond. In this case = = 3. In Table II., column 
 v 8 
 
 TT 
 
 for violent shock, line 6, we find = 3.04 ; and the 
 
 TT 
 
 corresponding pitch (found opposite this value of 
 
 V 
 
 in the pitch column) is p 2 J". 
 
 Example 5. A certain cast-iron gear transmits 75- 
 horse power. The pitch of the gear is 3|". Required 
 the circumferential velocity safe for the gear at mod- 
 
TOOTHED GEARING. 1 05 
 
 erate shock. We have from Table II., column for 
 
 TT 
 
 moderate shock, the value 8.95, corresponding to 
 
 /=3j. Hence ^^8.95, v = ^= 8.38 feet per 
 v 8.95 
 
 second. 
 
 Example 6. Required the pitch for a cast-iron gear 
 to transmit safely 5O-horse power, violent shock, at 100 
 revolutions per minute ; the diameter of the gear being 
 16". We have 
 
 H 50 i 
 
 In Table III., column for violent shock, line 11, we find 
 
 TT 
 
 - = 0.0321. The corresponding pitch is/ = 3^". 
 
 The following table will be found very convenient in 
 converting'decimals into fractions: 
 
io6 
 
 TOOTHED GEARING. 
 TABLE IV. 
 
 V) 
 
 <n 
 
 ui 
 
 
 tn 
 
 
 t/i 
 
 -/-: 
 
 M 
 
 
 
 _c 
 
 
 tj. 
 
 00 
 
 vO 
 
 *w 
 
 ^J. 
 
 
 "-J. 
 
 00 
 
 O 
 
 i 
 
 3. 
 
 
 
 
 
 CO 
 
 VO 
 
 
 
 
 " 
 
 m 
 
 VO 
 
 
 
 
 
 
 I 
 
 0.015625 
 
 
 
 
 
 33 
 
 0.515625 
 
 
 
 
 1 
 
 
 0.031250 
 
 
 
 
 17 
 
 
 0.531250 
 
 
 
 
 
 3 
 
 0.046875 
 
 
 
 
 
 35 
 
 0.546875 
 
 
 
 I 
 
 
 
 0.062500 
 
 
 
 9 
 
 
 
 0.562500 
 
 
 
 
 
 5 
 
 0.078125 
 
 
 
 
 
 37 
 
 0.578125 
 
 
 
 
 3 
 
 
 0.093750 
 
 
 
 
 19 
 
 
 o.59375o 
 
 
 
 
 
 7 
 
 0.109375 
 
 
 
 
 
 39 
 
 0.609375 
 
 
 I 
 
 
 
 
 0.125000 
 
 
 5 
 
 
 
 
 0.625000 
 
 
 
 
 
 9 
 
 0.140625 
 
 
 
 
 
 41 
 
 0.640625 
 
 
 
 
 5 
 
 
 0.156250 
 
 
 
 
 21 
 
 
 0.656250 
 
 
 
 
 
 ii 
 
 0.171875 
 
 
 
 
 
 43 
 
 0.671875 
 
 
 
 3 
 
 
 
 0.187500 
 
 
 
 ii 
 
 
 
 0.687500 
 
 
 
 
 
 13 
 
 0.203125 
 
 
 
 
 
 45 
 
 0.703125 
 
 
 
 
 7 
 
 
 0.218750 
 
 
 
 
 23 
 
 
 0.718750 
 
 
 
 
 
 15 
 
 0.234375 
 
 
 
 
 
 47 
 
 0-734375 
 
 I 
 
 
 
 
 
 O.25OOOO 
 
 3 
 
 
 
 
 
 0.750000 
 
 
 
 
 
 17 
 
 0.265625 
 
 
 
 
 
 49 
 
 0.765625 
 
 
 
 
 9 
 
 
 0.281250 
 
 
 
 
 25 
 
 
 0.781250 
 
 
 
 
 
 19 
 
 0.296875 
 
 
 
 
 
 51 
 
 0.796875 
 
 
 
 5 
 
 
 
 0.312500 
 
 
 
 13 
 
 
 
 0.812500 
 
 
 
 
 
 21 
 
 0.328125 
 
 
 
 
 
 53 
 
 0.828125 
 
 
 
 
 ii 
 
 
 0.343750 
 
 
 
 
 27 
 
 
 0.843750 
 
 
 
 
 
 23 
 
 0-359375 
 
 
 
 
 
 55 
 
 0.859375 
 
 
 3 
 
 
 
 
 0.375000 
 
 
 7 
 
 
 
 
 0.875000 
 
 
 
 
 
 25 
 
 0.390625 
 
 
 
 
 
 57 
 
 0.890625 
 
 
 
 
 13 
 
 
 0.406250 
 
 
 
 
 29 
 
 
 0.906250 
 
 
 
 
 
 27 
 
 0.421875 
 
 
 
 
 
 59 
 
 0.921875 
 
 
 
 7 
 
 
 
 0.437500 
 
 
 
 15 
 
 
 
 0.937500 
 
 
 
 
 
 2 9 
 
 0.453125 
 
 
 
 
 
 61 
 
 0.953125 
 
 
 
 
 15 
 
 
 0.468750 
 
 
 
 
 31 
 
 
 0.968750 
 
 
 
 
 
 31 
 
 0.484375 
 
 
 
 
 
 63 
 
 0.984375 
 
 2 
 
 
 
 
 
 0.500000 
 
 4 
 
 
 
 
 
 1. 000000 
 
TOOTHED GEARING. IO? 
 
 For very high speed gears, we may take, at the sug 
 gestion of Professor Reuleaux and Mr. W. C. Unwin, 
 
 f=- , v being the circumferential velocity in feet 
 
 Jv 
 per second. For example, suppose it was required to 
 
 determine the pitch of a cast-iron gear, the velocity of 
 which is 35 feet per second, and the force to be trans- 
 mitted is 5,000 pounds. We have for our safe working- 
 stress 
 
 . I 0000 I 0000 
 
 From formula (10), taking *-~ = -, we have 
 
 / 4 
 
 /CQOO I / - 
 
 = 4 ' 93 V 3058 X 4 = 4-93Y0.4088 = 4.93 x 0.639 
 
 XIV. Strength of Arms, Rim, Nave, Shafts, etc. 
 
 The arms of a gear-wheel being symmetrically placed 
 with reference to each other and to the rim of the 
 wheel, we may assume, without demonstration, that 
 each arm bears an equal share of the total strain upon 
 the rim : in other words, the strain upon each arm is 
 equal to the total strain upon the rim divided by the 
 number of arms in the gear. This is a bending-strain 
 acting in a direction perpendicular to the axis of the 
 arm and in the plane of the wheel. The proper strength 
 for the arms may therefore be calculated similarly to 
 that of the teeth. 
 
 Fig. 8 1 represents a portion of a gear showing one 
 
io8 
 
 TOOTHED GEARING. 
 
 arm. R is the radius of the pitch circle in inches ; P, 
 the total strain upon the rim (the total force transmit- 
 ted in pounds) ; //, and /; t respectively, the width (in the 
 plane of the wheel) and thickness of the arm in inches ; 
 and ;//, the number of arms.* Denoting by / the 
 greatest safe working-stress for 
 the material, the equation for 
 equilibrium is 
 
 P _ fbji? 
 
 Fig. 81 
 
 We may take, in this case, for 
 cast-iron, / = 3,000 pounds. Con- 
 sequently 
 
 P 
 
 P / 6R 
 
 From this, by transposing, we have 
 
 which may be easily solved by assuming a value for b l 
 
 in terms of //, { = -, -r = -> etc.), and finding the 
 \/t l 2 ft t 4 / 
 
 corresponding value of h v 
 
 For convenience, we may write formula (17) in the 
 form of an equation having one unknown quantity, 
 thus : 
 
 (18) 
 
 * The dimensions bi and 7/ x are taken at the rim of the wheel, and 
 tapered, as shown in Fig. 81. 
 
TOOTHED GEARING. 
 
 109 
 
 and find values of the co-efficient x for different values 
 of b t and /.* 
 
 The following table gives values of x for different 
 
 values of -^ and / : 
 h* 
 
 TABLE V. 
 
 1 
 
 ;rfor x '= 
 
 k\ 
 
 4 
 
 5 
 
 6 
 
 8 
 
 IO 
 
 i 
 
 O.I 00 
 
 0.093 
 
 0.087 
 
 0.079 
 
 0.074 
 
 1 
 
 0.126 
 
 0.117 
 
 O.IIO 
 
 O.I 00 
 
 0.093 
 
 1 
 
 0.144 
 
 0.134 
 
 0.126 
 
 0.114 
 
 O.I 06 
 
 1 
 
 0.159 
 
 0.147 
 
 0.139 
 
 0.126 
 
 0.117 
 
 Rule. To determine the width of cast-iron gear 
 arms in the plane of the wheel, multiply the force 
 transmitted by the radius of the pitch circle, extract 
 the cube root of this product, and multiply the result 
 by the tabular number corresponding to the given values 
 
 of ^ and /. 
 h, 
 
 Example i. A cast-iron gear the diameter of which 
 is 48" transmits a force of 5,000 pounds. Required 
 the width and thickness of the arms, of which there are 
 
 5. If we assume 7 1 = -, Table V. gives, for the value 
 
 of the co-efficient, x =. o. 1 1 7. 
 becomes 
 
 Hence formula (18) 
 
 * This form is given by Umvin in Elements of Machine Design. 
 
10 TOOTHED GEARING. 
 
 h l = o.i 171/7^ = o.uysooo x 24 = o.ny' 12000 = 0.117 
 
 X 49-324 = 5-77" 
 or in fractions, from Table IV., //, = 5ff": hence 
 
 *, = #, = i X 5-77 = 14425" = i&". 
 
 Example 2. A cast-iron 72" gear transmits a force 
 of 15,000 pounds. Taking n' = 6, and - -, required 
 
 //! 2 
 
 the dimensions of the arms. From Table V., x = 0.087 : 
 hence, from formula (18), we have 
 
 h l = 0.08 7 V^# = 0.08 7'V 1 5 ooo x 36 = 0.087 X 81.433 
 
 = 7-0847" =7*". 
 For the thickness we have 
 
 *, = fa = j x 7.0847 = 3-54235" = 3H"- - 
 
 If, instead of rectangular, we have circular cross- 
 sections for gear arms, and represent the diameter by 
 d r , the equation for equilibrium becomes 
 
 P _ /x 0.0982^3 
 ?" ~~R~ 
 or, for cast-iron, 
 
 P _ 3000 x 0.0982^'* 
 7~ ^ 
 
 Reducing and transposing this equation gives 
 
 PR 
 
TOOTHED GEARING. Ill 
 
 Rule. To determine the diameter for cast-iron gear 
 arms having circular cross-sections, multiply the force 
 transmitted by the pitch radius, divide this product by 
 the number of arms, extract the cube root of the quo- 
 tient thus obtained, and multiply the result by 0.15. 
 
 Example 3. A cast-iron gear of $6" diameter has 
 5 arms (circular cross-sections), and transmits a force 
 of 600 pounds. Required the diameter for the arms. 
 From formula (19) we have 
 
 ,, * 3/i 8 X 600 R/ - 
 
 d = o.i5y --- = o.i5V2i6o = 0.15 x 12.927 = 1.939 
 
 or, from Table IV., </'= i^". 
 
 For elliptical cross-sections, representing by a the 
 major and by b' the minor axis, the equation for equilib- 
 rium is 
 
 P _ f x 0.0982^2 
 _____ _____ 
 
 or, for cast-iron, 
 
 P _ 3000 X 0.0982^2 
 
 Hence, by reducing and transposing, we obtain 
 
 PR 
 
 0.003397- (20). 
 
 Rule. To determine the dimensions for cast-iron 
 gear arms having elliptical cross-sections, multiply the 
 force transmitted by the pitch radius, multiply the prod- 
 uct thus obtained by 0.00339, and divide the result by 
 the number of arms. This gives the product of the 
 
112 TOOTHED GEARING. 
 
 minor into the square of the major axis (b'a 2 ) : the axes 
 may then be found as in formula (17). 
 
 Example 4. Required the axes for the cross-sections 
 of the elliptical arms of a cast-iron gear, the diameter 
 of which is 24". The force transmitted is 800 pounds, 
 and the number of arms 3. Let us assume a relation 
 
 between the cross-section axes, say ' = -. Formula 
 (20) then gives 
 
 a* 800 x 12 
 
 or 
 
 - = 0.00339 
 
 Q-00339 X 800 x 12 X 2 = 
 3 
 
 Hence 
 
 
 
 
 a = V 21. 696 
 
 = 2.789" =2 
 
 Also 
 
 
 
 
 j, = g = g-y89 
 
 2 ~~ 2 
 
 = 1-3945" = 
 
 For arms having flanged cross-sections, such as is 
 shown in Fig. 82, the equation for equilibrium becomes 
 
 '* 4- BhJ 
 
 -7 = ^X 
 
 7/ t A ujj 
 
 Substituting for / its value of 3,000 pounds, and re- 
 ducing, we obtain, for cast-iron, 
 
 P _ 5oo(,,//' 3 4- 
 *7~ RH' 
 
 or 
 
 - Bh,t PR 
 
TOOTHED GEARING. 
 
 Example 5. Required the dimensions for the arms 
 of a 36" cast-iron gear which transmits a force of 800 
 pounds ; the arms to be flanged, as in Fig. 82, and to 
 be 4 in number. Let us assume relations between the 
 
 Fig.82 
 
 Fig. 83 
 
 several unknown quantities in formula (21). Thus, 
 suppose 
 
 h J, B H ' 
 b,,= h,,= -~. 
 
 By substitution the formula becomes 
 
 800 X 18 
 
 5 * 4 ' 
 
 H' 
 
 Reducing, we have 
 
 H'* 20'* _ I 2 iff'* _ $6 
 
 5 625 625 ' 5* 
 
 Hence 
 
 5X127 
 
114 TOOTHED GEARING. 
 
 For the other dimensions, 
 
 and 
 
 B-tZ. -'-*&*- w&. 
 
 Converted into fractions by means of Table IV., the arm 
 dimensions are H' = 3^2 "> b,, = /i,,= fi", and #~ i T 5 g". 
 For arms with cross-sections, flanged as in Fig. 83, 
 the equation for equilibrium is 
 
 P___f_ BIT* - b,,h,* 
 n/ ~~ 7? X 6H' ' 
 
 = 3,000 in this equation, we have, for cast-iron, 
 P 
 
 (22) * 
 
 n!~ RH' 
 
 or 
 
 b,,h,t PR 
 
 Example 6. A 48" cast-iron gear transmits a force 
 of i ,000 pounds, and has 5 arms, the cross-sections 
 being flanged as in Fig. 83. Required the arm dimenr 
 
 TTf J 
 
 sions. Let us take B= t //// = //', and ~\H'. 
 
 2 2 
 
 These values, substituted in formula (22), give 
 
 1000 X 24 
 H 1 5 X 5 * 
 
 Reducing, we have 
 
 256 
 
TOOTHED GEARING. 
 
 and 
 
 = 0.3623" = 
 
 The number of arms in a gear-wheel is often deter- 
 mined, according to the pitch diameter, by the following 
 table : 
 
 For a gear of i J to 3^ feet diameter, 4 arms. 
 For a gear of 3^ to 5 feet diameter, 5 arms. 
 For a gear of 5 to 8 feet diameter, 6 arms. 
 For a gear of 8| to 16 feet diameter, 8 arms. 
 For a gear of 16 to 25 feet diameter, 10 arms. 
 
 Reuleaux gives for the number of arms the formula 
 
 (23) 
 
 in which ;// is the number of arms, ^V the number of 
 teeth in the gear, and/ the pitch.* 
 
 Ride. To determine the number of arms for a gear- 
 wheel, extract the square root of the number of teeth 
 and the fourth root of the pitch, multiply the roots 
 together and the product by 0.56. 
 
 Example 6 a. A gear-wheel has 100 teeth and a 
 
 * Small pinions, and sometimes narrow-faced gears, are made without 
 arms; i.e., having a continuous web cast between the rim and nave. 
 
Il6 TOOTHED GEARING. 
 
 pitch of i". Required the number of arms. From for 
 mula (23) 
 
 / = 0.56^100 V7 = 0.56 X 10 X i = 5.6 or 6. 
 
 A convenient formula for the arm dimensions, in 
 terms of the horse-power transmitted and the revolu- 
 tions, may be obtained as follows. As explained in 
 XIII., we have the expressions 
 
 v = 0.008 Rn and P= 
 
 v being the circumferential velocity in feet per second, 
 H the horse-power, and n the revolutions per minute. 
 By combining these we obtain 
 
 H 
 
 This value of P substituted in formula (17) give's 
 63000;? R 
 
 s i *fr i n s\ t 
 
 Rn SOCK/ 
 or 
 
 IT 
 
 M, a =i26 , (24). 
 Tin i 
 
 Rule. To determine the quantity bji? (the thick- 
 ness multiplied by the square of the width) for cast-iron 
 gear arms, from the horse-power and revolutions, mul- 
 tiply the horse-power by 126, and divide by the product 
 of the number of revolutions into the number of arms. 
 
 Example 7. A 36" cast-iron gear makes 80 revolu- 
 tions per minute, and transmits 15 -horse power. Re- 
 
TOOTHED GEARING. 1 1/ 
 
 quired the dimensions of the arms. From the table we 
 have for the number of arms ;// = 4, and from for- 
 mula (24) 
 
 126 x 15 
 
 ^ = -8^r := s - 9 6 - 
 
 We may now assume b l = : hence 
 
 7/ 3 
 Mi 2 = = 5-906 
 
 h* = ^23.624 = 2.869" = 2 J" 
 and 
 
 _/Z r _ 2.869 _ 23" 
 
 *,---- - : 0.717 
 
 For arms having circular cross-sections we have, as 
 above, 
 
 P = 63000^ 
 
 which, substituted in formula (19), gives, for the diame- 
 ter of the arm cross-section, 
 
 or 
 
 ' H r, (25). 
 
 Rule. To determine the diameter for cast-iron gear 
 arms having circular cross-sections, from the horse- 
 power and revolutions, divide the horse-power by the 
 product of the number of revolutions per minute into 
 the number of arms, extract the cube root of this quo- 
 tient, and multiply the result by 5.969. 
 
 Example 8. The diameter of a cast-iron gear is 48", 
 
Il8 TOOTHED GEARING. 
 
 the horse-power transmitted 15, and the number of revo- 
 lutions per minute 40. Required the diameter for the 
 circular cross-sections of the arms. From the table, 
 the number of arms is 5 : hence, from formula (25), 
 
 5-969 X 0.4217 = 2.517 = 2 ff". 
 
 For elliptical cross-sections, of which a and b' are 
 respectively the major and minor axes, we have, by 
 substituting in formula (20), the value 
 
 />= 63000 -g, 
 
 H R 
 
 b'a? = 0.00339 x 63000 =- x > 
 Rn n' 
 
 or 
 
 * V = 2I3 - 57 J? (26) - 
 
 Rule. To determine the quantity b'a? (the minor 
 axis multiplied by the square of the major), for cast-iron 
 gear arms having elliptical cross-sections, from the 
 horse-power and revolutions, multiply the horse-power 
 by 213.57, an d divide by the product of the number of 
 revolutions per minute into the number of arms. 
 
 Example 9. A 48" cast-iron gear makes 40 revolu- 
 tions per minute, and transmits 2O-horse power. Re- 
 quired the arm dimensions for elliptical cross-sections. 
 In this case, n' = 4, and hence formula (26) gives 
 
TOOTHED GEARING. 119 
 
 If we take b' = \a, we shall have 
 
 a* 
 * = -- = 21.357 
 
 = ai.357X 2 = 3496"= 3*" 
 and 
 
 For arms having cross-sections flanged, as shown in 
 Fig. 82, we obtain, by substituting in formula (21) the 
 value of P determined above, 
 
 b.,H'* + Bh,t H R 
 
 - - -- = 63000 -5- X -- 7 
 H' Rn soo, 
 
 or 
 
 (27) 
 
 which may be solved as explained in Example 5 of this 
 section. 
 
 Similarly, for arms having cross-sections flanged, as 
 in Fig. 83, we obtain 
 
 (28) * 
 
 It is often convenient to calculate the dimensions 
 of the arms from the pitch and radius of the gear. 
 Formulas for the arm dimensions, in terms of these 
 quantities, may be obtained as follows : 
 
 From formula (12, b) we may write 
 
 ~" 0.0025 
 
I2O TOOTHED GEARING. 
 
 which, substituted in formula (17), gives 
 
 or 
 
 ?. To determine the quantity bji? (the thick- 
 ness of the arm multiplied by the square of its width) 
 from the pitch and radius of the gear, divide the con 
 tinned product of 0.8 into the square of the pitch into 
 the radius, by the number of arms. 
 
 Example 10. Required the dimensions for the arms 
 of a gear-wheel, the diameter of which is 24", and the 
 pitch i". In this case, n^-=.^\ hence, from formula 
 
 0.8 X i X 12 
 <M< 2 =- = 2.4. 
 
 If we take b l = -^, 
 o 
 
 h W 
 *' ~ 6 ~ OT ' 
 
 By substituting /> = f- in formulas (19), (20), 
 0.0025 
 
 (21), and (22), the following formulas may be obtained. 
 For arms having circular cross-sections, of which d r is 
 the diameter, 
 
 ( 3 o). 
 
TOOTHED GEARING. 121 
 
 For elliptical cross-sections, a and b' being the major 
 and minor axes respectively, 
 
 (3i). 
 
 "i 
 
 For cross-sections, as shown in Fig. 82, 
 
 H f n{ 
 
 For cross-sections, as shown in Fig. 83, 
 
 (32)- 
 
 H' 
 
 (33)- 
 
 Example n. Taking the data of Example 10, re- 
 quired the diameter for arms having circular cross- 
 sections. Formula (30) gives, by substituting the 
 numerical data, 
 
 = i.io S VJ= i.i5937"= itt". 
 
 Example 12. With the same data, required the 
 dimensions for arms having elliptical cross-sections. 
 From formula (31) we have 
 
 I X 1 2 
 
 tfa 2 1.356 = I-356 X 3 = 4.068. 
 
 4 
 
 Assuming b f = \a 
 
 b'a? = = 4.068 
 
 a \/4^o68~X*2 =s 2" 
 fi'=ia= i". 
 
122 TOOTHED GEARING. 
 
 Example 13. Using the same data, it is required to 
 determine the dimensions for flanged arms having cross- 
 sections, such as shown in Fig. 83. 
 
 From formula (33) 
 
 b,,h,t 0.8 X i X 12 
 
 H' 
 
 = 2.4. 
 
 Let us take B = ///,= #"' and -= j/f: hence 
 
 = h,, = iff' =0.862" = 
 
 _. A 
 
 and 
 
 More often than otherwise, the arms of gear-wheels 
 are made straight, as in Fig. 81 : sometimes, however, 
 especially in large gears and in gears subjected to 
 violent shock and strain, curved arms are preferred, as 
 tending to stiffen and support the rim better. Also 
 curved arms, as a general rule, cast better. When 
 single curved arms are used, they may be constructed 
 as follows : 
 
 After having determined the number of arms by one 
 of the foregoing rules, and having marked their cen- 
 tres A, C (Fig. 84), upon the circumference ABC, take 
 the arc AB = f arc A C, and draw the radial line OB. 
 From the centre O of the wheel, erect the line OD per- 
 pendicular to OB, and find upon OD, by trial, the centre 
 
TOOTHED GEARING. 
 
 123 
 
 a for a circular arc passing through the points O and A. 
 This arc is the axis of the arm. Lay off, as shown in 
 the figure, // (</, a, or //', according as the cross- 
 sections are to be rectangular, circular, elliptical, or 
 flanged*) at the rim, and not less than \h 
 -//') at the nave. Find 
 upon OD, by trial, the 
 centres b and c for the Fig.84 
 arcs gk and df, which 
 determine the form of 
 the arms. 
 
 Fig. 85 shows anoth- 
 er method for drawing 
 curved arms. Through 
 the centre o of the wheel 
 draw the line oA, making 30 with the horizontal. 
 Draw also the line AB, making 60 with the horizon- 
 tal. The point B is the centre for the axis oA of the 
 
 Fig.85 
 
 arm. Lay off, as before, 
 // and |/r, and find upon 
 the line oB the centres 
 for the arcs df and gk' . 
 
 Double curved arms 
 are sometimes used for 
 large gears. Fig. 86 
 shows a simple method ~ 
 for their construction. 
 Draw the radial line oA, making 30 with the horizon- 
 tal. Take oc = \oA, and through the point c draw 
 the line pD, making 60 with the horizontal. Intersect 
 
 * The cross-sections of curved arms are generally elliptical, the curved 
 form giving sufficient stiffness to dispense with flanges, etc. 
 
124 
 
 TOOTHED GEARING. 
 
 the line pD by a horizontal line through the point 
 A : the points D and / are respectively the centres 
 for the arcs oc and cA, which together form the axis 
 
 Fig. 86 
 
 of the arm. Lay off the arm widths as shown in the 
 figure. From the point / as a centre strike the arcs 
 ab and cf, and find upon the line oD the centres for the 
 remaining arcs bd and//&'. 
 
 Fig. 87 
 
 Another very similar method for laying out double 
 curved arms is shown in Fig. 87. Draw the radial line 
 oA, making 45 with the horizontal. Take oc=.^oA, 
 and through the point c draw the vertical line pD. In- 
 
TOOTHED GEARING. 125 
 
 tersect the line pD by the horizontal line Ap. The 
 points / and D are the centres for the arcs of the axis. 
 Lay off // and |//, as shown in the figure, and proceed, 
 as in Fig. 86, to strike the arcs ab, ef, bd, and fk f . 
 
 Rim : For the thickness of the rim in the plane of 
 the wheel, t (Fig. 87), Reuleaux gives the formula 
 
 /=0.12-f0.4/ (34) 
 
 in which t is the rim thickness, and/ the pitch. 
 
 Rule. To determine the thickness of the rim of a 
 cast-iron gear-wheel, multiply the pitch by 0.4, and to 
 this product add o. 12". 
 
 Example 14. Required the thickness of rim for a 
 gear having a pitch of 3^". From formula (34) 
 
 /= 0.12 -f 0.4 x 3.5 = 0.12 4- 1.40 = 1.52" = i|". 
 
 A simple and not very accurate rule in use in the 
 shops is to take the rim thickness equal to f the pitch. 
 
 Nave : The old formulas for the thickness of the 
 nave (k, Fig. 85) k\p and k=^d, in which k is 
 the nave thickness,/ the pitch, and </the diameter of the 
 eye of the wheel are probably nearly correct, notwith- 
 standing their simplicity. Unwin gives the formula 
 
 * = o. 4 ^ + i (35) 
 
 in which/ is the pitch of the teeth, and R the radius of 
 the wheel. 
 
 Ride. To determine the thickness of the nave of a 
 cast-iron gear wheel, multiply the square of the pitch by 
 the radius of the wheel, extract the cube root of this 
 product, multiply the result by 0.4, and add ". 
 
126 TOOTHED GEARING. 
 
 Example 15. The diameter of a gear is 36" and the 
 pitch ij". Required the thickness of the nave. From 
 formula (35) we have 
 
 k = o.4'V(i-|) 2 X 18 + 0.5 = 0.4/40-5 + 0.5 
 
 = 0.4 x 3.434 + 0.5 = 1.874" = i J". 
 By the formula k |/, we would have 
 
 = 0.75 x 1.5 = 1.125"= if- 
 
 Thus, in this case the difference between the results of 
 the two formulas is J". 
 
 For the length of the nave we may use the formula 
 
 /'='+! (36) 
 
 in which I' is the length of the nave, / the face width of 
 the teeth, and D the diameter of the gear. 
 
 Rule. To determine the nave length of a gear, 
 divide the diameter of the gear by 30 and to the quo- 
 tient add the face width of the teeth. 
 
 Example 16. The diameter of a gear is 60" and the 
 face width of the teeth 8". Required the length of 
 the nave. Formula (36) gives 
 
 /' = 8 + |$ = 8 + 2 = 10". 
 
 According to Unwin, the length of the nave should 
 never be less than three times its thickness. He gives, 
 for the length, the formula /' / 4- o.o6R, which agrees 
 very nearly with formula (36). 
 
TOOTHED GEARING. 
 
 127 
 
 Fig.88 
 
 Shafts : When a shaft is so supported by its bear- 
 ings as to be subjected to a torsional strain only, as is 
 almost invariably the case in gear shafts (the bending- 
 strain, due to the weight of the gear and the pressure 
 between the gears in the direction of a line joining 
 their centres, being ordinarily slight enough to be safely 
 neglected), the calculation of the proper strength for 
 the shaft may be made as follows : 
 
 In Fig. 88, P represents the total force tending to 
 twist the shaft, i.e., the total 
 force transmitted by the gear ; 
 R> the distance from the cen- 
 tre of the shaft to the point 
 at which the force acts, i.e., 
 the radius of the gear ; and 
 d, the diameter of the shaft. 
 The greatest safe torsional 
 strain which can be sustained 
 by the shaft is given by the 
 expression 
 
 _ *?** _ 
 
 (ZED 
 
 in which f is the greatest safe shearing-stress in 
 pounds per square inch for the material of the shaft. 
 
 From this, 
 
 PR 
 
 - T 9 6 35/ 
 
 or 
 
 , 
 d i. 
 
 (37). 
 
 Rule. To determine the diameter of a gear shaft of 
 any material, multiply the total force transmitted by 
 
128 TOOTHED GEARING. 
 
 the gear by the radius of the gear, divide this product 
 by the greatest safe shearing-stress in pounds per 
 square inch for the material of the shaft, extract the 
 cube root of the quotient thus obtained, and multiply 
 the result by 1.720. 
 
 Example 1 7. Required the diameter for an oak 
 shaft, upon which is a 60" gear transmitting a force of 
 1,000 pounds, taking /' 500 pounds. From formula 
 (37), 
 
 = 1.720 x 3-915 = 6.734" = 6". 
 
 We propose to take, for steel, /' = 12,000 pounds; 
 for wrought-iron, f = 8,000 pounds ; and, for cast-iron, 
 /' = 4,000 pounds. These values of f are nearly mean 
 between those used by Stoney, Haswell, and Unwin, 
 which differ far more than is conducive to any degree 
 of accuracy. Substituting the above values of f suc- 
 cessively in formula (37), and reducing, we obtain; 
 
 For steel, d = v.v\$PR (38) 
 
 For wrought-iron, d = o.o86'V/^ (39) 
 For cast-iron, d '= 0.108 ^fPR (40)* 
 
 Rule. To determine the diameter for a gear shaft 
 of steel, wrought or cast iron, multiply the total force 
 transmitted by the radius of the gear, extract the cube 
 root of the product, and multiply the result by 0.075 for 
 steel, 0.086 for wrought-iron, and o. 108 for cast-iron. 
 
 Example 18. A 48" gear transmits a force of 
 100,000 pounds. Required the diameter for a steel 
 
TOOTHED GEARING. 1 2g 
 
 shaft. From formula (38) we have 
 d= 0.075^100000 x 24 = 0.075 x 62.145 = 4-66 v = 4ff". 
 
 Example 19. Taking the data of Example 18, re- 
 quired the diameter for a shaft of cast-iron. Formula 
 (40) gives 
 
 d = o.ioSViooooo x 24 = 0.108 X 62.145 = 6.712"= 6|-f". 
 
 Formulas for the diameters of gear shafts, in terms of 
 the horse-power transmitted and the revolutions per 
 minute, may be obtained as follows : 
 
 As before explained, we have the expression 
 
 ^=63000^ 
 
 H representing the horse-power, R the radius of the 
 gear, and ;/ the number of revolutions per minute. 
 Substituting this value of P in formulas (37), (38), (39), 
 and (40), and reducing, we obtain the following : 
 
 fjr 
 
 General formula, d 68.44 y-y/ (4 1 ) 
 
 .984^-^ 
 
 For steel, </= 2.984- (42) 
 
 For wrought-iron, d 3.422^ (43) 
 
 js 
 
 = 4.297V" 
 
 For cast-iron, d= 4.297V- (44) 
 
I3O TOOTHED GEARING. 
 
 Rule. To determine the diameter for a gear shaft 
 of any material, from the horse-power and number of 
 revolutions per minute, divide the horse-power by the 
 product of the number of revolutions into the greatest 
 safe shearing-stress in pounds per square inch for the 
 material of the shaft, extract the cube root of the quo- 
 tient thus obtained, and multiply the result by 68.44. 
 
 To determine the diameter for a gear shaft of steel, 
 wrought or cast iron, from the horse-power and number 
 of revolutions per minute, divide the horse-power by 
 the number of revolutions, extract the cube root of the 
 quotient, and multiply the result by 2.984 for steel, 
 3.422 for wrought-iron, and 4.297 for cast-iron. 
 
 Example 20. Required the diameter for an oak 
 gear shaft which transmits a force of zo-horse power, 
 and makes 40 revolutions per minute. If we take for 
 the greatest safe shearing-stress for oak f = 500 pounds 
 per square inch, we shall have, from formula (41), 
 
 d = 68.44V 5 / = 68.44V 3 / ^ = 68.44 X 4r 
 
 "V 4 * 500 V 2000 1 2.60 
 
 = 5432" = 5*" nearly. 
 
 Example 21. Taking the data of Example 20, re- 
 quired the diameters for shafts of steel and wrought- 
 iron. From formula (42), 
 
 </= 2.984^ = 2.984^0^5" = 2.984x0.62996= 1.88" = iff" 
 for steel. From formula (43), 
 
 d= 3422\/H = 3-422 x 0.62996 = 2.1557"= 2-gs" 
 for wrought-iron. 
 
TOOTHED GEARING. 131 
 
 Convenient formulas for gear-shaft diameters in terms 
 of the pitch and radius, may be obtained in the following 
 manner. From formula (12, b) we have, as before, 
 
 0.0025 
 
 = 400/ 2 
 
 which value, substituted in formulas (37), (38), (39), and 
 (40), gives the following formulas : 
 
 General formula, ^=12.673^/^-77- (45) 
 
 For steel, d= o. $$$#* (46) 
 
 For wrought-iron, <t = o.634V / / 2 ^ (47) 
 
 For cast-iron, d 0.796^^ (48). 
 
 Rule. To determine the diameter of a gear shaft of 
 any material, from the pitch and radius of the gear, mul- 
 tiply the square of the pitch by the radius, divide the 
 product by the greatest safe shearing-stress in pounds 
 per square inch for the material of the shaft, extract 
 the cube root of the quotient thus obtained, and mul- 
 tiply the result by 12.673. To determine the diameter 
 of a gear shaft of steel, wrought or cast iron, from the 
 pitch and radius of the gear, multiply the square of 
 the pitch by the radius, extract the cube root of the 
 product, and multiply the result by 0.553 for steel, 0.634 
 for wrought-iron, and 0.796 for cast-iron.* 
 
 * The expression P = 4oo/ 2 is true only for cast-iron gears : hence the 
 value of p z in formulas (45), (46), (47), and (48), must be for a cast-iron 
 gear. 
 
132 TOOTHED GEARING. 
 
 Example 22. A cast-iron gear has a diameter ot 
 12" and a pitch of J' 1 '. Required the diameter for a 
 brass shaft, supposing f = 3,000 pounds for brass. 
 From formula (45) 
 
 = i2.673'</o.ooo5 = 12.673 X 0.07937= i 
 
 Example 23. The diameter of a cast-iron gear is 
 60" and the pitch 2". Required the diameters for 
 shafts of steel and wrought-iron. From formula (46) 
 
 <^= -553'V4 X 30 = o.553Vi20 = 0.553 X 4-93 2 
 
 = 2.727" = 2 jr 
 
 for steel. From formula (47) we have 
 
 </= 0.634^4 X 30 = 0.634 x 4-932 = 3- 7" = 3t" 
 
 for wrought-iron. 
 
 Gear shafts are most commonly of wrought-iron : 
 when, however, wrought-iron shafts, in order to give 
 the necessary strength, become so large as to be in- 
 convenient, steel shafts are used. Cast-iron shafts are, 
 as a rule, unreliable and treacherous ; they are there- 
 fore seldom used, except for the transmission of slight 
 powers, and in cheap, inferior machinery. The follow- 
 ing tables, calculated from formulas (38), (39), (42), and 
 (43), to the nearest ^", will be found very convenient 
 in designing gear shafts of steel and wrought-iron : 
 
TOOTHED GEARING, 
 
 133 
 
 TABLE VI. 
 
 PR 
 
 //for 
 steel. 
 
 //for 
 wrought 
 iron. 
 
 PR 
 
 //for 
 steel. 
 
 //for 
 wrought 
 iron. 
 
 250 
 
 H" 
 
 ft" 
 
 60000 
 
 2H" 
 
 3ir 
 
 500 
 
 ti 
 
 ti 
 
 70000 
 
 3& 
 
 3H 
 
 IOOO 
 
 ! 
 
 ft 
 
 80000 
 
 3H 
 
 3tt 
 
 1500 
 
 if 
 
 H 
 
 90000 
 
 3ft 
 
 3tt 
 
 2000 
 
 IT 
 
 T /T 
 
 IOOOOO 
 
 3H 
 
 4 
 
 2500 
 
 *A 
 
 tt 
 
 1 1 0000 
 
 3li 
 
 4i 
 
 3000 
 
 '& 
 
 '54 
 
 120000 
 
 3 
 
 4H 
 
 3500 
 
 *& 
 
 J ft 
 
 130000 
 
 3ff 
 
 41! 
 
 4000 
 
 lT 3 6 
 
 r l! 
 
 140000 
 
 3il 
 
 4H 
 
 4500 
 
 IB! 
 
 IJ 
 
 1 50000 
 
 3f! 
 
 4ft 
 
 5000 
 
 1^ 
 
 1 1| 
 
 175000 
 
 4ef 
 
 411 
 
 6000 
 
 if! 
 
 i A 
 
 200000 
 
 4|f 
 
 5^ 
 
 7000 
 
 i* 
 
 
 250000 
 
 4M 
 
 sH 
 
 8000 
 
 '1 
 
 iff 
 
 500000 
 
 sH 
 
 ^f! 
 
 IOOOO 
 
 Iff 
 
 iff 
 
 750000 
 
 6 if 
 
 7ii 
 
 12500 
 
 '! 
 
 2 
 
 IOOOOOO 
 
 7! 
 
 8 fl 
 
 15000 
 
 \\\ 
 
 2 i 
 
 1 500000 
 
 ^11 
 
 9J 
 
 20000 
 
 2^2 
 
 2 i 
 
 2000000 
 
 91! 
 
 io|| 
 
 25000 
 
 2 ft 
 
 2ff 
 
 2500000 
 
 
 n|! 
 
 30000 
 
 * 
 
 2 e! 
 
 3000000 
 
 ioif 
 
 I2 M 
 
 35000 
 
 
 2 if 
 
 3500000 
 
 nit 
 
 !3ft 
 
 40000 
 
 2ft 
 
 4f 
 
 4000000 
 
 "H 
 
 r 3ii 
 
 45000 
 
 Hf 
 
 3ft 
 
 4500000 
 
 I2| 
 
 I 4ii 
 
 50000 
 
 Hi 
 
 3H 
 
 5000000 
 
 I2|f 
 
 4H 
 
134 
 
 TOOTHED GEARING. 
 
 TABLE VII. 
 
 H 
 
 n 
 
 dTfor 
 steel. 
 
 dTfor 
 
 wrought 
 iron. 
 
 H 
 
 n 
 
 steel. 
 
 ^for 
 wrought 
 iron. 
 
 0.025 
 
 i" 
 
 i " 
 
 3-75 
 
 411" 
 
 5T 5 e" 
 
 0.050 
 
 '& 
 
 l \\ 
 
 4 
 
 4ef 
 
 sA 
 
 0.075 
 
 'H 
 
 "A 
 
 4.25 
 
 4e4 
 
 5*4 
 
 O.I 00 
 
 1 IT 
 
 T M 
 
 4.50 
 
 4l6 
 
 sft 
 
 0.150 
 
 '*i 
 
 i ^^ 
 
 4-75 
 
 sA 
 
 54 
 
 O.2OO 
 
 if 
 
 2 
 
 5 
 
 sA 
 
 511 
 
 0.250 
 
 \\ 
 
 ~3 
 
 5.50 
 
 sH 
 
 6 A 
 
 0.300 
 
 2 
 
 2 el 
 
 6 
 
 5*4 
 
 6/ 5 
 
 0.350 
 
 2 A 
 
 "3"2 
 
 6.50 
 
 5i 9 6 
 
 6|| 
 
 0.400 
 
 2 i4 
 
 2| 
 
 7 
 
 514 
 
 ^6T 
 
 O.5OO 
 
 2| 
 
 2f| 
 
 8 
 
 5fi 
 
 6fJ 
 
 0.600 
 
 2f| 
 
 2 if 
 
 9 
 
 6^j 
 
 71 
 
 0.700 
 
 2 3'2 
 
 3 A 
 
 10 
 
 52.7 
 
 71 
 
 0.800 
 
 2|| 
 
 3H 
 
 ii 
 
 6|| 
 
 7*4 
 
 O.pOO 
 
 2 6T 
 
 3A 
 
 12 
 
 6^T 
 
 ' 7*4 
 
 I 
 
 2|| 
 
 3il 
 
 14 
 
 7l6" 
 
 ST 
 
 1.25 
 
 3A 
 
 3ii 
 
 16 
 
 7*4 
 
 8| 
 
 1.50 
 
 364 
 
 3 1| 
 
 18 
 
 7M 
 
 8 M 
 
 i-75 
 
 3H 
 
 4i 
 
 20 
 
 8 A 
 
 93% 
 
 2 
 
 311 
 
 4^ 
 
 22 
 
 8 ff 
 
 911 
 
 2.25 
 
 3f| 
 
 4*1 
 
 25 
 
 m 
 
 10 
 
 2.50 
 
 4A 
 
 4H 
 
 27 
 
 8 fi 
 
 IO H 
 
 2.75 
 
 4H 
 
 4eT 
 
 30 
 
 9^T 
 
 i of 
 
 3 
 
 4A 
 
 41! 
 
 32 
 
 9M 
 
 io|| 
 
 3-25 
 
 4|| 
 
 sA 
 
 35 
 
 9*4 
 
 "A 
 
 3-50 
 
 4^ 
 
 5A 
 
 40 
 
 Itt 
 
 *4 
 
TOOTHED GEARING. 135 
 
 Example 24. Required the diameter for a wrought- 
 iron shaft for a 40" gear which transmits a force of 
 10,000 pounds. In this case 
 
 PR = IOOOO X 2O = 2OOOOO 
 
 and, from Table VI., the value of d for wrought-iron 
 corresponding to PR = 200,000 is d = S^"- 
 
 Example 25. The diameter of a wrought-iron gear 
 shaft is 4^". Required the force which the shaft can 
 safely transmit by means of a 24" gear. From Table 
 VI. the value of PR corresponding to d 4^" for 
 wrought-iron is 1 10,000 : hence we will have 
 
 I IOOOO I IOOOO 
 
 p = = = 9167 pounds nearly. 
 
 Example 26. A gear transmitting a force of 2O-horse 
 power makes 200 revolutions per minute. Required the 
 diameter for a shaft of steel. We have 
 
 H 20 I 
 
 = = = o.ioo 
 
 n 200 10 
 
 and, from Table VII., the value of d for steel corre- 
 
 TT 
 
 spending to =o. 100 is d = I |f ". 
 
 Example 27. A 2" steel shaft transmits a force of 
 2 5 -horse power. It is required to determine the proper 
 number of revolutions per minute. From Table VII. 
 
 TT 
 
 the value of which corresponds to d = 2!' for steel is 
 ;/ 
 
 TT 
 
 = 0.300 : hence we have 
 
 
 H 25 
 
 = = 0.300 
 n n 
 
 or 
 
 n = 83^ revolutions per minutt. 
 
136 TOOTHED GEARING. 
 
 Keys : We may take for the mean width of the key 
 which fixes the gear upon its shaft S = o.2&/, and, for 
 the thickness, S' = 0.014^; vS and S' being respectively 
 the mean width and thickness of the key, and d the 
 diameter of the shaft. More accurately, according to 
 Reuleaux, 
 
 S= o.i6 + | (49) 
 and 
 
 S' = o.i6 + ^ (50). 
 
 Rule. To determine the mean width of the fixing- 
 key, divide the diameter of the shaft by 5, and to the 
 result add o. 16". To determine the key thickness, di- 
 vide the diameter of the shaft by 10, and to the result 
 add o.i 6". 
 
 Example 28. Required the mean width and thick- 
 ness for a fixing-key of sufficient strength for the gear 
 and shaft given in Example 24. From formula (49) we 
 have 
 
 S= 0.16 + 5 ' 3125 = 0.16 + 1.00625 = 1.16625" = itt"- 
 From formula (50), 
 S'= 0.16 + 5 ' 3125 = 0.16 + 0.503125 = 0.663125"= |f". 
 
 Weight of Gears : The approximate weight of a 
 spur-wheel may be calculated by the following formula, 
 given by Reuleaux. G represents the approximate 
 weight in pounds, N the number of teeth, and / and / 
 respectively the pitch and face width : 
 
 0.0014^*) (51). 
 
TOOTHED GEARING. 137. 
 
 Rule. To determine the approximate weight of a 
 spur wheel, add 0.215 times the number of teeth to 
 0.0014 times the square of the number of teeth, and 
 multiply the sum by the product of the square of the 
 pitch into the face width. 
 
 Example 29. Required the approximate weight of a 
 spur wheel having 50 teeth, a pitch of 2", and a face 
 width of 4j". From formula (51) we have 
 
 G = 4j x 2 2 (o.2i5 x 50 -f- 0.0014 x so 2 ) 
 
 = 18(10.75 + 3-5) = 18 x 14-25 = 256.50 pounds. 
 
 When the face width is twice the pitch (/==2/), for- 
 mula (51) becomes 
 
 G = 2/3 (0.2 1 57V + O.OOI47V 2 ) 
 or 
 
 G =/3(o.430^+ 0.0028^) (52). 
 
 Rule. To determine the approximate weight of a 
 spur wheel when the face width is equal to twice the 
 pitch, add 0.430 times the number of teeth to 0.0028 
 times the square of the number of teeth, and multiply 
 the sum by the cube of the pitch. 
 
 Example 30. Required the approximate weight of 
 a spur wheel having 50 teeth, a pitch of 2", and a face 
 width of 4". From formula (52) we have 
 
 G = 2 (0.430 X 50 + 0.0028 X 5Q 2 ) = 8 X 28.50= 228 pounds. 
 
 The following table, computed from formula (51), 
 
 /" 
 gives values of for different numbers of teeth : 
 
138 
 
 TOOTHED GEARING. 
 TABLE VIII. 
 
 N 
 
 o 
 
 2 
 
 4 
 
 6 
 
 8 
 
 20 
 
 4.86 
 
 541 
 
 5-97 
 
 6.54 
 
 7.12 
 
 30 
 
 7.71 
 
 8.3 r 
 
 8-93 
 
 9-55 
 
 10.19 
 
 40 
 
 10.84 
 
 11.50 
 
 12.17 
 
 12.85 
 
 13-55 
 
 50 
 
 14.25 
 
 14.97 
 
 15.69 
 
 16.43 
 
 17.18 
 
 60 
 
 17.94 
 
 18.71 
 
 19.49 
 
 20.29 
 
 21.09 
 
 70 
 
 21.91 
 
 22.74 
 
 23.58 
 
 2443 
 
 25.29 
 
 80 
 
 26.16 
 
 27.04 
 
 27.94 
 
 28.84 
 
 29.76 
 
 90 
 
 30.69 
 
 31-63 
 
 32.58 
 
 33-54 
 
 34.52 
 
 100 
 
 35-50 
 
 36.50 
 
 37-50 
 
 38.52 
 
 39-55 
 
 120 
 
 45.96 
 
 47.07 
 
 48.19 
 
 49-32 
 
 50.46 
 
 140 
 
 57-54 
 
 58.76 
 
 59-99 
 
 61.23 
 
 62.49 
 
 160 
 
 70.24 
 
 71-57 
 
 72.91 
 
 74.27 
 
 75-63 
 
 1 80 
 
 84.06 
 
 85.50 
 
 86.96 
 
 88.42 
 
 89.90 
 
 2OO 
 
 99.00 
 
 100.56 
 
 102.12 
 
 103.70 
 
 105.29 
 
 220 
 
 115.06 
 
 116.73 
 
 Il8.4I 
 
 120.10 
 
 121.80 
 
 Example 31. Required the approximate weight of a 
 spur wheel having 126 teeth, the pitch being 3" and the 
 
 face width f. From Table VIII. the value of ~, 
 which corresponds to N= 126, is 49.32 : hence 
 
 G 
 
 7X 3 2 ~ 
 (7=49.32 X 7 X 9 = 3107.16 pounds. 
 
 To determine the approximate weight of a bevel gear, 
 proceed as explained in the above example for a spur 
 wheel, except that the . tabular number must be 
 multiplied by 0.855. 
 
TOOTHED GEARING. 139 
 
 Example 32. Required to determine the approxi- 
 mate weight of a bevel wheel, for which N 48, 
 
 /* 
 p 3", and / = 7". From the table the value of ~ 
 
 corresponding to ^=48 is 13.55. This multiplied by 
 0.855 gives 1 1.585 : hence 
 
 =11.585 
 
 7 X 9 
 
 G 11.585 x 7 X 9 = 729.855 pounds. 
 
 XV. Recapitulation of Formulas and Rules. 
 
 For convenience in designing, the various rules and 
 formulas developed in the foregoing pages have been 
 gathered together in the following recapitulation : 
 
 NOTATION. 
 
 R = radius of the pitch circle. 
 D = diameter of the pitch circle. 
 C = circumference of the pitch circle. 
 TT = constant 3.14159. 
 / = circumferential pitch. 
 pd diametral pitch. 
 N number of teeth. 
 I" = length of chord subtending the pitch. 
 n = number of revolutions per minute. 
 P = total force transmitted. 
 W = total force transmitted. 
 v = circumferential velocity in feet per second. 
 V = circumferential velocity in feet per second. 
 / = greatest safe working-stress in pounds per square inch 
 
 for the material. 
 / = face width. 
 
140 TOOTHED GEARING. 
 
 h = total height of teeth. 
 
 h' height of teeth below pitch circle. 
 
 h" = height of teeth above pitch circle. 
 
 b = breadth of teeth at pitch circle. 
 
 H = horse-power transmitted. 
 
 72 / = number of arms. 
 
 Jit = width of rectangular arms in plane of the pitch circle. 
 
 b l = thickness of rectangular arms. 
 
 x variable co-efficient. 
 
 d' diameter of circular arms. 
 
 //', B, h,,, and b,, = dimensions for Figs. 82 and 83. 
 
 a = major axis for elliptical arms. 
 
 b' = minor axis for elliptical arms. 
 
 / = thickness of rim. 
 
 k = thickness of nave. 
 
 /' = length of nave. 
 
 d diameter of shaft. 
 
 S = mean width of fixing-key. 
 
 S' = thickness of fixing-key. 
 
 G = approximate weight of spur wheel. 
 
 Dimensions are in inches, forces and weights in pounds, 
 unless otherwise stated. 
 
 C=TrD=2irR (l). 
 
 Rule. To find the circumference of the pitch circle, 
 multiply the diameter by 3.14159, or the radius by 
 6.28318. 
 
 Z>=-, R~- (2). 
 
 7T 27T 
 
 Rule. To find the diameter of the pitch circle, di- 
 vide the circumference by 3.14159. To find the radius, 
 divide the circumference by 6.28318. 
 
TOOTHED GEARING. 141 
 
 C C 
 
 W=, C=Nj>, P^T (3). 
 
 Rule. To find the number of teeth, divide the cir- 
 cumference by the pitch. To find the circumference, 
 multiply the number of teeth by the pitch. To find 
 the pitch, divide the circumference by the number of 
 teeth. 
 
 _N_ir_ _ jr_ 
 
 Rule. To find the diametral pitch, divide the num- 
 ber of teeth by the diameter, or divide 3.14159 by the 
 pitch. To find the pitch, divide 3.14159 by the diame- 
 tral pitch. 
 
 Rule. To find the length of the chord which sub- 
 tends the pitch, multiply twice the radius by the natural 
 sine of half the angle limited by the pitch. 
 
 (6). 
 
 Rule. To find the length of the chord which sub- 
 tends the pitch, divide 180 by the number of teeth, 
 take the natural sine of the angle thus obtained, and 
 multiply by the diameter. 
 
 'n'~~N~~R~T>~~~C '"" 
 
 Rule. The ratio of the numbers of revolutions of a 
 pair of gears is inversely proportional to the ratio of 
 their numbers of teeth to the ratio of their radii, 
 diameters, or circumferences. 
 
142 
 
 TOOTHED GEARING, 
 
 =- r (8). 
 
 Rule. The ratio of the powers of two gears on the 
 same shaft is inversely proportional to the ratio of their 
 radii. 
 
 (9). 
 
 W V PV Wv 
 
 - = , W= ) P= -jy 
 P v v V 
 
 R^ile. The ratio of the powers of two gears on the 
 same shaft is inversely proportional to the ratio of their 
 circumferential velocities. 
 
 (10). 
 
 Rule. To find the pitch for a gear of any material, 
 divide the force transmitted by the greatest safe work- 
 ing-stress in pounds per square inch for the material, 
 multiply the quotient by the ratio of the pitch to the 
 face width, extract the square root of the product, and 
 multiply the result by 4.93. 
 
 For cast-iron.* 
 
 Violent shock, / = o.oySy/ P x ~ (a) 
 Moderate shock, / = 0.07 y P X ^ (b) 
 Little or no shock, / = 0.05 y P x ~ (c) 
 
 (ii). 
 
 Rule. To find the pitch for a cast-iron gear, multi- 
 ply the force transmitted by the ratio of the pitch to the 
 face width, extract the square root of the product, and 
 
 * h = o.;/, ti o-4/, h" 0.3^, and b 
 
TOO THED GEA AYA'C. 
 
 143 
 
 multiply the result by 0.078 for violent shock, 0.07 for 
 moderate shock, or 0.05 for little or no shock. 
 
 When / = 2p, 
 
 Violent shock, / = 0.055^ (a) 
 
 Moderate shock, / = 0.05 ^P (b) (12). 
 
 Little or no shock, / = 0.035^ (c) 
 
 Rule. To find the pitch for a cast-iron gear when 
 the face width is twice the pitch, multiply the square 
 root of the force transmitted by 0.055 for violent shock, 
 0.05 for moderate shock, or 0.35 for little or no shock. 
 
 Violent shock, / = 1.83! 
 
 f 
 
 / J-f *h 
 
 Moderate shock, / = i.64y x -, (b} 
 
 Little or no shock 
 
 c,/=i.i7\/f x^ 
 
 7 (<) 
 
 (13) 
 
 Rule. To find the pitch for a cast-iron gear from 
 the horse-power transmitted and circumferential velocity 
 in feet per second, divide the horse-power by the cir- 
 cumferential velocity, multiply the quotient by the ratio 
 of the pitch to the face width, extract the square root of 
 the product, and multiply the result by 1.83 for violent 
 shock, 1.64 for moderate shock, or 1.17 for little or no 
 
 shock. 
 
 When / = 2/, 
 
 I H 
 Violent shock, /=i.29y (a) 
 
 Moderate shock, / = 
 Little or no shock, / = 
 
 f- 
 
144 
 
 TOOTHED GEARING. 
 
 Rule. To find the pitch for a cast-iron gear, from 
 the horse-power and velocity, when the face width is 
 twice the pitch, divide the horse-power by the velocity, 
 extract the square root of the quotient, and multiply 
 the result by 1.29 for violent shock, 1.17 for moderate 
 shock, or 0.82 for little or no shock. 
 
 Violent shock, / = sy.yiy -=- x j (a) 
 
 Moderate shock, p = 24.84^7 - x (V) 
 
 Little or no shock, / = I 7-7 2 y ^ x / ( c ) 
 
 (15) 
 
 Rule. To find the pitch for a cast-iron gear from 
 the horse-power and number of revolutions per minute, 
 divide the horse-power by the product of the diameter 
 into the number of revolutions, multiply the quotient 
 by the ratio of the pitch to the face width, extract the 
 square root of the product, and multiply the result by 
 
 27.71 for violent shock, 24.84 for moderate shock, or 
 
 17.72 for little or no shock. 
 
 When / = 2/, 
 Violent shock, / = 19. 
 
 72\/ 
 
 Moderate shock, /= 17 
 
 Little or no shock, p = 12.42^ 
 
 H_ 
 Dn 
 Jf 
 
 (16). 
 
 Rule. To find the pitch for a cast-iron gear, from 
 the horse-power and number of revolutions per minute, 
 when the face width is twice the pitch, divide the horse- 
 
TOOTHED GEARING. 145 
 
 power by the product of the diameter into the number 
 of revolutions, extract the square root of the quotient, 
 and multiply the result by 19.54 for violent shock, 17.72 
 for moderate shock, or 12.42 for little or no shock. 
 
 /, i ;, I a = -^- 7 (17). 
 5oo;// 
 
 Rule. To find the quantity^//, 2 (the thickness of 
 the arm multiplied by the square of the width) for cast- 
 iron arms, multiply the force transmitted by the radius 
 of the pitch circle, and divide the product by 500 times 
 the number of arms. 
 
 (18). 
 
 Rule. To find the width of the arms in the plane 
 of the pitch circle, multiply the force transmitted by 
 the radius of the pitch circle, extract the cube root of 
 the product, and multiply the result by the tabular 
 number (in Table V.) corresponding to the required 
 
 number of arms and value of -i. 
 
 Rule. To find the diameter for cast-iron arms 
 having circular cross-sections, multiply the force trans- 
 mitted by the radius of the pitch circle, divide the 
 product by the number of arms, extract the cube root 
 of the quotient, and multiply the result by 0.15. 
 
 PR 
 
 b'a? = 0.00339 7 (20). 
 n l 
 
 . To find the quantity of b'a 2 (the minor axis 
 
146 TOOTHED GEARING. 
 
 of elliptical cross-section multiplied by the square of the 
 major axis) for cast-iron arms, multiply the force trans- 
 mitted by the radius of the pitch circle, divide the 
 product by the number of arms, and multiply the result 
 by 0.00339. 
 
 b,,H's + Bh,t PR 
 
 (21) * 
 
 H' ~ 500;;, 
 
 - b,,h,t PR 
 
 soon 
 
 (22) f 
 
 (23). 
 
 Rule. To find the number of arms, extract the 
 fourth root of the pitch and the square root of the num- 
 ber of teeth, multiply the two roots together, and the 
 product by 0.56. 
 
 (24). 
 
 Rule. To find the quantity bji? (see formula 17) 
 for cast-iron arms, from the horse-power and number of 
 revolutions per minute, multiply the horse-power by 
 1 26, and divide by the product of the number of revolu- 
 tions into the number of arms. 
 
 Rule. To find the diameter for cast-iron arms 
 having circular cross-sections, from the horse-power and 
 number of revolutions per minute, divide the horse- 
 power by the product of the number of revolutions into 
 
 * See Fig. 82. t See Fig. 83. 
 
TOOTHED GEARING. 147 
 
 the number of arms, extract the cube root of the quo- 
 tient, and multiply the result by 5.969. 
 
 TT 
 
 t'a* = 2iwj (26). 
 
 Rule. To find the quantity b f a 2 (see formula 20) 
 for cast-iron arms, from the horse-power and number of 
 revolutions per minute, divide the horse-power by the 
 product of the number of revolutions into the number 
 of arms, and multiply the quotient by 213.57. 
 
 H' ' nnT 
 
 -b,,h,t I26H 
 
 
 . 
 ^A 2 = -^r- (29). 
 
 Ride. To find the quantity bji? (see formula 17) 
 for cast-iron arms, from the pitch, multiply o.S times 
 the square of the pitch by the radius of the pitch circle, 
 and divide the product by the number of arms. 
 
 (30). 
 
 Rule. To find the diameter of cast-iron arms having 
 circular cross-sections, from the pitch, multiply the 
 square of the pitch by the radius of the pitch circle, 
 divide the product by the number of arms, extract the 
 cube root of the quotient, and multiply the result by 
 1.105. 
 
 * See Fig. 82. t See Fig. 83. 
 
148 TOOTHED GEARING, 
 
 b'a* = 1.356^ (3')- 
 
 "I 
 
 Rule. To find the quantity b'a 2 (see formula 20) 
 from the pitch, multiply the square of the pitch by the 
 radius of the pitch circle, divide the product by the num- 
 ber of arms, and multiply the result by 1.356. 
 
 (3*) 
 
 (33) t 
 
 H' 
 
 H' 
 
 / = o.i2+o.4/ (34). 
 
 Rule. To find the thickness of the rim, add 0.12" to 
 0.4 times the pitch. 
 
 * = o. 4 V/^ + t (35). 
 
 Rule. To find the thickness of the nave, multiply 
 the square of the pitch by the radius of the pitch circle, 
 extract the cube root of the product, multiply the root 
 by 0.4, and to the result add J". 
 
 Rule. To find the length of the nave, divide the 
 diameter of the pitch circle by 30, and to the result add 
 the face width of the teeth. 
 
 (37). 
 Rule. To find the diameter of a gear shaft of any 
 
 * See Fig. 82. t See Fig. 83. 
 
TOOTHED GEARING. 149 
 
 material, multiply the force transmitted by the radius 
 of the pitch circle, divide the product by the greatest 
 safe shearing-stress in pounds per square inch for the 
 material, extract the cube root of the quotient, and mul- 
 tiply the result by 1.720. 
 
 For steel, d = 0.0751^? (38) 
 
 For wrought-iron, d 0.086'^^ (39) 
 
 For cast-iron, d o.io8 3 V ' PR (40). 
 Ride. To find the diameter of a gear shaft, multiply 
 the force transmitted by the radius of the pitch circle, 
 extract the cube root of the product, and multiply the 
 result by 0.075 f r steel, 0.086 for wrought-iron, and 
 0.108 for cast-iron. 
 
 = 68.44^1 
 
 , x 
 (40. 
 
 Rule. To find the diameter of a gear shaft of any 
 material from the horse-power and number of revolu- 
 tions, divide the horse-power by the product of the 
 number of revolutions into the greatest safe shearing- 
 stress in pounds per square inch for the material, ex- 
 tract the cube root of the quotient, and multiply the 
 result by 68.44. 
 
 fff 
 
 For steel, ^=2.984^ (42) 
 
 For wrought-iron, d = 3.422^- (43) 
 
 3/5" 
 For cast-iron, ^=4.297^ (44) 
 
150 TOOTHED GEARING. 
 
 Rule. To find the diameter of a gear shaft from the 
 horse-power and number of revolutions, divide the horse- 
 power by the number of revolutions, extract the cube 
 root of the quotient, and multiply the result by 2.984 
 for steel, 3.422 for wrought-iron, and 4.297 for cast-iron. 
 
 (45). 
 
 Ride. To find the diameter of a gear shaft of any 
 material from the pitch, multiply the square of the pitch 
 by the radius of the pitch circle, divide the product by 
 the greatest safe shearing-stress in pounds per square 
 inch for the material used, extract the cube root of the 
 quotient, and multiply the result by 12.673. 
 
 For steel, d = o.P 2R (46) 
 
 For wrought-iron, d=o.6i ) $p 2 R (47) 
 
 For cast-iron, dQ.^lp^R (48) 
 
 Rule. To find the diameter of a gear shaft from the 
 pitch, multiply the square of the pitch by the radius of 
 the pitch circle, extract the cube root of the product, 
 and multiply the result by 0.553 for steel, 0.634 for 
 wrought-iron, and 0.796 for cast-iron. 
 
 (49) 
 
 S'=o.i6 + (50). 
 
 Rule. To find the mean width of a fixing-key, 
 divide the diameter of the shaft by 5, and to the result 
 
TOOTHED GEARING. 151 
 
 add o. 16". To find the thickness of the key, divide the 
 diameter of the shaft by ro, and to the result add 0.16". 
 
 O.OOI4/V 72 ) (si). 
 
 Rule. To find the approximate weight of a spin- 
 wheel, add 0.215 times the number of teeth to 0.0014, 
 the square of the number of teeth, and multiply the 
 sum by the product of the face width into the square 
 of the pitch. 
 
 When / = 2p, 
 G=p* (0.4307V -f 0.00287V 2 ) (5 2) . 
 
 Rule. To find the approximate weight of a spur 
 wheel when the face width is twice the pitch, add 0.430 
 times the number of teeth to 0.0028 times the square 
 of the number of teeth, and multiply the sum by the 
 cube of the pitch. 
 
 XVI. Complete Design of Spur-Wheel, Bevels, Worm, Screw 
 Gear, etc. 
 
 Example I. Required to design and make full work- 
 ing drawings for a 36" cast-iron spur wheel to transmit 
 a force of 5,000 pounds, violent shock. 
 
 For the pitch we have, from formula (12, a), 
 
 p = o.o55\/5ooo = 0.055 X 70-71 = 3-889" 
 for the face width, 
 
 /= 2 p= 2 x 3.889= 7.778". 
 As explained in XIII. , we have for the total height 
 
152 TOOTHED GEARING. 
 
 of the teeth, and heights below and above the pitch 
 circle, 
 
 h = h' + h" = o.4/ -f o.3/ = 0.7 x 3.889 = 2.7223" 
 ^'=0.4 x 3-889= 1.5556" 
 ^"=0.3 x 3.889= 1.1667". 
 
 We may take, for the breadth of the teeth on the pitch 
 circle, b = o.^p= i. 75". From formulas (i) and (3), 
 for the circumference and number of teeth, 
 
 (7=3.14159 x 36 = 113.10 
 and 
 
 113.10 _ 
 
 'IW 
 From formula (23), the number of arms is 
 
 /= 0.56^29 ^3.889 = 0.56 x 5.385 x 1.40 = 4. 
 
 If we wish to have elliptical cross-sections for the arms, 
 we have, from formula (20), 
 
 ,, qooo x 1 8 
 
 b a* = 0.00339 x = 0.00339 X 22500 = 76.275. 
 
 4 
 
 Taking 
 
 *>'="-> ^ 2 = ^= 76.275; 
 
 or, for the major axis of the cross-section, 
 
 a = ^152.55 = 5.343" 
 and, for the minor axis, 
 
 '=^=2.6715". 
 
TOOTHED GEARING. 153 
 
 For the thickness of the rim, from formula (34), 
 /= 0.12 -f 0.4 x 3.889 = 0.12 4- 1.5556 = 1.6756." 
 Formula (35) gives, for the thickness of the nave, 
 
 k = 0.4V3-889 2 x 18 4- \ = 0.4 x 6.481 -f \ = 3.092". 
 The length of the nave is, from formula (36), 
 
 /' = 7.778 -f f- = 7.778 4- 1.2 = 8.978". 
 
 Formula (39) gives, for the diameter of the wrought-iron 
 shaft, 
 
 d 0.086^5000 x 18 = 0.086 x 44.814 = 3.854". 
 
 For the mean width and thickness of the fixing-key we 
 have, from formulas (49) and (50), 
 
 s = 0.16 -f l- = O .i6 4- 0.7708 = 0.9308" 
 and 
 
 s t = 0.16 4- ^-^ = 0.16 -|- 0.3854 = 0.5454". 
 
 We may now recapitulate our dimensions, and by means 
 of Table IV. convert the decimals into convenient 
 fractions : 
 
 Diameter, D = 36" 
 
 Pitch, / = 3^ 
 
 Face width, / = yf f" 
 
 Total tooth height, h = 2ff " 
 
 Height below pitch circle, ti = iff" 
 
 Height above pitch circle, h" iJ" 
 
 Breadth of tooth on pitch circle, b = i f" 
 
154 TOOTHED GEARING. 
 
 Number of teeth, N = 29 
 
 Number of arms, n/ = 4 
 
 Axes of arm cross-sections, < 7 , ~ ff,, 
 
 ( ^ 2 F4 
 
 Thickness of rim, / = iff'' 
 
 Nave length, I' = 8JJ" 
 
 Nave thickness, k = 3-^" 
 
 Diameter of shaft, d = 3fJ" 
 
 Key width, s \%"' 
 
 Key thickness, s, = f }". 
 
 Fig. 89 shows the working drawings for the above 
 spur wheel. Fig. (/?) is a simple horizontal projection of 
 the gear, showing the pitch, tooth dimensions, thick- 
 ness of rim and nave, dimensions of arms, number of 
 teeth, arms, etc. Fig. (c) is a vertical projection taken 
 from Fig. (b), as shown by the dotted lines, and Fig. (a) 
 is a sectional, vertical projection taken from Fig. (b) on 
 the line AB, and showing the face width, nave length, 
 etc. The profiles were drawn by the method of IV., 
 Fig. 26. 
 
 Example 2. Required to design and make full work- 
 ing drawings for a pair of cast-iron bevel wheels to 
 transmit a force of lO-horse power from a smoothly 
 running turbine wheel (moderate shock), the smaller 
 bevel to be fixed upon the 3" shaft of the turbine wheel, 
 which makes 30 revolutions per minute, the bevel 
 wheels to be 15" and 30" diameters. The circumfer- 
 ential velocity of the smaller bevel (as also that of the 
 larger) is 
 
 ^o x TT x is ^o x 47.124 
 v = - = = 2 feet per second nearly. 
 
 12 X 60 12 X 60 
 
TOOTHED GEARING. 
 ttg.89 
 
 155 
 
 fe) 
 
 * The scale of all working drawings should be , |, i, -rV, TjV, etc. The 
 scale of - 4 3 U - is taken here in order to bring the drawings of convenient size. 
 
1 56 TOOTHED GEARING. 
 
 For the smaller bevel, from formula (14, b), we have, 
 therefore, for the pitch, 
 
 /= i..i7y = 1.17 x 2.236 = 2.616". 
 
 For the face width, 
 
 /= 2 x 2.616 = 5.232". 
 
 For the total height of the teeth, 
 
 h = 0.7 x 2.616 = 1.8312". 
 For the heights below and above the pitch circle, 
 
 h' ' = 0.4 X 2.616 = 1.0464" 
 and 
 
 #'=0.3 x 2.616 = 0.7848". 
 
 Taking, for the breadth of the teeth at the pitch circle, 
 b = 0.48^, we have 
 
 b = 0.48 x 2.616 = 1.25568". 
 
 The bevel, being so small, may be made without rim 
 or arms, i.e., cast solid, as shown in the drawing (Fig. 
 91, a). From formula (3) the number of teeth is 
 
 2.616 
 For the thickness of the nave, from formula (35), 
 
 k = 0.4V2.62 2 x 7i 4- \ = 2". 
 
 From formula (36), for the length of the nave, we have 
 /' = 5.232 -Hi = 5.732"- 
 
TOOTHED GEARING. 
 
 157 
 
 The diameter of the shaft is that of the turbine, or 
 d = 3". From formulas (49) and (50) the mean width 
 of the key which fixes the bevel to its shaft is 
 
 0.16 + 1 = 0.76" 
 
 and the thickness, 
 
 = o.i 6 
 
 = 0.46". 
 
 For the larger bevel the pitch and tooth dimensions 
 are the same as for the smaller bevel. From formula 
 (3) the number of teeth is 
 
 TTX 30 _ 94.25 _ 
 " 2.616 2.616 ~ 3 ' 
 
 From formula (34) the thickness of the rim is 
 
 *= 0.12 -f- 0.4 x 2.616 =s 1.1664". 
 Formula (23) gives for the number of arms, 
 
 n s '= 0.56^36^2.616 = 4. 
 For the number of revolutions per minute, we have, 
 
 Fig.90 
 
 from formula (7), n = 15. For 
 the flanged cross-sections of the 
 arms, such as that represented 
 
 in Fig. 90, taking b,, equal to the 
 rim-thickness = 1.1664", //// = i", 
 and B = //"', we have, from for- 
 mula (27), 
 
 1.1664 X ff f * + H* X i _ 126 X io 
 H 1 15X4 
 
158 TOOTHED GEARING. 
 
 or 
 
 1.1664^3 + 1 = 21. 
 
 Hence 
 
 and 
 
 B = H f =4.141". 
 
 For the thickness of the nave, from formula (35) we 
 have _ 
 
 k = 0.4V2.62 2 X 15 + i = 2.36". 
 
 Formula (36) gives, for the length of the. nave, 
 /'= 5-232+18 = 6.232". 
 
 For the diameter of the wrought-iron shaft we have, 
 from formula (43), 
 
 ,/= 3.422^4 = 3 ". 
 
 Formulas (49) and (50) give, for the mean width and 
 thickness of the fixing-key, 
 
 j = o.i6 + =0.76" 
 and 
 
 / = o.i 6 + T 3 o = 0.46". 
 
 Our dimensions in fractions instead of decimals are as 
 follows : 
 
 For smaller bevel. 
 
 Diameter, D = 15" 
 
 Pitch, / = 2ft" 
 
 Face width, / = sjf" 
 
 Total height of teeth, h = iff" 
 Height below pitch circle, h' = i&" 
 Height above pitch circle, h" f }" 
 
TOO THED GEA RING. 
 
 159 
 
 Breadth on pitch circle, 
 Number of teeth, 
 Thickness of nave, 
 Length of nave, 
 Diameter of shaft, 
 Key width, 
 Key thickness, 
 
 For smaller bevel. 
 
 b = itf" 
 
 N = 18 
 k = 2" 
 
 i' = sir 
 
 d = 3" 
 
 s = ' 
 
 For larger bevel. 
 
 D = 30- 
 
 A = i 
 
 Diameter, 
 
 Pitch, 
 
 Face width, 
 
 Total height of teeth, 
 
 Height below pitch circle, 
 
 Height above pitch circle, 
 
 Breadth of teeth at pitch circle, b = i JJ" 
 
 Number of teeth, N = 36 
 
 Rim thickness, t = iJJ" 
 
 Number of arms, /= 4 
 
 (ff= 4 A" 
 
 Arm dimensions. See Fig. 90. 
 
 B = 4&" 
 
 *- t" 
 
 /<= i" 
 
 Thickness of nave, 
 Length of nave, 
 Diameter of shaft, 
 Key width, 
 Key thickness, 
 
 /' = 6H" 
 
 </ = 3" 
 
 * = r 
 
 / = ". 
 
 Fig. 91 gives the working drawings, drawn to a scale 
 of -f-Q. Fig. (<?) is. a sectional drawing of both bevels 
 in gear, showing teeth, rim, nave thickness, etc., and at 
 x the true form of the profiles and true tooth dimen- 
 sions. Fig. (b) is a partial projection of the smaller 
 
i6o 
 
 TOOTHED GEARING. 
 
 bevel; and Fig. (r), a projection of the larger bevel, 
 showing the arms, fixing-key, etc. 
 
 Example 3. Required to design, and make complete 
 working drawings for, a worm and wheel to transmit a 
 
TOOTHED GEARING. l6l 
 
 force of 850 pounds, little or no shock, the wheel to be 
 12" in diameter. From formula (12, c) we have, for the 
 pitch, 
 
 p = 0.035^850 = 0.035 x 2 9- I 5 = 1.02", 
 
 for the heights of the teeth, 
 
 h 0.7 x 1.02 = 0.714" 
 
 h' = 0.4 x 1.02 = 0.408'' 
 and 
 
 h" = 0.3 x 1.02 = 0.306". 
 
 The breadth of the teeth at the pitch circle is 
 
 b 0.48 x 1.02 = 0.4896". 
 For the number of teeth in the wheel, from formula (3), 
 
 JV=^ = 37 . 
 
 1.02 
 
 Face width of wheel, 
 
 /= 2 x 1.02 = 2.04". 
 
 From formula (23) the number of arms is 
 
 n' = 0.56^37 v/i.02 = 0.56 x 6.08 x 1.005 = 4' 
 For the thickness and width of the arms* we have, 
 from formula (17), taking l\ =. , 
 
 h? _ 850 X 6 _ 1275 
 
 2 ~~ 500 x 4 ~~~ 500 
 
 * Ordinarily so small a gear would be made without arms. For 
 the purpose of illustrating, however, we use four arms, as given by the 
 formula. 
 
 "of \ 
 
1 62 TOOTHED GEARING. 
 
 or 
 
 *,_VsS-i.7" 
 
 = -^ = 0.86". 
 
 2 
 
 Formula (34) gives, for the thickness of the rim, 
 
 /= 0.12 4- 0.408 = 0.528". 
 From formula (35) the thickness of the nave is 
 
 k = 0.4^1. 02 2 x 6 + i = 0.4 X 1.841 -f |- = 1.236". 
 
 The length of the nave is, from formula (36), 
 /'= 2.04 -f- J-g- = 2.44". 
 
 Formula (39) gives, for the diameter of the wrought-iron 
 shaft of the wheel, 
 
 d o.o86'V85o x 6 = 0.086 X 17.29 = 1.48". 
 
 From formulas (49) and (50) the width of the fixing- 
 key is 
 
 , = 0.16 + ^? = 0.456" 
 
 and the thickness 
 
 / = 0.16+^ = 0.308". 
 
 From VIII., taking the radius of the worm equal to 
 I \ times the pitch, we have 
 
 /r=iix 1.02 = 1.53" 
 and, for the angle (X) of the teeth, 
 
 1.02 
 
 tan A = 0.159 = 0.159 X 0.6667 = 0.106 
 
TOOTHED GEARING. 163 
 
 or X = 6 3 . From formula (39) the shaft diameter for 
 the worm is 
 
 d 0.086^850 x 1.53 = 0.939". 
 
 Dimensions. 
 
 Diameter of wheel, D 12" 
 
 Pitch, 
 
 Total height of teeth, 
 
 Height below pitch circle, ti -|f 
 Height above pitch circle, 
 
 Breadth of teeth on pitch circle, b = - 
 
 Face width, / = 2 -fa 
 
 Number of teeth on wheel, N = 37 
 
 Number of arms on wheel, / = 4 
 
 Thickness of rim, / = -J J 
 
 Width of arms, h, = iff 
 
 Thickness of arms, b t = f f 
 
 Thickness of nave, k = iJJ 
 
 Length of nave, /' = 2 T 
 
 Diameter of shaft, d = if 
 
 Width of key, s = |J 
 
 Thickness of key, / = J{ 
 
 Radius of worm, R' = i if 
 
 Angle of the teeth, A = 6 3 
 
 Shaft diameter for worm, d = -JJ 
 
 // 
 
 The working drawings, with dimensions, are given in 
 Fig. 92, of which Fig. (b) x is a full projection, showing 
 the arms, rim, nave thickness, tooth dimensions in sec- 
 tion, angle (A.) 'of inclination of the teeth, etc. Fig. (c) is 
 a sectional projection of Fig. (b), showing the shape of 
 the wheel teeth, arm thickness, nave length, etc. ; and 
 Fig. (a) is a full projection taken from Fig. (b). 
 
1 64 
 
 TOOTHED GEARING. 
 
 Example 4. Required to design, and make full work- 
 ing drawings for, a pair of screw gears to transmit a 
 force of 2-horse power, little or no shock ; the larger 
 
 gear to be fixed upon a i J" wrought-iron shaft, which 
 makes 20 revolutions per minute, and the smaller gear 
 to make 40 revolutions per minute. The angle included 
 between the two gear shafts to be 60. 
 
TOOTHED GEARING. 165 
 
 Suppose we take, for the diameter of the smaller 
 gear, 6": hence, from formula (7), the diameter of the 
 larger gear is 1 2". 
 
 The circumferential velocity is 
 
 irDn 37.7 x 20 
 
 " = 17*15 = 12x60 == '- 47 feet P er second - 
 
 From VIII. we have for the angles of inclination 
 (< and <') of the teeth, the angle () included between 
 the axes of the shafts being 60, 
 
 < + $ + = 180, </> + <' = 180 - 60 = 120. 
 If we assume <f> = 60, we have, 
 
 ^= 1 20 -60 =60.* 
 
 For the larger wheel the dimensions are calculated as 
 follows : The pitch, from formula (14, c), is 
 
 / = o.82\/ ^ = 0.82^91 = 0.82 x 1.382 = 1.133". 
 ? 1.047 
 
 The face width is 
 
 /= 2 x 1.133 = 2.266". 
 The heights of the teeth are 
 
 h = 0.7 x 1.133 = 0.793" 
 
 ^'= 0.4 x 1.133 = -453 2 " 
 and 
 
 A" =0.3 X 1.133 = 0.3399". 
 
 * We can assume 90, in which case the gear upon which the 
 inclination of the teeth is <j> = 90 is a spur wheel, and then have 
 f = 120 90 = 30 for the inclination of the teeth of the other gear. 
 
1 66 TOOTHED GEARING. 
 
 For the breadth of the teeth at the pitch circle we may 
 take 
 
 b = o.48/ = 0.48 x 1.133 = 0.54384"- 
 
 From formula (3) we have, for the number of teeth, 
 
 Ar_ 37-7 _ 
 
 ~' 
 
 Formula (34) gives, for the rim thickness, 
 
 /= 0.12 + (0.4 X I.I33) = 0.573 2 ". 
 
 From formula (35), the thickness of the nave is 
 
 k = 0.4VI.I33 2 X 6 + i = 0.4V7.70 + \ 
 
 = 0.4 x i. 9 7-f|= 1.29". 
 
 Formula (36) gives, for the nave length, 
 /' = 2.266 + j#= 2.666". 
 
 The fixing-key width and thickness are, from formulas 
 (49) and (50), 
 
 s = 0.16 -f- - = 0.41" 
 and 
 
 /=o.i6 + = 0.285". 
 
 The gear is small enough to be made without arms. 
 The thickness of the web between the nave and rim 
 may be calculated from formula (24), by assuming the 
 gear to have 10 arms, the width of each being one- 
 tenth the outer circumference of the nave. Thus the 
 
TOOTHED GEARING. 1 67 
 
 shaft diameter is 1.2$", and the nave thickness 1.29": 
 hence the diameter across the nave is 
 
 1.25 -f (2 x 1.29) =3.83" 
 
 and the circumference 12.052". The width of the 
 assumed arms is therefore - ' , or//, = 1.203". For- 
 mula (24) becomes 
 
 b, x i.2O3 2 
 or 
 
 *, = -^ = 0.87". 
 1.447 
 
 For the smaller gear the pitch and tooth dimensions 
 are the same as for the larger gear, as is also the rim 
 thickness. The thickness of the nave is, from formula 
 (35), 
 
 k = o. 4 Vi.i33 2 X 3 + J = o. 4 V 3 .8 5 i + J 
 
 = 0.4 X 1.567 + I = I.I268". 
 
 From formula (36) we have, for the length of the nave, 
 
 /'= 2.2664- & = 2.466". 
 From formula (3), for the number of teeth, we have 
 
 18.85 
 
 N= 5.= 17. 
 
 i-i33 
 
 The diameter of the wrought-iron shaft is, from for- 
 mula (43), 
 
 <t= 3.42 2^ = 3.422^.050 = 3.422 x 0.368 = 1.2593". 
 
and 
 
 1 68 TOOTHED GEARING. 
 
 Formulas (49) and (50) give, for the mean width and 
 thickness of the fixing-key, 
 
 = 0.41186 
 
 /= o.i 6 + = 0.28593". 
 
 For the thickness of the web between the rim and 
 nave, the diameter across the nave is 
 
 i. 2593 + (2 x 1.1268) = 3.51" 
 
 and the circumference 11.03": hence //, i.io". And 
 formula (24) gives 
 
 126 x 2 
 
 40 x 10 
 or 
 
 b l X i.io 2 = 
 b 
 
 
 
 Dimensions for larger gear. 
 
 Diameter, d =12" 
 
 Pitch, / = i&" 
 
 Face width, / = 2}%' 
 
 Total height of teeth, h = -|J" 
 Height below pitch circle, h' = ff " 
 Height above pitch circle, h" %% f 
 Breadth at pitch circle, b = ff " 
 Rim thickness, / = ff" 
 
 Number of teeth, N = 33 
 
 Thickness of web, b l = |" 
 
 Thickness of nave, k = i^/' 
 
 Length of nave, /' = 2|f " 
 
TOOTHED GEARING. 169 
 
 Dimensions for larger gear. 
 
 Diameter of shaft, d = ij" 
 
 Width of fixing-key, s -J-f " 
 
 Thickness of fixing-key, / = %" 
 
 Angle of the teeth, < = 60. 
 
 Dimensions for smaller gear. 
 
 Diameter, D = 6" 
 
 Pitch, / = i^" 
 
 Face width, / = 2^-J" 
 
 Total height of teeth, h = f" 
 
 Height below pitch circle, h' ff" 
 
 Height above pitch circle, //' = J" 
 
 Breadth of teeth, b = f f " 
 
 Rim thickness, / = JJ" 
 
 Number of teeth, N = 1 7 
 
 Thickness of web, b l = f" 
 
 Thickness of nave, k i" 
 
 Length of nave, /' = 2 Jf " 
 
 Diameter of shaft, // = i J" 
 
 Width of fixing-key, s = f " 
 
 Thickness of fixing-key, / = ^" 
 
 Angle of the teeth, <' = 60. 
 
 The working drawings with marked dimensions are 
 given in Fig. 92 A. Fig. (a) is a full projection of the 
 larger gear, showing the pitch, tooth dimensions, rim, 
 etc. The right half of Fig. (b) is a full projection of 
 the larger gear, taken from Fig. (a) ; and the left half 
 is a sectional projection taken from Fig. (a), showing 
 the web thickness, etc. Similarly, for the smaller gear, 
 Fig. (c) is a full projection, and Fig. (d) a full and sec- 
 tional projection taken from Fig. (c). 
 
 Example 5. Required to design, and make complete 
 working drawings for, a cast-iron internal spur gear and 
 
I/O 
 
 TOOTHED GEARING. 
 
 pinion which will safely transmit a force of 6,197 pounds 
 
 Fig. 92 A 
 
 SCALE = 
 
 moderate shock, the pinion to be fixed upon a 
 wrought-iron shaft, the face width to be 2j times the 
 
TOOTHED GEARING. I/I 
 
 pitch, and the revolution ratio 3 to i. From formula 
 (n, b) we have, for the pitch, 
 
 / = o.o7y 6197 x ~ = 0.07^2478.80 = 0.07x49.8 = 3.486". 
 
 For the face width, 
 
 /=2^ x 3.486 = 8.715". 
 The heights are, 
 
 h 0.7 x 3.486 = 2.44" 
 
 //=o.4 x 3486 = 1.394" 
 and 
 
 #' = 0.3 x 3.486 = 1.046". 
 
 Taking the breadth of the teeth equal to 0.45^, we have 
 = 0.45 x 3486= 1.569". 
 
 If we take for the diameter of the pinion 25^", we shall 
 have for the number of teeth, from formula (3), 
 
 80.10 
 
 From formula (7) the diameter of the internal gear is 
 
 2 5i X 3 = 7 6 i"> 
 and from formula (3) the number of teeth is 
 
 = 69. 
 
1 72 TOOTHED GEARING. 
 
 Formula (34) gives, for the rim thickness of the pinion, 
 /= 0.12 + (0.4 x 3.486) = 1.51". 
 
 Since in an internal gear the rim is not supported by 
 the arms, as in an external gear (see Fig. 93, #), we may 
 take the rim thickness for the internal gear equal to 
 2t = 3". From formula (35), the thickness of the nave 
 for the pinion is 
 
 k = o. 4 3 v/3.486 2 x 12.75 + \ = 0.4^/12.15 X 12.75 4- } 
 
 = 0.4 x 5.371+^=2.6484", 
 and, for the internal gear, 
 
 k = o.4'V 3 .486 2 x 38.25 + \ = 0.4^12.15 x 38.25 + 1 
 
 = 0.4 x 7.746+1= 3.598". 
 
 Formula (36) gives for the nave lengths of the pinion 
 and internal gear respectively, 
 
 /' = 8.715 + 2 -~ = 8.715 + 0.85 = 9.565" 
 and 
 
 /'= 8.715 + ^ = 8.715 + 2.55 = 11.265". 
 
 The pinion may be without arms, and the thickness of 
 the web calculated from formula (29) by assuming the 
 pinion to have 10 arms, each having a width of one- 
 tenth the outer circumference of the nave. Thus the 
 diameter of the shaft is 3.6875", and the nave thickness 
 2.6484": hence the diameter across the nave is 
 
 3.6875 + (2 x 2.6484) = 9", 
 
TOOTHED GEARING. 
 
 and the circumference 28.27". We therefore have 
 //, = 2.827"; anc l formula (29) gives 
 
 0.80 x 3.486* x 12.715 
 ^X2.82 7 2 =- ^_ ^=12.393, 
 
 or 
 
 12,393 
 
 7-99 
 
 For the number of arms for the internal gear, formula 
 (23) gives 
 
 n! = 0.56^ >/3.486 = 0.56 x 8.307 x 1.366 = 6.35, 
 
 say ;// 7. If we wish to have elliptical arm cross- 
 sections, we have from formula (31), taking the minor 
 axis equal to one-half the major, 
 
 03 3.486* x 38.25 
 
 *, = - = 1.356 6 - ^ * = 90.03. 
 
 Hence _ 
 
 a = V9-3 x 2 = 5.647" 
 and 
 
 * I = S^47 = 2i8 . 
 
 From formula (39), the diameter of the wrought-iron 
 shaft for the internal gear is 
 
 </= o.o86V6i97 X 38.25 = 0.086 X 61.888 = 5.322". 
 
 Formulas (49) and (50) give, for the mean width and 
 thickness of the fixing-key for the pinion, 
 
 s = 0.16 + = 0.16 + 0.7375 = 0-8975" 
 
 o.i6 + ' Q ' J = 0.16 + 0.36875 = 0.52875' 
 
1/4 TOOTHED GEARING. 
 
 and the same for the internal gear, 
 
 * = o.i6 + ^^= 1 .22 4 4" 
 
 and 
 
 s' = 0.16 + - = 0.6922". 
 
 Dimensions for pinion. 
 
 Diameter, D = 25!" 
 
 Pitch, / = 3 fA" 
 
 Face width, / = 8f|" 
 
 Total height of teeth, h = 2 T y 
 
 Height below pitch circle, ti = if f" 
 
 Height above pitch circle, h" i^" 
 
 Number of teeth, N = 23 
 
 Breadth of teeth, b = i T y 
 
 Thickness of rim, / = iff" 
 
 Thickness of nave, k = 2%%" 
 
 Length of nave, /' = 9 T y 
 
 Thickness of web, b, = iff" 
 
 Diameter of shaft, d = 3^" 
 
 Mean width of fixing-key, s = f " 
 
 Thickness of fixing-key, / = | " 
 
 Dimensions for internal gear. 
 
 Diameter, D = 76^" 
 
 Pitch, / = 3U" 
 
 Number of teeth, N = 69 
 
 Total height of teeth, h 2 T V' 
 Height below pitch circle, h' = iff'' 
 Height above pitch circle, h" = i$$" 
 Face width, / = 8ff" 
 
 Breadth of teeth, b = i^" 
 
 Rim thickness, 2/ = 3" 
 
 Nave thickness, k = 
 
TOOTHED GEARING. 175 
 
 Dimensions for internal gear 
 
 Nave length, /' = nJ" 
 
 Number of arms, 
 
 Major axis of arm cross-sections, a 
 
 Minor axis of arm cross-sections, // = 2ff " 
 
 Diameter of shaft, d 5 J" 
 
 Width of fixing-key, s = ifa" 
 
 Thickness of fixing-key, / = JJ" 
 
 Fig. 93 shows the working drawings for the pair of 
 gears, to the scale of -^ ; Fig. (a) being a full projection 
 of both gears in position for action, showing the pitch, 
 tooth dimensions, number of arms, etc., and Fig. (b) 
 being a sectional projection of both gears, taken from 
 Fig. (a), on the line xy, showing the shape of the arms 
 of the larger gear necessary to the proper action of the 
 pair, etc. 
 
 Example 6. Required to design, and make full 
 working drawings for, a cast-iron rack and pinion to 
 transmit a force of 1,000 pounds, moderate shock ; the 
 pinion to make 20 revolutions per minute, and the rack 
 (which is to be 9 feet long) to move at the rate of 63 \ 
 feet (762") per minute. From formula (12, b), for the 
 pitch we have 
 
 p = 0.05^1000 =s 0.05 x 31.62 == 1.581". 
 The face width is, consequently, 
 
 /= 2 x 1.581 = 3.162". 
 For the heights of the teeth, 
 
 h 0.7 x 1.581 = 1.1067" 
 
 W '= 0.4 x 1.581 = 0.6324" 
 and 
 
 #' = 0.3 x 1.581 = 0.4743". 
 
TOOTHED GEARING. 
 
 Taking the breadth of the teeth equal to 0.48 times the 
 pitch gives 
 
 b = 0.48 x 1.581 = 0.7589". 
 
 The circumferential velocity of the pinion (which is equal 
 to the velocity of the rack) is 762" per minute : hence 
 the circumference of the pinion must be J 2 ^=38.i". 
 
TOOTHED GEARING. If? 
 
 From formula (3), the number of teeth is 
 
 and from formula (2), the diameter is 
 
 3-I4I59 
 
 Formula (34) gives, for the thickness of the rim, 
 
 /= 0.12 -f (0.4 x 1.581) = 0.7524", 
 and, from formula (35), the nave thickness is 
 
 k = o.4Vi.58i 2 x 6.0625 + -J = o.4Vi5.i56 + 
 
 = 0.4 x 2.475 + i = M9 
 
 The nave length is, from formula (36), 
 
 e =-162 = .66". 
 
 From formula (39) we have, for the diameter of the 
 wrought-iron pinion shaft, 
 
 d= 0.086 AOOO x 6.0625 = o.o86'V6o62.5 
 
 = 0.086 X 18.234 = 1.5 f. 
 
 Formulas (49) and (50) give, for the width and thickness 
 of the pinion fixing-key, 
 
 * = 0.16 + ^ = 0.474" 
 and 
 
 / 0.16 + 
 
178 TOOTHED GEARING. 
 
 The pinion is small enough to be made without arms. 
 For the thickness of the web we have the following. 
 The diameter across the nave is 
 
 1.5 7 + (2 x 1.49) =4-55"> 
 and the circumference is 
 
 4.55 x 3.14159=-= 14.294". 
 Hence //, = 1.43", and formula (17) becomes 
 
 1000 x 6.0621% 
 
 bth? = 2.045^, = - 
 
 500 x 10 
 
 1.2 I 2 S 
 
 b l = 0.599". 
 
 2.045 
 
 The dimensions, converted into fractions, are as fol- 
 lows : 
 
 Diameter of pinion, D = i2|" 
 
 Length of rack, 9' 
 
 Pitch, / = iff" 
 
 Number of teeth, N = 24 
 
 Face width, / = 3&" 
 
 Total height of teeth, h = !&" 
 Height below pitch circle, h' f" 
 Height above pitch circle, h" f " 
 Breadth of teeth, b = ft" 
 
 Thickness of rim, / = f " 
 
 Thickness of nave, k i" 
 
 Length of nave, /' = 
 
 Diameter of shaft, d =; 
 
 Width of key, s 
 
 Thickness of key, / = 
 
 Thickness of web, , = 
 
TOOTHED GEARING. 
 
 179 
 
 The working drawings are shown in Fig. 94, drawn 
 to a scale of T 3 6 -, and dimensions marked. Fig. (a) is a 
 
 full projection of the rack and pinion in gear, the rack 
 being broken in order to save space ; Fig. (c), a full 
 projection taken from Fig. (a) ; and Fig. (#), a sectional 
 
180 TOOTHED GEARING. 
 
 projection of the rack and pinion, taken from Fig. (a), 
 on the line xy. The cycloidal profiles of the teeth were 
 drawn by the method given under Fig. 26 for the pinion, 
 and under Fig. 35 for the rack. 
 
 Example 7. Required to design, and make working 
 drawings for, a cast-iron lantern gear and pinion to 
 transmit a force of 1,600 pounds, moderate shock, the 
 revolution ratio of the lantern to the pinion being . 
 
 From formula (12, b), for the pitch, 
 
 p = 0.05^1600 = 0.05 x 40 = 2". 
 The total height of the teeth is 
 
 h = 0.7 x 2 = 1.4", 
 and the breadth may be 
 
 b = 0.46 x 2 = 0.92". 
 The face width is 
 
 1=2 X 2= 4 ". 
 
 If we take, for the diameter of the lantern, 19^", we 
 have for the number of teeth, from formula (3), 
 
 60.08 
 ;V=- - = 3 o 
 
 and, from formula (7), the diameter of the pinion is 
 
 a*-6r. 
 
 The number of teeth for the pinion is 
 
 20.02 
 = 10. 
 
 2 
 
 Formula (34) gives, for the rim thickness, 
 /= 0.12 -f- (0.4 x 2) = 0.92". 
 
TOOTHED GEARING. l8l 
 
 Formula (35) gives for the nave thickness, for the lan- 
 tern, 
 
 k = 0.41/2- x 9.5625 + | = 0.4-^38.25 + i 
 
 = 0.4 x 3.369 + = 1.8476", 
 and for the pinion, 
 
 k = o.4'V2 2 X 3.1875 -f- 1= o.4'Vi2.75 -h | 
 
 = 0.4 x 2.336 + 1 = 1.4344"- 
 
 For the nave length of the pinion we have, from for- 
 mula (36), 
 
 and for the lantern, 
 
 r- 4 + ^ = 4.6375". 
 
 The diameter of the pinion shaft, from formula (39), is 
 
 </= o.o86Vi6oo x 3.1875 = 0.086^5100 
 
 = 0.086 X 17.213 = 1.48", 
 
 and the diameter of the lantern shaft, 
 
 d o.o86Vi6oo x 9.5625 = o.o86Vi53QO 
 
 = 0.086 X 24.826 = 2.135". 
 
 For the pinion, the width and thickness of the fixing- 
 key, from formulas (49) and (50), are 
 
 s = 0.164- = 0.456" 
 and 
 
 0.164- = 0.308". 
 
1 82 TOOTHED GEARING. 
 
 For the lantern, 
 
 s 0.16 + ^12$ = 0.587" 
 
 3 
 
 and 
 
 /= o.i 6 + 2 -^ = 0.3735". 
 
 The pinion is small enough to be made without arms. 
 Formula (23) gives, for the number of arms in the lan- 
 tern, 
 
 n ; = 0.56^30 V2 = 0.56 x 5.48 x 1.19 = 4. 
 
 For arms having circular cross-sections, formula (19) 
 gives a diameter of 
 
 4 3/1600x0. 562^ 3/r 
 
 d =0.15^ - v:> * = 0.15^/3825 = 0.15 x 15.64= 2.346". 
 4 
 
 As explained in VI., under Fig. 42, the radius for the 
 lantern rungs is j$ X 2 = 0.475". 
 
 Dimensions for the lantern. 
 
 Diameter, D = ipj" 
 
 Pitch, / = 2" 
 
 Face width, / = 4"* 
 
 Radius of rungs, = Jf" 
 
 Number of rungs, N = 30 
 
 Thickness of rim, / = fj" 
 
 Number of arms, / = 4 
 
 Diameter of arm cross-section, d r = 2 JJ" 
 
 Thickness of nave, k = iff" 
 
 Length of nave, /' = 4f" 
 
 Diameter of shaft, d = 2^" 
 
 Width of fixing-key, j = J|" 
 
 Thickness of fixing-key, / = f " 
 
 * See Fig. 95 (c). 
 
TOOTHED GEARING. 183 
 
 Dimensions for the pinion. 
 
 Diameter, D = 6f " 
 
 Pitch, / = 2" 
 
 Face width, / = 4" 
 
 Total height of teeth, h if" 
 
 Breadth of teeth, b = f J" 
 
 Number of teeth, JV 10 
 
 Thickness of nave, k i-fr" 
 
 Length of nave, /' = 
 
 Diameter of shaft, d = 
 
 Width of fixing-key, s = 
 
 Thickness of fixing-key, / = T 5 /'. 
 
 Fig. 95 gives the working drawings of the lantern and 
 pinion, drawn to a scale of -%- One of the lantern rungs 
 is shown in section in Fig. (c) in order to show that the 
 rungs are to be cast on the lantern, instead of being 
 made separately, and driven into holes along the lantern 
 rim, as is ordinarily the case. The arrangement of the 
 rim, etc., is sufficiently explained by the figure. The 
 teeth of the pinion are drawn according to the explana- 
 tion given in VI., under Fig. 42. 
 
 Example 8. Given the data and dimensions of the 
 pinion of Example 7, it is required to design an internal 
 lantern, the revolution ratio of which to the pinion shall 
 be \ ; the rungs of the lantern to be of wrought-iron, 
 and to be driven into holes along the rim. 
 
 The radius for the rungs is the same as in Example 7, 
 as is also the calculated rim thickness. But for an in- 
 ternal gear we take the rim thickness from ij times to 
 twice that of an external gear (see Fig. 96, b). 
 
 From formula (7), the diameter of the lantern is 
 
 D = 6 X 4 = 25 J' ', 
 
1 84 
 
 TOOTHED GEARING. 
 
 and, from formula (3), the number of rungs is 
 
 .. 80.1 
 N= = 40. 
 
 Formula (23) gives, for the number of arms, 
 
 n' 0.56^40 \/2 = 0.56 x 6.32 x 1.187 = 5 
 
 From formula (19) the diameter for the circular cross- 
 section of the arms is 
 
 .3/1600X12.75 s/ r~ 
 
 = o.i5y- - = 0.15^4080 0.15x15.98 = 2.40. 
 
TOOTHED GEARING. 185 
 
 For the nave thickness, formula (35) gives, 
 
 = 0.4^2* x 12.75 +i = 0-4^5 r +4 = 0.4x3.7 -f J = 1.98", 
 and the nave length is, from formula (36), 
 /'= 4 + 2 S = 4.8 S ". 
 
 Formula (39) gives, for the diameter of the wrought-iron 
 lantern shaft, 
 
 d o.o86'Vi6oo x 12.75 = o.o86Y20400 = 0.086 X 27.32 
 
 = 2.349"- 
 
 The width and thickness of the fixing-key are, from for- 
 mulas (49) and (50), 
 
 and 
 
 = 0.16 + = 0.395 
 
 " 
 
 Dimensions for lantern. 
 
 " 
 
 Diameter, D 25% 
 
 Pitch, / = 2" 
 
 Face width, / = 4" 
 
 Radius of rungs, -Jf " 
 
 Number of rungs, N = 40 
 
 Thickness of rim, / = ff-" 
 
 Number of arms, #/ = 5 
 
 Diameter of arms, d' = 2\%' 
 
 Thickness of nave, k = iff" 
 
 Length of nave, /' = 4f|" 
 
 Diameter of shaft, d = 2f" 
 
 Width of fixing-key, j = JJ" 
 
 Thickness of fixing-key, / = |f" 
 
1 86 TOOTHED GEARING. 
 
 Dimensions for pinion. 
 
 Diameter, D = 6f" 
 
 Pitch, / = 2" 
 
 Face width, / == 4" 
 
 Total height of teeth, h = i-|f" 
 
 Breadth of teeth, = f" 
 
 Number of teeth, ^V = 10 
 
 Thickness of nave, k = i-&" 
 
 Length of nave, /' = 4-^" 
 
 Diameter of shaft, // = ifj" 
 
 Width of fixing-key, j = f" 
 
 Thickness of fixing-key, / = ^". 
 
 The working drawings for Example 8 (shown in Fig. 
 96, drawn to a scale of ^) need but little explanation. 
 The dimensions are marked on the drawings ; and the 
 arrangement of the lantern arms, proportions of the 
 rim, etc., will be sufficiently explained by a glance at 
 Fig. (b). The teeth of the pinion were drawn by the 
 method explained in VI., under Fig. 44. 
 
 Example 9. Required to design a train of cast-iron 
 gears to lift a weight of 8,000 pounds (say, moderate 
 shock) by means of a drum and cord as outlined in 
 Fig. 97- " The circumferential force of the driving-gear r 
 is 1,000 pounds, and the diameter of the driver 12". Let 
 us assume that ten per cent of the driving-force is lost 
 in overcoming the friction of the gear teeth, shaft bear- 
 ings, etc. We have, therefore, an effectual force of 
 10001000X0.10 = 900 pounds, with which to lift 
 the weight of 8,000 pounds. We must gear our power 
 from 900 pounds to 8,000 pounds, or, in other words, we 
 must gear our power up -^o - = 9 times. Since the 
 powers of the gears are inversely proportional to their 
 
TOOTHED CKAKIXG. 
 
 IS 7 
 
 radii (formula 8), we must gear down our radii 9 times. 
 We can gear from R to r' 2\ times, and from R f to 
 
 the drum r" 4 times (2\ x 4 = 9). If, therefore, we 
 take R = 13^", we have 
 
1 88 
 
 TOOTHED GEARING. 
 
 and, if we take R' 28", we have, for the radius of the 
 drum, 
 
 Fig. 97 
 
 The power (or circumferential force) of the gear R is, of 
 course, that of the driver r, 1,000 pounds;* and from 
 
 formula (8) the power of 
 the gear r f (and conse- 
 quently that of the gear 
 R') is 1000 X 2\ = 2250 
 pounds. The total power 
 of the drum is 2250 X 4 
 = 9000 pounds. Our ex- 
 ample is now reduced to 
 two very simple ones ; 
 viz., first to design a pair 
 of gears (r and R) to 
 transmit a force of 1,000 pounds (moderate shock), the 
 diameters to be 2r = 1 2", and 2R = 27" ; and, second, 
 to design a pair of gears (r' and R') to transmit a force 
 of 2,250 pounds (moderate shock), the diameters being 
 2r'=i2", and 2R' $6". Let us take them in the 
 order given. From formula (12, b), the pitch for the 
 gears r and R is 
 
 / = 0.05^1000 = 0.05 x 31.62 = 1.581" 
 
 for the face width, 
 
 /= 2 x 1.581 = 3.162". 
 
 * We do not take the lost power into account in calculating the 
 strength of the gears. 
 
TOOTHED GEARING. 189 
 
 The heights are, 
 
 h = 0.7 x 1.581 = 1.1067" 
 
 h' 0.4 x 1.581 = 0.6324" 
 and 
 
 h" = 0.3 x 1.581 = 0.4743". 
 
 Taking the breadth of the teeth equal to 0.45^ gives 
 b = 0.45 x 1.581 = 0.7115". 
 
 From formula (3), the number of teeth for r is 
 
 ^=37^9 
 1.581 
 and for R, 
 
 Formula (34) gives, for the rim thickness, 
 
 /= 0.12 4- (0.4 x 1.581) = 0.7524". 
 
 The gear r is without arms. For the gear R the num- 
 ber of arms, from formula (23), is 
 
 / = 0.56^54 Vi.sSi = 0.56 X 7.348 x 1. 121 = 5. 
 
 For elliptical cross-sections, taking b' = -, formula (20) 
 
 gives 
 
 a* 1000 X 13.5 
 
 b'a* = ~ = 0.00339 -- = 9-153 
 
 or 
 
 a = Vi8.so6 = 2.636" 
 
19 TOOTHED GEARING. 
 
 Formula (35) gives, for the nave thickness for r, 
 
 k = o. 4 Vi.58i a x 6 + J = o. 4 Vi5 + | 
 
 = 0.4 x 2.466 4- \ 1.486" 
 and for R, 
 
 k = o. 4 Vi.58i 2 X 13.5 4- i = 0.4Y/33-744 + I 
 
 = 0.4 X 3.231 + = 1.794", 
 
 From formula (36), the nave length for r is 
 
 /'= 3.162 4-t = 3.562", 
 and for J?, 
 
 /' =3.1624-1$ = 4-062". 
 
 The diameter of the shaft for r is, from formula (39), 
 
 d 0.086^1000 x 6 = 0.086 x 18.17 = 1.5626" 
 and for R, 
 
 d 0.086^1000 x 13.5 = 0.086 x 23.81 = 2.0477". 
 
 Formulas (49) and (50) give, for the width and thick- 
 ness of the fixing-key for r, 
 
 1.^626 
 s = 0.16 H = 0.4725" 
 
 and 
 
 /= 0.16 + ^^ = 0.3163", 
 
 and for R, 
 
 ,= . l6 + ^ = 
 
 and 
 
 /= 0.164-^^ = 0.3648". 
 For the thickness of the web between the nave and 
 
TOOTHED GEARING. 191 
 
 rim of the gear r, the calculations are as follows. The 
 diameter across the nave is equal to 
 
 d+ 2k = 1.5626 4- (2 X 1.486) = 4.535" 
 
 and the circumference is 14.2$". Supposing the gear to 
 have 10 arms, each having a width of one-tenth the 
 nave circumference, we have 
 
 Formula (17) therefore gives, for the web thickness, 
 
 1000 x 6 
 bji* 2.03/>, = 
 
 500 x 10 
 
 or 
 
 1000 x 6 
 
 b l = = 0.591". 
 
 500 X 10 X 2.03 
 
 For the second pair of gears, r f and R f , formula (12, b) 
 gives a pitch of 
 
 / = o.o5V / l2~50 = 0.05 x 47434 = 2.3717"- 
 
 The face width is 
 
 /= 2 x 2.3717 = 4.7434". 
 
 For the heights of the teeth we have 
 
 h = 0.7 x 2.3717 = 1.6602" 
 
 /&'= 0.4 x 2.3717 = 0.9487" 
 and 
 
 A" =0.3 x 2.3717 = 0.7115". 
 
 The breadth of the teeth at the pitch circle is 
 ^ = 0.45 x 2.3717= 1.0673". 
 
192 TOOTHED GEAK1XG. 
 
 From formula (3), the number of teeth for r 1 is 
 
 and for R' t 
 
 N = 115^3 = 
 2-3717 
 
 The small gear r f is without arms. From formula (23), 
 the number of arms for R f is 
 
 /= 0.56^74 V2.37I7 = 0.56 x 8.60 x 1.241 = 6. 
 
 For elliptical cross-sections, taking tf = -, formula (20) 
 gives 
 
 7, , 3 2250 X 28 
 
 * * = ~ = 0.00 3 39 - 5-_- - = 35.60, 
 
 or 
 
 a = ^71.20 = 4.1447" 
 and 
 
 j,_ 4.H47 = 2 ,/ 
 
 2 
 
 The thickness of the rim is, from formula (34), 
 /= 0.12 -h (0.4 x 2.3717) = 1.0687". 
 
 Formula (35) gives, for the nave thickness for /, 
 
 k = 0.4/2.37 2 x 6 -f \ = 0.4^33.701 + 
 
 = 0.4 x 3.23 +|= 1.792" 
 and for the gear R f , 
 
 k = o. 4 V2. 3 7 2 x 28 + = 0.4^157.273 + J 
 
 = 0.4 x 5.398 + 1= 2.6592". 
 
 From formula (36), the nave length for r' is 
 /'= 4.7434 -f if = 5.1434", 
 
TOOTHED GEARING. 193 
 
 and for R\ 
 
 /'= 4-7434 4-18 = 6.61". 
 
 For the shaft diameter for /-', formula (39) gives 
 
 d = 0.086^2250 x 6 =0.086^13500 = 0.086 x 23.81 = 2.048" 
 
 and for R' t 
 
 //= 0.086^2250 x 28 =0.086 v 63000 = 0.086 x 39.79 = 3.42 1 9". 
 
 For the width and thickness of the fixing-key for r' 
 formulas (49) and (50) give 
 
 ,r = o.i6 + - - 0.5696'' 
 and 
 
 /= o.i 6 + ^^ = 0.3648", 
 and for R' t 
 
 and 
 
 10 
 
 For the web thickness for ;-', as before, the nave 
 
 diameter is 
 
 d + 2k = 2.048 + (2 x 1.792) = 5.63", 
 
 and the circumference is 17. 69": hence 
 
 10 
 From formula (17), 
 
 2250 x 6 
 i i 3- J 3 i ^ 00 x I0 
 
 or 
 
 500 x 10 x 3.13 
 
194 TOOTHED GEARING. 
 
 Dimensions for gear r. 
 
 Diameter, D = 12" 
 
 Pitch, / = ifj" 
 
 Face width, / = 3^" 
 
 Total height of teeth, h = i-fa" 
 
 Height below pitch circle, h' J|" 
 
 Height above pitch circle, h" = if" 
 Breadth of teeth on pitch circle, b = if" 
 
 Number of teeth, N = 24 
 
 Rim thickness, / = -" 
 
 Nave thickness, k = ifj" 
 
 Nave length, /' = 3yV 
 
 Shaft diameter, d = r^-" 
 
 Width of fixing-key, s = ^f " 
 
 Thickness of fixing-key, / = -$" 
 
 Thickness of web, b, = \\ " 
 
 Dimensions for gear R. 
 
 Diameter, D = 27" 
 
 Pitch, p = ifj" 
 
 Face width, / = 33%" 
 
 Total height of teeth, h i^" 
 
 Height below pitch circle, h' |-J" 
 
 Height above pitch circle, h" = tt" 
 Breadth of teeth on pitch circle, b = f|" 
 
 Number of teeth, N = 54 
 
 Rim thickness, / = f" 
 
 Number of arms, n/ = 5 
 
 Major axis of cross-sections, a = 2^" 
 
 Minor axis of cross-sections, b' = i A" 
 
 Nave thickness, = ifj-" 
 
 Nave length, /' = 4 T y 
 
 Shaft diameter, d = 2-^" 
 
 Width of fixing-key, s = |f" 
 
 Thickness of fixing-key, / = f J" 
 
TOOTHED GEARING. 195 
 
 Dimensions for gear r'. 
 
 Diameter, D = 12" 
 
 Pitch, / = 2f" 
 
 Face width, / = 4j" 
 
 Total height of teeth, h = if|" 
 
 Height below pitch circle, h' f J" 
 
 Height above pitch circle, h" = jj" 
 
 Breadth of teeth on pitch circle, ^ = i^" 
 
 Number of teeth, N = 16 
 
 Rim thickness, t = i^V' 
 
 Nave thickness, / = if J" 
 
 Nave length, /' = 5^:" 
 
 Shaft diameter, d = 2^" 
 
 Width of fixing-key, j = fj" 
 
 Thickness of fixing-key, / = f f" 
 
 Thickness of web, b^ f }' r 
 
 Dimensions for gear /?'. 
 
 Diameter, Z> = 56" 
 
 Pitch, / = 2|" 
 
 Face width, / = 4j" 
 
 Total height of teeth, h ifj" 
 
 Height below pitch circle, # = f J" 
 
 Height above pitch circle, //' = ^}" 
 Breadth of teeth on pitch circle, b i^" 
 
 Number of teeth, N = 74 
 
 Rim thickness, / = i^r" 
 
 Number of arms, n{ 6 
 
 Major axis for cross-sections, a 4^" 
 
 Minor axis for cross-sections, b' = 2-faf' 
 
 Nave thickness, k 2%%" 
 
 Nave length, /' = 6}|" 
 
 Shaft diameter, d = 
 
 Width of fixing-key, s 
 
 Thickness of fixing-key, / = 
 
I 9 6 
 
 TOOTHED GEARING. 
 
 The working drawings for the train are given in 
 Fig. 98, drawn to a scale of g 3 ^. Fig. (a) is a full projec- 
 
 tion of the whole train, showing the pairs in gear ; and 
 Fig. (6) is a sectional projection of the whole train, taken 
 
TOOTHED GEARING. 197 
 
 from Fig. (a), on the line AB, The double curved arms 
 of the large ($6") gear were drawn by the method ex- 
 plained in XIV., under Fig. 87. It may be remarked 
 here that very often, perhaps in the majority of cases, 
 in order to save calculation and work, the pitches for all 
 the gears of a train are taken the same. Obviously, 
 when such is the case, the common pitch must be taken 
 equal to that of the gear which transmits the greatest 
 force ; in the last example, that of the gear R f (or r' 
 which transmits the same force). Suppose the driving- 
 gear r to make 120 revolutions per minute ; then, from 
 formula (7), the number of revolutions per minute made 
 by R is 120 X Jf 53^. The gear r' y being fixed to 
 the same shaft, makes the same number of revolutions 
 as R ; and the number of revolutions per minute of R f , 
 and consequently of the drum r" , is 53^ X \\ 11.43. 
 The diameter of the drum is 14", and its circumference 
 43.98" : hence the circumferential velocity of the drum, 
 or the velocity with which the weight will be lifted, is 
 43.98 X H.43 _ 4 , i8g feet per minute 
 
 XVII. Special Applications of the Principles of Toothed Gearing. 
 
 In the foregoing pages the subject of toothed gearing 
 has been treated in so far as it relates to ordinary ma- 
 chinery only. The simple, uniform, rotary motion of 
 the spur wheel, bevel, or screw gear, the continuous 
 rectilinear movement of the rack these are met with 
 daily in almost every shop and factory. But there are 
 many special cases in which these simple, uniform mo- 
 tions are not sufficient. According to the work which 
 is to be performed, we need, in one case, an intermittent 
 
198 
 
 TOOTHED GEARING. 
 
 rotary or rectilinear motion ; in another, a gradually in- 
 creasing or decreasing speed ; and, in another, a recip- 
 rocating movement. These variations must be obtained 
 from the uniformly rotating shop-shaft ; and there arc 
 few fields in which the ingenuity of man has had wider 
 scope, or produced more variety and beauty of mechan- 
 ism, than in that of special gear-contrivances. Some of 
 the more useful and common of these many special 
 mechanisms will be found explained in the following 
 pages. 
 
 Fig.99 
 
 (i) Spur Gearing. Fig. 99 represents a pair of 
 "square" or "rectangular" gears, the object of which 
 is to obtain a varying speed for the driven gear <f from 
 the uniform rotary motion of the driver c. As explained 
 in XL, we have the expression, 
 
 n R . Rn 
 
 , = TT, or n = -&-, 
 
 in which R and n are the radius and number of revolu- 
 tions of the driver, and R r and ;/ the same for the 
 driven gear. From this last expression it is plain, that, 
 if we increase R ', we decrease the value of ;/; if we 
 
TOO Til ED GEA RING. 
 
 199 
 
 decrease R f , we increase ;/; if we increase R, we in- 
 crease ;/; and, if we decrease R, we decrease .'... In 
 Fig. 99, while the gears are in the position shown, the 
 greatest radius of the driver gears with the smallest; 
 radius of the driven gear : the speed of the driven gear 
 is, therefore, then at its maximum. As the gears revolve 
 in the directions indicated by the arrows, the radius of 
 the driver gradually decreases, and that of the driven 
 gear gradually increases, until the points x and a' are in 
 contact. The speed of the driven gear, therefore, grad- 
 
 Fig.lOO 
 
 ually decreases during this eighth of a revolution. 
 From the moment of contact between the points x and 
 a', the reverse action takes place, and the speed of the 
 driven gear gradually increases until the points b and k 
 are in contact. Thus, during the entire revolution, the 
 driven gear continues to alternate from a gradually 
 decreasing to a gradually increasing speed, and vice 
 versa. In order that rectangular gears shall work 
 properly together, it is necessary, first, that the pitch 
 peripheries of the two gears be equal in length, and, 
 second, that the sum of the radii of each pair of points 
 
200 TOO 7^1 ED GEARING. 
 
 (points which come into contact with each other) on the 
 two pitch peripheries be equal to the distance between 
 the centres of the gears. These two conditions sug- 
 gest the method for finding the pitch periphery (or 
 pitch line) of a rectangular gear which shall properly 
 gear with a given driver, shown in Fig. 100, which is as 
 follows. O is the given driver. Since the smallest 
 radius of the driver gears with the greatest radius of 
 the driven gear, and the sum of these two radii is equal 
 to the distance between the centres, make aO r = R, and 
 O' is the centre for the required gear. Divide the peri- 
 phery of O into a number of small parts, ai, 12, 23, etc. ; 
 and from the point O as a centre, and Oi, 6>2, O$, etc., 
 as radii, strike arcs cutting the line of centres in the 
 points b, c, d, etc. With the centre O' and radii O f b, O'c, 
 O'd, etc., strike circle arcs, and lay off the arcs ax, xy, 
 yz, etc., equal respectively to ai, 12, 23, etc., taking 
 care that the points x, y, z., etc., fall upon the corre- 
 sponding arcs, of which the point O' is the centre ; so 
 on, until the entire required pitch line is determined 
 by the points thus found. As may be at once seen by 
 comparing Figs. 99 and 100, the shape of the driven 
 periphery depends upon the amount of curvature of the 
 " corners " of the driver. Thus, for very slightly curved 
 corners, the driven periphery becomes more nearly star 
 shaped, as in Fig. 100. If we take the radius of curva- 
 ture for the corners (b'd ', Fig. 99 ) equal to b'c, the gear 
 peripheries become equal and similar, and the gears 
 square, with rounded corners. 
 
 Fig. 101 represents a pair of "triangular" gears, the 
 object of which is to obtain an alternating, varying 
 speed from the uniformly rotating driver, as in rec- 
 
TOOTHED GEARING. 
 
 20 T 
 
 tangular gears. Triangular gears give fewer changes 
 of speed per revolution than rectangular gears. In 
 Fig. 101, C being the driver and C the driven gear, the 
 speed of the latter is at its minimum when the gears 
 are in the positions shown in the figure. Since, from 
 these positions, the radius of the driver gradually in- 
 creases, and that of the driven gear decreases, as far 
 as the points b and b r , the speed of the driven gear 
 will gradually increase until the points b and b' are in 
 contact, or for one-sixth of an entire revolution. The 
 
 reverse action will then take place until the points c 
 and c' are in contact, and so on. Thus while, in rec- 
 tangular gears, each gradually increasing or decreasing 
 period takes place during one-eighth of a revolution, in 
 triangular gears each of these periods occupies one- 
 sixth of a revolution ; that is, in rectangular gears there 
 are eight alternately increasing and decreasing periods 
 in one entire revolution of the driven gear, and in trian- 
 gular gearing there are but six. 
 
 In "elliptical" gears (shown in Fig. 102) we have 
 still another means of obtaining the same result, with 
 the difference, that, in elliptical gears, each period of 
 
202 
 
 TOOTHED GEARING. 
 
 gradually increasing and decreasing speed takes place 
 during one-fourth of a revolution : in other words, 
 there are but four periods of increasing and decreasing 
 
 Fig. 102 
 
 speed during one entire revolution of the driver. To 
 construct the pitch lines of triangular and elliptical 
 gears, we proceed as already explained, under Fig. 100, 
 for rectangular gears. Fig. 103 represents a pair of 
 
 Fig. 103 
 
 "scroll" gears ; c being the driver, and c the driven 
 gear. From the positions shown in the figure (in which 
 the greatest radius of the driver gears with the smallest 
 radius of the driven gear), as the gears revolve in the 
 
TOOTHED GEARING. 
 
 203 
 
 directions indicated by the arrows, the radius of the 
 driver gradually and uniformly decreases, while that of 
 the driven gear gradually and uniformly increases. The 
 speed of the driven gear is therefore at its maximum 
 when the gears are in the positions shown, and gradu- 
 ally and uniformly decreases during the entire revolu- 
 tion. The moment before the positions shown in the 
 figure are reached, the smallest radius of the driver 
 gears with the greatest radius of the driven gear : the 
 speed of the latter is then at its minimum, and sud- 
 denly (as the gears assume the positions in the figure) 
 
 Fig, 104 
 
 changes to its maximum. To construct the pitch lines 
 for a pair of scroll gears, proceed as follows. Con- 
 struct the square 1234 (Fig. 104), each side of which 
 is equal to one-fourth the distance <?/, which determines 
 the rapidity of variation in the speed of the driven gear. 
 Produce the sides of the square, as shown in the figure. 
 From the point I as a centre, and a radius \a, strike 
 the arc ab ; with the point 2 as a centre, and 2b as a 
 radius, strike the arc be; with centre 3, and radius 3^, 
 strike the arc cd ; and with centre 4, and radius 4</, 
 strike the arc df. These four arcs together form the 
 pitch line of the driver, the axis of which is at the cen- 
 
204 
 
 TOOTHED GEARING. 
 
 tre x of the square 1234. Make aC'=fx, and C r is 
 the centre for the driven pitch line ; after which pro- 
 ceed to find points, and construct the pitch line ab'c'd' 
 as explained, for rectangular gears, under Fig. 100. 
 
 The mechanism represented in Fig. 105 is known as 
 " sector " gears, and the object is to obtain a series of 
 
 Fig. 105 
 
 different uniform speeds. In the figure, C is the driver, 
 and C' the driven gear. As long as the arcs ab and ab' 
 are in gear, the speed of the driven gear is the same. 
 When the arcs cd and cd' come into gear, the speed of 
 the driven gear becomes slower, but remains the same 
 throughout the gearing of these two arcs. Similarly, 
 when the arcs ef and e'f come into gear, the speed of 
 the driven gear becomes still slower, but uniform during 
 the gearing of these arcs. Thus, during each revolu- 
 tion, the driven gear has three periods of uniform speed, 
 each differing from the others. In order that sector 
 gears shall work properly together, it is necessary that 
 the arcs which gear together be equal in length (ab = ab', 
 cd c'd', etc.), and that the sum of the arc lengths upon 
 one gear be equal to the sum of the arc lengths upon the 
 other (ab + cd+ef = ab'+ c'd'+ e'f). Also the sum of 
 
TOOTHED GEARING. 205 
 
 the radii of each two arcs which gear together must be 
 equal to the distance between the centres of the gears. 
 Sector gears are somewhat difficult to construct, be- 
 cause considerable care must be taken that no two 
 sectors of the driver gear at the same time with the 
 driven gear. To illustrate, suppose (Fig. 105) that the 
 arcs ab and ab' gear together at the same time as do 
 the arcs ef and e'f'\ that is, that the last few teeth of 
 ab gear with the driven gear at the moment when the 
 first few teeth of efdo the same. The driver will then 
 strive to drive the driven gear at its maximum and 
 minimum speeds at the same time, an attempt which 
 must obviously result in a fracture. In the figure, the 
 arc </ ceases to gear with the. driven gear at the mo- 
 ment when the arc ab begins to gear. Thus each arc 
 of the driver must escape gear just in time for its suc- 
 cessor to begin gear, and yet leave between these events 
 no appreciable interval to disturb the uniformity of mo- 
 tion. Fig. 106 represents a peculiar kind of spur wheel 
 and pinion. The wheel has two sets of teeth, one set 
 being on each side ; and the teeth of the two sets alter- 
 nating in position, as shown in the figure. The pinion 
 consists of two heart-cams, so arranged that each gears, 
 in turn, with one set of teeth of the spur wheel. By 
 this means a very slow motion is obtained for the spur 
 wheel, which is moved through a distance of two teeth 
 at each revolution of the cam-pinion. In the figure, the 
 cam a leaves the tooth a' some time before the cam /; 
 comes into contact with the tooth b f : during this time, 
 therefore, the spur wheel remains motionless, or, in 
 other words, the motion of the spur wheel is intermit- 
 tent. The length of time during which the driven gear 
 
2O6 
 
 TOOTHED GEARING. 
 
 remains motionless depends upon the shape of the cams. 
 Thus, if we give to the cam b the shape indicated by 
 the dotted outline, the cam will engage sooner with the 
 tooth //: consequently the period of rest will be shorter, 
 and the period of motion longer. Also, if the cams 
 differ from each other in shape, the periods of rest pre- 
 
 Fig. 106 
 
 Fig. 107 
 
 ceding the engagement of the two cams will be of 
 different lengths, and the motion of the driven gear 
 will be rendered still more variable. Fig. 107 repre- 
 sents the device known as " stepped " gears. This ar- 
 rangement is used when very heavy powers are to be 
 transmitted, and is met with sometimes in large and 
 powerful machine tools. In the figure, each of the 
 shafts c and / bears three spur wheels ; the pitches, 
 diameters, etc., being equal, and the three gears being 
 keyed firmly to the shaft. The gears are so fixed upon 
 the shaft that their teeth are arranged in steps along the 
 combined face, as shown in the figure ; i.e., each gear is 
 turned round upon the shaft slightly farther than the 
 
TOOTHED CEARfA r G. 
 
 207 
 
 Fig. (08 
 
 preceding one, so that instead of there being, say, \\ 
 teeth gearing with the driver at one time, as is the case 
 in a pair of ordinary spur wheels, there are 3 X i \ = 4j. 
 The strain is thus divided among three gears, and 
 the contrivance is capable of transmitting three times 
 the power which can be transmitted by one pair of the 
 gears. The device represented in Fig. 108 consists of a 
 mutilated spur driver c, a spur pinion /, and a mutilated 
 internal gear d; the gears c and d are fixed upon the 
 same shaft. The mutilated 
 spur wheel c drives the pinion 
 c in the direction shown by 
 the arrow, until the point a 
 is reached, when the gear c 
 ceases to be the driver, and 
 the mutilated internal gear 
 takes its place. This drives 
 the pinion c in a contrary 
 direction, until the point k 
 is reached, when c again be- 
 comes the driver, and again 
 reverses the direction of rotation of the pinion. The 
 gear c being small in diameter, and the gear d large, 
 the former drives the pinion c' at a slow speed, and the 
 latter gives to it a high speed. The mechanism is there- 
 fore useful where a slow forward motion and a quick 
 return are needed, as in the planer, and other machine 
 tools. 
 
 The arrangement represented in Fig. 109, which con- 
 sists of two spur wheels and a mutilated spur driver, is 
 intended to give to the spur wheels c' and d an alter- 
 nating, intermittent motion. The driver c, rotating in 
 
208 
 
 TOOTHED GEARING. 
 
 Fig.109 
 
 the direction indicated by the arrow, drives the spur 
 wheel c in the direction shown, until the tooth x comes 
 into gear. From that moment the driver acts upon the 
 spur-wheel d, which it drives in the same 
 direction as that given to the gear d ' , 
 When the tooth x comes into contact 
 with the gear d, the driver ceases to act 
 upon this gear, and returns to the gear /. 
 Thus the intermittent motions of the 
 two spur wheels are made to alternate ; 
 the gear cf remaining at rest while the 
 gear d is in motion, and contrariwise. 
 
 In Fig. i JO the mutilated driving pin- 
 ion engages alternately with the racks / 
 and d, which it drives, at the same speed, 
 in opposite directions. The two racks 
 being rigidly fixed to one frame, a recip- 
 rocating rectilinear motion is given to 
 the frame ; the forward and return mo- 
 tions being the same in velocity. 
 
 If the two racks are mutilated and the pinion entire, 
 as in Fig. 1 1 1, the mutilations being alternately situated 
 Fig.no on the two racks, a con- 
 
 ^ tinuous rectilinear mo- 
 & ^\\ tion of the rack frame 
 \A will give to the pinion 
 J J an alternating rotary 
 yj motion ; the speeds of 
 S advance and return mo- 
 
 tion being the same. If the rack mutilations are of 
 different lengths, the motion of the pinion will be vari- 
 able ; the pinion moving over a greater distance when 
 
TOOTHED GEARING. 2OQ 
 
 engaging with a long toothed part, and a less distance 
 when engaging with a part of the rack containing a few 
 teeth only. Fig. 1 12 represents a device for obtaining a 
 uniform rectilinear motion in one direction, and a sudden 
 return motion. The mutilated pinion, rotating continu- 
 ously in the direction shown by the arrow, imparts a 
 downward motion to the rack until the toothless part of 
 the pinion is reached. The rack, being then free, is lifted 
 quickly into its original position by means of the weight 
 Wj cord and pulley K. This arrangement is sometimes 
 used on special auto-mat- Fig. in 
 
 ic drills, in which case 
 the rack is fixed upon a 
 frame within which the 
 drill spindle works. The 
 spindle bears a raised 
 ring, which fits into an 
 annular depression within the frame. This allows the 
 spindle to revolve freely, still enables the pinion and 
 rack to give to the spindle the rectilinear motion neces- 
 sary for the feed ; and at the proper time the weight 
 returns the spindle to its original position in readiness 
 to repeat the desired operation. 
 
 In Fig. 113 the mutilated driver C acts upon the 
 gear C', driving it uniformly in the direction shown by 
 the arrow, until the toothless parts are opposite each 
 other, when, the gear C' being free, the weight W falls, 
 and quickly carries the gear C' into such a position that 
 the driver again gears with it ; and the same action 
 again takes place. Thus a variable rotary motion is 
 imparted to the driven gear, slow when the driver 
 acts upon it, and fast when it is acted upon by the 
 
 /A/WV\J 
 
2IO 
 
 TOOTHED GEARING. 
 
 weight W. If we change the numbers of teeth, so that, 
 when the teeth of the driver and driven gear cease con- 
 tact, the weight has the position W (the directions of 
 rotation being the same as in the figure), the weight, in 
 falling, will carry the gear C' in a direction contrary to 
 that imparted by the driver, and the motion of the driven 
 gear will be an alternating or oscillating one, made up 
 of a slow forward and a quick return movement. 
 
 Fig. 1 12 
 
 Fig. 113 
 
 (2) Bevel Gearing. Under the head of special appli- 
 cations of bevel gearing we propose to include some 
 pairs of gears which are not, strictly speaking, bevel 
 gears, since the teeth are not "bevelled," but which 
 resemble bevels in that their shafts are not parallel, but 
 form either oblique or right angles with each other. 
 An example of such a pair is seen in Fig. 114. The 
 gear C is an ordinary spur wheel, and ^'what is termed 
 a "crown gear." The teeth of the latter gear are made 
 so thin that their sides are practically parallel, and 
 
TOOTHED GEARING. 
 
 21 i 
 
 Fi g .ii4 
 
 hence gear with the spur wheel, notwithstanding the 
 
 fact that their side lines all intersect at the centre of 
 
 the pitch circle. Because 
 
 of the necessarily thin teeth 
 
 of the crown gear, such a 
 
 pair as is shown in the 
 
 figure can be used only for 
 
 the transmission of very 
 
 slight powers. They are 
 
 very seldom seen in prac- 
 
 tice, except in models, 
 
 mathematical instruments, 
 
 and such like. 
 
 Fig. 115 represents a 
 crown gear, C', which en- 
 gages with a wide-faced spur driver, C. The shaft of the 
 crown gear is set eccentrically, instead of in the centre 
 of the gear : hence a variable motion is given to it by 
 
 Fig, 115 
 
 the uniform rotary motion of the driver. The motion 
 of the crown gear is fast when the gears are in the posi- 
 tions shown in the figure, and gradually decreases until 
 the largest radius comes into gear, when the reverse 
 
212 
 
 TOOTHED GEARING. 
 
 action takes place for the remaining half-revolution. 
 The face width, ab, of the driver, must be at least equal 
 to the difference between the greatest and smallest 
 inner radii of the crown gear plus the thickness of the 
 crown gear teeth, xy. 
 
 Fig. 1 1 6 represents a contrivance for obtaining three 
 different uniform rotary motions for the shaft C' from 
 a uniformly revolving shaft C, the two shafts being at 
 
 Fig. 116 
 
 Fig.117 
 
 right angles with each other. The wheel C has three 
 sets of projecting pins, arranged in circles of different 
 diameters, as shown in the figure. The pitches (dis- 
 tance between the centres of two adjacent pins) of all 
 the circles are equal. The gear C' has a slotted face, 
 the slots being slightly larger than the pins of the wheel 
 C, and equally distant from each other. By sliding the 
 gear C' along its shaft, it may be made to engage at 
 will with either of the three circles of the wheel C, thus 
 obtaining a quick or slow motion as may be required. 
 
TOOTHED GEARING. 
 
 213 
 
 Fig. 117, which represents a device for giving two 
 different velocities to the same shaft, consists of a 
 driving bevel c> and two driven bevels c and d, of differ- 
 ent diameters, and running on the same shaft. The 
 bevel d, being smaller than the bevel c', is driven at a 
 greater speed, and in a direction contrary to that of /. 
 The bevel c is fixed to a collar or hollow shaft, g, which 
 fits over the shaft k, thus allowing it to revolve in a con- 
 trary direction. If the driving bevel c is mutilated, and 
 
 Fig. 118 
 
 the bevels c' and d fixed to the shaft k, an alternating 
 rotary motion will be given to the shaft, the alternations 
 being at different speeds. The same result may be 
 obtained for the shaft c by mutilating the gears c' and d 
 so that the toothed part of one is opposite the toothless 
 part of the other, and making the bevel c the driven 
 gear. 
 
 Fig. 118 represents a method of obtaining an alter* 
 nating rotary motion from a uniformly rotating shaft, 
 the driving and driven shafts being at right angles with 
 
214 
 
 TOOTHED GEARING. 
 
 Fig. 119 
 
 each other. The mutilated driving bevel c drives the 
 shaft c'd alternately in opposite directions, according as 
 it gears with the bevel c' or d. The speeds of the for- 
 ward and return motions are the same, since the bevels 
 /and d are of the same diameter. This contrivance 
 was once used to give the reciprocating motion to 
 
 planer-beds; a thread on the 
 shaft c'd, which worked in a 
 female thread in the bed, pro- 
 ducing the rectilinear motion. 
 The arrangement soon fell 
 into disuse, for the reason that 
 as much time was required for 
 the return as for the forward 
 motion, a waste which is now 
 obviated by the more modern 
 "quick return." 
 
 The device represented in 
 Fig. 119 is intended to trans- 
 mit a gradually increasing speed to a shaft from the 
 uniform rotary motion of a shaft at right or oblique 
 angles. The scroll bevel C is the driver, and the ordi- 
 nary bevel C the driven gear. Starting with the small- 
 est radius of the scroll bevel (at the point a) in gear with 
 the driven bevel, and rotating in the direction indicated 
 by the arrow, the radius gradually and steadily increases 
 until the bevels assume the positions shown in the 
 figure : consequently the speed of the driven bevel 
 gradually and steadily increases during the entire revo- 
 lution. The toothed part of the scroll bevel may be 
 carried farther than in the figure, as indicated by the 
 dotted lines, and the described action thus made to take 
 
TOOTHED GEARING. 
 
 215 
 
 place during more than one revolution. The shaft of 
 the driven bevel carries a feather, which allows the 
 bevel to slide along it without interfering with the ro- 
 tary motion. In the figure, when the described action 
 begins, the driven bevel C is in its highest position on 
 the shaft ; and, as during the rotation the radius of the 
 driving bevel increases, the former bevel is forced down- 
 ward upon its shaft until the positions shown in the 
 figure are reached. If the scroll bevel be made to rotate 
 in a direction opposite to that indicated in the figure, 
 it is plain that the teeth, not being prevented from so 
 
 Fig.120 
 
 doing by the converging of their lines, will lift out of 
 gear as the radius decreases, and thus destroy the 
 action. 
 
 Fig. 1 20 represents a peculiar kind of bevel gear, 
 more properly a pair of right-angle gears. The driving 
 gear C bears upon its circumference small rollers, which 
 gear into curved projections, or grooves, in the face of 
 the driven gear C' t and, by rolling down these curves, 
 give to the driven gear a rotary motion at right angles 
 with that of the driver. The motion of the driven gear 
 depends upon the shape of the projections. If these 
 are curved, the curves being more oblique to the verti' 
 
216 
 
 TOOTHED GEARING. 
 
 cal at the bottoms than at the tops, as in the figure, the 
 motion of the driven gear will be variable, slow when 
 each roller of the driver gears with the upper part of a 
 projection, and gradually faster as the roller progresses 
 downward. If, instead of being curved, the profiles of 
 the projections are straight lines, the motion of the 
 driven gear will be nearly uniform. 
 
 The motions described under Fig. 113 may be trans- 
 mitted from one shaft to another at right or oblique 
 
 Fig. 121 
 
 angles, by using mutilated bevels in place of the spur 
 gears shown in that figure. Fig. 121 represents an ar- 
 rangement of bevels known as the " mangle wheel" and 
 pinion, the object of which is to obtain an alternating 
 rotary motion for the mangle wheel C f . This wheel 
 has teeth upon both sides, one side only being shown in 
 the figure. As the driving bevel C rotates, it drives the 
 mangle wheel in the direction indicated by the arrow, 
 until the opening ef is reached. At this point the guide 
 a comes into contact with the shaft of the driver, which 
 
TOOTHED GEARING. 
 
 it forces downward through the opening, and into such 
 a position, that the driver gears with the teeth on the 
 other side of the mangle wheel. The latter is then 
 driven in an opposite direction, until the opening cf is 
 again reached, when the guide b lifts the driver up 
 through the opening into gear with the first-mentioned 
 side of the mangle wheel. This operation is repeated 
 indefinitely; the mangle wheel making one entire revo- 
 lution alternately in each direction. The shaft of the 
 driving bevel carries a universal joint, x, which allows 
 
 Fig. 122 
 
 it enough freedom of motion to fall and rise through 
 the opening in the mangle wheel. 
 
 Fig. 122 represents an arrangement of bevel gears, 
 the object of which is to produce a double or half 
 speed ; the three bevel gears having the same diameter. 
 The bevel c is rigidly fixed (so that it cannot rotate) to 
 the bed of the mechanism, and the shaft ab runs loosely 
 through it. The bevel c r runs loose upon the shaft ab, 
 which carries a short, right-angle shaft, cf. Upon this 
 right-angle shaft the bevel d runs loose. If, now, a 
 rotary motion be given to the shaft ab, the right-angle 
 
2l8 TOOTHED GEARING. 
 
 shaft ef, and with it the bevel d, will be made to revolve 
 in a vertical plane about the axis ab. The bevel d will 
 also, by its gearing with the fixed bevel c, be made to 
 rotate upon its own axis, ef. Since the bevels c and d 
 are of the same diameter, the speeds of these two rota- 
 tions will be the same : therefore the bevel d will trans- 
 mit to the bevel c' the effect of two speeds, each equal 
 to that of the shaft ab. And, since the speeds are in 
 the same direction, the bevel c' will be made to rotate 
 about the shaft ab with a speed equal to twice that of 
 the shaft ; that is, while the shaft ab makes one entire 
 revolution in a given direction, the bevel c' will make 
 two revolutions in the same direction. If the bevel c' 
 be made the driver, its rotary motion will transmit to 
 the bevel d a rotary motion about its axis ef, and, by 
 means of the fixed bevel c, also a revolving motion in 
 a vertical plane about the axis ab. The bevels having 
 equal diameters, half the speed of the driver is trans- 
 mitted in the rotation of the bevel d about its axis ef, 
 and half in the vertical rotation about the axis ab ; that 
 is, the shaft ab will be made to rotate with a speed 
 equal to one-half that of the driver : while the driver c' 
 makes two entire revolutions in a given direction, the 
 shaft ab will make one revolution in the same direction. 
 The relative speeds of the shaft ab and the bevel c may 
 be varied by changing the relative diameters of the 
 bevels. 
 
 (3) Screw Gearing. Fig. 123 represents a very com- 
 mon mode of transforming uniform rotary into uniform 
 rectilinear motion. The threaded shaft ab, rotating 
 upon its axis, and restrained from other motion by the 
 collars xy and fg, works in a female thread in the piece 
 
TOOTHED GEARING. 
 
 Fig. (23 
 
 r, thus giving to the latter piece a rectilinear motion 
 upon the slides k, k. By reversing the direction of rota- 
 tion of the shaft ab, the 
 direction of the motion of 
 the piece C will also be 
 changed. This device is 
 seen in the leading-screws 
 of lathes, in the arrange- 
 ment for feeding the tool 
 
 holders in planing ma- 
 
 chines, drills, etc. 
 
 In Fig. 1 24 the cylinder C has right and left spiral 
 grooves cut in its surface, as shown in the figure. The 
 
 Fi g .i24 
 
 Fig. 125 
 
 tooth k of the slide / fits 
 into the grooves. Upon 
 giving to the cylinder a 
 rotary motion about its axis 
 ab (supposing the tooth k < 
 to be working in the right- 
 hand groove), the slide / is made to move along the 
 frame d, upon which it rests, until the end of the groove 
 is reached, when the tooth runs into 
 the left-hand groove, and the slide f 
 returns in the opposite direction. 
 Thus a reciprocating rectilinear mo- 
 tion is obtained from the uniform 
 rotary motion of the cylinder. 
 
 In Fig. 125 a uniform rotary motion 
 of the pulley C gives to the slide / 
 a reciprocating rectilinear motion 
 along the frame d, by means of the zigzag groove upon 
 the pulley surface, in which the tooth k of the slide 
 
220 
 
 TOOTHED GEARING. 
 
 works. By giving to the groove in the surface of the 
 pulley the proper shape, the motion of the slide f may 
 be made uniform, variable, or intermittent. 
 
 Fig. 126 represents a device for transforming uniform 
 rotary motion into two rectilinear motions in opposite 
 Fj g . 126 directions. The shaft 
 
 ab, which carries the 
 right and left screw- 
 threads shown in the 
 figure, rotates within 
 its bearing O. The 
 right and left screws 
 work in female screws within the pieces C and C' : 
 consequently these pieces are driven in contrary direc- 
 tions, approaching each other, or receding from each 
 other, according to the direction of rotation of the shaft 
 
 ab. This arrangement is 
 used in presses of .various 
 kinds, the arms indicated 
 by the dotted lines being 
 drawn together at their 
 tops by the action of the 
 screws, and the point x 
 being forced slowly down- 
 ward with great force and 
 steadiness. The arrange- 
 ment of worm wheels rep- 
 resented in Fig. 127 is 
 intended to produce two uniform rotary motions in oppo- 
 site directions. The right and left worms on the shaft 
 ab cause the worm wheels C and C' to rotate in opposite 
 directions when the shaft is given a rotary motion. The 
 
 Fig. 127 
 
TO O THE!) GEA R L \ T G. 
 
 221 
 
 Fig. 129 
 
 same effect may be obtained with one worm, by gearing 
 with it two worm wheels, C and d, on opposite sides of 
 the shaft, as indicated by the dotted circle. 
 
 Fig. 128 represents a peculiar example of screw gear- 
 ing. The disk C carries upon its side Fig.i28 
 an elevated spiral, as shown in the 
 figure. This spiral gears with an ordi- 
 nary spur gear C f , the shaft of which 
 is at right angles with that of the disk. 
 At each revolution of the disk C, the 
 constantly changing radius of the spi- 
 ral causes the spur gear to rotate for 
 a distance equal to one tooth ; the pitch / of the spiral 
 being equal to the pitch of the spur gear. By gearing 
 with the spiral two spur gears (the 
 second is indicated in the figure by 
 the dotted lines), motion may be 
 transmitted from the spiral to two 
 shafts at right angles with each 
 other. In a like manner the spiral 
 may be made to drive several spur 
 gears at once, the shafts making 
 oblique angles with each other. 
 
 In Fig. 129 we have represented a 
 " side " worm wheel C', and worm. 
 The former carries upon its side pro- 
 jections or teeth, as shown in the fig- 
 ure ; and the worm on the shaft ab t 
 gearing with these teeth, causes the 
 wheel C' to rotate uniformly, the ac- 
 tion being similar to that of an ordinary worm and wheel. 
 By gearing with the worm two side worm wheels (the 
 second being indicated in the figure by the dotted circle 
 
222 
 
 TOOTHED GEARING. 
 
 d\ the teeth being on the sides of the wheels which 
 face towards each other, two uniform rotary motions in 
 opposite directions may be obtained, as in Fig. 127. 
 The motions described under Fig. 113 may be obtained 
 for shafts at right angles with each other by substitut- 
 ing for the mutilated spur gears a worm and mutilated 
 worm wheel. 
 
 Fig. 130 represents a kind of worm and worm wheel 
 sometimes used to transmit very heavy powers. The 
 Pig.,30 primitive surface cdef, of the 
 
 worm, instead of being a right 
 cylinder, as in ordinary worms, 
 is a solid of revolution gener- 
 ated by the revolution of the 
 circle arc cf about the axis ab. 
 The object of this is to obtain 
 a contact of several teeth at 
 one time. In the figure* seven 
 teeth of the worm are in gear 
 at the same time with the teeth 
 of the worm wheel, and each 
 tooth sustains an equal share 
 of the transmitted strain. In Fig. 127 only two teeth 
 of the worm are in gear at one time with the teeth of the 
 driven wheel C. If, therefore, we have to transmit such 
 a force that the strain on the teeth is 10,000 pounds, for 
 example, each tooth of the worm in Fig. 127 will sustain 
 
 a strain of -i.Q-.2-0 _ 
 
 pounds ; while under similar 
 
 circumstances each tooth of the worm in Fig. 130 will 
 sustain a strain of -1M.Q.& z= 1,430 pounds : in other 
 words, the latter worm is capable of transmitting | = 3| 
 times the force of the former worm with the same strain 
 upon each tooth. 
 
APPENDIX. 
 
 THE present tendency among mechanical men in favor 
 of the use of the diametral instead of the older and more 
 widely known circumferential pitch, together with the 
 increasing importance of cut gears (in the construction 
 of which the diametral pitch seems to be especially con- 
 venient), has induced the author to devote an appendix 
 to the brief discussion of the relative values of the two 
 kinds of pitch, to a brief explanation of the method of 
 constructing cut gears, and to the working-out of simple 
 rules and formulas, by means of which all the necessary 
 calculations may be made without the use of the circum- 
 ferential pitch. From X we have the expression 
 
 pd = -, in which p d and / represent respectively the 
 
 / 
 
 diametral and circumferential pitch, and TT the irrational 
 constant 3.14159-}-. The following table gives values 
 for the diametral pitch, for different circumferential 
 pitches, in inches. A glance at the table will show, that, 
 in the list of most common circumferential pitches, not 
 one corresponds to a diametral pitch of whole numbers, 
 or even exact eighths, sixteenths, thirty-seconds, etc. 
 In fact, the diametral pitch can be a whole number only 
 
 223 
 
224 
 
 TOOTHED GEARING. 
 
 when the corresponding circumferential pitch is an exact 
 divisor of the irrational constant TT, a condition which 
 is not at all likely to be fulfilled. 
 
 p 
 
 A* 
 
 P 
 
 A* 
 
 I 
 
 25.1327 
 
 3* 
 
 0.8976 
 
 I 
 
 12.5664 
 
 4 
 
 0.7854 
 
 i 
 
 6.2832 
 
 4* 
 
 0.6981 
 
 } 
 
 4.1888 
 
 5 
 
 0.6283 
 
 I 
 
 3.1416 
 
 5* 
 
 0.5711 
 
 II 
 
 2.7926 
 
 6 
 
 0.5236 
 
 I* 
 
 2.5132 
 
 6 
 
 0.4833 
 
 I 
 
 2.0944 
 
 7 
 
 0.4488 
 
 If 
 
 1.7952 
 
 7* 
 
 0.4188 
 
 2 
 
 1.5708 
 
 8 
 
 0.3927 
 
 2* 
 
 i -3963 
 
 9 
 
 0.3491 
 
 2 
 
 1.2566 
 
 10 
 
 0.3142 
 
 2| 
 
 1.1424 
 
 12 
 
 0.2618 
 
 3 
 
 1.0472 
 
 H 
 
 0.2244 
 
 For this reason, in all gears which have to be laid out, 
 as cast gears, in the construction of which the pitch 
 must be stepped off around the pitch circumference in 
 the drawings and pattern, the circumferential pitch 
 only can be conveniently used. In such cases, even if 
 we have given the diametral pitch, we must practically 
 find the circumferential pitch before we can properly 
 divide our pitch circumference, and lay out the teeth. 
 At this point of the construction, the important ques- 
 tions are, " How many teeth is the gear to have?" and 
 " How much space on the pitch circle does each tooth 
 need?" We care as little how many teeth there are per 
 
TOOTHED GEARfNG. 22$ 
 
 inch of diameter as how many teeth there may be per 
 pound of metal. In performing the calculations neces- 
 sary to the laying-out of gears, the diametral pitch offers 
 no advantages over the circumferential. Thus, to ob- 
 tain the number of teeth with the latter pitch, we divide 
 the pitch circumference (an irrational quantity) by the 
 pitch ; while in using the former pitch the case is no 
 better, for, to find the number of teeth in the gear, we 
 must multiply the pitch diameter by the diametral pitch 
 (itself an irrational quantity). Again : the rules and 
 formulas for the tooth dimensions at present in use in 
 the shops are in terms of the circumferential pitch, for 
 example, the formulas /= 2/, or /= 2\p, h = o.?/, etc., 
 given in the preceding pages, and, while using the 
 diametral pitch, we must either obtain the circumferen- 
 tial pitch in order to find our tooth dimensions, or devise 
 and introduce new rules and formulas in terms of the 
 diametral pitch. The author having taken the pains to 
 ask a considerable number (68) of draughtsmen and 
 pattern-makers in the States of Jtfew York, Pennsylvania, 
 New Jersey, and Connecticut, their preferences, finds 
 that a very large majority (61 to 7) of those spoken to 
 favor the use of the old circumferential pitch. This 
 would seem to indicate, that, while the same theorists 
 who are striving to force upon the American mechanic 
 the French metric system are clamoring for an absolute 
 discontinuance of the use of the old pitch, the practical 
 mechanic, who does the measuring and constructing, 
 goes steadily on with his work, looking neither to the 
 right for a "centimeter," nor to the left for a "diametral 
 pitch." 
 
 But while, according to the opinion and experience of 
 
226 TOOTHED GEARING. 
 
 the author, the diametral pitch is of no practical use in 
 cast gears, it cannot be reasonably disputed, that, in the 
 construction of cut gears, this pitch has, indeed, advan- 
 tages over the circumferential, and for this reason 
 deserves the attention and respect of every intelligent 
 mechanic. 
 
 In the construction of cut gears the wheels are first 
 cast without teeth, the entire thickness of the rim being 
 its own thickness when finished plus the height of the 
 teeth (t -f- //). The spaces between the teeth are then 
 cut out by means of revolving circular cutters, the 
 blades of the cutters being as nearly. as possible the 
 shape of the required spaces. In order to properly con- 
 struct cut gears, a shop must be provided with different 
 sets of cutters, corresponding to the different pitches 
 and diameters of gears. The principle of the gear-cutter 
 series may be illustrated as follows. Suppose we wish 
 to construct a set of cutters for a No. i pitch.. The 
 extreme variation in the shape of the cutters must obvi- 
 ously be between the cutter for the gear having the 
 greatest diameter (the rack) and that for the gear having 
 the smallest diameter (say the pinion having eleven 
 teeth). Between these two we must have a sufficient 
 number of cutters to cut No. I teeth for a gear of any 
 diameter without serious error. Similarly, for each other 
 necessary pitch, we must have a set of cutters composed 
 of a sufficient number to make our errors unimportant. 
 Of course the greater number of cutters we have in each 
 set, the more accurate will be our work. Thus, if we 
 have a cutter of each pitch for a gear of eleven teeth, 
 another for a gear of twelve teeth, another for thirteen 
 teeth, and so on, our gears will be theoretically accurate. 
 
TOOTHED GEARING. 22 / 
 
 But gear-cutters are expensive tools, and it is therefore 
 important to reduce the number to the minimum which 
 can be used without making the errors so great as to do 
 practical harm. Mr. George B. Grant, in an article pub- 
 lished some months ago in the "American Machinist," 
 points out the fact, that, since the extreme variation in 
 the shape of the cutters is less for fine than for coarse 
 pitches, the number of cutters necessary for the same 
 degree of accuracy is less in the former than in the 
 latter. He gives for the proper number of cutters in 
 the different sets the following table : 
 
 For a 1 6 pitch or finer, 6 cutters 
 
 For an 8 to 1 6 pitch, 1 2 cutters 
 
 For a 4 to 8 pitch, 24 cutters 
 
 For a 2 to 4 pitch, 48 cutters. 
 
 If we substitute for the circumferential pitch, / in 
 formula (10), its value in terms of the diametral pitch, 
 
 L v 3-i4i59\' 
 
 \ p =j*=-*r) 
 
 we shall have 
 
 or 
 
 From this, by transposing, we have, 
 
 or 
 
 "* (') 
 
228 T007WED GEARING. 
 
 Rule. To find the diametral pitch for a gear of 
 any material, divide the greatest safe working-stress in 
 pounds per square inch for the material used by the 
 force transmitted, multiply the quotient by the assumed 
 ratio of the face width to the circumferential pitch, ex- 
 tract the square root of the product thus obtained, and 
 multiply the result by 0.637. 
 
 By substituting 
 
 . * = 3^4159 
 P Pd pd 
 
 in formulas (12, a t b, c\ we obtain, 
 
 , = 3.14.59 yj? 
 pd p.t 
 
 and 
 
 From these, by transposing, we have, 
 _ 3.14159. /i 
 
 *-*= 
 
 and 
 
 ^3- I 4i594/i 
 
 * : 0.035 \ p 
 
 or, reducing the three last found equations;, we have, 
 
TOOTHED GEARING. 2 29 
 
 For violent shock, pd= ST- 12 \ ~p ( a ) 
 
 i 
 For moderate shock, pd 62.83^ -p (b) 
 
 i 
 For little or no shock, pd = 89. 76V/ (c) 
 
 Formulas (12, a, b, c) were determined upon the 
 condition that the face width equals twice the circumfer- 
 ential pitch : hence substituting/ = in the expression 
 
 Pd 
 l2p gives, 
 
 27T 6.283 
 
 After determining the diametral pitch p d from formula 
 
 (2), the face width must not be taken less than . 
 
 Pd 
 
 Ride. To determine the diametral pitch for a cast- 
 iron gear, when / = -II, extract the square root of 
 
 Pd 
 
 the reciprocal of the force transmitted, and multiply 
 the result by 57.12 for violent shock, 62.83 for moder- 
 ate shock, or 89.76 for little or no shock. 
 
 The above value of the circumferential in terms of the 
 diametral pitch, substituted in formulas (14, a, b, c), gives 
 
 ^-'W? 
 
 pd 
 
 * = 3-I4'59 = Lx-i/fl 
 
 Pd Pd ' V V 
 
 and 
 
 Pd Pd 
 
 3.14159 = 0.82^. 
 
230 TOOTHED GEARING. 
 
 By transposing, 
 
 and 
 
 or, reducing, we have, 
 
 For violent shock, p d = 2.435X7 -^ (a) 
 
 f 77 
 
 For moderate shock, /?= 2.685X7^ (^) 
 
 For little or no shock, pd = 3-83 iy -73. (<:) 
 
 (3). 
 
 As before, the condition / ^ -^- must be fulfilled. 
 
 /^ 
 
 Rule. To determine the diametral pitch for a cast- 
 iron gear from the horse-power and circumferential 
 
 velocity in feet per second, when / = , divide the 
 
 Pd 
 
 velocity by the horse-power, extract the square root of 
 the quotient, and multiply the result by 2.435 f r v ^- 
 lent shock, 2.685 f r moderate shock, or 3.831 for little 
 or no shock. 
 
 In a similar manner, by substituting p =. in formulas 
 
 Pd 
 (16, a, b, c), we obtain, 
 
TOOTHED GEARING. 
 
 H 
 
 and 
 
 p d 
 
 Transposing and reducing these three equations, as with 
 the preceding, we have 
 
 For violent shock, pd = o.i6i\/-^ (a) 
 
 - _ 
 
 For moderate shock, pd = o. 1 7 yV -^ 
 
 .// 
 
 For little or no shock, /^ = 0.253^^ 
 
 (4). 
 
 ?. To determine the diametral pitch for a cast- 
 iron gear from the horse-power and number of revolu- 
 
 tfans per minute, when / = - ^, multiply the diameter 
 
 Pd 
 
 of the gear by the number of revolutions, divide thc 
 product by the horse-power, extract the square root of 
 the quotient thus obtained, and multiply the result by 
 o. 161 for violent shock, 0.177 for moderate shock, or 
 0.253 for little or no shock. 
 
 Example i. Required the diametral pitch for a steel 
 gear which will transmit a force of 30,000 pounds, as- 
 
 suming -=13; the greatest safe working-stress per 
 P 
 
232 TOOTHED GEARING. 
 
 square inch being 20,000 pounds. From formula (i) 
 we have, 
 
 /- 
 
 = 0.637^2 = 0.637 x 1.41 -0.898. 
 
 2OOOO 
 
 Example 2. Required the diametral pitch for a cast- 
 iron gear to transmit a force of 900 pounds, moderate 
 shock. From formula (2, b) we have, 
 
 /~7~ 62.83 
 
 pd 62.83V/ ~ = = 2 -9- 
 
 ^V 900 30 
 
 Hence 
 
 /= 6.283 = 6.283 = ,/ 
 p d ~ 2.09 
 
 Example 3. The horse-power to be transmitted by 
 a cast-iron gear is 10, moderate shock, and the circum- 
 ferential velocity 5 feet per second. Required the 
 diametral pitch. Formula (3, b) gives 
 
 '- 90 
 
 Arms : If, in formula (23), we substitute for/ its value 
 of , we will have 
 
 pd 
 
 Extracting the fourth root of * in this equation gives 
 
 x i. 
 
 or _ 
 
 I ~~" */T' ' V ft. \JJ* 
 
TOOTHED GEARING. 233 
 
 Rule. To determine the number of arms in a gear, 
 extract the square root of the number of teeth and the 
 fourth root of the reciprocal of the diametral pitch ; 
 multiply these two roots together, and their product by 
 0.746. 
 
 From the expression / = t-^ we may obtain, by 
 
 Pd 
 
 squaring both sides, p 2 = 5-A This, substituted in 
 
 Pd 
 formula (29), gives 
 
 - ' S x 9.86965;? 
 
 or 
 
 Rule. To determine the quantity bji? (the thick- 
 ness of the arm multiplied by the square of the width), 
 divide the radius of the gear by the product of the 
 square of the diametral pitch into the number of arms, 
 and multiply the quotient by 7.896. 
 
 By substituting/ 2 = , in formula (30), we ob- 
 tain, 
 
 d' = 1.105^/9.86965^-, 
 
 which reduces to 
 
 Rule. To determine the diameter for arms having 
 circular cross-sections, divide the radius of the gear by 
 the product of the square of the diametral pitch into the 
 number of arms, extract the cube root of the quotient, 
 and multiply the result by 2.37. 
 
234 TOOTHED GEARING. 
 
 In a similar manner we may obtain from formulas 
 (31), (32), and (33), the expressions, 
 
 D 
 b'a* = 1.356 X 9.86965--^-, 
 
 b,,H'* + ,,3 7? 
 
 -7T- 0.8x9.86965, 
 
 and 
 
 BH'* - b,,h,t R 
 
 -- = o.8x 9 .86 9 6 5 
 
 / , 
 which reduce respectively to the following : 
 
 and 
 
 BH'* - b.,hj 
 
 Rim, Nave, etc. : The total rim thickness before the 
 spaces between the teeth are cut out is equal to / + / / - 
 If we add together formula (34) and the expression for 
 the total height of the teeth, // = o.//, we shall have, 
 
 and, by substituting for/ its value of 
 
 = 0:12 
 
 Pd 
 0-7 X 3.14159 
 
 pd Pd 
 
 * See Fig. 82. t See Fig. 83. 
 
TOOTHED GEARING. 235 
 
 Or, calling the total thickness of the rim /, and reducing, 
 we obtain 
 
 (i,). 
 
 . 
 
 Rule. To determine the total thickness of the rim 
 (the height of the teeth plus the true rim thickness), 
 divide 3.46 by the diametral pitch, and to the quotient 
 add o.i 2". 
 
 The expression p 2 ^-^, substituted in formula 
 
 Pd 
 (35)i S ives 
 
 ,3/9.8696 
 
 = 4 V /~ + = 4 X 
 
 .86965^ R 
 
 ^~ 
 or 
 
 * = .858^| + i (12). 
 
 Rule. To determine the thickness of the nave, 
 divide the radius of the gear by the square of the di- 
 ametral pitch, extract the cube root of the quotient, 
 multiply the root by 0.858, and to the result add J inch. 
 
 For the length of the nave we have formula (36), 
 which is, 
 
 Formula (45) becomes, on substituting for p* its value 
 in terms of the diametral pitch, 
 
 which reduces to 
 
236 TOOTHED GEARING. 
 
 Rule. To determine the diameter of a gear shaft of 
 any material, divide the radius of the gear by the prod- 
 uct of the square of the diametral pitch into the greatest 
 safe shearing-stress in pounds per square inch for the 
 material of the shaft, extract the cube root of the prod- 
 uct thus obtained, and multiply the result by 27.184. 
 
 Similarly we may obtain from formulas (46), (47), and 
 (48) the equations, 
 
 8/ R 
 
 ^=0-553^9-86965 
 
 </= 0.634^9-86965^ 
 and _ 
 
 ^=0.796^9.86965^. 
 From which, by reducing, we obtain the following : 
 
 For steel, ^= i.i86y- 2 (15) 
 
 For wrought-iron, d = i-36y (16) 
 
 /-ft 
 
 For cast-iron, ^=1.707^7- 
 
 Pd 
 
 Rule. To determine the diameter of a gear shaft, 
 divide the radius of the gear by the square of the diame- 
 tral pitch, extract the cube root of the quotient, and 
 multiply the result by 1.186 for steel, 1.36 for wrought- 
 iron, or 1.707 for cast-iron. 
 
 The formulas for the mean width and thickness of 
 
TOOTHED GEARING. 
 
 the fixing-key are, as before explained for formulas (49) 
 and (50), 
 
 S=o.i6 + (18) 
 and 
 
 S>=o.i6+* Q (19). 
 
 Example 4. Required to design a 24" cut gear-wheel 
 (of cast-iron) which will safely transmit a force of 1,000 
 pounds, moderate shock. 
 
 From formula (2, b) we have, for the diametral pitch, 
 
 / i 62.83 
 
 pd 6 2.83V/ - ~ == ~^~ = 2 ver Y nearly. 
 V 1000 31.62 
 
 The face width is consequently 
 
 For the number of teeth in the gear we have the ex- 
 pression, 
 
 N = pdD= 2 x 24 = 48. 
 Therefore formula (5) gives, for the number of arms, 
 
 n' 0.746^48 Vj = 0.746 X 6.928 X - - = 4. 
 
 If we wish to have rectangular cross-sections for our 
 arms, and take the thickness equal to one-half the width, 
 formula (6) gives 
 
 hf 7-896 X 12 
 
 O !/J , 2 = ---- = - . 
 
 2 4X4 
 
 Hence 
 
 h, = ^ 1 1. 844 = 2.28" 
 and 
 
 2.28 
 
 " 
 
238 TOOTHED GEARING. 
 
 From formula (11) we have, for the total thickness of 
 the rim, 
 
 /' = 0.12 + - = 0.12 + 1.73 = 1.85". 
 
 he thickness of the nave is, from formula (12), 
 
 k = 0.858^- + | = 0.858 x 1.44 + | = 1.736" 
 and the length, from formula (13), is 
 
 /' = 3.14 + |-J = 3.94". 
 
 Formula (16) gives, for the diameter of the wrought- 
 iron shaft, 
 
 d= 1.36?^- = 1.36 x 1.44 = 1.96", say 2". 
 
 Formulas (18) and (19) give, for the mean width and 
 thickness of the fixing-key, 
 
 5=0.16+ f =0.56" 
 and 
 
 5' =0.1 6+ ^ = 0.36". 
 
 The following table will be found convenient in 
 constructing cut gears of cast-iron. To illustrate its 
 application, suppose we have to construct a cut gear 
 which will transmit a force of 4,000 pounds, moderate 
 shock. We find in the table, column for moderate 
 shock, P = 3,948 pounds, which corresponds to a No. I 
 diametral pitch. We also find in the table the face 
 width of 6.28", and the total rim thickness of 3.58". 
 
TOO 7^HED GEA RING. 
 
 239 
 
 
 
 
 
 "V 
 
 
 V 
 
 
 
 
 P 
 
 P 
 
 P 
 
 77 
 
 H 
 
 77 
 
 /' 
 
 / 
 
 p* 
 
 
 Moder- 
 
 Little 
 
 
 
 Little 
 
 
 
 
 Violent 
 
 
 
 Violent 
 
 Moderate 
 
 
 In 
 
 
 
 shock. 
 
 ate 
 shock. 
 
 or no 
 shock. 
 
 shock. 
 
 shock. 
 
 or no 
 shock. 
 
 inches. 
 
 In inches. 
 
 
 
 
 
 
 
 
 
 i 
 
 52203 
 
 6 3 l66 
 
 128910 
 
 0.0106 
 
 0.0087 
 
 0.0043 
 
 13.96 
 
 25-I3 
 
 ! 
 
 23201 
 
 28074 
 
 57293 
 
 0.0237 
 
 0.0195 
 
 0.0095 
 
 9-36 
 
 16.76 
 
 i 
 
 I305I 
 
 I579I 
 
 32227 
 
 0.0423 
 
 0.0350 
 
 0.0170 
 
 7.04 
 
 12.57 
 
 1 
 
 8352 
 
 IOI06 20625 
 
 0.066 
 
 0.054 
 
 0.027 
 
 5 .66 
 
 10.05 
 
 I 
 
 5800 
 
 7018 
 
 H323 
 
 0.095 
 
 0.078 
 
 0.038 
 
 4-73 
 
 8.38 
 
 i 
 
 3263 
 
 3948 
 
 8057 
 
 0.169 
 
 0.139 
 
 0.068 
 
 3-58 
 
 6.28 
 
 'i 
 
 2088 
 
 2527 
 
 5156 
 
 0.265 
 
 0.217 
 
 0.107 
 
 2.89 
 
 5-03 
 
 ii 
 
 1450 
 
 1754 
 
 3581 
 
 0.380 
 
 0.312 
 
 0.153 
 
 243 
 
 4.19 
 
 'i 
 
 1066 
 
 1290 
 
 2631 
 
 0.519 
 
 0.425 
 
 0.208 
 
 2.10 
 
 3-59 
 
 2 
 
 816 
 
 987 
 
 2014 
 
 0.676 
 
 o-555 
 
 O.272 
 
 1.85 
 
 3-14 
 
 a* 
 
 644 
 
 779 
 
 1591 
 
 0.858 
 
 0.702 
 
 0-344 
 
 1.66 
 
 2.79 
 
 4 
 
 522 
 
 632 
 
 I28 9 
 
 1.056 
 
 0.867 
 
 0.425 
 
 1.50 
 
 2.51 
 
 2} 
 
 43i 
 
 522 
 
 1065 
 
 1.283 
 
 1.049 
 
 0.514 
 
 1-37 
 
 2.28 
 
 3 
 
 362 
 
 439 
 
 8 95 
 
 I.52I 
 
 1.248 
 
 0.612 
 
 1.27 
 
 2.09 
 
 4 
 
 204 
 
 247 
 
 504 
 
 2.704 
 
 2.219 
 
 1. 088 
 
 0.99 1.57 
 
 5 
 
 131 
 
 158 
 
 322 
 
 4.225 
 
 3467 
 
 1.700 
 
 0.81 
 
 1.26 
 
 6 
 
 9i 
 
 no 224 
 
 6.084 
 
 4-993 
 
 2.448 
 
 0.70 
 
 1.05 
 
 7 
 
 67 
 
 81 
 
 164 
 
 9.281 
 
 6.796 
 
 3-332 
 
 0.61 
 
 0.90 
 
 8 
 
 5i 
 
 62 
 
 126 
 
 10.816 
 
 8.877 
 
 4-352 
 
 0.55 
 
 0.79 
 
 9 
 
 40 
 
 49 
 
 99 
 
 13.689 
 
 11.235 
 
 5.508 
 
 0.50 
 
 0.70 
 
 10 
 
 33 
 
 40 
 
 8r 
 
 16.863 
 
 13-870 
 
 6.812 
 
 0.47 
 
 0.63 
 
 12 
 
 23 
 
 27 
 
 56 
 
 24-336 
 
 19-973 
 
 9.792 
 
 0.41 
 
 0.52 
 
INDEX. 
 
 [The numbers refer to the pages.] 
 
 A. 
 
 Actual pitch, 52. 
 Angle of repose, 57. 
 Arc of approach, 89. 
 of contact, 44, 89. 
 of recess, 89. 
 Arms, circular sections, no. 
 
 curved, 122. 
 
 elliptical sections, HI. 
 
 flanged sections, 112. 
 
 methods for drawing, 122. 
 
 number of, 115. 
 
 rectangular sections, 108. 
 
 straight, 122. 
 
 strength of, 107. 
 
 B. 
 
 Bastard gears, 54. 
 Bevel gears, 49. 
 
 angle of shafts of, 49. 
 
 design of, 154. 
 
 drawings of, 160. 
 
 internal, 53. 
 
 method for drawing, 52. 
 
 mutilated, 213. 
 
 scroll, 214. 
 Bevel rack, 54. 
 Breadth of teeth, 89, 97. 
 Breaking-weight, 96. 
 
 C. 
 
 Cam-pinion, 206. 
 
 Circle, generating, 15, 20, 33. 
 
 of centres, 32. 
 
 of the gorge, 68. 
 
 pitch, 15, 17,49. 
 
 primitive, 17, 19. 
 
 rolling, ii, 28, 31. 
 
 root, 36. 
 
 top, 36. 
 
 Circumference, 72. 
 Circumferential pitch, 73. 
 Conditions for minimum friction, icx 
 for uniform velocity, 1 5. 
 Cone, pitch, 49. 
 
 supplementary, 50. 
 Constant TT, 72. 
 Crown gears, 210. . 
 Cycloid, 37. 
 Cycloidal teeth, 22. 
 Cylindrical gears, 49, 54. 
 
 D. 
 
 Decimals, table of, 106. 
 Design of bevel gears, 154. 
 
 of gear train, 186. 
 
 of internal lantern, 183. 
 
 of internal spur gear, 169. 
 
 of 1-antern gear, 180. 
 241 
 
242 
 
 INDEX. 
 
 Design of rack and pinion, 175. 
 of screw gears, 164. 
 of spur gear, 151. 
 of worm and wheel, 160. 
 Diameter, 72. 
 
 Diametral formulas, arms, 232. 
 cutters, 227. 
 nave, 234. 
 rim, 234. 
 shafts, 235. 
 Diametral pitch, 74. 
 Dimensions for bevel gears, 158. 
 for gear train, 194. 
 for internal lantern, 185. 
 for internal spur gear, 
 
 174. 
 
 for lantern gear, 182. 
 for rack and pinion, 178. 
 for screw gears, 168. 
 for spur gear, 153. 
 for worm and wheel, 
 
 163. 
 
 Disk wheel, 53. 
 Drawings of bevel gears, 160. 
 of gear train, 196. 
 of internal lantern, 187. 
 of internal spur gear, 176. 
 of lantern gear, 184. 
 of rack and pinion, 179. 
 of screw gears, 170. 
 of spur gear, 1 55. 
 of worm and wheel, 164. 
 
 E. 
 
 Elliptical gears, 201. 
 Epicycloid, u, 17. 
 Epicycloidal faces, 13, 16. 
 Examples, arms, 109-120. 
 
 bevel gears, 154. 
 
 diameter, 80. 
 
 face width, 97-100. 
 
 gear train, 186. 
 
 Examples, hyperbolic gears, 68. 
 
 internal "lantern, 183. 
 
 internal spur gear, 169. 
 
 keys, 136. 
 
 lantern gear, 180. 
 
 nave, 126. 
 
 number of teeth, 80. 
 
 pitch, 74, 96-100. 
 
 pitch, diametral, 74. 
 
 power, 84. 
 
 rack and pinion, 175. 
 
 revolutions, 81. 
 
 rim, 125. 
 
 screw gears, 164. 
 
 shafts, 128-135. 
 
 spur gear, 151. 
 
 velocity, 84. 
 
 weight of gears, 137. 
 
 worm and wheel, 160. 
 Experiments v/ith involute teeth, 24. 
 
 F. 
 
 Face, epicycloidal, 13, 16, 
 involute, 22. 
 width, 89-93. 
 
 Flank, hypocycloidal, 13, 16. 
 radial, 19, 48. 
 straight, 19, 48. 
 Formulas for arms, circular, no, 117, 
 
 1 20. 
 for arms, elliptical, in, 
 
 118, 121. 
 
 for arms, flanged, 112, 114, 
 
 119, 121. 
 
 for arms, rectangular, 108, 
 
 116, 120. 
 
 for chord of the pitch, 75. 
 for circumference, 72. 
 for diameter, 72. 
 for diametral pitch, 73. 
 for fixing-keys, 136. 
 for nave kugth, 126. 
 
INDEX. 
 
 243 
 
 Formulas for nave thickness, 125. 
 
 for number of arms, 115. 
 
 for number of revolutions, 
 80. 
 
 for number of teeth, 73. 
 
 for pitch, from force trans- 
 mitted, 91-93- 
 
 for pitch, from horse-pow- 
 er, 94-96. 
 
 for pitch, from revolu- 
 tions, 95. 
 
 for power, 83. 
 
 for radius, 72. 
 
 for rim, 125. 
 
 for shafts, 127-131. 
 
 for velocity, 83. 
 
 for weight of gears, 136. 
 Fractions, table of, 106. 
 Friction, minimum, 10. 
 Fundamental principle, 2. 
 
 G. 
 
 Gears, bastard, 54. 
 bevel, 49. 
 cast, 224. 
 crown, 210. 
 cut, 226. 
 cylindrical, 54. 
 elliptical, 201. 
 high-speed, 107. 
 hyperbolic, 65. 
 internal, 40, 169. 
 lantern, 43. 
 mangle, 216. 
 mixed, 47. 
 mutilated, 208, 214. 
 pin, 212. 
 rectangular, 198. 
 screw, 54. 
 scroll, 202. 
 sector, 204. 
 spur, 49. 
 
 Gears, square, 198. 
 stepped, 206. 
 triangular, 200. 
 Gear at two points, 46. 
 Generating circle, 15, 17, 33. 
 Generating of epicycloid, n. 
 of hypocycloid, i: 
 of involute, 21. 
 
 H. 
 
 Height of teeth, 90. 
 working, 35. 
 High-speed gears, 107. 
 Horse-power, 94. 
 Hyperbolic gears, 65. 
 Hypocycloid, 12, 18. 
 Hypocycloidal flanks, 13, 16. 
 
 I. 
 
 Infinite radius, 28, 37. 
 Intermittent motion, 205. 
 Internal bevels, 53. 
 
 lantern gears, 44. 
 
 spur gears, 169. 
 
 worm wheel, 62. 
 Introduction, i. 
 Involute, 21. 
 
 faces, 22. 
 profiles, 21. 
 Irregular motion, 208. 
 
 K. 
 
 Keys, formulas for, 136. 
 rules for, 136. 
 
 L. 
 
 Lantern gears, 43. 
 
 internal, 44. 
 Line of contact, 87. 
 
 M. 
 
 Mangle wbel, 
 
244 
 
 INDEX. 
 
 Method for drawing bevels, 52. 
 
 for drawing curved arms, 
 
 122. 
 
 for drawing cycloidal pro- 
 files, 28, 31. 
 
 for drawing involute pro- 
 files, 35. 
 for stepping off the pitch, 
 
 76. 
 
 Minimum friction, 10. 
 Mixed gears, 47. 
 Motion, intermittent, 205. 
 irregular, 208. 
 quick return, 207. 
 reciprocating, 208. 
 rectilinear, 197. 
 rotary, 3, 198. 
 \ uniform, 10. 
 variable, 212. 
 
 N. 
 
 Nave, 125. 
 Notation, 139. 
 Number of arms, 1 1 5. 
 
 of revolutions, 80. 
 
 of teeth, 73. 
 
 P. 
 
 Pcricycloid, 45. 
 Pin wheel, 212. 
 Pi-rule, 77. 
 Pitch, actual, 52. 
 
 circle, 15, 17, 49. 
 
 circumferential, 73. 
 
 cone, 49. 
 
 diametral, 74. 
 
 frusta, 49. 
 
 point, 15. 
 
 virtual, 52. 
 Plane wheel, 53. 
 Power ratio, 82. 
 Primitive circle, 17, 19. 
 
 Primitive gear wheel, 4, 7. 
 Profiles, cycloidal, 37. 
 
 epicycloidal, 13. 
 
 hypocycloidal, 13. 
 
 involute, 21. 
 
 Q- 
 
 Quick return motion, 207. 
 
 R. 
 
 Rack, 37. 
 Radius, 72. 
 
 infinite, 28, 37. 
 Ratio, power, 82. 
 
 revolution, 79. 
 
 velocity, 78. 
 Recapitulation, 139. 
 Reciprocating motion, 208. 
 Rectilinear motion, 197. 
 Rolling circle, n. 
 Root circle, 36. 
 Rotary motion, 3, 198. 
 Rules for arms, circular, in, 117. 
 
 for arms, elliptical, in, 118. 
 
 for arms, rectangular, 109. 
 
 for circumference, 72. 
 
 for diameter, 72. 
 
 for fixing-keys, 136. 
 
 for nave length, 126. 
 
 for nave thickness, 125. 
 
 for number of arms, 115. 
 
 for number of revolutions, So. 
 
 for number of teeth, 73. 
 
 for pitch, from force transmit- 
 ted, 92, 93. 
 
 for pitch, from horse-power, 
 
 94, 95- 
 for pitch, from revolutions, 
 
 95,96. 
 
 for power, 84. 
 for radius, 72. 
 for rim, 125. 
 
INDEX. 
 
 245 
 
 Rules for shafts, 127, 130. 
 
 for weight of gears, 136. 
 
 S. 
 
 Safe shearing-stress, 127. 
 
 working-stress, 90. 
 Screw gears, 54. 
 rack, 58. 
 
 Scroll gears, 202. 
 Sector gears, 204. 
 Shafts, cast-iron, 132. 
 
 formulas for, 127, 131. 
 rules for, 127, 131. 
 steel, 132. 
 tables for, 133. 
 wrought-iron, 132. 
 Special forms, 45. 
 Spur gears, 49. 
 Square gears, 198. 
 Stepped gears, 206. 
 Straight flanks, 19, 48. 
 Strength of arms, 107. 
 of keys, 136. 
 of nave, 125. 
 of rim, 125. 
 of shafts, 127. 
 of teeth, 89. 
 
 Supplementary angle, 66. 
 cones, 50. 
 
 T. 
 
 Tables for arm widths, 109. 
 
 for decimals and fractions, 
 
 106. 
 for diametral pitches, 224. 
 
 Tables for number of arms, 1 1 5. 
 
 for number of gear cutters, 
 227. 
 
 for pitch, 101. 
 
 for shaft diameters, 133. 
 
 for weight of gears, 138. 
 Teeth, cast, 224. 
 
 cut, 226. 
 
 cycloidal, 22. 
 
 involute, 22. 
 
 of bevels, 52. 
 
 of hyperbolic gears, 71. 
 
 of screw gears, 60. 
 Top circle, 36. 
 Train of gears, 81, 186. 
 Triangular gears, 200. 
 
 U. 
 
 Uniform motion, 10. 
 velocity, 15. 
 
 V. 
 
 Variable motion, 212. 
 Velocity ratio, 78. 
 Virtual pitch, 52. 
 
 W. 
 
 Wear on teeth, 8, 63. 
 Weight of gears, 136. 
 Working height, 35. 
 stress, 90. 
 
 Worm, internal, 62. 
 and rack, 62. 
 and wheel, 61. 
 
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 (Trautwiue.) 8vo, 2 00 
 
 Bovey 's Treatise on Hydraulics 8vo, 4 00 
 
 Church's Mechanics of Engineering, Solids, and Fluids. . . .8vo, 6 00 
 
 Coffin's Graphical Solution of Hydraulic Problems 12mo, 2 50 
 
 Fen-el's Treatise on the Winds, Cyclones, and Tornadoes. . .8vo, 4 00 
 
 Folwel'i's Water Supply Engineering 8vo, 4 00 
 
 Frizell's Water-power 8vo, 5 00 
 
 Fuertes s Water and Public Health 12mo, 1 50 
 
 Water Filtration Works 12ino ? 2 50 
 
 Ganguillet & Kutter's Flow of Water. (Hering & Trautwine.) 
 
 8vo, 4 00 
 
 Hazeu's Filtration of Public Water Supply 8vo, 3 00 
 
 Hazlehurst s Towers and Tanks for Cities aud Towns 8vo, 2 50 
 
 Herschel's 115 Experiments 8vo, 2 00 
 
 Kiersted s Sewage Disposal 12mo, 1 25 
 
 Mason s Water Supply 8vo. 5 00 
 
 " Examination of Water , 12mo, 1 25 
 
 Merrimau's Treatise on Hydraulics. 8vo, 4 00 
 
 9 
 
Kichols's Water Supply (Chemical aud Sanitary) 8vo, $2 50 
 
 Schuyler's Reservoirs for Irrigation Large 8vo, 5 00 
 
 Turneaure and Russell's Public Water-supplies 8vo, 5 00 
 
 Wegmaun's Water Supply of the City of New York 4to, 10 00 
 
 Weisbach's Hydraulics. (Du Bois.) 8vo, 5 00 
 
 Whipple's Microscopy of Drinking Water 8vo, 3 50 
 
 Wilson's Irrigation Engineering ... .8vo, 4 00 
 
 " Hydraulic and Placer Mining 12mo, 2 00 
 
 Wolff's Windmill as a Prime Mover 8vo, 3 00 
 
 Wood's Theory of Turbines 8vo, 2 50 
 
 LAW. 
 
 Davis's Elements of Law 8vo, 2 50 
 
 ' ' Treatise on Military Law 8vo, 7 00 
 
 Sheep, 7 50 
 
 Manual for Courts-martial 16mo, morocco, 1 50 
 
 Wait's Engineering and Architectural Jurisprudence 8vo, 6 00 
 
 Sheep, 6 50 
 
 " Law of Contracts 8vo, 300 
 
 " Law of Operations Preliminary to Construction in En- 
 gineering and Architecture 8vo, 5 00 
 
 Sheep, 5 50 
 
 Winthrop's Abridgment of Military Law 12mo, 2 50 
 
 MANUFACTURES. 
 
 Allen's Tables for Iron Analysis 8vo, 3 00 
 
 Beaumont's Woollen and Worsted Manufacture 12mo, 1 50 
 
 Bolland's Encyclopaedia of Founding Terms 12mo. 3 00 
 
 " The Iron Founder 12mo, 250 
 
 Supplement 12mo, 250 
 
 Eissler's Explosives, Nitroglycerine and Dynamite 8vo, 4 00 
 
 Ford s Boiler Making for Boiler Makers 18mo, 1 00 
 
 Metcalfe's Cost of Manufactures 8vo, 5 00 
 
 Metcalf 's Steel A Manual for Steel Users 12mo, 2 00 
 
 * Reisig's Guide to Piece Dyeing 8vo, 25 00 
 
 Spencer's Sugar Manufacturer's Handbook . . . .16rno, morocco, 2 00 
 Handbook for Chemists of Beet Sugar Houses. 
 
 16mo, morocco, 3 00 
 
 Thurston's Manual of Steam Boilers 8vo, 5 00 
 
 Walke's Lectures on Explosives 8vo, 4 00 
 
 W est's American Foundry Practice 12mo, 2 50 
 
 Moulder's Text book 12mo. 2 50 
 
 Wiechmaun's Sugar Analysis Small 8vo, 2 50 
 
 Woodbury's Fire Protection of Mills 8vo, 2 50 
 
 10 
 
MATERIALS OF ENGINEERING. 
 
 (See also ENGINEERING, p. 7.) 
 
 Baker's Masonry Construction 8vo, $5 00 
 
 Bovey's Strength of Materials 8vo, 7 50 
 
 Burr's Elasticity and Resistance of Materials 8vo, 5 00 
 
 By rue's Highway Construction 8vo, 5 00 
 
 Church's Mechanics of Engineering Solids and Fluids 8vo, 6 00 
 
 Du Bois's Stresses iu Framed Structures Siiiull 4to, 10 00 
 
 Johnson's Materials of Construction 8vo, 6 00 
 
 Lanza's Applied Mechanics 3vo, 7 50 
 
 Marteus's Testing Materials. (Heuning.) 2 vols., 8vo, 7 50 
 
 Merrill's Stones for Building and Decoration 8vo, 5 00 
 
 Merriman's Mechanics of Materials 8vo, 4 00 
 
 " Strength of Materials 12mo, 1 00 
 
 Pattou's Treatise on Foundations 8vo, 5 00 
 
 Rockwell's Roads and Pavements in France 12mo, 1 25 
 
 Spalding's Roads and Pavements 12mo, 2 00 
 
 Thurstou's Materials of Construction , 8vo, 5 00 
 
 Materials of Engineering 3 vols. , 8vo, 8 00 
 
 Vol. I. , Non-metallic 8vo, 2 00 
 
 Vol. II., Iron and Steel 8vo, 3 50 
 
 Vol. III., Alloys, Brasses, and Bronzes 8vo, 2 50 
 
 Wood's Resistance of Materials 8vo, 2 00 
 
 MATHEMATICS. 
 
 Baker's Elliptic Functions 8vo, 1 50 
 
 *Bass's Differential Calculus 12mo, 4 00 
 
 Briggs's Plane Analytical Geometry 12mo, 1 00 
 
 Chapman's Theory of Equations 12mo, 1 50 
 
 Comptou's Logarithmic Computations 12mo, 1 50 
 
 Davis's Introduction to the Logic of Algebra 8vo, 1 50 
 
 Halsted's Elements of Geometry Svo, 1 75 
 
 " Synthetic Geometry 8vo, 1 50 
 
 Johnson's Curve Tracing 12mo, 1 00 
 
 " Differential Equations Ordinary and Partial. 
 
 Small Svo, 3 50 
 
 " Integral Calculus 12mo, 150 
 
 " " Unabridged. Small Svo. (In press.) 
 
 " Least Squares , 12mo, 1 50 
 
 *Ludlow's Logarithmic and Other Tables. (Bass.) Svo, 2 00 
 
 * " Trigonometry with Tables. (Bass.) Svo, 300 
 
 *Mahan's Descriptive Geometry (Stone Cutting) Svo, 1 50 
 
 Merrimaii and Woodward's Higher Mathematics. Svo, 5 00 
 
 11 
 
Merriinan's Method of Least Squares 8vo, $2 00 
 
 Rice and Johnson's Differential and Integral Calculus, 
 
 2 vols. in 1, small 8vo, 2 50 
 
 " Differential Calculus Small 8vo, 3 00 
 
 " Abridgment of Differential Calculus. 
 
 Small 8vo, 1 50 
 
 Totten's Metrology 8vo, 2 50 
 
 Warren's Descriptive Geometry 2 vols., 8vo, 3 50 
 
 " Drafting Instruments .12mo, 1 25 
 
 " Free-hand Drawing 12mo, 100 
 
 " Linear Perspective 12mo, 100 
 
 " Primary Geometry 12mo, 75 
 
 Plane Problems 12mo, 1 25 
 
 " Problems and Theorems 8vo, 2 50 
 
 " Projection Drawing 12mo, 1 50 
 
 Wood's Co-ordinate Geometry 8vo, 2 00 
 
 " Trigonometry 12mo, 1 00 
 
 Woolf's Descriptive Geometry Large 8vo, 3 00 
 
 MECHANICS-MACHINERY. 
 
 (See also ENGINEERING, p. 7.) 
 
 Baldwin's Steam Heating for Buildings .12mo, 2 50 
 
 Barr's Kinematics of Machinery 8vo, 2 50 
 
 Benjamin's Wrinkles and Eecipes 12ino, 2 00 
 
 Chordal's Letters to Mechanics 12mo, 2 00 
 
 Church's Mechanics of Engineering 8vo, 6 00 
 
 " Notes and Examples in Mechanics 8vo, 2 00 
 
 Crehore's Mechanics of the Girder 8vo, 5 00 
 
 Cromwell's Belts and Pulleys 12mo, 1 50 
 
 Toothed Gearing 12mo, 1 50 
 
 Compton's First Lessons in Metal Working 12mo, 1 50 
 
 Compton and De Groodt's Speed Lathe 12mo, 1 50 
 
 Dana's Elementary Mechanics 12mo, 1 50 
 
 Dingey's Machinery Pattern Making .12mo, 2 00 
 
 * Dredge's Trans. Exhibits Building, World Exposition. 
 
 Large 4to, half morocco, 5 00 
 
 Du Bois's Mechanics. Yol. I., Kinematics 8vo, 3 50 
 
 " " Vol. II., Statics 8vo, 400 
 
 " " Vol. III., Kinetics 8vo, 350 
 
 Fitzgerald's Boston Machinist 18mo, 1 00 
 
 Flather's Dynamometers 12mo, 2 00 
 
 Rope Driving 12mo, 200 
 
 Hall's Car Lubrication 12mo, 1 00 
 
 HoMy's Saw Filing 18mo, 75 
 
 12 
 
* Johnson's Theoretical Mechanics. An Elementary Treatise. 
 
 12mo, $3 00 
 
 Jones's Machine Design. Part I., Kinematics 8vo, 1 50 
 
 " Part II., Strength and Proportion of 
 
 Machine Parts 8vo, 3 00 
 
 Lanza's Applied Mechanics 8vo, 7 50 
 
 MacCord's Kinematics 8vo, 5 00 
 
 Merrimau's Mechanics of Materials 8vo, 4 00 
 
 Metcalfe's Cost of Manufactures 8vo, 5 00 
 
 *Michie's Analytical Mechanics 8vo, 4 00 
 
 Richards's Compressed Air 12mo, 1 50 
 
 Robinson's Principles of Mechanism 8vo, 3 00 
 
 Smith's Press-working of Metals 8vo, H 00 
 
 Thurston's Friction and Lost Work 8vo, 3 00 
 
 The Animal as a Machine 12mo, 1 00 
 
 Warren's Machine Construction 2 vols., 8vo, 7 50 
 
 Weisbaclrs Hydraulics and Hydraulic Motors. (Du Bois.)..8vo, 5 00 
 Mechanics of Engineering. Vol. III., Part I., 
 
 Sec. I. (Klein.) 8vo, 500 
 
 Weisbach's Mechanics of Engineering. Vol. III., Part I., 
 
 Sec. II. (Klein.) 8vo, 5 00 
 
 Weisbach's Steam Engines. (Du Bois.) 8vo, 500 
 
 Wood's Analytical Mechanics 8vo, 3 00 
 
 " Elementary Mechanics 12mo, 125 
 
 " " Supplement and Key 12ino, 1 25 
 
 METALLURGY. 
 
 Allen's Tables for Iron Analysis 8vo, 3 00 
 
 Egleston's Gold and Mercury Large 8vo, 7 50 
 
 Metallurgy of Silver Large 8vo, 7 50 
 
 * Kerl's Metallurgy Steel, Fuel, etc 8vo, 15 00 
 
 Kunhardt's Ore Dressing in Europe 8vo, 1 50 
 
 Metcalf's Steel A Manual for Steel Users 12mo, 2 00 
 
 O'Driscoll's Treatment of Gold Ores 8vo, 2 00 
 
 Thurston's Iron and Steel 8vo, 3 50 
 
 Alloys 8vo, 250 
 
 Wilson's Cyanide Processes '. 12mo, 1 50 
 
 MINERALOGY AND MINING. 
 
 Barringer's Minerals of Commercial Value Oblong morocco, 2 50 
 
 Beard's Ventilation of Mines 12mo, 2 50 
 
 Boyd's Resources of South Western Virginia 8vo, 3 00 
 
 Map of South Western Virginia Pocket-book form, 2 00 
 
 Brush and Penfield's Determinative Mineralogy. New Ed. 8vo, 4 00 
 
 13 
 
Chester's Catalogue of Minerals 8vo, $1 25 
 
 Paper, 50 
 
 " Dictionary of the Names of Minerals .8vo, 3 00 
 
 Dana's American Localities of Minerals Large 8vo, 1 00 
 
 " Descriptive Mineralogy. (E.S.) Large 8vo. half morocco, 12 50 
 
 " First Appendix to System of Mineralogy. . . .Large 8vo, 1 00 
 
 " Mineralogy and Petrography. (J. D.) 12mo, 2 00 
 
 " Minerals and How to Study Them. (E. S.).. 12mo, 1 50 
 
 " Text-book of Mineralogy. (E. S.).. .New Edition. 8vo, 400 
 
 * Drinker's Tunnelling, Explosives, Compounds, and Rock Drills. 
 
 4to, half morocco, 25 00 
 
 Egleston's Catalogue of Minerals and Synonyms 8vo, 2 50 
 
 Eissler's Explosives Nitroglycerine and Dynamite 8vo, 4 00 
 
 Hussak's Rock forming Minerals. (Smith.) Small 8vo, 2 00 
 
 Ihlseng's Manual of Mining . . 8vo, 4 00 
 
 Kunhardt's Ore Dressing in Europe 8vo, 1 50 
 
 O'Driscoll's Treatment of Gold Ores 8vo, 2 00 
 
 * Penfield's Record of Mineral Tests Paper, 8vo, 50 
 
 Rosenbusch's Microscopical Physiography of Minerals and 
 
 Rocks. (Idduigs.) 8vo, 500 
 
 Sawyer's Accidents in Mines Large 8vo, 7 00 
 
 Stockbridge's Rocks and Soils 8vo, 2 50 
 
 *Tillman's Important Minerals and Rocks 8vo, 2 00 
 
 "Walke's Lectures on Explosives 8vo, 4 00 
 
 Williams's Lithology 8vo, 3 00 
 
 Wilson's Mine Ventilation 12mo, 125 
 
 Hydraulic and Placer Mining ...... 12mo, 2 50 
 
 STEAM AND ELECTRICAL ENGINES, BOILERS, Etc. 
 
 (See also ENGINEERING, p. 7.) 
 
 Baldwin's Steam Heating for Buildings 12mo 2 50 
 
 Clerk's Gas Engine Small 8vo, 4 00 
 
 Ford's Boiler Making for Boiler Makers 18mo, 1 00 
 
 Hemen way's Indicator Practice 12mo, 2 00 
 
 Kent's Steam-boiler Economy , .'. 8vo, 4 00 
 
 Kneass's Practice and Theory of the Injector 8vo, 1 50 
 
 MacCord's Slide Valve 8vo, 2 00 
 
 Meyer's Modern Locomotive Construction 4to, 10 00 
 
 Peabody and Miller's Steam-boilers 8vo, 4 00 
 
 Peabody's Tables of Saturated Steam 8vo, 1 00 
 
 " Thermodynamics of the Steam Engine 8vo, 5 00 
 
 Valve Gears for the Steam Engine 8vo, 250 
 
 " Manual of the Steam-engine Indicator 12mo, 1 50 
 
 Fray's Twenty Years with the Indicator , , , .Large 8vo, 2 50 
 
 14 
 
Pupin and Ostcrberg's Thermodynamics 12mo, $1 25 
 
 Reagan's Steam and Electric Locomotives . .12mo, 2 00 
 
 Rontgen's Thermodynamics. (Du Bois. ) 8vo, 5 00 
 
 Sinclair's Locomotive Running 12mo, 2 00 
 
 Snow 's Steam-boiler Practice 8vo. 3 00 
 
 Thurston's Boiler Explosions 12mo, 1 50 
 
 Engine and Boiler Trials 8vo, 500 
 
 " Manual of the Steam Engine. Part I., Structure 
 
 and Theory 8vo, 6 00 
 
 Manual of the Steam Engine. Part II., Design, 
 
 Construction, and Operation 8vo, 6 00 
 
 2 parts, 10 00 
 
 " Philosophy of the Steam Engine 12mo, 75 
 
 " Reflection, on the Motive Power of Heat. (Caruot.) 
 
 12mo, 1 50 
 
 Stationary Steam Engines 8vo, 2 50 
 
 " Steam-boiler Construction and Operation 8vo, 5 00 
 
 Spangler's Valve Gears 8vo, 2 50 
 
 Notes on Thermodynamics 12mo, 1 00 
 
 Weisbach's Steam Engine. (Du Bois.) 8vo ; 500 
 
 Whitham's Steam-engine Design.., .-, 8vo, 5 00 
 
 Wilson's Steam Boilers. (Flather.) 12mo, 250 
 
 Wood's Thermodynamics, Heat Motors, etc 8vo, 4 00 
 
 TABLES, WEIGHTS, AND MEASURES. 
 
 Adrian ce's Laboratory Calculations 12nio, 1 25 
 
 Allen's Tables for Iron Analysis , . .8vo, 3 00 
 
 Bixby's Graphical Computing Tables Sheet, 25 
 
 Cornptou's Logarithms 12mo, 1 50 
 
 Crandall's Railway and Earthwork Tables 8vo, 1 50 
 
 Davis's Elevation and Stadia Tables Small 8vo, 1 00 
 
 Fisher's Table of Cubic Yards Cardboard, 25 
 
 Hudson's Excavation Tables. Vol. II 8vo, 1 00 
 
 Johnson's Stadia and Earthwork Tables 8vo, 1 25 
 
 Ludlow's Logarithmic and Other Tables. (Bass.) 12mo, 2 00 
 
 Totten's Metrology 8vo, 2 50 
 
 VENTILATION. 
 
 Baldwin's Steam Heating 12rno, 2 50 
 
 Beard's Ventilation of Mines. 12mo, 2 50 
 
 Carpenter's Heating and Ventilating of Buildings 8vo, 3 00 
 
 Gerhard's Sanitary House Inspection 12mo, 1 00 
 
 Wilson's Mine Ventilation 12mo, I 25 
 
 15 
 
MISCELLANEOUS PUBLICATIONS. 
 
 Alcott's Gems, Sentiment, Language Gilt edges, $5 00 
 
 Emmou's Geological Guide-book of the Rocky Mountains. .8vo, 1 50 
 
 Ferrel' s Treatise ou the Winds 8vo, 4 00 
 
 Haines's Addresses Delivered before the Am. Ry. Assn. ..12mo, 2 50 
 
 Mott's The Fallacy of the Present Theory of Sound. .Sq. 16mo, 1 00 
 
 Richards's Cost of Living 12mo, 1 00 
 
 Ricketts's History of Rensselaer Polytechnic Institute 8vo, 3 00 
 
 Rotherham's The New Testament Critically Emphasized. 
 
 12mo, 1 50 
 " The Emphasized New Test. A new translation. 
 
 Large 8vo, 2 00 
 
 Totten's An Important Question in Metrology 8vo, 2 50 
 
 HEBREW AND CHALDEE TEXT-BOOKS. 
 
 FOR SCHOOLS AND THEOLOGICAL SEMINARIES. 
 
 Gesenius's Hebrew and Chaldee Lexicon to Old Testament. 
 
 (Tregelles.) Small 4to, half morocco, 5 00 
 
 Green's Elementary Hebrew Grammar 12mo, 1 25 
 
 " Grammar of the Hebrew Language (New Edition). 8 vo, 3 00 
 
 " Hebrew Chrestomathy 8vo, 2 00 
 
 Letteris's Hebrew Bible (Massoretic Notes in English). 
 
 8vo, arabesque, 2 25 
 
 MEDICAL. 
 
 Hammarsten's Physiological Chemistry. (Mandel.) 8vo, 4 00 
 
 Mott's Composition, Digestibility, and Nutritive Value of Food. 
 
 Large mounted chart, 1 25 
 
 Ruddiman's Incompatibilities in Prescriptions 8vo, 2 00 
 
 Steel's Treatise on the Diseases of the Dog 8vo, 3 50 
 
 WoodhulPs Military Hygiene 16mo, 1 50 
 
 Worcester's Small Hospitals Establishment and Maintenance, 
 including Atkinson's Suggestions for Hospital Archi- 
 tecture 12mo, 1 25 
 
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