ENGIN. 
 LIBRARY 
 
 UC-NRLF 
 
 B 3 HE 
 

 
 Mechanics Department 
 
 Engineering 
 Library 
 
' 
 


 FORMULAS 
 
 IN 
 
 GEAR ING 
 
 THIRD EDITION. 
 
 WITH PRACTICAL SUGGESTIONS. 
 
 PROVIDENCE, R. I. 
 
 BROWN & SHARPE MANUFACTURING COMPANY 
 
 1900. 
 
Engineering- 
 Library 
 
 , 
 
 ":/; V ': 
 
 DEPT. 
 
 Entered according to Act of Congress, in the year 1900 by 
 
 BROWN & SHARPE MF(r. CO., 
 
 In the Office of the Librarian of Congress at Washington. 
 
 Registered at Stationers' Hall, London, Eng. 
 
 All rights reserved. 
 
.PREFACE. 
 
 It is the aim, in the following pages, to condense as much 
 as possible the solution of all problems in gearing which in the 
 ordinary practice may be met with, to the exclusion of prob- 
 lems dealing with transmission of power and strength of 
 gearing. The simplest and briefest being the symbolical 
 expression, it has, whenever available, been resorted to. The 
 mathematics employed are of a simple kind, and will present 
 no difficulty to anyone familiar with ordinary Algebra and 
 the elements of Trigonometry. 
 
 735767 
 
CONTENTS. 
 
 FORMULAS IN GEARING. 
 
 CHAPTER I. 
 
 PAGB 
 Systems of Gearing ... . . i 
 
 CHAPTER II. 
 
 Spur Gearing Formulas Table of Tooth Parts Comparative Sizes 
 
 of Gear Teeth 4 
 
 CHAPTER III. 
 
 Bevel Gears, Axes at Right Angles Formulas Bevel Gears, Axes at 
 any Angle Formulas Undercut in Bevel Gears Diameter Incre- 
 ment Tables for Angles of Edge and Angles of Face Tables of 
 Natural Lines 1 1 
 
 CHAPTER IV. 
 
 Worm and Worm Wheel, Formulas Undercut in Worm Wheels 
 
 Table for gashing- Worm Wheels 34 
 
 CHAPTER V. 
 
 Spiral or Screw Gearing Axes Parallel Axes at Right Angles 
 Axes at any Angle General Formulas Table of Prime Num- 
 bers and Factors 4 
 
 CHAPTER VI. 
 Internal Gearing Internal Spur Gearing Internal Bevel Gears 58 
 
 CHAPTER VII. 
 Gear Patterns 64 
 
 CHAPTER VIII. 
 Dimensions and Form for Bevel Gear Cutters 67 
 
 CHAPTER IX. 
 Directions for cutting Bevel Gears with Rotary Cutter 70 
 
 CHAPTER X. 
 The Indexing of any Whole or Fractional Number 73 
 
 CHAPTER XL 
 
 The Gearing of Lathes for Screw Cutting Simple Gearing Compound 
 
 Gearing Cutting a Multiple Screw 77 
 
FORMULAS IN GEARING. 
 
 I. 
 
 SYSTEMS OF GEARING. 
 
 (Figs, i, 2.) 
 
 There are in common use two systems of gearing, viz.: the 
 involute and the epicycloidal. 
 
 In the involute system the-outlines of the working parts of a 
 tooth are single curves, which may be traced by a point in a 
 flexible, inextensible cord being unwound from a circular disk 
 the circumference of which is called the base circle, the disk 
 being concentric with the pitch circle of the gear. 
 
 In Fig. i the two base circles are represented as tangent to 
 the line P P. This line (P P) is variously called " the line of 
 pressure," '" the line of contact," or " the line of action." 
 
BROWN & SHARPE MFG. CO. 
 
 In our practice this is drawn so as to make with a normal 
 to the center line (O O') 14/4, or with the center line 75%. 
 
 The rack of this system has teeth with straight sides, the two 
 sides of a tooth making, together, an angle of 29 (twice 
 
 This applies to gears having 30 teeth or more. For gears 
 having less than 30 teeth special rules are followed, which are 
 explained in our " Practical Treatise on Gearing." 
 
 Fig. 2. 
 
 In epicycloidal, or double-curve teeth, the formation of the 
 curve changes at the pitch circle. The outline of the faces of 
 epicycloidal teeth may be traced by a point in a circle rolling 
 on the outside of pitch circle of a gear, and the flanks by a point 
 in a circle rolling on the inside of the pitch circle. The faces 
 of one gear must be traced by the same circle that traces the 
 flanks of the engaging gear. 
 
 In our practice the diameter of the rolling or describing 
 circle is equal to the radius of a i5-tooth gear of the pitch 
 required ; this is the base of the system. The same describing 
 circle being used for all gears of the same pitch. 
 
PROVIDENCE, R. I. 3 
 
 The teeth of the rack of this system have double curves, 
 which may be traced by the base circle rolling alternately on 
 each side of the pitch line. 
 
 An advantage of the involute over the epicycloidal tooth is, 
 that in action gears having involute teeth may be separated a 
 little from their normal positions without interfering with the 
 angular velocity, which is not possible in any other kind of 
 tooth. 
 
 The obliquity of action is sometimes urged as an objection 
 to involute teeth, but a full consideration of the subject will 
 show that the importance of this has been greatly over-esti- 
 mated. 
 
 The tooth dimensions for both the involute and epicycloidal 
 gears may be calculated from the formulas in Chapter II. 
 
BROWN & SHARPE MFG. CO. 
 
 n. 
 
 SPUR GEARING. 
 
 (Figs. *, 4.) 
 
 Two spur gears in action are comparable to two correspond- 
 ing plain rollers whose surfaces are in contact, these surfaces 
 representing the pitch circles of the gears. 
 
 PITCH OF GEARS. 
 
 For convenience of expression the pitch of gears may be 
 stated as follows : 
 
 Circular pitch is the distance from the center of one tooth to 
 the center of the next tooth, measured on the pitch line. 
 
 Diametral pitch is the number of teeth in a gear per inch of 
 pitch diameter. That is, a gear that has, say, six teeth for each 
 inch in pitch diameter is six diametral pitch, or, as the expres- 
 sion is universally abbreviated, it is " six pitch." This is by 
 far the most convenient way of expressing the relation of 
 diameter to number of teeth. 
 
 Module is the pitch diameter of a gear divided by the 
 number of teeth. 
 
 Chordal pitch is a term but little employed. It is the dis- 
 tance from center to center of two adjacent teeth measured in 
 a straight line. 
 
Gear Tooth i F>. 
 
 Ctiordal Thickness of Teetn for Gears on 
 a Basis of i Diametral Pitch. 
 
 S=Distance from pitch line to top- of teeth. 
 
 S Corrected=H+S. 
 
 N= Number of teeth in gear. 
 
 T=Chordal thickness of Tooth. T=D' sin. ft' 
 
 H = Height of Arc. H = R ( i cos. ft') 
 
 D'= Pitch Diameter. 
 
 R= Pitch Radius. 
 
 ^'=90 divided by the number of teeth. 
 
 NOTE When tin 1 tooth of H Rear is measured, add the height of arc to iSi. 
 
Chordal Thickness 
 
 OF 
 
 GEAR TEETH. 
 
 The dimensions of Tooth Parts as given in the tables, pages 6 to 9, 
 are correct according to the definition of Tooth Parts ; but, as the pitch 
 line of gears is curved, the thickness of a tooth will not be measured on 
 the pitch line if the Caliper is set to the figures given in Tables of Tooth 
 Parts. 
 
 To measure the tooth accurately the 
 Caliper must be set to the 
 
 Chorclal 
 
 See Formula on reverse page 
 
 Gear 
 
 Tooth Calipei 
 
PROVIDENCE, R. I. 
 
 FORMULAS. 
 
 N = number of teeth. 
 
 s = addendum and module. 
 
 / = thickness of tooth on pitch line. 
 
 /= clearance at bottom of tooth. 
 D" = working depth of tooth. 
 D" + / = whole depth of tocJi. 
 
 d pitch diameter. 
 (f = outside diameter. 
 P' = circular pitch. 
 P^ = chord pitch. 
 
 P = diametral pitch, 
 C = center distance. 
 
 P _N 
 
 P' = - 
 
 N N + 2 
 
 2 2 P 
 
 10 
 
 D" = 2 S 
 
 p-I^lf 
 
 * 
 
 P' = dn where sin d 
 
 d' - d + 2 s 
 
BROWN & SHARPE MFG. CO. 
 
 GEAR WHEELS. 
 
 TABLE OF TOOTH PARTS CIRCULAR PITCH IN FIRST COLUMN. 
 
 Circular 
 Pitch. 
 
 Threads or 
 Teeth per inch 
 Linear . 
 
 Diametral 
 Pitch. 
 
 Thickness of 
 Tooth on 
 Pitch Line. 
 
 Addendum 
 and Module. 
 
 Working Depth 
 of Tooth. 
 
 & o5 
 
 213 
 
 121 
 
 < 
 
 
 Whole Depth 
 of Tooth. 
 
 =L 
 
 fiw 
 Fj. 
 
 PI 
 
 oS 
 
 a is 
 
 T3 TJ 
 
 *l 
 
 P' 
 
 i" 
 f 
 
 P 
 
 t 
 
 s 
 
 D" 
 
 +/ 
 
 D"+/ 
 
 P'X.31 
 
 P'X.335 
 
 2 
 
 i 
 
 2 
 
 1.5708 
 
 1.0000 
 
 .6366 
 
 1.2732 
 
 .7366 
 
 1.3732 
 
 .6200 
 
 .6700 
 
 If 
 
 8 
 IB" 
 
 1.6755 
 
 .9375 
 
 .5968 
 
 1.1937 
 
 .6906 
 
 1.2874 
 
 .5813 
 
 .6281 
 
 H 
 
 4 
 
 T 
 
 1.7952 
 
 .8750 
 
 .5570 
 
 1.1141 
 
 .6445 
 
 1.2016 
 
 .5425 
 
 .5863 
 
 H 
 
 8 
 
 laT 
 
 1.9333 
 
 .8125 
 
 .5173 
 
 1.0345 
 
 .5985 
 
 1.1158 
 
 .5038 
 
 .5444 
 
 11 
 
 2 
 3 
 
 2.0944 
 
 .7500 
 
 .4775 
 
 .9549 
 
 .5525 
 
 1.0299 
 
 .4650 
 
 .5025 
 
 Ifr 
 
 16 
 23 
 
 2.1855 
 
 .7187 
 
 .4576 
 
 .9151 
 
 .5294 
 
 .9870 
 
 .4456 
 
 .4816 
 
 11 
 
 JL 
 11 
 
 2.2848 
 
 .6875 
 
 4377 
 
 .8754 
 
 .5064 
 
 .9441 
 
 .4262 
 
 .4606 
 
 11 
 
 3 
 4 
 
 2.3562 
 
 .6666 
 
 .4244 
 
 .8488 
 
 .4910 
 
 .9154 
 
 .4133 
 
 .4466 
 
 1* 
 
 16 
 21 
 
 2.3936 
 
 .6562 
 
 4178 
 
 .8356 
 
 .4834 
 
 .9012 
 
 .4069 
 
 .4397 
 
 H 
 
 4 
 5 
 
 2.5133 
 
 .6250 
 
 .3979 
 
 .7958 
 
 .4604 
 
 .8583 
 
 .3875 
 
 .4188 
 
 i* 
 
 16 
 19 
 
 2.6456 
 
 .5937 
 
 .3780 
 
 .7560 
 
 .4374 
 
 .8156 
 
 .3681 
 
 .3978 
 
 il 
 
 8 
 9 
 
 2.7925 
 
 .5625 
 
 3581 
 
 .7162 
 
 .4143 
 
 .7724 
 
 .3488 
 
 .3769 
 
 i* 
 
 if 
 
 2.9568 
 
 .5312 
 
 3382 
 
 .6764 
 
 .3913 
 
 .7295 
 
 .3294 
 
 .3559 
 
 i 
 
 1 
 
 3.1416 
 
 .5000 
 
 3183 
 
 .6366 
 
 .3683 
 
 .6866 
 
 .3100 
 
 .3350 
 
 15 
 10 
 
 1* 
 
 3.3510 
 
 .4687 
 
 2984 
 
 .5968 
 
 .3453 
 
 .6437 
 
 .2906 
 
 .3141 
 
 7 
 8 
 
 1 1 
 
 AT 
 
 3.5904 
 
 .4375 
 
 2785 
 
 .5570 
 
 .3223 
 
 .6007 
 
 2713 
 
 .2931 
 
 
 
 ia 
 
 3.8666 
 
 .4062 
 
 2586 
 
 -.5173 
 
 .2993 
 
 .5579 
 
 .2519 
 
 .2722 
 
 f 
 
 11 
 
 3.9270 
 
 .4000 
 
 2546 
 
 .5092 
 
 2946 
 
 .5492 
 
 2480 
 
 .2680 
 
 _L 
 
 4 
 
 n 
 
 4.1888 
 
 .3750 
 
 2387 
 
 .4775 
 
 2762 
 
 .5150 
 
 2325 
 
 .2513 
 
 11 
 
 16 
 
 if 
 
 4.5696 
 
 .3437 
 
 2189 
 
 .4377 
 
 .2532 
 
 .4720 
 
 2131 
 
 .2303 
 
 2 
 3 
 
 il 
 
 4.7124 
 
 .3333 
 
 2122 
 
 .4244 
 
 2455 
 
 .4577 
 
 2066 
 
 .2233 
 
 5 
 
 8 
 
 il 
 
 5.0265 
 
 .3125 
 
 1989 
 
 .3979 
 
 2301 
 
 .4291 
 
 1938 
 
 .2094 
 
 3 
 
 ' 5 
 
 il 
 
 5.2360 
 
 .3000 
 
 1910 
 
 .3820 
 
 2210 
 
 .4120 
 
 1860 
 
 .2010 
 
 JL. 
 
 7 
 
 if 
 
 5.4978 
 
 .2857 
 
 1819 
 
 .3638 
 
 2105 
 
 .3923 
 
 1771 
 
 .1914 
 
 J_ 
 16 
 
 11 
 
 5.5851 
 
 .2812 
 
 1790 
 
 .3581 
 
 2071 
 
 .3862 
 
 1744 
 
 .1884 
 
PROVIDENCE, R. I. 
 
 TABLE OF TOOTH PARTS. Continued. 
 
 CIRCULAR PITCH IN FIRST COLUMN. 
 
 Circular 
 Pitch. 
 
 Threads or 
 Teeth per inch 
 Linear. 
 
 Diametral 
 Pitch. 
 
 Thickness of 
 Tooth on 
 Pitch Line. 
 
 Addendum 
 and Module. 
 
 l~ 
 
 fil 
 
 bo o 
 |H 
 
 fo 
 
 
 
 o3 $ 
 
 i*'\ 
 <~ J2 ^ 
 
 ^ A 
 
 -d r 
 
 J* 
 
 
 1 
 
 3* 
 
 *!* 
 
 li 
 
 1-1 O> -M 
 
 *r 
 
 *l 
 
 3^ 
 
 1 
 2 
 
 i 
 
 P' 
 
 i" 
 "p 7 
 
 P 
 
 t 
 
 s s 
 
 D" 
 
 +/ 
 
 D'4-/. 
 
 PX.31 
 
 PX.335 
 
 1 
 
 2 
 
 6.2832 
 
 .2500 
 
 .1592 
 
 .3183 
 
 .1842 
 
 .3433 
 
 .1550 
 
 .1675 
 
 k 
 e 
 
 2| 
 
 7.0685 
 
 .2222 
 
 .1415 
 
 .2830 
 
 .1637 
 
 .3052 
 
 .1378 
 
 .1489 
 
 7 
 1C 
 
 2f 
 
 7.1808 
 
 .2187 
 
 .1393 
 
 .2785 
 
 .1611 
 
 .3003 
 
 .1356 
 
 .1466 
 
 A 
 
 7 
 
 21 
 
 7.3304 
 
 .2143 
 
 .1364 
 
 .2728 
 
 .1578 
 
 .2942 
 
 .1328 
 
 .1436 
 
 2 
 
 T 
 
 21 
 
 7.8540 
 
 .2000 
 
 .1273 
 
 .2546 
 
 .1473 
 
 .2746 
 
 .1240 
 
 .1340 
 
 3 
 8 
 
 2f 
 
 8.3776 
 
 .1875 
 
 .1194 
 
 .2387 
 
 .1381 
 
 .2575 
 
 .1163 
 
 .1256 
 
 4 
 
 11 
 
 2f 
 
 8.6394 
 
 .1818 
 
 .1158 
 
 .2316 
 
 .1340 
 
 .2498 
 
 .1127 
 
 .1218 
 
 1 
 T 
 
 3 
 
 9.4248 
 
 .1666 
 
 .1061 
 
 .2122 
 
 .1228 
 
 .2289 
 
 .1033 
 
 .1117 
 
 5 
 16 
 
 31 
 
 10.0531 
 
 .1562 
 
 .0995 
 
 .1989 
 
 .1151 
 
 .2146 
 
 .0969 
 
 .1047 
 
 To 
 
 31 
 
 10.4719 
 
 .1500 
 
 .0955 
 
 .1910 
 
 .1105 
 
 .2060 
 
 .0930 
 
 .1005 
 
 2 
 
 7 
 
 31 
 
 10.9956 
 
 .1429 
 
 .0909 
 
 .1819 
 
 .1052 
 
 .1962 
 
 .0886 
 
 .0957 
 
 1 
 i 
 
 4 
 
 12.5664 
 
 .1250 
 
 .0796 
 
 .1591 
 
 .0921 
 
 .1716 
 
 .0775 
 
 .0838 
 
 T 
 
 41 
 
 14.1372 
 
 .1111 
 
 .0707 
 
 .1415 
 
 .0818 
 
 .1526 
 
 .0689 
 
 .0744 
 
 T 
 
 5 
 
 15.7080 
 
 .1000 
 
 .0637 
 
 .1273 
 
 .0737 
 
 .1373 
 
 .0620 
 
 .0670 
 
 3 
 10 
 
 5i 
 
 16.7552 
 
 .0937 
 
 .0597 
 
 .1194 
 
 .0690 
 
 .1287 
 
 .0581 
 
 .0628 
 
 2 
 11 
 
 51 
 
 17.2788 
 
 .0909 
 
 .0579 
 
 .1158 
 
 .0670 
 
 .1249 
 
 .0564 
 
 .0609 
 
 1 
 
 T 
 
 6 
 
 18.8496 
 
 .0833 
 
 .0531 
 
 .1061 
 
 .0614 
 
 .1144 
 
 .0517 
 
 .0558 
 
 2 
 13 
 
 61 
 
 20.4203 
 
 .0769 
 
 .0489 
 
 .0978 
 
 .0566 
 
 .1055 
 
 .0477 
 
 .0515 
 
 1 
 
 7 
 
 7 
 
 21.9911 
 
 .0714 
 
 .0455 
 
 .0910 
 
 .0526 
 
 .0981 
 
 .0443 
 
 .0479 
 
 2 
 
 15 
 
 71 
 
 23.5619 
 
 .0666 
 
 .0425 
 
 .0850 
 
 .0492 
 
 .0917 
 
 .0414 
 
 .0446 
 
 1 
 
 T 
 
 8 
 
 25.1327 
 
 .0625 
 
 .0398 
 
 .0796 
 
 .0460 
 
 .0858 
 
 .0388 
 
 .0419 
 
 1 
 
 9 
 
 9 
 
 28.2743 
 
 .0555 
 
 .0354 
 
 .0707 
 
 .0409 
 
 .0763 
 
 .0344 
 
 .0372 
 
 1 
 
 10 
 
 10 
 
 31.4159 
 
 .0500 
 
 .0318 
 
 .0637 
 
 .0368 
 
 .0687 
 
 .0310 
 
 .0335 
 
 1 
 10 
 
 16 
 
 50.2655 
 
 .0312 
 
 .0199 
 
 .0398 
 
 .0230 
 
 .0429 
 
 .0194 
 
 .0209 
 
 1 
 
 20 
 
 20 
 
 62.8318 
 
 .0250 
 
 .0159 
 
 .0318 
 
 .0184 
 
 .0343 
 
 .0155 
 
 .0167 
 
BROWN & SHARPE MFG. CO. 
 
 GEAR WHEELS. 
 
 TABLE OF TOOTH PARTS DIAMETRAL PITCH IN FIRST COLUMN. 
 
 Diametral 
 Pitch. 
 
 Circular 
 Pitch. 
 
 Thickness 
 of Tooth on 
 Pitch Line. 
 
 Addendum 
 and Module. 
 
 Working Depth 
 of Tooth. 
 
 09 
 
 fl 
 
 Q 
 
 I* 
 
 P 
 
 P' 
 
 t 
 
 s 
 
 D" 
 
 +/. 
 
 D"+/. 
 
 i 
 
 6.2832 
 
 3.1416 
 
 2.0000 
 
 4.0000 
 
 2.3142 
 
 4.3142 
 
 I 
 
 4.1888 
 
 2.0944 
 
 1.3333 
 
 2.6666 
 
 1.5428 
 
 2.8761 
 
 1 
 
 3.1416 
 
 1.5708. 
 
 1.0000 
 
 2.0000 
 
 1.1571 
 
 2.1571 
 
 1J 
 
 2.5133 
 
 1.2566 
 
 .8000 
 
 1.6000 
 
 .9257 
 
 1.7257 
 
 !i 
 
 2.0944 
 
 1.0472 
 
 .6666 
 
 1.3333 
 
 .7714 
 
 1.4381 
 
 If 
 
 1.7952 
 
 .8976 
 
 .5714 
 
 1.1429 
 
 .6612 
 
 1.2326 
 
 2 
 
 1.5708 
 
 .7854 
 
 .5000 
 
 1.0000 
 
 .5785 
 
 1.0785 
 
 2J 
 
 1.3963 
 
 .6981 
 
 .4444 
 
 .8888 
 
 .5143 
 
 .9587 
 
 2i 
 
 1.2566 
 
 .6283 
 
 .4000 
 
 .8000 
 
 .4628 
 
 .8628 
 
 2f 
 
 1.1424 
 
 .5712 
 
 .3636 
 
 .7273 
 
 .4208 
 
 .7844 
 
 3 
 
 1.0472 
 
 .5236 
 
 .3333 
 
 .6666 
 
 .3857 
 
 .7190 
 
 8J 
 
 .8976 
 
 .4488 
 
 .2857 
 
 .5714 
 
 .3306 
 
 .6163 
 
 4 
 
 .7854 
 
 .3927 
 
 .2500 
 
 .5000 
 
 .2893 
 
 .5393 
 
 5 
 
 .6283 
 
 .3142 
 
 .2000 
 
 .4000 
 
 .2314 
 
 .4314 
 
 6 
 
 .5236 
 
 .2618 
 
 .1666 
 
 .3333 
 
 .1928 
 
 .3595 
 
 7 
 
 .4488 
 
 .2244 
 
 .1429 
 
 .2857 
 
 .1653 
 
 .3081 
 
 8 
 
 .3927 
 
 .1963 
 
 .1250 
 
 .2500 
 
 .1446 
 
 .2696 
 
 9 
 
 .3491 
 
 .1745 
 
 .1111 
 
 .2222 
 
 .1286 
 
 .2397 
 
 10 
 
 .3142 
 
 .1571 
 
 .1000 
 
 .2000 
 
 .1157 
 
 .2157 
 
 11 
 
 .2856 
 
 .1428 
 
 .0909 
 
 .1818 
 
 .1052 
 
 .1961 
 
 12 
 
 .2618 
 
 .1309 
 
 0833 
 
 .1666 
 
 .0964 
 
 .1798 
 
 13 
 
 .2417 
 
 .1208 
 
 .0769 
 
 .1538 
 
 .0890 
 
 .1659 
 
 14 
 
 .2244 
 
 .1122 
 
 .0714 
 
 .1429 
 
 .0826 
 
 .1541 
 
PROVIDENCE, R. I. 
 
 TABLE OF TOOTH PAKTS Continued. 
 
 DIAMETRAL PITCH IN FIRST COLUMN. 
 
 M Diametral 
 Pitch. 
 
 Circular 
 Pitch. 
 
 Thickness 
 of Tooth on 
 Pitch Line. 
 
 Addendum 
 and Module. 
 
 Working Depth 
 of Tooth. 
 
 rv <U 
 
 "** 
 
 0^ 
 &> 
 
 ^^ 
 
 & 
 
 Q 
 
 tl 
 SI 
 
 0^ 
 
 r 3 
 
 
 P'. 
 
 t. 
 
 8. 
 
 D". 
 
 +/. 
 
 D" + / 
 
 15 
 
 .2094 
 
 .1047 
 
 .0866 
 
 .1333 
 
 .0771 
 
 .1438 
 
 16 
 
 .1963 
 
 .0982 
 
 .0625 
 
 .1250 
 
 .0723 
 
 .1348 
 
 17 
 
 .1848 
 
 .0924 
 
 .0588 
 
 .1176 
 
 .0681 
 
 .1269 
 
 18 
 
 .1745 
 
 .0873 
 
 .0555 
 
 .1111 
 
 .C643 
 
 .1198 
 
 19 
 
 .1653 
 
 .0827 
 
 .0526 
 
 .1053 
 
 .0609 
 
 .1135 
 
 20 
 
 .1571 
 
 .0785 
 
 .0500 
 
 .1000 
 
 .0579 
 
 .1079 
 
 22 
 
 .1428 
 
 .0714 
 
 .0455 
 
 .0909 
 
 .0526 
 
 .0980 
 
 24 
 
 .1309 
 
 .0654 
 
 .0417 
 
 .0833 
 
 .0482 
 
 .0898 
 
 26 
 
 .1208 
 
 .0604 
 
 .0385 
 
 .0769 
 
 .0445 
 
 .0829 
 
 28 
 
 .1122 
 
 .0561 
 
 .0357 
 
 .0714 
 
 .0413 
 
 .0770 
 
 30 
 
 .1047 
 
 .0524 
 
 .0333 
 
 .0666 
 
 .0386 
 
 .0719 
 
 32 
 
 .0982 
 
 .0491 
 
 .0312 
 
 .0625 
 
 .0362 
 
 .0674 
 
 34 
 
 .0924 
 
 .0462 
 
 .0294 
 
 .0588 
 
 .0340 
 
 .0634 
 
 36 
 
 .0873 
 
 .0436 
 
 .0278 
 
 .0555 
 
 .0321 
 
 .0599 
 
 38 
 
 .0827 
 
 .0413 
 
 .0263 
 
 .0526 
 
 .0304 
 
 .0568 
 
 40 
 
 .0785 
 
 .0393 
 
 .0250 
 
 .0500 
 
 .0289 
 
 .0539 
 
 42 
 
 .0748 
 
 .0374 
 
 .0238 
 
 .0476 
 
 .0275 
 
 .0514 
 
 44 
 
 .0714 
 
 .0357 
 
 .0227 
 
 .0455 
 
 .0263 
 
 .0490 
 
 46 
 
 .0683 
 
 .0341 
 
 .0217 
 
 .0435 
 
 .0252 
 
 .0469 
 
 48 
 
 .0654 
 
 .0327 
 
 .0208 
 
 .0417 
 
 .0241 
 
 .0449 
 
 50 
 
 .0628 
 
 .0314 
 
 .0200 
 
 .0400 
 
 .0231 
 
 .0431 
 
 56 
 
 .0561 
 
 .0280 
 
 .0178 
 
 .0357 
 
 .0207 
 
 .0385 
 
 60 
 
 .0524 
 
 .0262 
 
 .0166 
 
 .0333 
 
 .0193 
 
 .0360 
 
10 
 
 BROWN & SHARPE MFG. CO. 
 
 Comparative Sizes of Gear Teeth. 
 Involute. 
 
 8 P 
 
 Fig. 4. 
 
PROVIDENCE, R. I. 
 
 IT 
 
 CHAPTER III. 
 
 BEVEL GEARS. AXES AT RIGHT ANGLES. 
 
 (Fig. 5.) 
 
12 BROWN & SHARPE MFG. CO. 
 
 FORMULAS. 
 
 N-= [Number of teeth j 
 
 P = diametral pitch. 
 P' = circular pitch. 
 
 a a = } center angle = angle of edge j gear. 
 a b \ or pitch angle ( pinion. 
 
 ft = angle of top. 
 fi' angle of bottom. 
 
 g = [angle of face 
 
 A = apex distance from pitch circle. 
 A' = apex distance from large bottom of tooth. 
 
 d = pitch diameter. 
 d' = outside diameter. 
 
 s = addendum and module. 
 
 / = thickness of tooth at pitch line. 
 / = clearance at bottom of tooth. 
 D" = working depth of tooth. 
 D" + /= whole depth of tooth. 
 2 a = diameter increment. 
 
 b distance from top of tooth to plane of pitch circle, 
 F = width of face. 
 
PROVIDENCE, R. I. 13 
 
 tan .=-*--; tan . = 
 
 ;or tan /? = - 
 
 N A 
 
 tan /S' = sng2 + T == 2.314 sn . tan ^ = 
 
 a - 90 - K + ft) \g b = 90 - 
 = a ft' (See Note, page 69.) 
 
 A= 
 
 2 P sin a 
 
 N 
 
 A'= ^ A' = 
 
 cos p' 2 P sin a cos 
 
 A',= - - * -, cos ft 
 sin ( + /?) 
 
 P, N 
 
 2 A sin a 
 
 P 7T 
 
 2 a = 2 s cos <* (Seepage 20.) 
 
 a for gear = b for pinion 
 
 tf for pinion = * for gear 
 
 p _ ?T_ 
 - 
 
 J= JL = = .3183?' j = A tan/? 
 
 P 71 
 
 s + / = . 3685 P' s + / = A tan ft' 
 
 N~ N~ ~A~ 
 
 F = 1 + _ or = 2 P' to 3 P' 
 P 7 
 
 NOTE. Formulas containing notations without the designating letters a and b 
 apply equally to either gear or pinion. If wanted for one or the other, the respective 
 letters are simply attached. 
 
BROWN & SHARPE MFG. CO. 
 
 BEVEL GEARS WITH AXES AT ANY ANGLE. 
 
 Pinion 
 
 Fly. 6. 
 
PROVIDENCE, R. I, 
 
 FORMULAS. 
 
 C = angle formed by axes of gears. 
 N ; = } number of teeth ) ^ on . 
 
 P = diametral pitch. 
 P' = circular pitch. 
 
 ' "l=\ an le of ed S e = P itch an S le | pfnior . 
 ft = angle of top. 
 ft angle of bottom. 
 
 = [angle of face | S^on. 
 = [cutting angle \^ on 
 
 A = apex distance from pitch circle. 
 A' = apex distance from large bottom of tooth. 
 
 d= pitch diameter. 
 d' = outside diameter. 
 2 a = diameter increment. 
 
 b = distance from top of tooth to plane of pitch circle. 
 
 NOTE. The formulas for tooth parts as given on page 5 apply equally to these 
 cases. 
 
1 6 BROWN & SHARPE MFG. CO. 
 
 tan a a = -^ - ; or cot a a = - 4- cot C 
 ^ + cos C Na sm C 
 
 N a 
 
 sin C N 
 
 tan tx b = - ; or cot a b = ^ + cot C 
 
 N,,, N 6 sm C 
 
 NOTE. The above formulas are correct only for values of C less than 90 
 If C is greater than 90, consult page 18. 
 
 a 2 sm a s 
 
 tan p - - ; or tan p = ; 
 
 tan ft' = **+ = 2.314 sn nr . tan ft , = 5+ . 
 
 9 o (^ tt + /?) for Cases I and II. 
 
 ga = ft, for Case III. 
 
 ga 9 o (- /?) for Case IV. 
 
 5-6 = 90 (-*,+/?) 
 
 h a ^ (See page 6 9 . ) 
 
 N 
 
 f for Cases I and II, 
 
 a = a -\- z a \ . . . TTT TTT 
 
 I and pinions in Cases III and IV. 
 
 d' = d, for gear in Case III. 
 
 d' = d 2 a, for gear in Case IV. 
 
 2 a -=2s cos a 
 
 b = s sin # 
 
 NOTE. Formulas containing notations without the designating letters a and t> 
 apply equally to either gear or pinion. If wanted for one or the other, the 
 respective letters are simply attached. 
 
PROVIDENCE, R. I. 
 
l8 BROWN & SHARPE MFG. CO. 
 
 The formulas given for a a and a b (when C, N a and N 6 are 
 known) undergo some modifications for values of C greater 
 than 90. 
 
 For bevel gears at any angle but 90 we may distinguish 
 four cases ; C, N , N 6 being given. 
 
 /. Case. See pages 14 and 16. 
 
 II. Case. C is greater than 90. 
 
 sin (180 C) sin (180 C 
 
 tan of a = L- ; tan a b = __ 
 
 _ 6 -cos(i8o-C) a -cos(iSo-C } 
 
 N a N 6 
 
 ///. Case. (x a = 90 ; a b = C 90 
 IV. Case. 
 
 sinE sin E 
 
 1ST NT 
 
 cos E - 1N - & _ a - cos E 
 
 N a N 6 
 
 For an example to apply to Case III., the following condi- 
 tion must be fulfilled : 
 
 N a sin (C - 90) = N b 
 
 To distinguish whether a given example belongs to Case II. 
 or case IV., we are guided by the following condition : 
 
 T ^j /p. \ j smaller than N 6 , we have Case II. 
 
 iM a sm ({, 90 ) -j ] +v,an XT r^ v,o^^ case IV. 
 
PROVIDENCE, R. I. 19 
 
 UNDERCUT IN BEVEL GEARS. 
 
 By undercut in gears is understood a special formation of 
 the tooth, which may be explained by saying that the elements 
 of the tooth below the pitch line are nearer the center line of 
 the tooth than those on the pitch line. Such a tooth outline is 
 to be found only in gears with few teeth. In a pair of bevel 
 gears where the pinion is low-numbered and the ratio high, we 
 are apt to have undercut. For a pair of running gears this 
 condition presents no objection. Should, however, these gears 
 be intended as patterns to cast from, they would be found use- 
 less, from the fact that they would not draw out of the sand. 
 We have stated on page 2 (see Fig. i) that the base of our 
 involute system is the 14^ pressure angle. If a pair of bevel 
 gears with teeth constructed on this basis have undercut, we 
 can nearly eliminate the undercut and for the practical work- 
 ing this is quite sufficient by taking as a basis for the con- 
 struction of the tooth outline a pressure angle of 20. 
 
 The question now is : When do we, and when do we not 
 have undercut ? Let there be : 
 
 N = number of teeth in gear. 
 n = number of teeth in pinion. 
 
 n* = 4 
 
 where we have undercut for/ less than 30. 
 
 This formula is strictly correct for epicycloidal gears only. 
 It is, however, used as a safe and efficient approximation for 
 the involute system. 
 
20 
 
 BROWN & SHARPE MFG. CO. 
 
 DIAMETER INCREMENT. 
 
 2 a. 
 
 RULE. The ratio being given or determined, to find the outside diameter 
 divide figures given in table for large and small gear by pitch (P; and add 
 quotient to pitch diameter. 
 
 RATIO. 
 
 GEARS. 
 
 RATIO. 
 
 GEARS. 
 
 RATIO. 
 
 GEARS. 
 
 Large 
 
 Small 
 
 Large 
 
 Small 
 
 Large 
 
 SmalJ 
 
 1 00 
 
 1:1 
 
 .41 
 
 1.41 
 
 1 65 
 
 
 1.05 
 
 1.70 
 
 4.40 
 
 
 .45 
 
 1.94 
 
 1.05 
 
 
 .37 
 
 1.42 
 
 1.67 
 
 5:3 
 
 1.03 
 
 1.72 
 
 4 50 
 
 9:2 
 
 .44 
 
 1.95 
 
 1.07 
 
 
 .36 
 
 .43 
 
 1.70 
 
 
 1.01 
 
 1.73 
 
 4.60 
 
 
 .42 
 
 1 95 
 
 1.10 
 
 
 .35 
 
 .44 
 
 1.75 
 
 7:4 
 
 .99 
 
 1.74 
 
 4.80 
 
 
 .41 
 
 1.96 
 
 1.11 
 
 10:9 
 
 .34 
 
 .46 
 
 1.80 
 
 9:5 
 
 .97 
 
 1.75 
 
 5 00 5:1 
 
 .39 
 
 1.96 
 
 1.12 
 
 
 .33 
 
 46 
 
 1.85 
 
 
 .95 
 
 1.76 
 
 5.20 
 
 .38 | 1.96 
 
 1.13 
 
 9:8 
 
 .33 
 
 .47 
 
 1 90 
 
 
 .93 
 
 1 77 
 
 5.40 
 
 .37 1.96 
 
 1.14 
 
 8:7 
 
 .32 
 
 .49 
 
 1 95 
 
 
 .91 
 
 1.78 
 
 5.60 
 
 
 .36 
 
 1.97 
 
 1.15 
 
 
 1.31 
 
 .50 
 
 2 00 
 
 2:1 
 
 .89 
 
 .79 
 
 5.80 
 
 
 .34 
 
 1.97 
 
 1.16 
 
 
 1.30 
 
 .51 
 
 2.10 
 
 
 .87 
 
 .80 
 
 6.00 
 
 6:1 
 
 .33 
 
 1.97 
 
 .17 
 
 7:6 
 
 1.30 
 
 .52 
 
 2.20 
 
 
 .84 
 
 .81 
 
 6.20 
 
 
 .32 
 
 1 97 
 
 .18 
 
 
 1.29 
 
 1.53 
 
 2 25 
 
 9:4 
 
 .82 
 
 .82 
 
 6.40 
 
 
 .31 
 
 1.97 
 
 .19 
 
 
 1.28 
 
 1 53 
 
 2.30 
 
 
 .80 
 
 .83 
 
 6.60 
 
 
 .30 
 
 1 97 
 
 .20 
 
 6:5 
 
 1.28 
 
 1 54 
 
 2.33 
 
 7:3 
 
 .78 
 
 .84 
 
 6.80 
 
 
 .29 
 
 1 98 
 
 .23 
 
 
 1.27 
 
 1.55 
 
 2.40 
 
 
 .76 
 
 .85 
 
 7 00 
 
 7:1 
 
 .28 1.98 
 
 .25 
 
 5:4 
 
 1.25 
 
 1.56 
 
 2.50 
 
 5:2 
 
 .75 
 
 86 
 
 7.20 
 
 
 .27 
 
 1.98 
 
 .27 
 
 
 1.25 
 
 1.57 
 
 2.60 
 
 
 .73 
 
 .86 
 
 7.40 
 
 
 .27 
 
 1 98 
 
 .29 
 
 9:7 
 
 1.24 
 
 1.58 
 
 2.67 
 
 8:3 
 
 .71 
 
 .87 
 
 7.60 
 
 
 .26 1 98 
 
 1.30 
 
 
 1.22 
 
 1.59 
 
 2.70 
 
 
 .69 
 
 .87 
 
 7 80 
 
 
 .26 1.98 
 
 1.33 
 
 4:3 
 
 1.20 
 
 1.60 
 
 2.80 
 
 
 .67 
 
 .88 
 
 8 00 
 
 8:1 
 
 .25 1.98 
 
 1.35 
 
 
 1.18 
 
 1.61 
 
 2.90 
 
 
 .65 
 
 1.89 
 
 8.20 
 
 
 .24 1.98 
 
 1 37 
 
 
 1.17 
 
 1.61 
 
 3.00 
 
 3:1 
 
 .63 
 
 1.91 
 
 8 40 
 
 
 .24 1.98 
 
 1.40 
 
 7:5 
 
 1.16 
 
 1.62 
 
 3.20 
 
 
 .60 
 
 1.92 
 
 8.60 
 
 
 .23 1.98 
 
 1.43 
 
 10:7 
 
 1.15 
 
 1.63 
 
 3.33 
 
 
 .58 
 
 1 92 
 
 8.80 
 
 
 .23 1.98 
 
 1.45 
 
 
 1.13 
 
 1.65 
 
 3.40 
 
 
 .56 
 
 1 92 
 
 9.00 
 
 9:1 
 
 .22 1.99 
 
 1.50 
 
 3:2 
 
 1.11 
 
 1.66 
 
 3.50 
 
 7:2 
 
 .54 
 
 1.93 
 
 9.20 
 
 
 .22 1.99 
 
 1.53 
 
 
 1.10 
 
 1.67 
 
 3.60 
 
 
 .52 
 
 1 93 
 
 9.40 
 
 
 .21 1.99 
 
 1 55 
 
 
 1.09 
 
 1 67 
 
 3 80 
 
 
 .50 
 
 1.94 
 
 9.60 
 
 
 .21 2.00 
 
 1.58 
 
 
 1.08 
 
 1.68 
 
 4.00 
 
 4:1 
 
 .49 
 
 1.94 
 
 9.80 
 
 
 .20 2.00 
 
 1.60 
 
 8:5 
 
 1.07 
 
 1.68 
 
 4.20 
 
 
 .47 
 
 1.94 
 
 10.00 10:1 
 
 .20 i 2 00 
 
 
 
 
 
 
 
 
 
 
 1 
 
 NOTE. To be used only for bevel gears with axes at right angle. 
 
PROVIDENCE, R. I. 21 
 
 TABLES FOR ANGLES OF EDGE AND ANGLES 
 
 OF FACE. 
 
 The following four tables have been computed for the 
 convenience in calculating datas for bevel gears with axes at 
 right angle. They do not hold QQQ>& for bevel gears with axes 
 at any other angle. 
 
 To use the tables the number of teeth in gear and pinion 
 must be known. 
 
 Having located the number of teeth in the gear on the 
 horizontal line of figures at the top of the table, and the num- 
 ber of teeth in the pinion on the vertical line of figures on the 
 left-hand side, we follow the two columns to the square formed 
 by their intersections. 
 
 The two angles found in the same square are the respective 
 angles for gear and pinion. The tables are so arranged that 
 the angle belonging to the gear is always placed above the 
 angle for the pinion. 
 
22 
 
 BROWN & SHARPE MFG. CO. 
 
 TABLE i 
 
 ANGLE OF EDGE. 
 GEAR. 
 
 
 
 41 
 
 40|39 
 
 38 
 
 37 
 
 36 
 
 35 
 
 34 
 
 33 
 
 32 
 
 31 
 
 30 
 
 29 
 
 28 
 
 27 
 
 
 12 
 
 734i 
 
 73*18 paV 
 
 72*38 
 
 72"e 
 
 7 13* 
 
 71 's' 
 
 703+ 
 
 70' i' 
 
 6926 
 
 6350' 
 2lV 
 
 68V 
 
 2 1 *48 
 
 6731 
 22 M 
 
 6648 
 
 23*12 
 
 66V 
 23*s 
 
 6*19 
 
 6V 17V 
 
 1732 
 
 17 58 
 
 1886 
 
 18*55 
 
 1926 
 
 1359 
 
 ^034 
 
 
 13 
 
 78*25 
 17*35 
 
 71 59 
 
 I8*' 
 
 71 M 
 
 1 8*26 
 
 71*7' 
 18*53' 
 
 /I) 39 
 
 I9*ef 
 
 70 9 
 
 19*51' 
 
 69*37' 
 
 20*23' 
 
 GS's 
 
 ao'ss 
 
 6830 
 
 2 1 *3li 
 
 6753 
 22*7' 
 
 67'| 6 
 
 22*45 
 
 66*34 
 
 23*86 
 
 65"sr 
 
 24*9 
 
 65*6 
 
 24*54 
 
 64*17' 
 
 25*43 
 
 
 14 
 
 71 9 
 
 I8*5l' 
 
 7043 
 13*17 
 
 70 is 
 
 19*45 
 
 6946 
 
 eoV 
 
 69 16 
 
 20*44 
 
 684S 
 
 21*5 
 
 68*12' 
 
 21*48 
 
 6737 
 22*Z3 
 
 6/ o 
 23o 
 
 6623 
 63*37 
 
 6S' 
 24*m' 
 
 64*59 
 25* T 
 
 64K 
 
 25*46 
 
 63*26 
 
 26V 
 
 62'36 
 
 27*24 
 
 
 15 
 
 69*54 
 
 2oV 
 
 69*e668s8 
 20*3412 iV 
 
 68*88 
 
 21 W 
 
 67 56 
 
 22*4 
 
 67*23 
 
 22*37' 
 
 66*48 
 23 M 
 
 66*12 
 
 23*48 
 
 65 3 
 
 24*27 
 
 64*53 
 
 esV 
 
 64 V 
 25*so' 
 
 63*2t 
 
 26*34' 
 
 62*39 
 
 272i 
 
 C.I49 
 28*n' 
 
 60*S7 
 29J1 
 
 
 16 
 
 68V 
 21 V 
 
 68V 
 
 8148 
 
 674! 
 
 22*16 
 
 67*0 
 22*5d 
 
 6637 
 
 23*E3 
 
 662' 
 
 23*58 
 
 6526 
 24*34 
 
 6448 
 2SI2 
 
 64" 8 
 
 25*58 
 
 63*e& 
 
 26M 
 
 62V 
 27* 
 
 6l*Sfc' 
 28*4' 
 
 61*7 
 28 
 
 60 a i5 
 
 29*45 
 
 59*2i' 
 
 30*33' 
 
 
 17 
 
 6729 
 
 66* 
 23Y 
 
 66 27 
 
 65V 
 
 65' 
 
 6*43 
 
 64" 6 
 
 6326 
 
 6245 
 
 62%' 
 
 61 "is 
 
 eo'tt 
 
 59"37 
 
 584* 
 
 5748' 
 
 2333 
 
 U46 
 
 244i 
 
 25'i7 
 
 25 54 
 
 2634 
 
 27 is 
 
 2/59 
 
 28*45 
 
 29 3z 
 
 30 n 
 
 31 16 
 
 32 iz 
 
 
 18 
 
 66"ie 
 
 2348 
 
 654* 
 24w 
 
 65V 
 24%6 
 
 64*S9 
 25V 
 
 fa4 J 4 
 25*ES 
 
 fea 1 ?*' 
 2b*M 
 
 6247 
 
 27 n 
 
 b26 
 
 87*54 
 
 6l't3 
 
 28*3?' 
 
 60 36 
 29*22 
 
 5951 
 30*9 
 
 59*2' 
 
 3058 
 
 58'w 
 3i*w 
 
 57 l |6 
 
 32*44 
 
 56*19 
 33V 
 
 
 19 
 
 65*8 
 24*si 
 
 64*1* 
 
 64"e 
 
 6326 
 
 6849 
 
 62' 10 
 
 61*30 
 
 6048 
 
 60'4 
 
 59 J m 
 
 58"3d 
 
 5739 
 
 56 
 
 55si' 
 
 54*52 
 
 2S4 
 
 B5*S8 
 
 26 
 
 C7 11 
 
 27 so 
 
 28*30 
 
 29 ti 
 
 2956 
 
 30V 
 
 31*30 
 
 32 ei 
 
 A3 (4 
 
 34 9 
 
 35'8' 
 
 PINION. 
 
 20 
 
 164V 
 26*o' 
 
 63'2 62*51 
 26*3*' 27V 
 
 62"i4 
 
 6I"S7 
 
 60 U S7 
 
 60 U iS 
 
 59 J 3' 
 
 5847' 
 
 58V 
 
 57 U K> 
 
 56 19 
 
 5524 
 
 54"28 
 
 53*28 
 
 
 
 
 
 3028 
 
 
 JU o 
 
 3250 
 
 33 41 
 
 i436 
 
 3b32 
 
 3637 
 
 21 
 
 68*53 
 
 27*7' 
 
 62*8 6I42 
 274d 28*18 
 
 61*4' 
 2856 
 
 60"es 
 
 29*35 
 
 sa^' 
 
 30*is 
 
 59* ' 
 
 30*58' 
 
 58" 
 
 31*42 
 
 57 L 32 
 
 32*28 
 
 56 '43 
 33*17 
 
 55-53 
 34*7' 
 
 55 J o 
 35*o' 
 
 54-5' 
 
 35*55 
 
 53*7 
 
 36*53' 
 
 52*8 
 37sz' 
 
 22 
 
 61*47 
 
 61* ' 
 
 60*34 
 
 53S6 
 
 59 IS 
 
 5834 
 
 5751 
 
 576' 
 
 56*13 
 
 55 y z9 
 
 54*38 
 
 53*45 
 
 52"43 
 
 51*50 
 
 50*49' 
 
 2fi3 
 
 2849 
 
 29 ?6 
 
 304 
 
 30 45 
 
 3)26 
 
 329 
 
 32 S4 
 
 3341 
 
 3431 
 
 3522 
 
 36i5 
 
 37H 
 
 38 10 
 
 39)1 
 
 23 
 
 60V 
 
 60V 
 
 59*28 
 
 5849 
 
 58*8 
 
 57*25 
 
 scV 
 
 55*55 
 
 5SV 
 
 54 e 
 
 53*26 
 
 52*3.' 
 
 51*35 
 
 5036 
 
 4^34 
 
 eg 5 * 
 
 29S4J3032 
 
 31 ti 
 
 31 52 
 
 32 35 
 
 33 13 
 
 34s 
 
 3453 
 
 3542 
 
 363* 
 
 37*89 
 
 3&8S 
 
 3924 
 
 4026 
 
 24 
 
 S9J9 
 30V 
 
 59Y 
 30 SB 
 
 5823 
 
 57 '44 
 
 57 l e 
 
 5619 
 
 55 U 33 
 
 5447 
 
 53*58 
 
 53*7 
 
 52*15 
 
 5I20 
 
 SOa 
 
 43 24 
 
 48 
 
 31 37 
 
 32 it 
 
 32 sa 
 
 334i 
 
 3427 
 
 3bi3 
 
 36*2 
 
 3653 
 
 3745 
 
 38*40' 
 
 3937 
 
 4036 
 
 41 3* 
 
 25 
 
 se 
 
 3lV 
 
 seV 
 
 32V 
 
 5726 
 %4o 
 
 56*40 
 33*eo' 
 
 55*57 
 
 34*a 
 
 ssV 
 
 34*47 
 
 54*88 
 35*32' 
 
 53 
 36*20 
 
 52*st 
 37*3* 
 
 52V 
 38V 
 
 51*7' 
 38*53' 
 
 50i2 
 3948' 
 
 49V 
 
 40*46 
 
 48" 14 
 4146 
 
 47*l 
 
 42*4 
 
 26 
 
 157*37 
 
 5658 
 
 56" 19 
 
 5537 
 
 54*54 
 
 54 16 
 
 53M 
 
 52"3fr 
 
 51*46 
 
 50 J S4 
 
 soV 
 
 49* & 
 
 48 U 7 
 
 47 U 7 
 
 46*5 
 
 3223 
 
 33 e 
 
 3341 
 
 3423 
 
 3b6 
 
 35so 
 
 36 3* 
 
 3/24 
 
 38)4 
 
 396 
 
 39 59 
 
 4055 
 
 4153 
 
 4253 
 
 43 K 
 
 27 
 
 56"JS 
 
 5559 
 
 ss"j 
 
 54 U 36 
 
 53"S3 
 
 53 U 7 
 
 52*21 
 
 51 ^ 
 
 SO'in 
 
 49 U 5I 
 
 48 U 57 
 
 4V 
 
 473' 
 42*57 
 
 46*2 
 43*58 
 
 4-5* 
 
 33 a 
 
 34i 
 
 344? 
 
 3!>24 
 
 367 
 
 3bs3 
 
 3/39 
 
 38e? 
 
 3317 
 
 409 
 
 41 3 
 
 42o 
 
 28 
 
 55*w 
 34* 
 
 55> 
 
 35 o' 
 
 54*19 
 
 3SV 
 
 53"37 
 
 52"w 
 
 52 J 8 
 
 5l*zo 
 
 5032 
 
 494l 
 
 48"48 
 
 47 U 5S 
 
 46^58 
 
 46*0 
 
 45* 
 
 
 3623 
 
 3/7 
 
 3/sz 
 
 3S40 
 
 &<* 
 
 40 19 
 
 41 12 
 
 42 S 
 
 43a 
 
 44" o 
 
 29 
 
 54V 
 
 54*3 
 
 53a 
 
 S2 1 39 
 
 5l55 
 
 51*9 
 
 soV 
 
 43 J 32 
 
 40*28 
 
 484i 
 4I* 
 
 4749 
 42* ' 
 
 46V 
 43*6' 
 
 45^8 
 44*z' 
 
 45* 
 
 
 35 16 
 
 35*57 36*38 
 
 37 ei 
 
 38 s 
 
 38 s> 
 
 3339 
 
 30 
 
 153*4* 
 36*,2 
 
 S3*7|S2*t6 
 36S3]37V 
 
 5142' 
 
 50S8 
 
 SO'e 
 
 4924 
 
 483S 
 
 47 
 
 465I 
 
 45"56 
 
 45 
 
 
 38)8 
 
 39 
 
 ddw 
 
 40 M 
 
 4(25 
 
 4217 
 
 439 
 
 44*4 
 
 31 
 
 5254 
 37V 
 
 52i3]5lV 
 
 5048 
 
 50Y 
 
 49",6 
 
 4828 
 
 47 
 
 4647 
 
 45 J S4 
 
 45* 
 
 
 3747 3*W 
 
 39 w 
 
 39 58 
 
 4044 
 
 4-1 '32 
 
 4221 
 
 43)3 
 
 446 
 
 32 
 
 52'z 
 
 5 1'wSO'w 
 
 49*54 
 
 49" 9 
 
 48H 
 
 4734 
 
 46*44 
 
 4S J 5J 
 
 45* 
 
 
 37 ss 
 
 384039*88 
 
 
 40 si 
 
 41*38 
 
 42 26 
 
 43 16 
 
 447 
 
 X 
 
 51*10 
 
 50*E9l4946 
 
 492 
 
 48 16 
 
 47* 
 
 4641 
 
 45si 
 44*9 
 
 45* 
 
 
 s\ 
 
 38so 
 
 393i 40*4 
 
 40 z 
 
 41 44 
 
 42zi 
 
 43i 
 
 34 
 
 50*20 
 
 49~48SS 
 
 48*' 
 
 47*25 
 
 <K>"3 
 
 4S"so 
 
 45* 
 
 
 3940 
 
 40'4lV 
 
 4I49 
 
 4V>35 
 
 43 6 22 
 
 44)0' 
 
 35 
 
 49*3i 
 
 48%8|4e*s 
 
 47"n 
 
 46"35 
 
 45 < 4S 
 
 45" 
 
 
 4023 
 
 4lie 
 
 41 55 
 
 4239 
 
 4345 
 
 44 12 
 
 36 
 
 48*43 
 41*17 
 
 48" 
 42 o 
 
 47 U (7 
 
 4243 
 
 46*33 
 43*27 
 
 4547 
 
 44* 
 
 45 
 
 
 37 
 
 47 
 42*4 
 
 47*14 
 
 42*44 
 
 463 
 433Q 
 
 454* 
 44V 
 
 45 
 
 
 38 
 
 47'io 
 
 42*55 
 
 46'w 
 43-3S 
 
 45"*s 
 44-*is 
 
 45* 
 
 
 39 
 
 46*26 
 4334 
 
 4543 
 44*7 
 
 45* 
 
 
 40 
 
 |45*4z 
 44*, 8 
 
 45 
 
 
 4 
 
 45' 
 
 
 
 
PROVIDENCE, R. I. 
 
 TABLE i. (Continued^ 
 
 ANGLE OF EDGE. 
 
 GEAR. 
 
 2625242322 
 
 21 
 
 20l9 
 
 18 
 
 17 
 
 16 
 
 5 
 
 I4 
 
 I3 
 
 I2 
 
 12 
 
 si 
 
 64*22 
 
 6326 
 
 6227 
 27*33 
 
 60.s 
 
 2837 
 
 2945 30 sal 32 16 
 
 537 
 
 36*53 
 
 51 80492447 17 
 38*40 
 
 403C 
 
 45 
 
 \f* 
 
 
 
 63 ee 6231 
 
 56*561 5 5 37 
 
 33 
 
 34 23 35 so 
 
 s 
 
 4055 
 
 47 7 
 42*53 
 
 45 
 
 594S 
 30*5 
 
 5840 
 
 31*20 
 
 7 x 56 19 55 o 
 
 32 28 33 41 
 
 5337 
 
 35"ol36*z3' 
 
 52 8 
 
 3752 
 
 43 e' 
 
 45 
 
 59*2 
 30*58 
 
 50V 
 
 32o 
 
 3532 
 
 38 18 3848 
 
 45 
 
 53 SB 58 42' 
 
 51 EO 43*5* 48Z2 
 
 43 16 
 
 45* 
 
 I7 5649 
 
 I/ 33 ii - 
 
 5547 
 
 5441 
 
 5332 
 
 36*28 
 
 51*0* 
 
 3742 
 
 38 o' 
 
 49 3* 48 ii 
 40"zj 41*43 
 
 18 
 
 *I5 
 
 35*4s 
 
 SVv 
 
 51*57 
 
 4036 
 
 4633 
 
 42"|4327 
 
 53SI 
 36*9 
 
 5I38 
 38E2 
 
 SOX 49n 
 
 4043 4? 8 
 
 20 SO R 
 
 45 
 
 2 I 
 
 1 
 
 38 se 
 
 4958 
 40* z' 
 
 4848' 
 
 473*1 46*i 
 4340 
 
 41*21 
 
 42*3i 
 
 43V 
 
 4-5 
 
 48 30 47 23 
 
 47 " 17 
 
 46'io 
 43*so' 
 
 45 
 
 45* 
 
BROWN & SHARPE MFG. CO. 
 
 TABLE 2. 
 
 ANGLE OF EDGE. 
 GEAR. 
 
 7271 
 
 686766656463 
 
 61 
 
 60595857 
 
 12 
 
 8033 
 9*? 
 
 80* 
 9*36 
 
 79* 
 I0*| 
 
 79 6t 73 
 
 79 3Z 79 23 79 13 
 
 0*37 10*47 
 
 78*52 70*1 
 
 78 3d 78 19 78 7 
 
 II 53 
 77V 
 I2*5i' 
 
 13 
 
 794* 
 10*14 
 
 7937 
 .0*23 
 
 7929 79zo 
 10*31 10*40 
 
 79*11 
 10*49 
 
 79 I 
 
 1851 
 
 784 
 
 IOW 
 
 78 31 78*20 
 il*JMll*4< 
 
 775 7746 77 
 
 12 2 
 
 14 
 
 79o 
 11*0 
 
 78*si 
 
 11*9 
 
 7641 7632 
 
 78*22 
 It* 38 
 
 77*51 
 
 II 59 
 
 77-40 77*28 
 12*20 12*32 
 
 76* 
 
 13*2/1 13*3* 
 
 76*26 76 ti. 
 
 348 
 
 rs^i 
 
 ]445 
 
 74 i< 
 
 15*4 
 
 78% 
 
 77V 
 
 12*6 
 
 77 44 77 3* 
 
 7723 
 
 12*37 
 
 77*12 
 
 12*48 
 
 76*48 76 e 
 
 7S58 7S*44]7S'3i 75*15 
 14* * 1430 
 
 16 
 
 77* 77*7 
 
 76S7 
 
 '2 S3 13 3 
 
 1315 
 
 763* 
 
 13*26 
 
 76*22 
 13*38 
 
 76 10 
 I3SO 
 
 75*58 
 
 75% 
 
 7449 74*35 74 19 
 
 I5"|||1S2S 
 73*56| 73*40 
 
 17 
 
 76*43 
 
 13V 
 
 76* 
 I3* 
 
 76*2i']76*id 
 I3*39'| 13*50 
 
 7S* 
 
 14" z 
 
 7S4S 75 33 
 
 758 
 
 74*5* 
 
 74..' 
 
 18 
 
 7558 
 14 Z 
 
 7546 
 14*14 
 
 75 
 I4*2i'| I4~37 
 
 75 M 75*io 
 
 74*56 
 
 15*2 
 
 7445 
 
 15*15 
 
 74*31 
 
 74 17 74*3 
 
 15431 15*58 
 
 73*ie 
 
 16*58 
 
 72*29 
 
 12 
 
 7 I 
 
 1826 
 7040 
 
 19*20 
 
 19 
 
 75*13 
 4*47 
 
 75 r 
 
 14*59 
 
 74*49 74 3* 74* 2 3 
 
 15 II ! 15 24 
 
 74* to 
 15*50 
 
 73*56 
 16*4 
 
 73*42 
 
 16*18 
 
 73 as) 73*t: 
 16*32 16*47 
 
 72*si 
 
 7242 
 
 17 16 
 
 7220 
 17*3* 
 
 72*9 71*52 
 I7*5l1 18*8 
 
 |74*t| 
 15*3 
 
 7416 
 
 IS 44 
 
 74*3 
 I5*S7| 16" 10 
 
 73sd 73*37 
 16*23 
 
 73*9 
 I6si 
 
 72*5* 
 17*6 
 
 72^72*23 
 17*21 17*37 
 
 72*7 
 
 7I5! 
 
 1753 
 
 71*16 7059 
 18*4* I9 f 
 
 21 
 
 73 .8 734' 
 
 I656 
 
 72*sp 
 17* K) 
 
 72 72 21 
 
 17 J* 
 
 17 is 
 
 726 
 17*5* 
 
 71 I 
 
 71 o 
 
 18 10 1826 
 
 18 43 
 
 19 o 
 
 7O43 
 19* ,7 
 
 70z*'|70 6 
 
 |73*T 
 16-59 
 
 72*47 
 
 17*3 
 
 72 19 
 17*41 
 
 72*4 
 175 
 
 71*49 
 
 71*34 
 
 1826 
 
 TIM 
 
 18*43 
 
 71* i 70*45 
 
 70*28 70*10 
 
 6952 69" 33 i 69 13 
 
 18 yj \9 is 
 
 1934 
 
 1950 
 
 68*54 
 
 20'27|2047|2I*6 
 68*4e68*t2|68*2 
 
 72a 
 
 I7*$7 
 
 I8n 
 
 71*9 
 
 18'si 
 
 70V 
 
 70 JO 
 I9* 
 
 70*14^ 69*57 
 I946l20*3 
 
 63 39 692C 
 
 202! 
 
 2040 
 
 . 
 
 18 tt 
 
 71*19 
 I8*4l' 
 
 71*5 70*49 
 
 70 e 34 70 17 
 I943 
 
 70* 
 
 I959 
 
 89V 
 20*16 
 
 20*3*1 20s 
 
 68 50 68 3 
 2lid 2l*N 
 
 68 iz 
 
 67 52 67 3i 
 
 21 48 22 8 \ZZZ9 22 50 
 
 70*51' 
 19*9 
 
 70*36 70 ti' 70*5' 
 I9*W 19*55 
 
 69 49 69 J2 69>5i 68V 68 
 
 20(1 
 
 202B204S2I 3 
 
 40 68 zi 
 2l'*20 1 21*39 
 
 33' 
 
 67 43 67*23 67 Z 66* 
 
 21 57 22 17 
 
 22 37 2258 23 19 2341 
 
 709 
 19V 
 
 693769V 
 
 69*4-' 
 
 207 
 
 20*0 20*39 20* 
 
 63 46 68*30 68 iz 
 
 21*30 
 
 2146 
 
 22*6 22* 
 
 67 34 67 is 6655 663*' 66 13 6Ss 
 
 2245 23 5 
 
 2326 2347 249 
 
 6529 
 
 24*31 
 
 27 
 
 V 69*10 
 
 66 54|68 e 38 6620 68*3 
 
 67 4i 67t6 67*8 664i 
 
 20*50 21*6 21*22' 
 
 21 57 22 IS 
 
 22*34 22*52 
 
 66*28 66*7 
 
 65*46 65*25 65 2 
 
 23*53 24*14 24*35 1 24*58 25*11 
 
 68 45 68*29 
 
 21*15 21*31 
 
 68*11 67*55' 
 21*4822*5' 
 
 67V 
 22Z3 22 
 
 67*19 67" | 
 22*59 
 
 MM 
 23*18 23 
 
 fc& Zi 64 53 
 24 
 
 63*sb 
 252i|25*46 26*io 
 
 6,747 
 
 67* ti 66*s* 66*36 66*17 6SS7 65*37 65*16 64 55 6434 64*iz 63*SO 63T26 Sfz 
 
 22*48 23*6 
 
 z 2343 
 
 243 
 
 25*5 
 
 2S 26 25 4 
 
 26io 26*3* 2658 
 
 6fe4 66*30 fetfit 65*51 65*33 6SV 64*si 6432 64* to 63*45 63 es'a' 
 
 22 S7 
 
 22V 
 
 23*12 23*SO J4 24* 
 
 2446 85 7 
 
 25*28 25*50 26 I 
 
 2634 
 
 26*57 27*21 
 
 2744 
 
 31 
 
 6*4Z 66' 
 
 23t8 
 
 23*35 
 
 666 
 23*1* 
 
 65466596510 
 24* li 24*31 
 
 64 SO 6430 64 9 1 63 48 63 63 3 
 
 24*6i 25*10 2S3(i 25*51 26* 
 
 62**b 
 
 16*3* Zb'si 27*20' 
 
 62 18 6I* 
 
 6544 6S* 65* 7 ' 
 
 235i 24*16 
 
 24*34 24*53 25V 
 
 64*46 64*2* ftt 
 
 25322SS2 
 
 6V*7 63 
 
 26*13 26*34|26*56 
 
 27I8 27*41 
 
 62*19 6I*M 61* 
 
 28V 
 
 617 
 28*53 
 
 60*4 
 29,9 
 5S6 
 
 6!>Z} 65V 
 
 24*37 
 
 62^43 6Z2i 61 st 6l*3&|6ru 
 24*56 25*isl2S*4l 25*53 26*13 26 S4|26S5 27*7 27 
 
 64*26 64*7 
 
 6347 63 26 63 S 
 
 6047|602i 
 
 3 
 
 28492913 2939304- 
 
 34 
 
 6443 fe425 
 25*.7 
 
 63*46' 
 2555 26*>4 
 
 265527 
 
 6245 
 
 62*13 
 
 62*,'6. 
 27 59 
 
 60 52 60 88 60 3 
 
 Z*w 2845 29*8 
 
 59V 
 57 30*23' 
 
 59*u 
 
 30*49' 
 58*27 
 31*33 
 
 63*4563' 
 
 26*3426*54 
 
 6246 6?. 25 62 4.' 
 
 6I4Z M 
 
 27*14 27* JS 27*56 28* 18 2841 29*3 
 
 t057 60T33 60 9 
 
 59 *5 59 i9 58 si 
 
 2927 
 
 2951 
 
 30 IS 3041 31 7 
 
 t*A*1i*rd 
 
 27*3 27 33 
 
 62*6 
 
 6145 6f 
 
 *ri 
 
 27S* 28*<5 2837 28*59 
 
 i|60 is 59 si 
 30*9 
 
 30*33 
 
 - S837 . 
 
 3058 31*23 31*50 32* ri 
 
 4 6228 62 8 61*48 
 
 6) 27 6) s 
 2833 26S^ 
 
 604*60ti 
 
 29.6 
 
 293 
 
 59 S% S9 is 59 10 5&4b 58 20 
 
 57 5*167 28 57 
 
 39302 3025 
 
 3050 31 14314032 
 
 3259 
 
 **> 
 
 428 Si 
 
 G048 602( 60*4 5941 
 
 29*i 
 
 59 18 58 S4 Stf 
 
 2934 295630i9 30 
 
 58S 
 3 Tad 3 1*55 
 
 57^39 
 32*21 
 
 57 is 1 56 46 56 19 
 
 32*47! 33*14 aiV 
 
 39 
 
 603I 
 
 6010 
 
 59*48 59*25 59 z 
 
 29 7 29 29 29 So 30 tt 30 35 
 
 30 
 
 58 39) 58%4) 57 so) 57 24 
 32>6 
 
 56 58 56 34 56 6 
 
 5537 
 
 40 
 
 60 57 6056 60 15 5953 
 
 29*3 
 
 93Z 59io 
 
 58*47 58 * 58 o 
 
 29V 
 
 2945 30 7 
 
 30283050 31 13 
 
 1 57 35 57jio 
 32*25 32*50 
 
 564*156 19 5552|5S2*|54S7 
 35*3 
 
 33*,6 
 
 34*8 3 ' 
 
 A I 
 
 41 
 
 29*40 
 
 5939 59JI7 
 30*Zi 30*43* 
 
 31 5 
 
 57*45 
 32*5 
 
 57^ 
 32*39 33" 3 
 
 33 2 3354-3421 
 
 55, -t. 5444- 
 34*4U5 P I6 
 
 54-J6 
 35*44 
 53*37 
 
 30 e ,i 
 
 S9*a*| 59*3 1 58*40 58ie 57;s5|5732 57" 8 
 32*5t 
 
 3141 
 
 3228 
 
 56*3)56 19 
 33*17 33*4i' 
 
 55 53 55 27 
 
PROVIDENCE, R. I. 
 
 TABLE 2. (Continued.) 
 
 ANGLE OF EDGE. 
 GEAR. 
 
 5655545352515049484746454443 
 
 42 
 
 12 
 
 77 54 77 42 
 
 77*5 
 I245 
 
 77V 
 I3*o' 
 
 7646|763o 76V 75 se 
 13*36 
 
 75*4 
 14*19 
 
 14*37 
 
 743 
 IS57 
 
 13 
 
 76*w 
 
 13*4 
 
 13 18 
 
 13 32 
 
 76'l3 
 13*47 
 
 756e 7^46 75*2* 7Se 
 
 14 18 
 
 74*5i' 
 IS*9' 
 
 74 
 15*28 
 
 74 
 15*47 
 
 (67 
 
 16 28 
 
 7246 
 
 17V 
 
 14 
 
 75*58 75*43 
 
 14*4 
 
 4I7' 
 
 75*28 
 I4*3i 
 
 75V 74*56 74*39 74"z 
 
 15*4 
 
 is'e 
 
 IS39 
 
 16*87 
 
 73*44173*25 
 16*45 
 
 17*39 
 
 15 
 
 75V 
 
 7A44 
 
 lS*lfa 
 
 74V 74*12 73S5 
 
 8 72 S3 7239 
 
 72*18 71 
 
 7lV 
 
 71*10 70 46 70V 
 
 1531 
 
 1742 
 
 19 14 
 
 \99 
 
 16 
 
 74-3 
 15*57 
 
 7347 
 
 6*.3 
 
 73V 73i 
 16V 
 
 72*54 7235 72 
 
 71" 12 
 
 1745 
 
 te's 
 
 70*49 TOV 
 9V 
 
 9*3S 69V 
 >9'59|2oV 20*51 
 
 17 
 
 737 
 
 7249 
 
 16 53 17 
 
 . 
 1729 
 
 T2"\i 
 
 17*47 
 
 71*54 71 
 
 70 52 7030 707 
 
 18*6 
 
 1826 
 
 930 
 
 1953 
 
 6943 69 17 
 2O*I7 20*43 
 
 68*5268 0,67*5* 
 
 18 
 
 749 
 
 71 53 
 
 18*7 
 
 71*34 
 
 7lis 
 
 I84S 
 
 70*54 
 19*6 
 
 70*33 
 19*27 
 
 70*2 
 
 69'so 
 
 69V 69 3 
 
 6838 68 iz 67 V 67 17 
 
 203* 
 
 2057 21 2148 
 
 22 
 
 2243 
 
 19 
 
 7l w is 
 18*4S 
 
 7057 
 
 19 I 
 
 70*17 
 
 69 34 69' 
 
 684868 
 
 1943 20 
 
 20 2048 
 
 2135 
 
 67 S9 67 34 67 6 
 22*2^ 22*54 
 
 663866 10 
 23*22 23*50 
 
 65*39 
 24*2,' 
 
 20 
 
 70*2 
 19V 
 
 69*4 
 20*.9 
 
 69 19 68*57 
 
 67 4d 67V 6657 66 V 6C2 
 
 21 25 2148 
 
 22 V 
 
 23 3 
 
 23 
 
 23V 
 
 65*33 65*3 
 
 64*32 
 
 25 
 
 21 
 
 69V 69* 
 2OV 20V 
 
 6737 67* 3 66*4B 66* 6Ssi 65*t8 64*59 64* 63*S8 
 
 22 2S 
 
 2247 23 12 
 
 2338 
 
 24 S 
 
 2432 
 
 25* 
 
 25*3 
 
 26 
 
 63*26 
 26*34 
 
 22 S 
 
 68 33 68 12 
 
 67 50 67 27 674 
 
 66*40 
 
 S 6549 65 13 64 55 
 
 22io 2233 22*56 23*20 
 
 24J7 
 
 25*5 
 
 64t6 63*57 
 
 25V 26*3 
 
 63* 62*54 
 26*3* 
 
 27*6 
 
 62*21 
 27V 
 
 67*18 
 
 22*42 
 
 23 S 
 
 65*44 65*18 64*51 
 24*16 24*42 25*9 
 
 64*34 63 55 63 26 62*56 
 25*36 26* s 26V 27*A 
 
 62V 6l*St 61* 
 
 23*12 23 V 
 
 6S|4 64*48 64*22 
 
 63V 63V 62*57 62*27 6l*s 61*2) 60 V 60is 
 
 2358 2422 
 
 25 12 
 
 2538 266 
 
 263* 
 
 27*3 
 
 65*57 6533 
 
 243 
 
 2*27 
 
 65*9 
 24*5.' 
 
 25*, 5 
 
 63*53(63*161 6ZVJ62V 61*59 61*29 6O*S7 
 28%i 39*3' 
 
 26*34 27*2 
 
 273l 
 
 28 
 
 29 ) 30 io 
 
 G5 6 6442 64*ie 63*52 63* 
 
 62* 62*34 
 
 27* . 
 
 62*3 
 
 60 31 59V 59*B 58 so 
 
 27 29 27 57 
 
 28 57 29 30 i 
 
 30V 3l'io 
 
 se 
 
 3I*4 
 
 ?Z 
 
 28 
 29 
 
 30 
 
 3] 
 32 
 33 
 34 
 35 
 36 
 37 
 38 
 39 
 40 
 41 
 
 64' ib 63V 
 25*4426*9' 
 
 62*34 
 
 61*3 
 
 61* ' 
 
 60V 60*7 
 
 26 3* 27 
 
 27 26 27 50 28 22 
 
 2V 29ti 29*! 
 
 S9V 59*a 
 
 soV 3oV 
 
 5e*i _. 
 
 l*32 32*7 
 
 S7i4 
 
 63*6 63*1 
 
 62*3* 62*9 
 
 26*34 26*S9 27V 27*si 
 
 61*42 
 
 28*8 
 
 6045 60*i5 594S 59*i3 SCT4t 58*7 
 
 57 S6*W 
 
 28*46 29*15 29*45 30i5 
 
 3047 3I*2 31*53 32* 33*4 
 
 33V 
 
 62V 62*iz 61*45 6) 
 27*23 27*48 28*15 
 
 29*9 
 
 60*23 59 53| 59 23 58 52 58* 57*46 57 e i2 56*37 56* 
 SI* 6 3l*4i 324: 32*48 
 
 29*37 30*7 
 
 3037 
 
 33 34*o 
 
 5523 
 34*7 
 
 5T27 se'ss 
 
 28)1 
 
 28 37 29 3 
 
 293.' 
 
 29 99 30 ta 30 s* 31 V 32o 
 
 32 
 
 33*7 
 
 33*4,' 
 
 *C28 
 34*C5 35*32 
 
 28*l 29*24 29V 30*19 50 V 3 
 
 59 ia &8 4t 58 >2 57*. 
 
 3lV 32* ' 32 
 
 57*6 
 
 rsci 
 
 5526 54V 54 it 
 
 53V 
 
 33*24 3359 34*34 35*l 5*48 36*26 
 
 60 is S9*48 59*i 
 
 29*4530", 
 
 30*3*3 
 
 56V 56 19 55 *s 55' 
 
 B4 
 
 H 32 6 
 
 3^37 33 8 
 
 33 41 34 IS 
 
 34*4935*2536 
 
 S3*M 53*2 
 
 36V 37ie 
 
 592959*2 
 
 58*34 58*5 
 
 30*31 30*58 31*26 31*55 3lV 
 
 57 36 57 6 563 56*2 
 
 32 54 33 26 
 
 33V 
 
 55*30 54*56 54*2 
 
 5345 53 8 
 
 52*29 5ISO 
 
 3430 
 
 3SV 
 
 35 39 36 is 36V 37 3 
 
 38' 
 
 5844 58 
 
 3*I6 3*4* 
 
 3^ 
 
 9 5547 55'is 54* 
 34*13 34*45 35*19 
 
 54V 
 
 53V 52V 5Z 5 
 
 3S S3 36 28 37 8 
 
 51* ' 
 3o' 
 
 58*0 
 32 o 
 
 S7*3i 57 
 
 32283257 3327 
 
 3357 
 
 5So 
 35*0 
 
 5428 53 54 53 20 
 36*40 
 
 52*44 
 37*16 
 
 37E7 
 
 5l3o 50 i 
 38*30 39*9 
 
 39*48 
 
 57 it 5648 56*19 5549 55*ia 54 47 54 IB 53V 53*8 
 
 32 44 33 w 3341 
 
 35*45 36*18 36*52 37*27 38*3 
 
 52 33 BIS? Bl*20 50*43 
 
 38 4o' 39 >7 
 
 49V 
 39*56 40*3^ 
 
 56 3t 56*4.' 
 
 55 M 35 s 
 
 33 EB 33 56 34 2S 34 sb 35 26 
 
 35 
 
 363i 37 A 
 
 37 37 38 13 38 48 39 2S 
 
 v. 50ss 49 s* 49 
 40V 
 
 4043 
 
 41*23 
 
 55si 55i 54*52 5423 53s 
 
 35*8 
 
 35*37 36*9 
 
 51*3 
 3 42| 37l4| 37481 38*22 
 
 50*27 494 
 
 38" 57 39*33-40' 
 
 49*i 
 
 42 * 
 
 559 
 
 3451 
 
 3S2i' 
 
 5339 53*7 
 
 52*3 
 
 52*3 
 r2i 36'53 37 4 37*57 38"3i 
 
 5lt 5054 50*19 434a 49S 
 
 396 
 
 39V,' 
 
 47 48 47*?' 
 
 40* 
 
 3S32 3* 
 
 3734386 
 
 3840 
 
 39)4 3948402441 
 
 SS 433* 
 
 5348J53I7 
 36*12 
 
 52)6 
 3643 37*2 37*44 
 
 5l*4s'Sl it SO 39 50 s' 
 
 49 30 48 54 48' 
 
 47*40 47* 
 
 46 22 45V 
 
 385 
 
 38483921 
 
 3955403041 f, 
 
 41 43 42 20 42 59 43 44 ,9 
 
 52*3* 52*8 
 
 'n 37*5Tj38'24|38'6|39'29|40'i 
 
 47*36 46*5S 
 42*z4 43* i' 
 
 46*20 
 
 41/40 
 
 4340 44-20 
 
26 
 
 BROWN & SHARPE MFG. CO, 
 
 TABLE 
 
 ANGLE OF FACE. 
 GEAR. 
 
 1140393837363534 333231 
 
 30292827 
 
 12 
 
 1 3'37113-s 
 
 70'ai 
 
 70' 
 
 14" IB 14*39 
 70*6' 69^ 
 
 15 
 69*s 
 
 15 4- 15 49 
 
 67*4 
 
 16 
 67is 
 
 1927 5' 
 
 J633'62-9 
 
 13 
 
 I53*||6' 
 68*4 
 
 I6"8 
 67*43 
 
 16 51 
 
 67' 9 
 
 
 
 66*33 
 
 65, 6 
 
 62*i 
 
 14 
 
 6 /5JI6 a 
 >3. 68*0 
 
 16*59 17*4.4 
 
 67*2966*5 
 
 65*8 
 
 9 
 64*30 
 
 63*6 
 
 2066 
 62*20 
 
 22*,.T 
 60*41 
 
 15 
 
 67*. 66 Va 
 
 14654 
 
 19 n 
 653 
 
 20 I 
 63V 
 
 23 10 
 60*2 
 
 23 S 
 
 59*JS83|S7'4. 
 
 16 
 
 18 V 19 
 
 I9'35 ZO * 
 
 64*5i 64*2i 
 
 21 ' 
 63*V 
 
 iS\ 
 57*< 
 
 17 
 
 205< 2lV 
 63*4 63% 
 
 t/Vi 
 
 62>. 
 
 22 
 6lVo 
 
 2353 
 60*15 
 
 24 10 
 
 275s 
 
 18 
 
 >|92I\7 
 
 6035 
 
 24 , 
 
 S95l 
 
 24 
 
 2657 
 56*39 
 
 28z 
 5449J5350I5247 
 
 19 
 
 [28 to] 22 4 
 >i"*t\e>l'\ 
 
 352 
 60*4<4 
 
 24t 
 
 25i 
 
 2537 
 
 58*37 
 
 4261 
 
 2738 28 
 
 574 
 
 S+, 
 
 29561304: 
 53 2f 
 
 51*14 
 
 ,23*aJ24 
 
 '6I*3060 < 5J 
 
 125*6 
 59^ 
 
 Ukt 
 
 2655 
 5725 
 
 2.71 
 
 28 15 28*58 
 
 5549 
 
 2944 
 
 Si"*. 
 
 5458 54* 
 
 33 ' 
 50* 
 
 21 
 
 s 594 
 
 26 is 
 
 58*26 
 
 26 53 
 
 574* 
 
 28*<o 
 552. 
 
 30*7 
 
 31 4- 
 
 543 53a SI 
 
 32 <> *|33atj34*a 
 5O*5J 
 
 I 254269 
 
 2653 
 
 272 
 
 28* 
 56*a< 
 
 28 
 
 6551 
 
 SM 23 
 
 30 s 
 54*7 
 
 49VJ 
 
 2836 
 
 55*3 
 
 29*4 35 a 
 544- 
 
 an 
 
 53*6 
 
 32' 
 
 333i 34-2? 
 
 36,*l37^t 
 
 4-ft-'5o|47 i7H>-62 
 
 275728*3 
 
 50 
 
 34"i 
 49*o|49l!i 
 
 36 
 47!, 
 
 rS 
 
 56 
 
 523*151 
 
 35 , 
 5pM49j 
 
 36*5 
 
 7 
 
 3 74-H 3 84 j| 3 941 
 46*i5 
 
 30 , \30*s 
 
 53e 
 
 35'a, 
 49% 
 
 38 46 39*55 40 SI 
 
 45 <> ,oU4 6 r|43 
 
 3I3 31 39 
 t/ 54J533 
 
 32V) 
 
 3331 
 
 353 
 
 1549 
 
 50 34 4 94 s 48*s$ 4^ 
 
 37Ss 
 47% 
 
 38 , 
 46*,o 
 
 404 41 i 
 
 J32"z 
 
 IssWstM 
 
 335 
 51,, 
 
 r. 
 
 \SZii 
 
 3338 
 
 51 
 
 34 n 
 
 3539 
 
 4750 
 
 4f,d 
 
 3373436 
 
 Slaa 50so 
 
 35i5 
 
 3 5 56 
 
 387 
 
 31 
 
 |34"S3|353 
 
 1504- (149 57 
 
 36 
 
 49i3 48t 
 
 3735 
 
 4739 
 
 3 6 Si 
 
 46*3 
 
 74S , 
 
 3941 
 45% 
 
 403Z 
 
 41 24 
 
 45*,7 
 
 42l 
 
 4t*M 
 
 1 354* 36*2 7 
 50 49 7 
 
 376 
 
 4049 4138 
 
 36 9 37 e 
 
 38 e o|38 C 4t 
 
 46*46 
 
 4558 
 
 40 
 45 
 
 45%. 
 
 40?; 
 
 42 2 e 
 
 3752 38" t 
 48*2 47 z 7 
 
 38sj 
 
 40"le 
 458 
 
 41 
 44t| 
 
 4149 
 
 4319 
 
 47lJ46a9 
 
 4554 45a 
 
 424i 
 
 ifa 
 
 43*34 
 
 |4S52J45*8 
 
 4248 
 
 J4047|4/ >a 
 145% 
 
 Jk ID I . u 
 ^4 142 " 
 
 41 
 
PROVIDENCE, R. I. 
 
 TABLE $. (Continued.) 
 
 ANGLE OF FACE. 
 GEAR. 
 
 = 90" - (ex b + 
 (See page 13.) 
 
28 
 
 TABLE 4 
 ANGLE OF FACE. GEAR. 
 
 V 54, 9* Z \9' 9 \y 18 9*26 9 3$ 3 
 
 17 si 77*42 rfmtrim if a' 
 
 ,, . . " 56 12 7 
 
 IS 31 75 20 75 7 74 M 74447427 
 
TABLE 4. (Continued.) 
 ANGLE OF FACE. GEAR. 
 
 29 
 
 
 56 
 
 55 
 
 54 
 
 53 
 
 52 
 
 51 
 
 50 
 
 49 
 
 48 
 
 47 
 
 46 
 
 45 
 
 44 
 
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 32*8" 
 5334 
 
 32*44133*19' 
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 31 
 
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 27'34 
 
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 28* Z y 
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 28*54 
 
 57*18 
 
 29*23 
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 55*4Z 
 
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 55*8 
 
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 5434 
 
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 52* 3' 51* 23 
 
 32 
 
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 31" 10 
 
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 32*14 
 
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 33*21 
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 32*3Z 
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 33"38 
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 34* iz 
 
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 34*47 
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 35*241 -46' 
 50 49*41 
 
 34 
 
 29*31 
 
 56*59 
 
 29*57 
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 30*51 
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 31*20 
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 32" 19 
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 35*6 
 
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 34 U 41 
 
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 35*n 
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 32 as 
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 37*3 
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 38*14 
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 39*28 40*7 
 
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 39 
 
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 36*53 
 
 49*3< 
 
 37*Z5 
 
 48*5-7 
 
 37*5 
 
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 38*3 
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 39*8 
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 3944 
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 402 
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 40*S'|4I*37 
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 35'3 
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 35-3 
 
 51*7 
 
 36*1 
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 36*3 
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 37*3 
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 37*35- 
 
 4%S_ 
 
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 4-817 
 
 38*4 
 
 47*41 
 
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 39'5 
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 40"n 
 45*4 
 
 4TS 
 457 
 
 41*44 42*Zt 
 4443'44 
 
 42 
 
 35*14 
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 36*1* 
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 36*42 
 49*54 
 
 37*13 
 
 49* a 
 
 37 e 44 
 48*4t 
 
 3^17 
 48*1 
 
 38*49 
 4737 
 
 39"z. 
 47*1 
 
 39"5 
 46 
 
 40*34 
 45'w 
 
 4*9 
 
 45 T 
 
 4l g 4 
 44*4 
 
 44 * 6 L* j 
 43*46| 43 4 
 
BROWN & SHARPE MFG. CO. 
 
 NATUKAL SINE. 
 
 Deg. 
 
 0' 
 
 10' 
 
 20' 
 
 SO' 
 
 40' 
 
 50' 
 
 60' 
 
 
 
 
 .00000 
 
 .00291 
 
 .00581 
 
 .00872 
 
 .01163 
 
 .01454 
 
 .01745 
 
 89 
 
 1 
 
 .01745 
 
 .02036 
 
 .02326 
 
 .02617 
 
 .02908 
 
 .03199 
 
 .03489 
 
 i 88 
 
 2 
 
 .03489 
 
 .03780 
 
 .04071 
 
 .04361 
 
 .04652 
 
 .04943 
 
 .05233 
 
 87 
 
 3 
 
 .05233 
 
 .05524 
 
 .05814 
 
 .06104 
 
 .06395 
 
 .06685 
 
 .06975 
 
 86 
 
 4 
 
 .06975 
 
 .07265 
 
 .07555 
 
 .07845 
 
 .08135 
 
 .08425 
 
 .08715 
 
 85 
 
 5 
 
 .08715 
 
 .09005 
 
 .09295 
 
 .09584 
 
 .09874 
 
 .10163 
 
 .10452 
 
 84 
 
 6 
 
 .10452 
 
 .10742 
 
 .11031 
 
 .11320 
 
 .11609 
 
 .11898 
 
 .12186 
 
 83 
 
 7 
 
 .12186 
 
 .12475 
 
 .12764 
 
 . 13052 
 
 . 13341 
 
 . 13629 
 
 .13917 
 
 i 82 
 
 8 
 
 .13917 
 
 .14205 
 
 .14493 
 
 . 14780 
 
 .15068 
 
 . 15356 
 
 .15643 
 
 1 81 
 
 9 
 
 .15643 
 
 .15930 
 
 .16217 
 
 .16504 
 
 .16791 
 
 .17078 
 
 .17364 
 
 i 80 
 
 10 
 
 .17364 
 
 .17651 
 
 .17937 
 
 .18223 
 
 .18509 
 
 .18795 
 
 .19080 
 
 ! 79 
 
 11 
 
 .19080 
 
 .19366 
 
 .19651 
 
 .19936 
 
 .20221 
 
 .20506 
 
 .20791 
 
 ! 78 
 
 12 
 
 .20791 
 
 .21075 
 
 .21359 
 
 .21644 
 
 .21927 
 
 .22211 
 
 .22495 
 
 77 
 
 13 
 
 .22495 
 
 .22778 
 
 .23061 
 
 .23344 
 
 .23627 
 
 .23909 
 
 .24192 
 
 ! 76 
 
 14 
 
 .24192 
 
 .24474 
 
 .24756 
 
 .25038 
 
 .25319 
 
 .25600 
 
 .25881 
 
 75 
 
 15 
 
 .25881 
 
 .26162 
 
 .26443 
 
 .26723 
 
 .27004 
 
 .27284 
 
 .27563 
 
 : 74 
 
 16 
 
 .27563 
 
 .27843 
 
 .28122 
 
 .28401 
 
 .28680 
 
 .28958 
 
 .29237 
 
 i 73 
 
 17 
 
 .29237 
 
 .29515 
 
 .29793 
 
 .30070 
 
 .30347 
 
 .30624 
 
 .30901 
 
 72 
 
 18 
 
 .30901 
 
 .31178 
 
 .31454 
 
 .31730 
 
 .32006 
 
 .32281 
 
 .32556 
 
 71 
 
 19 
 
 .32556 
 
 .32831 
 
 .33106 
 
 .33380 
 
 .33654 
 
 .33928 
 
 .34202 
 
 1 70 
 
 20 
 
 .34202 
 
 .34475 
 
 .34748 
 
 .35020 
 
 35293 
 
 .35565 
 
 .35836 
 
 69 
 
 21 
 
 .35836 
 
 .36108 
 
 .36379 
 
 .36650 
 
 .36920 
 
 .37190 
 
 .37460 
 
 ! 68 
 
 22 
 
 .37460 
 
 .37730 
 
 .37999 
 
 .38268 
 
 .38536 
 
 .38805 
 
 .39073 
 
 67 
 
 23 
 
 .39073 
 
 .39340 
 
 .39607 
 
 .39874 
 
 .40141 
 
 .40407 
 
 .40673 
 
 I 66 
 
 24 
 
 .40673 
 
 .40939 
 
 .41204 
 
 .41469 
 
 .41733 
 
 .41998 
 
 .42261 
 
 i 65 
 
 25 
 
 .42261 
 
 .42525 
 
 .42788 
 
 .43051 
 
 .43313 
 
 .43575 
 
 .43837 
 
 ! 64 
 
 26 
 
 .43837 
 
 .44098 
 
 .44359 
 
 .44619 
 
 .44879 
 
 .45139 
 
 .45399 
 
 | 63 
 
 27 
 
 .45399 
 
 .45658 
 
 .45916 
 
 .46174 
 
 .46432 
 
 .46690 
 
 .46947 
 
 62 
 
 28 
 
 .46947 
 
 .47203 
 
 .47460 
 
 .47715 
 
 .47971 
 
 .48226 
 
 .48481 
 
 61 
 
 29 
 
 .48481 
 
 .48735 
 
 .48989 
 
 .49242 
 
 .49495 
 
 .49747 
 
 .50000 
 
 ! 60 
 
 30 
 
 .50000 
 
 .50251 
 
 .50503 
 
 .50753 
 
 .51004 
 
 .51254 
 
 .51503 
 
 59 
 
 31 
 
 .51503 
 
 .51752 
 
 .52001 
 
 .52249 
 
 .52497 
 
 .52745 
 
 .52991 
 
 58 
 
 32 
 
 .52991 
 
 .53238 
 
 .53484 
 
 .53730 
 
 .53975 
 
 .54219 
 
 .54463 
 
 57 
 
 33 
 
 .54463 
 
 .54707 
 
 .54950 
 
 .55193 
 
 . 55436 
 
 .55677 
 
 .55919 
 
 56 
 
 34 
 
 .55919 
 
 .56160 
 
 .56400 
 
 .56640 
 
 .56880 
 
 .57119 
 
 .57357 
 
 55 
 
 35 
 
 .57357 
 
 .57595 
 
 .57833 
 
 .58070 
 
 .58306 
 
 .58542 
 
 .58778 
 
 54 
 
 36 
 
 .58778 
 
 .59013 
 
 .59248 
 
 .59482 
 
 .59715 
 
 .59948 
 
 .60181 
 
 53 
 
 37 
 
 .60181 
 
 .60413 
 
 .60645 
 
 .60876 
 
 .61106 
 
 .61336 
 
 .61566 
 
 52 
 
 38 
 
 .61566 
 
 .61795 
 
 .62023 
 
 .62251 
 
 .62478 
 
 .62705 
 
 .62932 
 
 51 
 
 39 
 
 .62932 
 
 .63157 
 
 .63383 
 
 .63607 
 
 .63832 
 
 .64055 
 
 .64278 
 
 . 50 
 
 40 
 
 .64278 
 
 .64501 
 
 .64723 
 
 .64944 
 
 .65165 
 
 .65386 
 
 .65605 
 
 i 49 
 
 41 
 
 .65605 
 
 .65825 
 
 .66043 
 
 .66262 
 
 .66479 
 
 .66696 
 
 .66913 
 
 48 
 
 42 
 
 .66913 
 
 .67128 
 
 .67344 
 
 .67559 
 
 .67773 
 
 .67986 
 
 .68199 
 
 47 
 
 43 
 
 .68199 
 
 .68412 
 
 .68624 
 
 68835 
 
 .69046 
 
 .69256 
 
 .69465 
 
 46 
 
 44 
 
 .69465 
 
 .69674 
 
 .69883 
 
 .70090 
 
 .70298 
 
 .70504 
 
 .70710 | 45 
 
 
 60' 
 
 50' 
 
 40' 
 
 30' 
 
 20' 
 
 10' 
 
 0' 
 
 re. 
 
 NATUBAL COSINE. 
 
PROVIDENCE, R. I. 
 
 NATUBAL SINE. 
 
 Deg. 
 
 0' 10' 
 
 20' 
 
 30' 
 
 40' 
 
 50' 
 
 w 
 
 
 45 
 
 .70710 
 
 .70916 
 
 .71120 
 
 .71325 
 
 .71528 
 
 .71731 
 
 .71934 
 
 44 
 
 46 
 
 1 .71934 
 
 .72135 
 
 .72336 
 
 .72537 
 
 .72737 
 
 .72936 
 
 .73135 
 
 43 
 
 47 
 
 .73135 
 
 .73333 
 
 73530 
 
 .73727 
 
 .73923 
 
 .74119 
 
 .74314 
 
 42 
 
 48 
 
 .74314 
 
 .74508 
 
 .74702 
 
 .74895 
 
 .75088 
 
 .75279 
 
 .75471 
 
 41 
 
 49 
 
 .75471 
 
 .75661 
 
 .75851 
 
 .76040 
 
 .76229 
 
 .76417 
 
 .76604 
 
 40 
 
 50 
 
 .76004 
 
 .76791 
 
 .76977 
 
 .77102 
 
 .77347 
 
 .77531 
 
 .77714 
 
 39 
 
 51 
 
 .77714 
 
 .77897 
 
 .78079 
 
 .78260 
 
 .78441 
 
 .78621 
 
 .78801 
 
 1 38 
 
 53 
 
 .78801 
 
 .78979 
 
 .79157 
 
 .79335 
 
 .79512 
 
 .79688 
 
 .79863 
 
 37 
 
 53 
 
 .79863 
 
 .80038 
 
 .80212 
 
 .80385 
 
 .80558 
 
 .80730 
 
 .80901 
 
 1 36 
 
 54 
 
 .80901- 
 
 .81072 
 
 .81242 
 
 .81411 
 
 .81580 
 
 .81748 
 
 .81915 
 
 35 
 
 55 
 
 .81915 
 
 .82081 
 
 .82247 
 
 .82412 
 
 .82577 
 
 .82740 
 
 .82903 
 
 34 
 
 56 
 
 .82903 
 
 .83066 
 
 .83227 
 
 .83388 
 
 .83548 
 
 .83708 
 
 .83867 
 
 33 
 
 57 
 
 .83867 
 
 .84025 
 
 .84182 
 
 .84339 
 
 .84495 
 
 .84650 
 
 .84804 
 
 32 
 
 58 
 
 .84804 
 
 .84958 
 
 .85111 
 
 .85264 
 
 .85415 
 
 .85566 
 
 .85716 
 
 31 
 
 59 
 
 .85716 
 
 .85866 
 
 .86014 
 
 .86162 
 
 .86310 
 
 .86456 
 
 .86602 
 
 30 
 
 60 
 
 .86602 
 
 .86747 
 
 .86892 
 
 .87035 
 
 .87178 
 
 .87320 
 
 .87462 
 
 29 
 
 61 
 
 .87462 
 
 .87602 
 
 .87742 
 
 .87881 
 
 .88020 
 
 .88157 
 
 .88294 
 
 28 
 
 62 
 
 .88294 
 
 .88430 
 
 .88566 
 
 .88701 
 
 .88835 
 
 .88968 
 
 .89100 
 
 27 
 
 63 
 
 .89100 
 
 .89232 
 
 .89363 
 
 .89493 
 
 .89622 
 
 .89751 
 
 .89879 
 
 26 
 
 64 
 
 .89879 
 
 .90006 
 
 .90132 
 
 .90258 
 
 .90383 
 
 .90507 
 
 .90630 
 
 25 
 
 65 
 
 .90630 
 
 .90753 
 
 .90875 
 
 .90996 
 
 .91116 
 
 .91235 
 
 .91354 
 
 24 
 
 66 
 
 .91354 
 
 .91472 
 
 .91589 
 
 .91706 
 
 .91821 
 
 .91936 
 
 .92050 
 
 23 
 
 67 
 
 .92050 
 
 .92163 
 
 .92276 
 
 .92388 
 
 .92498 
 
 .92609 
 
 .92718 
 
 22 
 
 68 
 
 .92718 
 
 .92827 
 
 .92934 
 
 .93041 
 
 .93148 
 
 .93253 
 
 .93358 
 
 21 
 
 69 
 
 .93358 
 
 .93461 
 
 .93565 
 
 .93667 
 
 .93768 
 
 .93869 
 
 .93969 
 
 20 
 
 70 
 
 .93969 
 
 .94068 
 
 .94166 
 
 .94264 
 
 .94360 
 
 .94456 
 
 .94551 
 
 19 
 
 71 
 
 .94551 
 
 .94646 
 
 .94789 
 
 .94832 
 
 .94924 
 
 .95015 
 
 .95105 
 
 18 
 
 '72 
 
 .95105 
 
 .95195 
 
 .95283 
 
 .95371 
 
 .95458 
 
 .95545 
 
 .95630 
 
 17 
 
 73 
 
 .95630 
 
 .95715 
 
 .95799 
 
 .95882 
 
 .95964 
 
 .96045 
 
 .96126 
 
 16 
 
 74 
 
 .96126 
 
 .96205 
 
 . 96284 
 
 .96363 
 
 .96440 
 
 .96516 
 
 .96592 
 
 15 
 
 75 
 
 .96592 
 
 .96667 
 
 .96741 
 
 .96814 
 
 .96887 
 
 .96958 
 
 .97029 
 
 14 
 
 76 
 
 .97029 
 
 .97099 
 
 .97168 
 
 .97237 
 
 .97304 
 
 .97371 
 
 .97437 
 
 13 
 
 77 
 
 .97437 
 
 .97502 
 
 .97566 
 
 .9762!) 
 
 .97692 
 
 .97753 
 
 .97814 
 
 12 
 
 78 
 
 .97814 
 
 .97874 
 
 .97934 
 
 .97992 
 
 .98050 
 
 .98106 
 
 .98162 
 
 11 
 
 79 
 
 .98162 
 
 .98217 
 
 . 98272 
 
 .98325 
 
 .98378 
 
 .98429 
 
 .98480 
 
 10 
 
 80 
 
 .98480 
 
 .98530 
 
 .98580 
 
 .98628 
 
 .98676 
 
 .98722 
 
 .98768 
 
 9 
 
 81 
 
 .98768 
 
 .98813 
 
 .98858 
 
 .98901 
 
 .98944 
 
 .98985 
 
 .99026 
 
 8 
 
 82 
 
 .99026 
 
 .99066 
 
 .09106 
 
 .99144 
 
 .99182 
 
 .99218 
 
 .99254 
 
 7 
 
 83 
 
 .99254 
 
 .99289 
 
 .99323 
 
 .99357 
 
 .99389 
 
 .99421 
 
 .99452 
 
 6 
 
 84 
 
 .99452 
 
 .99482 
 
 .99511 
 
 .99539 
 
 .99567 
 
 .99593 
 
 .99619 
 
 5 
 
 85 
 
 .99619 
 
 .99644 
 
 .99668 
 
 .99691 
 
 .99714 
 
 .99735 
 
 .99756 
 
 4 
 
 86 
 
 .99756 
 
 .99776 
 
 .99795 
 
 99813 
 
 .99830 
 
 .99847 
 
 .99863 
 
 3 
 
 87 
 
 .99863 
 
 .99877 
 
 .99891 
 
 .99904 
 
 .99917 
 
 .99928 
 
 .99939 
 
 2 
 
 88 
 
 .99939 
 
 .99948 
 
 .99957 ! .99965 
 
 .99972 
 
 .99979 
 
 .99984 
 
 1 
 
 89 
 
 .99984 
 
 .99989 
 
 .99993 | .99996 
 
 .99998 
 
 .99999 
 
 1.0000 
 
 
 
 - 
 
 60' 
 
 50' 
 
 40' 
 
 30' 
 
 20' 
 
 10' 
 
 0' 
 
 Deg. 
 
 NATUKAL COSINE. 
 
BROWN & bHARPE MFG. CO. 
 
 NATUKAL TANGENT. 
 
 Deg. 
 
 0' 
 
 10' 
 
 20' 
 
 30' 
 
 40' 
 
 50' 
 
 60' 
 
 
 
 
 .00000 
 
 .00290 
 
 .00581 
 
 .00872 
 
 .01163 
 
 .01454 
 
 .01745 
 
 89 
 
 1 
 
 .01745 
 
 .02036 
 
 .02327 
 
 .02618 
 
 .02909 
 
 .03200 
 
 .03492 
 
 88 
 
 2 
 
 .03492 
 
 .03783 
 
 .04074 
 
 .04366 
 
 .04657 
 
 .04949 
 
 .05240 
 
 87 
 
 3 
 
 .05240 
 
 .05532 
 
 .05824 
 
 .06116 
 
 .06408 
 
 .06700 
 
 .06992 
 
 86 
 
 4 
 
 .06992 
 
 .07285 
 
 .07577 
 
 .07870 
 
 .08162 
 
 .08455 
 
 .08748 
 
 85 
 
 5 
 
 .08748 
 
 .09042 
 
 .09335 
 
 .09628 
 
 .09922 
 
 .10216 
 
 .10510 
 
 84 
 
 6 
 
 .10510 
 
 . 10804 
 
 .11099 
 
 .11393 
 
 .11688 
 
 .11983 
 
 .12278 
 
 83 
 
 7 
 
 .12278 
 
 .12573 
 
 .12869 
 
 .13165 
 
 .13461 
 
 . 13757 
 
 .14054 
 
 ! 82 
 
 8 
 
 .14054 
 
 .14350 
 
 .14647 
 
 . 14945 
 
 .15242 
 
 .15540 
 
 .15838 
 
 81 
 
 9 
 
 .15838 
 
 .16136 
 
 .16435 
 
 .16734 
 
 .17033 
 
 .17332 
 
 M7632 
 
 : 80 
 
 10 
 
 .17632 
 
 .17932 
 
 .18233 
 
 .18533 
 
 .18834 
 
 .19136 
 
 . 19438 
 
 79 
 
 11 
 
 .19438 
 
 . 19740 
 
 .20042 
 
 .20345 
 
 .20648 
 
 .20951 
 
 .21255 
 
 78 
 
 12 
 
 .21255 
 
 .21559 
 
 .21864 
 
 .22169 
 
 .22474 
 
 .22780 
 
 .23086 
 
 77 
 
 13 
 
 .23086 
 
 .23393 
 
 .23700 
 
 .24007 
 
 .24315 
 
 .24624 
 
 .24932 
 
 76 
 
 14 
 
 .24932 
 
 .25242 
 
 .25551 
 
 .25861 
 
 .26172 
 
 .26483 
 
 .26794 
 
 75 
 
 15 
 
 .26794 
 
 .27106 
 
 .27419 
 
 .27732 
 
 .28046 
 
 .28360 
 
 .28674 
 
 74 
 
 16 
 
 .28674 
 
 .28989 
 
 .29305 
 
 .29621 
 
 .29938 
 
 .30255 
 
 .30573 
 
 73 
 
 17 
 
 .30573 
 
 .30891 
 
 .31210 
 
 .31529 
 
 .31850 
 
 32170 
 
 . 32492 
 
 72 
 
 18 
 
 .32492 
 
 .32813 
 
 .33136 
 
 .33459 
 
 .33783 
 
 .34107 
 
 .34432 
 
 71 
 
 19 
 
 .34432 
 
 .34758 
 
 .35084 
 
 .35411 
 
 .35739 
 
 .36067 
 
 .36397 
 
 70 
 
 20 
 
 .36397 
 
 .36726 
 
 .37057 
 
 .37388 
 
 .37720 
 
 .38053 
 
 .38386 
 
 69 
 
 21 
 
 .38386 
 
 .38720 
 
 .39055 
 
 .39391 
 
 .39727 
 
 .40064 
 
 .40402 
 
 68 
 
 22 
 
 .40402 
 
 .40741 
 
 .41080 
 
 .41421 
 
 .41762 
 
 .42104 
 
 .42447 
 
 67 
 
 23 
 
 .42447 
 
 .42791 
 
 .43135 
 
 .43481 
 
 .43827 
 
 .44174 
 
 .44522 
 
 66 
 
 24 
 
 .44522 
 
 .44871 
 
 .45221 
 
 .45572 
 
 .45924 
 
 .46277 
 
 .46630 
 
 65 
 
 25 
 
 .46630 
 
 .46985 
 
 .47341 
 
 .47697 
 
 .48055 
 
 .48413 
 
 .48773 
 
 64 
 
 26 
 
 .48773 
 
 .49133 
 
 .49495 
 
 .49858 
 
 .50221 
 
 .50586 
 
 .50952 
 
 63 
 
 27 
 
 .50952 
 
 .51319 
 
 .51687 
 
 .52056 
 
 .52427 
 
 .52798 
 
 .53170 
 
 62 
 
 28 
 
 .53170 
 
 .53544 
 
 .53919 
 
 .54295 
 
 .54672 
 
 .55051 
 
 .55430 
 
 61 
 
 29 
 
 .55430 
 
 .55811 
 
 .56193 
 
 .56577 
 
 .56961 
 
 .57847 
 
 .57735 
 
 60 
 
 30 
 
 .57735 
 
 .58123 
 
 .58513 
 
 .58904 
 
 .59297 
 
 .59690 
 
 .60086 
 
 59 
 
 31 
 
 .60086 
 
 .60482 
 
 .60880 
 
 .61280 
 
 .61680 
 
 .62083 
 
 .62486 
 
 58 
 
 32 
 
 .62486 
 
 .62892 
 
 .63298 
 
 .63707 
 
 .64116 
 
 .64528 
 
 .64940 
 
 57 
 
 33 
 
 .64940 
 
 .65355 
 
 .65771 
 
 .66188 
 
 .66607 
 
 .67028 
 
 .67450 
 
 56 
 
 34 
 
 .67450 
 
 .67874 
 
 .68300 
 
 .68728 
 
 .69157 
 
 .69588 
 
 .70020 
 
 55 
 
 35 
 
 .70020 
 
 .70455 
 
 .70891 
 
 .71329 
 
 .71769 
 
 .72210 
 
 .72654 
 
 54 
 
 36 
 
 .72654 
 
 .73099 
 
 .73546 
 
 .73996 
 
 .74447 
 
 .74900 
 
 .75355 
 
 53 
 
 37 
 
 .75355 
 
 .75812 
 
 .76271 
 
 .76732 
 
 .77195 
 
 .77661 
 
 .78128 
 
 52 
 
 38 
 
 .78128 
 
 .78598 
 
 .79069 
 
 .79543 
 
 .80019 
 
 .80497 
 
 80978 
 
 51 
 
 39 
 
 .80978 
 
 .81461 
 
 .81946 
 
 .82433 
 
 .82923 
 
 .83415 
 
 .83910 
 
 50 
 
 .40 
 
 .83910 
 
 .84406 
 
 .84906 
 
 .85408 
 
 .85912 
 
 .86419 
 
 . 86928 
 
 49 
 
 41 
 
 .86928 
 
 .87440 
 
 . 87955 
 
 .88472 
 
 .88992 
 
 .89515 
 
 .90040 
 
 48 
 
 42 
 
 .90040 
 
 .90568 
 
 .91099 
 
 .91633 
 
 .92169 
 
 .92709 
 
 .93S51 
 
 47 
 
 43 
 
 .93251 
 
 .93796 
 
 .94345 
 
 .94896 
 
 .95450 
 
 .96008 
 
 .96568 
 
 46 
 
 44 
 
 .96568 
 
 .97132 
 
 .97699 
 
 .98269 
 
 .98843 
 
 .99419 
 
 1.0000 
 
 45 
 
 - 
 
 60' 
 
 50' 
 
 40' 
 
 30' 
 
 20' 
 
 10' 
 
 0' 
 
 Deg. 
 
 NATURAL COTANGENT. 
 
PROVIDENCE, R. I, 
 
 33 
 
 NATURAL TANGENT. 
 
 Deg. 
 
 0' 
 
 10' 
 
 20' 
 
 30' 
 
 40' 
 
 * 
 
 60 
 
 
 45 
 
 1.0000 
 
 1.0058 
 
 1.0117 
 
 1.0176 
 
 1.0235 
 
 1 0295 
 
 1.0355 
 
 44 
 
 46 
 
 1.0355 
 
 1.0415 
 
 1.0476 
 
 .0537 
 
 1.0599 
 
 1.0661 
 
 1.0723 
 
 ,43 
 
 47 
 
 1.0723 
 
 1.0786 
 
 1.0849 
 
 .0913 
 
 1.0977 
 
 1.1041 
 
 1.1106 
 
 42 
 
 48 
 
 1.1106 
 
 1.1171 
 
 1.1236 
 
 .1302 
 
 1.1369 
 
 1.1436 
 
 1.1503 
 
 41 
 
 49 
 
 1.1503 
 
 1.1571 
 
 1.1639 
 
 .1708 
 
 1.1777 
 
 1.1847 
 
 1.1917 
 
 40 
 
 50 
 
 1.1917 
 
 1.1988 
 
 1.2059 
 
 .2131 
 
 1.2203 
 
 1.2275 
 
 1 2349 
 
 39 
 
 51 
 
 1.2349 
 
 1.2422 
 
 1.2496 
 
 .2571 
 
 1.2647 
 
 1.2723 
 
 .2799 
 
 38 
 
 52 
 
 1.2799 
 
 1.2876 
 
 1.2954 
 
 .3032 
 
 1.3111 
 
 1.3190 
 
 .3270 
 
 37 
 
 53 
 
 1.3270 
 
 1.3351 
 
 1.3432 
 
 1.3514 
 
 1.3596 
 
 1.3680 
 
 .3763 
 
 36 
 
 54 
 
 1.3763 
 
 1.3848 
 
 1.3933 
 
 1.4019 
 
 1.4106 
 
 1.4193 
 
 .4281 
 
 35 
 
 55 
 
 1.4281 
 
 1.4370 
 
 1.4459 
 
 1.4550 
 
 1.4641 
 
 1.4733 
 
 .4825 
 
 34 
 
 56 
 
 1.4825 
 
 1.4919 
 
 1.5013 
 
 1.5108 
 
 1.5204 
 
 1.5301 
 
 .5398 
 
 33 
 
 57 
 
 1.5398 
 
 1.5497 
 
 1.5596 
 
 1.5696 
 
 1.5798 
 
 1.5900 
 
 .6003 
 
 32 
 
 58 
 
 1.6003 
 
 1.6107 
 
 1.6212 
 
 1.6318 
 
 1.6425 
 
 1.6533 
 
 1.6642 
 
 31 
 
 59 
 
 1.6642 
 
 1.6753 
 
 1.6864 
 
 1.6976 
 
 1.7090 
 
 1.7204 
 
 1.7320 
 
 30 
 
 60 
 
 1.7320 
 
 1.7437 
 
 1.7555 
 
 1 . 7674 
 
 1.7795 
 
 1.7917 
 
 1.8040 
 
 29 
 
 61 
 
 1.8040 
 
 1.8164 
 
 1.8290 
 
 1.8417 
 
 1.8546 
 
 1.8676 
 
 1.8807 
 
 28 
 
 62 
 
 1.8807 
 
 1.8940 
 
 1.9074 
 
 1.9209 
 
 1.9347 
 
 1.9485 
 
 1.9626 
 
 27 
 
 63 
 
 1.9626 
 
 1.9768 
 
 1.9911 
 
 2.0056 
 
 2.0203 
 
 2.0352 
 
 2.0503 
 
 26 
 
 64 
 
 2.0503 
 
 2.0655 
 
 2.0809 
 
 2.0965 
 
 2.1123 
 
 2.1283 
 
 2.1445 
 
 25 
 
 65 
 
 2.1445 
 
 2.1609 
 
 2.1774 
 
 2.1943 
 
 2.2113 
 
 2.2285 
 
 2.2460 
 
 24 
 
 66 
 
 2.2460 
 
 2.2637 
 
 2.2816 
 
 2.2998 
 
 2.3183 
 
 2.3369 
 
 2.3558 
 
 23 
 
 67 
 
 2.3558 
 
 2.3750 
 
 2.3944 
 
 2.4142 
 
 2.4342 
 
 2.4545 
 
 2.4750 
 
 22 
 
 68 
 
 2.4750 
 
 2.4959 
 
 2.5171 
 
 2.5386 
 
 2.5604 
 
 2.5826 
 
 2.6050 
 
 21 
 
 69 
 
 2.6050 
 
 2.6279 
 
 2.6510 
 
 2.6746 
 
 2.6985 
 
 2.7228 
 
 2.7474 
 
 20 
 
 70 
 
 2.7474 
 
 2.7725 
 
 2.7980 
 
 2.8239 
 
 2.8502 
 
 2.8770 
 
 2.9042 
 
 19 
 
 71 
 
 2.9042 
 
 2.9318 
 
 2.9600 
 
 2.9886 
 
 3.0178 
 
 3.0474 
 
 3.0776 
 
 18 
 
 72 
 
 3.0776 
 
 3.1084 
 
 3.1397 
 
 3.1715 
 
 3.2040 
 
 3.2371 
 
 3.2708 
 
 17 
 
 73 
 
 3.2708 
 
 3.3052 
 
 3.3402 
 
 3.3759 
 
 3.4123 
 
 3.4495 
 
 3.4874 
 
 16 
 
 74 
 
 3.4874 
 
 3.5260 
 
 3.5655 
 
 3.6058 
 
 3.6470 
 
 3.6890 
 
 3.7320 
 
 15 
 
 75 
 
 3.7320 
 
 3.7759 
 
 3.8208 
 
 3.8667 
 
 3.9136 
 
 3.9616 
 
 4.0107 
 
 14 
 
 76 
 
 4.0107 
 
 4.0610 
 
 4.1125 
 
 4.1653 
 
 4.2193 
 
 4.2747 
 
 4.3314 
 
 13 
 
 77 
 
 4.3814 
 
 4.3896 
 
 4.4494 
 
 4.5107 
 
 4.5736 
 
 4.6382 
 
 4.7046 
 
 12 
 
 78 
 
 4.7046 
 
 4.7728 
 
 4.8430 
 
 4.9151 
 
 4.9894 
 
 5.0653 
 
 5.1445 
 
 11 
 
 79 
 
 5.1445 
 
 5.2256 
 
 5.3092 
 
 5.3955 
 
 5.4845 
 
 5.5763 
 
 5.6712 
 
 10 
 
 80 
 
 5.6712 
 
 5.7693 
 
 5.8708 
 
 5.9757 
 
 6.0844 
 
 6.1970 
 
 6.3137 
 
 9 
 
 81 
 
 6.3137 
 
 6.4348 
 
 6.5605 
 
 6.6911 
 
 6.8269 
 
 6.9682 
 
 7.1153 
 
 8 
 
 82 
 
 7.1153 
 
 7.2687 
 
 7.4287 
 
 7.5957 
 
 7.7703 
 
 7.9530 
 
 8.1443 
 
 7 
 
 83 
 
 S.1443 
 
 8.3449 
 
 8.5555 
 
 8.7768 
 
 9.0098 
 
 9.2553 
 
 9.5143 
 
 6 
 
 84 
 
 9.5143 
 
 9.7881 
 
 10.078 
 
 10.385 
 
 10.711 
 
 11 059 
 
 11.430 
 
 5 
 
 85 
 
 11.430 
 
 11.826 
 
 12.250 
 
 12.706 
 
 13.196 
 
 13 726 
 
 14.300 
 
 4 
 
 86 
 
 14.300 
 
 14.924 
 
 15.604 
 
 16.349 
 
 17.169 
 
 18.075 
 
 19.081 
 
 3 
 
 87 
 
 19.081 
 
 20.205 
 
 21.470 
 
 22.904 
 
 24.541 
 
 26.431 
 
 28.636 
 
 2 
 
 88 
 
 28.636 
 
 31.241 
 
 34.367 
 
 38.188 
 
 42.964 
 
 49.103 
 
 57.290 
 
 1 
 
 89 
 
 57.290 
 
 68.750 
 
 85.939 
 
 114.58 
 
 171.88 
 
 343.77 
 
 00 
 
 
 
 
 60' 
 
 50' 
 
 40' 
 
 30' 
 
 20' 
 
 10' 
 
 0' 
 
 Deg. 
 
 NATURAL COTANGENT. 
 
34 
 
 BROWN & SHARPE MFG. CO. 
 
 CHAPTER IV. 
 
 WORM AND WORM WHEEL 
 
 (Fig. 8.) 
 
PROVIDENCE, R. I. 35 
 
 FORMULAS. 
 
 L = lead of worm. 
 N = number of teeth in gear. 
 m = threads or turns per inch in worm, 
 d= diameter of worm. 
 d' = diameter of hob. 
 T = throat diameter. 
 B = blank diameter (to sharp corners). 
 C = distance between centers. 
 o = thickness of hob-slotting cutter. 
 /= width of lands at bottom. 
 b = pitch circumference of worm. 
 v width of worm thread tool at end. 
 w = width of worm thread at top. 
 P = diametral pitch. 
 P 1 = circular pitch. 
 .$ = addendum and module. 
 / = thickness of tooth at pitch line. 
 t n = normal thickness of tooth. 
 /= clearance at bottom of tooth. 
 D" = working depth of toolh. 
 D" +/ = whole depth of tooth. 
 
 tf = angle of tooth of worm wheel with its axis, or the 
 angle of thread of worm with a line at right angles to its axis. 
 If the lead is for single, double, triple, etc., thread, then 
 I, = P, 2 P', 3 P, etc. 
 
36 BROWN & SHARPE MFG. CO. 
 
 a = 60 to 9<D C 
 
 m 
 
 N + 2 
 
 D = NP/ = N 
 
 7T 
 
 (</ _ 2 j) 
 
 _ L j Practical only when width of wheel on wheel pitch circle 
 
 _ L j Practi 
 ~fr \ is 
 
 ~ not more than j/ 3 pitch diameter of worm. 
 
 2 
 
 TD _ -p , /!_ i Of\ A measurement of sketch is generally 
 
 tf - 1 + 2 I r - r COS -I sufficient. 
 
 ^ = ^+ 2 / 
 
 * = .3iP' 
 
 ^-.335?' 
 
 NOTE. The notations and formulas referring to tooth parts, given on page 5 for 
 spur gears, apply to worm wheels, and are here used. 
 
 NOTE. Hob and worm should be marked, as per example : 
 4 turns per i'' single .25 P'; .25 L. 
 2 turns per i" double .25 P'; .50 L. 
 
PROVIDENCE, R. I. 
 
 37 
 
 UNDERCUT IN WORM WHEELS. 
 
 In worm wheels of less than 30 teeth the thread of the worm 
 (being 29) interferes with the flank of the gear tooth. Such 
 a wheel finished with a hob will have its teeth undercut. To 
 avoid this interference two methods may be employed. 
 
 First Method. Make throat diameter of wheel 
 
 cos 
 
 + 4.$- 
 
 or 
 
 T = -937 N 
 
 P 
 
 This formula increases the throat diameter, and conse- 
 quently the center distance. The amount of the increase can 
 be found by comparing this value of T with the one as obtained 
 by formula on page 36. To keep the original center distance, 
 the outside diameter of the worm must be reduced by the 
 same amount the throat diameter is increased. 
 
 Second Method. Without changing any of the dimensions 
 we found by the formulas given on page 36, we can avoid the 
 interference to be found in worm wheels of less than 30 teeth 
 by simply increasing the angle of worm thread. We find the 
 value of this angle by the following formula : 
 Let there be 
 
 2 y = angle o worm tnreaa. 
 
 N = number of teeth in worm wheel. 
 
 cos y 
 
 From this formula we obtain the following values : 
 
 N 
 
 29 
 
 28 
 
 2 7 
 
 26 
 
 25 
 
 24 
 
 23 
 
 22 
 
 21 
 
 2 y 
 
 3oK 
 
 3i 
 
 $ 
 
 32^ 
 
 32^ 
 
 33^ 
 
 34K 
 
 35 
 
 36 
 
 N 
 
 19 
 
 18 
 
 17 
 
 16 
 
 15 
 
 14 
 
 13 
 
 12 
 
 2 y 
 
 38 
 
 39 
 
 40 
 
 4I y 2 
 
 42% 
 
 44/2 
 
 46^ 
 
 4 8 
 
 20 
 
 37 
 
 As this latter formula involves the making of new hobs in 
 many cases, on account of change of angle, we prefer to reduce 
 the diameter of worm as indicated by first method, if the dis- 
 tance of centers must be absolute. 
 
BROWN & SHARPS MFG. CO. 
 
 
 SO 
 
 99 
 
 CD 
 
 TURNS 
 PER INCH 
 
 HOlId 
 
PROVIDENCE, 11. I. 
 
 39 
 
 4, 
 
 
 
 fcl 
 
 o? 
 
 CO 
 
 CO 
 
 CD 
 
 1 
 
 H* 
 
 CO 
 
 HOiid 
 
BROWN & SHARPK MFG. CO. 
 
 CHAPTER v. 
 
 SPIRAL OR SCREW GEARING, 
 
 (Figs. 9, 10, n.) 
 
 RIGHT HAND SPIRAL GEARS. 
 
 In spiral gearing the wheels have cylindrical pitch surfaces, 
 but the teeth are not parallel to the axis. The line in which 
 the pitch surface intersects the face of a tooth is part of a 
 screw line, or helix, drawn at the pitch surface. A screw 
 wheel may have one or any number of teeth. A one-toothed 
 wheel corresponds to a one-threaded screw, a many-toothed 
 wheel to a many- threaded screw. The axes may be placed at 
 any angle. 
 
 Consider spiral gears with : 
 
 I. Axes parallel. 
 II. Axes at right angles. 
 III. Axes any angle. 
 
PROVIDENCE, R. I. 
 
 " S- number of teeth in gears ! * 
 
 Fig. 10. 
 
 LEFT HAND SPIRAL GEAR. 
 
 Let there be : 
 
 N,= 
 
 N, 
 
 C = center distance. 
 
 P = diametral pitch 
 
 P' = circular pitch. 
 P n = normal diametral pitch. 
 P' n = normal circular pitch. 
 
 y = angle of axes. 
 
 Lj = exact lead of spiral on pitch surface. 
 L Q = approximate lead of spiral on pitch surface. 
 
 T = number of teeth marked on cutter to be used when 
 teeth are to be cut on milling machine. 
 
 D = pitch diameter. 
 
 B = blank diameter. 
 
 ^ a = [ an gl e of teeth with axis 
 
 / = thickness of tooth. 
 s = addendum and module. 
 D" + / = whole depth of tooth. 
 
 NOTE. Letters a and b occurring at bottom of notations refer to gears a and b. 
 
 I. AXES PARALLEL. 
 
 Gears of this class are called twisted gears. The angle of 
 teeth with axes in both gears must be equal and the spirals 
 run in opposite directions. The angles are generally chosen 
 small (seldom over 20) to avoid excessive end thrust. End 
 thrust may, however, be entirely avoided by combining two 
 pairs of wheels with right and left-hand obliquity. Gears of 
 this class are known as Herringbone gears. They are com- 
 paratively noiseless running at high speed. 
 
42 BROWN & SHARPK MFG. CO. 
 
 II. AXES AT RIGHT ANGLES. 
 Here we must always have : 
 
 1. The teeth of same hand spiral ; 
 
 2. The normal pitches equal in both gears ; and 
 
 3. The sum of the angles of teeth with axes = 90. 
 
 CHOOSING ANGLE OF TEETH WITH AXES. 
 
 1. If in a pair of gears the ratio of the number of teeth is 
 equal to the direct ratio of the diameters, /. *., if the number of 
 teeth in the two gears are to each other as their pitch diame- 
 ters, then the angles of the spirals will be 45 and 45 ; for, this 
 condition being fulfilled, the circular pitches of the two gears 
 must be alike, which is only possible with angles of 45. In 
 such a combination either gear may be the driver. 
 
 2. If the ratio of the diameters determined upon is larger 
 or smaller than the ratio of the number of teeth, then the 
 angles are : 
 
 In such gears the velocity ratio is measured by the number 
 of teeth, and not by the diameters. 
 3. Given N a , N & and C : 
 If P ' is made = P 6 ', then we have case " i " and 
 
 p, _ 7t C 
 
 But if P a ' is assumed, then : 
 
 and 
 
 tan a a = ^ tan a b = ?1 
 
 " b * a 
 
 The gear whose P' or a is larger will ordinarily be the 
 driver, on account of the greater obliquity of the teeth. 
 
 4. Given N , N 6 and C or D. 
 See case " 7 " under III., considering ;/ = 90. 
 
 III. Axis AT ANY ANGLE (y}. 
 
 5. Given case " i," under II., then angles of spirals = }4 ^, 
 for the same reason. 
 
 6. Analogous cases to "2" and "3," under II., may be 
 worked out, when angles of axes = 7, but they have been 
 
PROVIDENCE, R. I. 
 
 43 
 
 omitted, partly because the formulas are too cumbersome, and 
 partly because they are to some extent covered by cases "5 " 
 and ' k 7." 
 
 7. Given N a , N 6 and C, or one of the pitch diameters. We 
 find the angles by a graphic method, which for all practical 
 purposes is accurate enough ; ro and v o are the axes of gears 
 forming angle y (see diagram, Fig. n.) On these axes we 
 lay off lines o r and o v representing the ratio of the number 
 of teeth (velocity ratio), so that N a : N 6 : : r s : s v, and 
 
 Fig. 11. 
 
 construct parallelogram o r s v. Then, according to Mc- 
 Cord,* the angles formed by the tangent s o in the pitch con- 
 tact o with the axes of the gears insures the least amount of 
 sliding. In bisecting angle y by tangent u o and using angles 
 produced in this manner we equally distribute the end thrust on 
 both shafts. Both methods have their advantages ; to profit 
 by .both we select angles cx a and a bt produced by tangent o x, 
 bisecting angle u o s. 
 
 Thus we have when angles are found and C given, 
 / 2 C n cos a n cos al 
 
 and when D a given 
 
 N a cos 
 
 D a 7t COS (X a 
 
 ~N7 
 
 and 
 
 7t cos a b 
 
 * McCord, Kinematics, page 278. 
 
44 BROWN & SHARPE MFG. CO. 
 
 GENERAL FORMULAS, 
 
 y ~ a a + a b 
 P a tn = P b ' n 
 
 TTCOSfX 
 
 B = D + 2 .r or = D + -. 
 
 or= 
 
 N cos a 
 
 p'n _ p' CO g a 
 
 P n = JL (Pitch of cutter.) 
 
 pm 
 
 D"+/ = 2 J -f 
 
 10 
 
 T = -5L- (5^ Note 7.) 
 
 cos 3 ** 
 
 NP' NTT 
 
 r 
 
 tan a P tan a [ L,i& = N& P' a 
 
 L a = I0 W ^ (6"^ A^^/^ 2 and examples^) 
 
 S G t 
 
 cos 45= 70711 
 cos 3 45= .3535 
 tan 45 = i.ooo 
 
 NOTE i. Cutters of regular involute system. 
 
 Use No. i cutter for T from 135 up. 
 
 2 i. . 55 to 134 
 
 26 to 34 
 
 No. 5 cutter for T from 21 to 25 
 
 6 '' " " " i? to 20 
 
 y i* ' " ** 
 
 35 to 54 " J 4 to 16 
 
 12 to 13 
 
 Note 2. Gears used on spiral head and bed for Brown & Sharpe milling 
 
 machine : 
 
 W = number of teeth in gear on worm. 
 
 Gi = " " ist " stud. 
 
 G 2 = " '* 2d " stud. 
 
 S = " *' '.' screw. 
 
 Should a spiral head of different construction be used, the formula might not 
 apply. 
 
PROVIDENCE, R. I. 
 
 45 
 
 The following data are usually required in cutting spiral 
 gears in a Universal Milling Machine, and it will be found 
 convenient to arrange them in tabular form as follows : 
 
 GEAR. 
 
 PINION. 
 
 No. of Teeth - - 
 
 Pitch Diameter - 
 
 Outside Diameter 
 
 Circular Pitch - - - - 
 
 Angle of Teeth with Axis 
 
 Normal Circular Pitch 
 
 Pitch of Cutter - - 
 
 Addendum s 
 
 Thickness of Tooth t 
 Whole Depth D"+f - 
 
 No. of Cutter 
 
 Exact Lead of Spiral ------- 
 
 Approximate Lead of Spiral - - - - 
 
 Gears on Milling Machine to Cut Spiral 
 
 Gear on Worm 
 
 ist Gear on Stud - 
 
 2nd Gear on Stud -------- 
 
 Gear on Screw - - 
 
 If the exact lead L! can be obtained by the gears at hand, 
 LI will equal L 2 and we shall have from the formula 
 io W G 2 
 
 S Gt 
 
 L, W Go 
 -J = Q-^r- (for B. & S. Milling Machine.) 
 
 I O o ^--^i 
 
 Example I. 
 
 Required the gears for cutting a spiral of 2%" lead. 
 
 ~- = - factoring, in the most simple way, we have 
 
 XI i X 28 32 x 28 W G 2 
 
 2X2 56 X 2 56 X 64 
 
46 BROWN & SHARPK MFG. CO. 
 
 Thus the gearing will be 32 T. on worm, 64 T. ist. on stud, 
 28 T. 2nd on stud, and 56 T. on screw. 
 
 Trying these gears on the Milling Machine we find that 
 they cannot be used, and as we have no other regular gears 
 in the ratio of 2 to I that can be used we must try, by factor- 
 ing, to get such ratios for the two pairs of gears as to be able 
 to use the gears at hand, bearing in mind that the combined 
 ratio must be J. 
 
 i 18. 3x6 24 x 6 24 x 48 
 
 4 = 72 = 9T8 == 9 x 64 == 72 x 64 
 
 These gears are at hand and the combination can be used 
 on the machine, giving the exact lead of 2$". 
 
 Example II. 
 
 Required the gears for cutting a spiral of 8.639" l ea -d. 
 
 8.639 = 8 T 6 % 9 u- ; reducing, by continued fractions, to a 
 smaller fraction of approximately the same value, as described 
 on pages 74 and 75 
 
 639 ) 1000 ( i 
 639 
 
 361 )639( i 
 361 
 
 278 ) 361 ( i 
 278 
 
 83 ) 278 ( 3 
 249 
 
 29 ) 8 3 ( 2 
 
 58 
 
 25 ) 29 ( I 
 25 
 
 4)25(6 
 24 
 
 1)4(4 
 4 
 
PROVIDENCE, R. I. 47 
 
 1113216 4 
 
 Selecting || as an approximation near enough for our 
 purpose, and in fact as near as we are likely to find gears for, 
 we have for our lead 8|f. Applying the formula as in Ex- 
 ample I. 
 
 _ W_G 2 
 
 10 " ~ S d 
 
 216 108 
 ijjjo == 7^5 
 
 9 X 12 9 X48 72 X 48 
 
 o ijjjo == 7^5 factorm g we have 
 
 the 
 
 these being regular gears furnished with the Milling Machine. 
 
 Proof : 
 
 72 x 48 x io 
 
 - = 0.040 = .Lo 
 
 100 x 40 
 
 8.639 =L! , 
 
 .001" error in lead. 
 
 In shops where much work is done in milling spirals it is 
 desirable to have a full set of gears for the milling machine, 
 from the smallest to the largest numbers of teeth that can be 
 used. This makes it possible, in most cases, to get closer 
 approximations than could be otherwise obtained, and often 
 saves a great deal of figuring. 
 
 When the use of continued fractions does not bring a 
 close enough approximation, one method to secure a closer 
 result is to add to or substract from the numerator and de- 
 nominator of the fraction to be reduced, any numbers nearly 
 in proportion to the given fraction, seeing that the numbers 
 added or substracted are such as to make the fraction reduc- 
 ible to lower terms. By a little ingenuity and patience ex- 
 tremely close approximations can generally be reached in 
 this way. 
 
 Take, as an illustration, the fraction in Example II. 
 
 _ 
 
 IO IOOOO 
 
 Adding 9 to the numerator and io to the denominator, these 
 
48 BROWN & SHARPK MFG. CO. 
 
 being in about the same ratio to each other as the numerator 
 and denominator of the fraction, we have 
 
 8639+9 = 8648 _ 4324 _ 47 x 92 
 10000+10 = 10010 5005 55 x 91 
 
 All of the gears in this case are special. 
 
 Applying the same proof as in Example II. we find that 
 this train of gears will give a lead of 8.6393+, making an 
 error of .0003" in the lead. 
 
 No doubt a much closer approximation even than this 
 could be obtained by further trial. 
 
 Another method is to multiply both terms of the fraction 
 by some number which will make one term of the fraction 
 easily reducible, and adding one to or subtracting it from the 
 other term to make it possible to reduce that also. 
 
 There is an element of uncertainty in both these methods, 
 as we never feel sure that we have obtained the best combina- 
 tion ; practical work, however, rarely requires accuracy beyond 
 a point that can readily be reached. 
 
 The accompanying list of prime numbers and factors will 
 be found useful in reducing and factoring fractions. 
 
PROVIDENCE, R. I. 
 
 49 
 
 PRIME NUMBERS AND FACTORS. 
 1 TO 1OOO. 
 
 1 
 
 
 26 
 
 2x13 
 
 51 
 
 3x17 
 
 76 
 
 2 2 xl9 
 
 2 
 
 
 27 
 
 3 3 
 
 52 
 
 2 2 xl3 
 
 77 
 
 7x11 
 
 3 
 
 
 28 
 
 2 2 x7 
 
 53 
 
 
 78 
 
 2x3x13 
 
 4 
 
 2 2 
 
 29 
 
 
 54 
 
 2x3 3 
 
 79 
 
 
 5 
 
 
 30 
 
 2x3x5 
 
 55 
 
 5x11 
 
 80 
 
 2 4 x5 
 
 6 
 
 2x3 
 
 31 
 
 
 56 
 
 2 3 x7 
 
 81 
 
 3 4 
 
 7 
 
 
 32 
 
 2 5 
 
 57 
 
 3x19 
 
 82 
 
 2x41 
 
 8 
 
 2 3 
 
 33 
 
 3x11 
 
 58 
 
 2x29 
 
 83 
 
 
 9 
 
 3 2 
 
 34 
 
 2x17 
 
 59 
 
 
 84 
 
 2 2 x3x7 
 
 10 
 
 2x5 
 
 35 
 
 5x7 
 
 60 
 
 2 2 x 3 x 5 
 
 85 
 
 5x17 
 
 11 
 
 
 36 
 
 2 2 x3 2 
 
 61 
 
 
 86 
 
 2x43 
 
 12 
 
 2 2 x3 
 
 37 
 
 
 62 
 
 2x31 
 
 87 
 
 3x29 
 
 13 
 
 
 38 
 
 2x19 
 
 63 
 
 3 2 x7 
 
 88 
 
 2 3 Xll 
 
 14 
 
 2x7 
 
 39 
 
 3x13 
 
 64 
 
 2 6 
 
 89 
 
 
 15 
 
 3x5 
 
 40 
 
 2 3 x5 
 
 65 
 
 5x13 
 
 90 
 
 2x3 2 x5 
 
 16 
 
 2 4 
 
 41 
 
 
 66 
 
 2x3x11 
 
 91 
 
 7x13 
 
 17 
 
 
 42 
 
 2x3x7 
 
 67 
 
 
 92 
 
 2 2 x23 
 
 18 
 
 2x3 2 
 
 43 
 
 
 68 
 
 2 2 xl7 
 
 93 
 
 3x31 
 
 19 
 
 
 44 
 
 2 2 xll 
 
 69 
 
 3x23 
 
 94 
 
 2x47 
 
 20 
 
 2 2 X5 
 
 45 
 
 3 2 x5 
 
 70 
 
 2x5x7 
 
 95 
 
 5x19 
 
 21 
 
 3x7 
 
 46 
 
 2x23 
 
 71 
 
 
 96 
 
 2 5 x3 
 
 22 
 
 2x11 
 
 47 
 
 
 72 
 
 2 3 X3 2 
 
 97 
 
 
 23 
 
 
 48 
 
 2 4 x3 
 
 73 
 
 
 98 
 
 2X7 2 
 
 24 
 
 2 3 x3 
 
 49 
 
 7 2 
 
 74 
 
 2x37 
 
 99 
 
 3 2 xll 
 
 25 
 
 5 2 
 
 50 
 
 2x5 2 
 
 75 
 
 3x5 2 
 
 100 
 
 2 2 X5 2 
 
BROWN & SHARPE MFG. CO. 
 
 101 
 
 
 131 
 
 
 161 
 
 7x23 
 
 191 
 
 
 102 
 
 2x3x17 
 
 132 
 
 2 2 x3xll 
 
 162 
 
 2x3 4 
 
 192 
 
 2 6 x3 
 
 103 
 
 
 133 
 
 7x19 
 
 163 
 
 
 193 
 
 
 104 
 
 2 3 xl3 
 
 134 
 
 2x67 
 
 164 
 
 2 2 X41 
 
 194 
 
 2x97 
 
 105 
 
 3x5x7 
 
 135 
 
 3 3 x5 
 
 165 
 
 3x5x11 
 
 195 
 
 3x5x13 
 
 106 
 
 2x53 
 
 136 
 
 2 3 xl7 
 
 166 
 
 2x83 
 
 196 
 
 2 2 x7 2 
 
 107 
 
 
 137 
 
 
 167 
 
 
 197 
 
 
 108 
 
 2 2 x3 8 
 
 138 
 
 2 x 3 x 23 
 
 168 
 
 2 3 x3x7 
 
 198 
 
 2x3 2 xll 
 
 109 
 
 
 139 
 
 
 109 
 
 13 2 
 
 199 
 
 
 110 
 
 2X5X11 
 
 140 
 
 2 2 x5x7 
 
 170 
 
 2x5x17 
 
 200 
 
 2 3 x5 2 
 
 111 
 
 3x37 
 
 141 
 
 3x47 
 
 171 
 
 3 2 xl9 
 
 201 
 
 3x67 
 
 112 
 
 2 4 x7 
 
 142 
 
 2x71 
 
 172 
 
 2 2 x43 
 
 202 
 
 2x101 
 
 113 
 
 
 143 
 
 11X13 
 
 173 
 
 
 203 
 
 7x29 
 
 114 
 
 2x3x19 
 
 144 
 
 2 4 x3 2 
 
 174 
 
 2x3x29 
 
 204 
 
 2 2 x3xl7 
 
 115 
 
 5x23 
 
 145 
 
 5x29 
 
 175 
 
 5 2 x 7 
 
 205 
 
 5x41 
 
 116 
 
 2 2 X29 
 
 146 
 
 2x73 
 
 176 
 
 2 4 xll 
 
 206 
 
 2x103 
 
 117 
 
 3 2 xl3 
 
 147 
 
 3x7 2 
 
 177 
 
 3x59 
 
 207 
 
 3-'x23 
 
 118 
 
 2x59 
 
 148 
 
 2 2 x37 
 
 178 
 
 2x89 
 
 208 
 
 2 4 xl3 
 
 119 
 
 7x17 
 
 149 
 
 
 179 
 
 
 209 
 
 11x19 
 
 120 
 
 2 3 x3x5 
 
 150 
 
 2 x 3 x 5 2 
 
 180 
 
 2 2 x3 2 x5 
 
 210 
 
 2x3x5x7 
 
 121 
 
 II 2 
 
 151 
 
 
 181 
 
 
 211 
 
 
 122 
 
 2x61 
 
 152 
 
 2 3 xl9 
 
 182 
 
 2x7x13 
 
 212 
 
 2 2 X53 
 
 123 
 
 3x41 
 
 153 
 
 3 2 xl7 
 
 183 
 
 3x61 
 
 213 
 
 3x71 
 
 124 
 
 2 2 x31 
 
 154 
 
 2x7x11 
 
 184 
 
 2 3 x.23 
 
 214 
 
 2x107 
 
 125 
 
 5 3 
 
 155 
 
 5x31 
 
 185 
 
 5x37 
 
 215 
 
 5x43 
 
 126 
 
 2x3 2 x7 
 
 156 
 
 2 2 x3xl3 
 
 186 
 
 2x3x31 
 
 216 
 
 2 3 x3 3 
 
 127 
 
 
 157 
 
 
 187 
 
 11 X17 
 
 217 
 
 7x31 
 
 128 
 
 2 7 
 
 158 
 
 2x79 
 
 188 
 
 2 2 x47 
 
 218 
 
 2x109 
 
 129 
 
 3x43 
 
 159 
 
 3x53 
 
 189 
 
 3 3 x7 
 
 219 
 
 3x73 
 
 130 
 
 2x5x13 
 
 160 
 
 2 5 X5 
 
 190 
 
 2x5x19 
 
 220 
 
 2 2 x5xll 
 
PROVIDENCE, R. I. 
 
 5 1 
 
 221 
 
 13x17 
 
 251 
 
 
 281 
 
 
 311 
 
 
 222 
 
 2x3x37 
 
 252 
 
 2 2 x3 2 x7 
 
 282 
 
 2x3x47 
 
 312 
 
 2 3 x3xl3 
 
 223 
 
 
 253 
 
 11x23 
 
 283 
 
 
 313 
 
 
 224 
 
 2 5 x7 
 
 254 
 
 2X127 
 
 284 
 
 2 2 x71 
 
 314 
 
 2x157 
 
 225 
 
 3 2 x5 2 
 
 255 
 
 3x5x17 
 
 285 
 
 3x5x19 
 
 315 
 
 3 2 x5x7 
 
 226 
 
 2x113 
 
 256 
 
 2 8 
 
 286 
 
 2x11 x!3 
 
 316 
 
 2 2 x79 
 
 227 
 
 
 257 
 
 
 287 
 
 7x41 
 
 317 
 
 
 228 
 
 2 2 x3xl9 
 
 258 
 
 2 X 3 x 43 
 
 288 
 
 2 5 X3 2 
 
 318 
 
 2x3x53 
 
 229 
 
 
 259 
 
 7x37 
 
 289 
 
 17 2 
 
 319 
 
 11X29 
 
 230 
 
 2 x 5 X 23 
 
 260 
 
 2 2 x5xl3 
 
 290 
 
 2x5x29 
 
 320 
 
 2 6 x5 
 
 231 
 
 3X7X11 
 
 261 
 
 3 2 x29 
 
 291 
 
 3x97 
 
 321 
 
 3x107 
 
 232 
 
 2 3 x29 
 
 262 
 
 2x131 
 
 292 
 
 2 2 x73 
 
 322 
 
 2x7x23 
 
 233 
 
 
 263 
 
 
 293 
 
 
 323 
 
 17x19 
 
 234 
 
 2x3 2 xl3 
 
 264 
 
 2 3 x3xll 
 
 294 
 
 2x3x7 2 
 
 324 
 
 2 2 x3 4 
 
 235 
 
 5x47 
 
 265 
 
 5x53 
 
 295 
 
 5x59 
 
 325 
 
 5 2 X13 
 
 236 
 
 2 2 x59 
 
 266 
 
 2x7x19 
 
 296 
 
 2 3 x37 
 
 326 
 
 2x163 
 
 237 
 
 Bx79 
 
 267 
 
 3x89 
 
 297 
 
 3 3 xll 
 
 327 
 
 3x109 
 
 238 
 
 2x7x17 
 
 268 
 
 2 2 x67 
 
 298 
 
 2x149 
 
 328 
 
 2 3 x41 
 
 239 
 
 
 269 
 
 
 299 
 
 13x23 
 
 329 
 
 7x47 
 
 240 
 
 2 4 x3xo 
 
 270 
 
 2x3 3 x5 
 
 300 
 
 2 2 x3x5 2 
 
 330 
 
 2X3X5X11 
 
 241 
 
 
 271 
 
 
 301 
 
 7x43 
 
 331 
 
 
 242 
 
 2xll 2 
 
 272 
 
 2 4 xl7 
 
 302 
 
 2x151 
 
 332 
 
 2 2 X83 
 
 243 
 
 ^ 
 
 273 
 
 3x7x13 
 
 303 
 
 3x101 
 
 333 
 
 3 2 X37 
 
 244 
 
 2 2 X61 
 
 274 
 
 2x137 
 
 304 
 
 2 4 X19 
 
 334 
 
 2X167 
 
 245 
 
 5X7 2 
 
 275 
 
 5 2 Xll 
 
 305 
 
 5x61 
 
 335 
 
 5x67 
 
 246 
 
 2x3x41 
 
 276 
 
 2 2 x3x23 
 
 306 
 
 2x3 2 xl7 
 
 336 
 
 2 4 x3x7 
 
 247 
 
 13x19 
 
 277 
 
 
 307 
 
 
 337 
 
 
 248 
 
 2 3 X31 
 
 278 
 
 2x139 
 
 308 
 
 2 2 x7xll 
 
 338 
 
 2X13 2 
 
 249 
 
 3x83 
 
 279 
 
 3 2 X31 
 
 309 
 
 3x103 
 
 339 
 
 3x113 
 
 250 
 
 2x5 3 
 
 280 
 
 2 3 x 5 x 7 
 
 310 
 
 2x5x31 
 
 340 
 
 2 2 xoxl7 
 
BROWN & SHARPK MFG. CO. 
 
 341 
 
 11x31 
 
 371 
 
 7x53 
 
 401 
 
 
 431 
 
 
 342 
 
 2x3 2 xl9 
 
 372 
 
 2 2 x3x31 
 
 402 
 
 2x3x67 
 
 432 
 
 2 4 x3 3 
 
 343 
 
 7 3 
 
 373 
 
 
 403 
 
 13x31 
 
 433 
 
 
 344 
 
 2 3 X43 
 
 374 
 
 2X11X17 
 
 404 
 
 2 2 xl01 
 
 434 
 
 2x7x31 
 
 345 
 
 3x5x23 
 
 375 
 
 3X5 3 
 
 405 
 
 3 4 X5 
 
 435 
 
 3x5x29 
 
 346 
 
 2x173 
 
 376 
 
 2 3 X47 
 
 406 
 
 2x7x29 
 
 436 
 
 2 2 xl09 
 
 347 
 
 
 377 
 
 13x29 
 
 407 
 
 11x37 
 
 437 
 
 19x23 
 
 348 
 
 2 2 x3x29 
 
 378 
 
 2x3 3 x7 
 
 408 
 
 2 3 x3xl7 
 
 438 
 
 2 X 3 x 73 
 
 349 
 
 
 879 
 
 
 409 
 
 
 439 
 
 
 350 
 
 2x5 2 x7 
 
 380 
 
 2 2 xoxl9 
 
 410 
 
 2x5x41 
 
 440 
 
 2 3 xoxll 
 
 351 
 
 3 3 X13 
 
 381 
 
 3x127 
 
 411 
 
 3x137 
 
 441 
 
 3 2 x7 2 
 
 352 
 
 2 5 Xll 
 
 382 
 
 2X191 
 
 412 
 
 2 2 X103 
 
 412 
 
 2x13x17 
 
 353 
 
 
 383 
 
 
 413 
 
 7x59 
 
 443 
 
 
 354 
 
 2x3x59 
 
 384 
 
 2 7 x3 
 
 414 
 
 2x3 2 x23 
 
 444 
 
 2 2 x3x37 
 
 355 
 
 5x71 
 
 385 
 
 5x7 X 11 
 
 415 
 
 5x83 
 
 445 
 
 5x89 
 
 356 
 
 2 2 x89 
 
 386 
 
 2x193 
 
 416 
 
 2 3 Xl3 
 
 446 
 
 2x223 
 
 357 
 
 3x7x17 
 
 387 
 
 3 2 X43 
 
 417 
 
 3x139 
 
 447 
 
 3x149 
 
 358 
 
 2x179 
 
 388 
 
 2 2 X97 
 
 418 
 
 2x11x19 
 
 448 
 
 2 6 x7 
 
 359 
 
 
 389 
 
 
 419 
 
 
 449 
 
 
 360 
 
 2 3 X3 2 X5 
 
 390 
 
 2X3X5X13 
 
 420 
 
 2 2 X3X5X7 
 
 450 
 
 2x3 2 x5 2 
 
 361 
 
 19 2 
 
 391 
 
 17x23 
 
 421 
 
 
 451 
 
 11X41 
 
 362 
 
 2x181 
 
 392 
 
 2 3 X7 2 
 
 422 
 
 2x211 
 
 452 
 
 2 2 X113 
 
 363 
 
 3xll 2 
 
 393 
 
 3x131 
 
 423 
 
 3 2 X47 
 
 453 
 
 3x151 
 
 364 
 
 2 2 x7xl3 
 
 394 
 
 2x197 
 
 424 
 
 2 3 x53 
 
 454 
 
 2x227 
 
 365 
 
 5x73 
 
 395 
 
 5x79 
 
 425 
 
 5 2 xl7 
 
 455 
 
 5x7x13 
 
 366 
 
 2x3x61 
 
 396 
 
 2 2 X3 2 XH 
 
 426 
 
 2x3x71 
 
 456 
 
 2 3 x3xl9 
 
 367 
 
 
 397 
 
 
 427 
 
 7x61 
 
 457 
 
 
 368 
 
 2 4 X23 
 
 398 
 
 2x199 
 
 428 
 
 2 2 Xl07 
 
 458 
 
 2x229 
 
 369 
 
 3 2 x41 
 
 399 
 
 3x7x19 
 
 429 
 
 3x11x13 
 
 459 
 
 -3 3 xl7 
 
 370 
 
 2x5x37 
 
 400 
 
 2 4 x5 2 
 
 430 
 
 2x5x43 
 
 460 
 
 2 2 x5x23 
 
PROVIDENCE, R. I. 
 
 53 
 
 461 
 
 
 491 
 
 
 521 
 
 
 551 
 
 19x29 
 
 462 
 
 2X3X7X11 
 
 492 
 
 2 2 x3x41 
 
 ,522 
 
 2x3 2 x29 
 
 552 
 
 2 3 x3x23 
 
 463 
 
 
 493 
 
 17x29 
 
 523 
 
 
 553 
 
 7x79 
 
 464 
 
 2 4 x29 
 
 494 
 
 2x13x19 
 
 524 
 
 2 2 xl31 
 
 554 
 
 2x277 
 
 465 
 
 3x5x31 
 
 495 
 
 3 2 x5xll 
 
 525 
 
 3x5 2 x7 
 
 555 
 
 3x5x37 
 
 466 
 
 2 X 233 
 
 496 
 
 2 4 X31 
 
 526 
 
 2x263 
 
 556 
 
 2 2 xl39 
 
 467 
 
 
 497 
 
 7x 71 
 
 527 
 
 17x31 
 
 557 
 
 
 468 
 
 2 2 x3 2 xl3 
 
 498 
 
 2x3x83 
 
 528 
 
 2 4 x3xll 
 
 558 
 
 2x3 2 x31 
 
 469 
 
 7x67 
 
 499 
 
 
 529 
 
 23 2 
 
 559 
 
 13x43 
 
 470 
 
 2x5x47 
 
 500 
 
 2 2 x5 3 
 
 530 
 
 2x5x53 
 
 560 
 
 2 4 x5x7 
 
 471 
 
 3x157 
 
 501 
 
 3x167 
 
 531 
 
 3 2 X59 
 
 561 
 
 3x11x17 
 
 472 
 
 2 3 x59 
 
 502 
 
 2x251 
 
 532 
 
 2 2 x7xl9 
 
 562 
 
 2x281 
 
 473 
 
 11X43 
 
 503 
 
 
 533 
 
 13X41 
 
 563 
 
 
 474 
 
 2x3x79 
 
 504 
 
 2 3 X3 2 X7 
 
 534 
 
 2x3x89 
 
 564 
 
 2 2 x3x47 
 
 475 
 
 5 2 X19 
 
 505 
 
 5x101 
 
 535 
 
 5x107 
 
 565 
 
 5x113 
 
 476 
 
 2 2 x7xl7 
 
 506 
 
 2 X 1 1 X 23 
 
 536 
 
 2 s x67 
 
 566 
 
 2x283 
 
 477 
 
 3 2 x53 
 
 507 
 
 3xl3 2 
 
 537 
 
 3x179 
 
 567 
 
 3 4 x7 
 
 478 
 
 2x239 
 
 508 
 
 2 2 xl27 
 
 538 
 
 2x269 
 
 568 
 
 2 3 X71 
 
 479 
 
 
 509 
 
 
 539 
 
 7 2 xll 
 
 569 
 
 
 480 
 
 2 5 x 3 x 5 
 
 510 
 
 2X3X5X17 
 
 540 
 
 2 2 x3 3 x5 
 
 570 
 
 2x3X5X19 
 
 481 
 
 13x37 
 
 511 
 
 7x73 
 
 541 
 
 
 571 
 
 
 482 
 
 2x241 
 
 512 
 
 2 9 
 
 542 
 
 2x271 
 
 572 
 
 2 2 xllXl3 
 
 483 
 
 3 X 7 X 23 
 
 513 
 
 3 3 xl9 
 
 543 
 
 3x181 
 
 573 
 
 3x191 
 
 484 
 
 2 2 xll 2 
 
 514 
 
 2 X 257 
 
 544 
 
 2 5 xl7 
 
 574 
 
 2x7x41 
 
 485 
 
 5x97 
 
 515 
 
 5x103 
 
 545 
 
 5x109 
 
 575 
 
 5 2 x23 
 
 486 
 
 2x3 5 
 
 516 
 
 2 2 x3x43 
 
 546 
 
 2X3X7X13 
 
 576 
 
 2 6 x3 2 
 
 487 
 
 
 517 
 
 11X47 
 
 547 
 
 
 577 
 
 
 488 
 
 2 3 x61 
 
 518 
 
 2x7x37 
 
 548 
 
 2 2 xl37 
 
 578 
 
 2X17 2 
 
 489 
 
 3x163 
 
 519 
 
 3x173 
 
 549 
 
 3 2 x61 
 
 579 
 
 3x193 
 
 490 
 
 2 X 5 X 7 2 
 
 520 
 
 2 3 x5xl3 
 
 550 
 
 2x5 2 xll 
 
 580 
 
 2 2 x5x29 
 
54 
 
 BROWN & SHARPE MFG. CO. 
 
 581 
 
 7x83 
 
 611 
 
 13x47 
 
 641 
 
 
 671 
 
 11X61 
 
 582 
 
 2x3x97 
 
 612 
 
 2 2 x3 2 xl7 
 
 642 
 
 2x3x107 
 
 672 
 
 2 5 x 3 x 7 
 
 583 
 
 11x53 
 
 613 
 
 
 643 
 
 
 673 
 
 
 584 
 
 2 3 x73 
 
 614 
 
 2x307 
 
 644 
 
 2 2 x7x23 
 
 674 
 
 2x337 
 
 585 
 
 3 2 xoxl3 
 
 615 
 
 3x5x41 
 
 645 
 
 3x5x43 
 
 675 
 
 3 3 x5 2 
 
 586 
 
 2x293 
 
 616 
 
 2 3 x7xll 
 
 646 
 
 2x17x19 
 
 676 
 
 2 2 xl3 2 
 
 587 
 
 
 617 
 
 
 647 
 
 
 677 
 
 
 588 
 
 2 2 x3x7 2 
 
 618 
 
 2x3x103 
 
 648 
 
 2 3 x3 4 
 
 678 
 
 2x3x113 
 
 589 
 
 19x31 
 
 619 
 
 
 649 
 
 11X59 
 
 679 
 
 7x97 
 
 590 
 
 2x5x59 
 
 620 
 
 2 2 x5x31 
 
 650 
 
 2x5 2 xl3 
 
 680 
 
 2 3 x5xl7 
 
 591 
 
 3x197 
 
 621 
 
 3 3 x23 
 
 651 
 
 3x7x31 
 
 681 
 
 3x227 
 
 592 
 
 2 4 x37 
 
 622 
 
 2x311 
 
 652 
 
 2 2 xl63 
 
 682 
 
 2x11x31 
 
 593 
 
 
 623 
 
 7x89 
 
 653 
 
 
 683 
 
 
 594 
 
 2x3 3 Xll 
 
 624 
 
 2 4 x3xl3 
 
 654 
 
 2x3x109 
 
 684 
 
 2 2 x3 2 xl9 
 
 595 
 
 5x7x 17 
 
 625 
 
 5 4 
 
 655 
 
 5x131 
 
 685 
 
 5x137 
 
 596 
 
 2 2 X149 
 
 626 
 
 2x313 
 
 656 
 
 2 4 x41 
 
 686 
 
 2x7 3 
 
 597 
 
 3x199 
 
 627 
 
 3x11x19 
 
 657 
 
 3 2 x73 
 
 687 
 
 3x229 
 
 598 
 
 2x13x23 
 
 628 
 
 2 2 xl57 
 
 658 
 
 2x7x47 
 
 688 
 
 2 4 x43 
 
 599 
 
 
 629 
 
 17x37 
 
 659 
 
 
 689 
 
 
 600 
 
 2 3 x3xo 2 
 
 630 
 
 2X3 2 X5X7 
 
 660 
 
 2 2 X3X5xll 
 
 690 
 
 2X3x5X23 
 
 601 
 
 
 631 
 
 
 661 
 
 
 691 
 
 
 602 
 
 2x7x43 
 
 632 
 
 2 3 x79 
 
 662 
 
 2x331 
 
 692 
 
 2 2 X173 
 
 603 
 
 3 2 x67 
 
 633 
 
 3x211 
 
 663 
 
 3x13x17 
 
 693 
 
 3 2 x7xll 
 
 604 
 
 2 2 X151 
 
 634 
 
 2x317 
 
 664 
 
 2 3 x83 
 
 694 
 
 2x347 
 
 605 
 
 5xll 2 
 
 635 
 
 5x127 
 
 665 
 
 5x7x19 
 
 695 
 
 5x139 
 
 606 
 
 2x3x101 
 
 636 
 
 2 2 x3x53 
 
 666 
 
 2x3 2 x37 
 
 696 
 
 2 3 x3x29 
 
 607 
 
 
 637 
 
 7 2 X13 
 
 667 
 
 23x29 
 
 697 
 
 17x41 
 
 608 
 
 2 5 xl9 
 
 638 
 
 2x11x29 
 
 668 
 
 2 2 xl67 
 
 698 
 
 2x349 
 
 609 
 
 3 X 7 X 29 
 
 639 
 
 3 2 X71 
 
 669 
 
 3x223 
 
 699 
 
 3x233 
 
 610 
 
 2x5x61 
 
 640 
 
 2 7 x5 
 
 670 
 
 2x5x67 
 
 700 
 
 2 2 x5 2 x7 
 
PROVIDENCE, R. I. 
 
 55 
 
 701 
 
 
 731 
 
 17x43 
 
 761 
 
 
 791 
 
 7x113 
 
 702 
 
 2x3 3 xl3 
 
 732 
 
 2 2 x3x61 
 
 762 
 
 2x3x127 
 
 792 
 
 2 3 x3 2 xll 
 
 703 
 
 19x37 
 
 733 
 
 
 763 
 
 7x109 
 
 793 
 
 13x61 
 
 704 
 
 2xll 
 
 734 
 
 2 X 367 
 
 764 
 
 2 2 xl91 
 
 794 
 
 2x397 
 
 705 
 
 3x5x47 
 
 735 
 
 3 X 5 x 7 2 
 
 765 
 
 3 2 x5xl7 
 
 795 
 
 3x5x53 
 
 706 
 
 2x353 
 
 736 
 
 2 5 x23 
 
 766 
 
 2 x 383 
 
 796 
 
 2 2 X199 
 
 707 
 
 7x101 
 
 737 
 
 11X67 
 
 767 
 
 13x59 
 
 797 
 
 
 708 
 
 2 2 x3x59 
 
 738 
 
 2x3 2 x41 
 
 768 
 
 2 8 x3 
 
 798 
 
 2X3X7X19 
 
 709 
 
 
 739 
 
 
 769 
 
 
 799 
 
 17x47 
 
 710 
 
 2x5x71 
 
 740 
 
 2 2 x5x37 
 
 770 
 
 2X5X7X11 
 
 800 
 
 2 5 x5 2 
 
 711 
 
 3 2 X79 
 
 741 
 
 3x13x19 
 
 771 
 
 3x257 
 
 801 
 
 3 2 x89 
 
 712 
 
 2 3 x89 
 
 742 
 
 2x7x53 
 
 772 
 
 2 2 xl93 
 
 802 
 
 2x401 
 
 713 
 
 23x31 
 
 743 
 
 
 773 
 
 
 803 
 
 11 X73 
 
 714 
 
 2X3X7X17 
 
 744 
 
 2"x3x31 
 
 774 
 
 2x3 2 x43 
 
 804 
 
 2 2 x3x67 
 
 715 
 
 5x11x13 
 
 745 
 
 5x149 
 
 775 
 
 5 2 x31 
 
 805 
 
 5 x 7 x 23 
 
 716 
 
 2 2 X179 
 
 746 
 
 2x373 
 
 776 
 
 2 3 x97 
 
 806 
 
 2x13x31 
 
 717 
 
 3 X 239 
 
 747 
 
 3 2 x83 
 
 777 
 
 3x7x37 
 
 807 
 
 3 x 269 
 
 718 
 
 2x359 
 
 748 
 
 2 2 xllxl7 
 
 778 
 
 2x389 
 
 808 
 
 2 3 X101 
 
 719 
 
 
 749 
 
 7x107 
 
 779 
 
 19x41 
 
 809 
 
 
 720 
 
 2 4 x3 2 x5 
 
 750 
 
 2 x 3 x 5 3 
 
 780 
 
 2 2 X3X5X13 
 
 810 
 
 2x3 4 x5 
 
 721 
 
 7x103 
 
 751 
 
 
 781 
 
 11x71 
 
 811 
 
 
 722 
 
 2xl9 2 
 
 752 
 
 2 4 x47 
 
 782 
 
 2 x 17x23 
 
 812 
 
 2 2 x7x29 
 
 723 
 
 3x241 
 
 753 
 
 3x251 
 
 783 
 
 3 3 x29 
 
 813 
 
 3x271 
 
 724 
 
 2 2 xl8l 
 
 754 
 
 2x13x29 
 
 784 
 
 2 4 x7 2 
 
 814 
 
 2x11x37 
 
 725 
 
 5 2 X29 
 
 755 
 
 5x151 
 
 785 
 
 5x ).">7 
 
 815 
 
 5x163 
 
 726 
 
 2x3xll 2 
 
 756 
 
 2 2 x3 3 x7 
 
 786 
 
 2x3x131 
 
 816 
 
 2 4 x3xl7 
 
 727 
 
 
 757 
 
 
 787 
 
 
 817 
 
 19x43 
 
 728 
 
 2 3 x7xl3 
 
 758 
 
 2x379 
 
 788 
 
 2 2 xl97 
 
 818 
 
 2x409 
 
 729 
 
 3 6 
 
 759 
 
 3x11x23 
 
 789 
 
 3x263 
 
 819 
 
 3 2 X7X13 
 
 730 
 
 2x5x73 
 
 700 
 
 2 3 x5xl9 
 
 790 
 
 2x5x79 
 
 820 
 
 2 2 x5x41 
 
BROWN & SHARPE MFG. CO. 
 
 821 
 
 
 851 
 
 23x37 
 
 881 
 
 
 911 
 
 
 822 
 
 2x3x137 
 
 852 
 
 2 2 x3x71 
 
 882 
 
 2x3 2 x7 2 
 
 912 
 
 2 4 x3xl9 
 
 823 
 
 
 853 
 
 
 883 
 
 
 913 
 
 11x83 
 
 824 
 
 2 3 xl03 
 
 854 
 
 2x7x61 
 
 884 
 
 2 2 xl3xl7 
 
 914 
 
 2x457 
 
 825 
 
 3x5 2 xll 
 
 855 
 
 3 2 x5xl9 
 
 885 
 
 3x5x59 
 
 915 
 
 3x5x61 
 
 826 
 
 2x7x59 
 
 856 
 
 2 3 xl07 
 
 886 
 
 2x443 
 
 916 
 
 2 2 x229 
 
 827 
 
 
 857 
 
 
 887 
 
 
 917 
 
 7x131 
 
 828 
 
 2 2 x3 2 x23 
 
 858 
 
 2x3x11x13 
 
 888 
 
 2 3 x3x37 
 
 918 
 
 2x3 3 xl7 
 
 829 
 
 
 859 
 
 
 889 
 
 7x127 
 
 919 
 
 
 830 
 
 2 X 5 x 83 
 
 860 
 
 2 2 x5x43 
 
 890 
 
 2 x 5 X 89 
 
 920 
 
 2 3 x5x23 
 
 831 
 
 3x277 
 
 861 
 
 3x7x41 
 
 891 
 
 3 4 xll 
 
 921 
 
 3x307 
 
 832 
 
 2 6 xl3 
 
 862 
 
 2x431 
 
 892 
 
 2 2 x223 
 
 922 
 
 2x461 
 
 833 
 
 7 2 xl7 
 
 863 
 
 
 893 
 
 19x47 
 
 923 
 
 13x71 
 
 834 
 
 2x3x 139 
 
 864 
 
 2 s X3 3 
 
 894 
 
 2x3x149 
 
 924 
 
 2 2 x3X7XH 
 
 835 
 
 5x167 
 
 865 
 
 5x173 
 
 895 
 
 5x179 
 
 925 
 
 5 2 X37 
 
 836 
 
 2 2 xllXl9 
 
 866 
 
 2x433 
 
 896 
 
 2 7 x7 
 
 926 
 
 2x463 
 
 837 
 
 3 3 x31 
 
 867 
 
 3xl7 2 
 
 897 
 
 3x13x23 
 
 927 
 
 3 2 xl03 
 
 838 
 
 2x419 
 
 868 
 
 2 2 x7x31 
 
 898 
 
 2x449 
 
 928 
 
 2 5 x29 
 
 839 
 
 
 869 
 
 11X79 
 
 899 
 
 29x31 
 
 929 
 
 
 840 
 
 2 3 X3X5X7 
 
 870 
 
 2X3X5X29 
 
 900 
 
 2 2 x3 2 x5 2 
 
 930 
 
 2X3X5X31 
 
 841 
 
 29 2 
 
 871 
 
 13x67 
 
 901 
 
 17x53 
 
 931 
 
 7 2 X19 
 
 842 
 
 2x421 
 
 872 
 
 2 3 X109 
 
 902 
 
 2x11x41 
 
 932 
 
 2 2 x233 
 
 843 
 
 3x281 
 
 873 
 
 3 2 x97 
 
 903 
 
 3x7x43 
 
 933 
 
 3x311 
 
 844 
 
 2 2 x211 
 
 874 
 
 2x19x23 
 
 904 
 
 2 3 Xll3 
 
 934 
 
 2x467 
 
 845 
 
 5X13 2 
 
 875 
 
 5 3 X7 
 
 905 
 
 5x181 
 
 935 
 
 5xllXl7 
 
 846 
 
 2x3 2 x47 
 
 876 
 
 2 2 x3x73 
 
 906 
 
 2x3x151 
 
 936 
 
 2 3 x3 2 X 13 
 
 847 
 
 7xll 2 
 
 877 
 
 
 907 
 
 
 937 
 
 
 848 
 
 2 4 X53 
 
 878 
 
 2x439 
 
 908 
 
 2 2 X227 
 
 938 
 
 2x7x67 
 
 849 
 
 3x283 
 
 879 
 
 3 x 293 
 
 909 
 
 3 2 xlOl 
 
 939 
 
 3x313 
 
 850 
 
 2x5 2 xl7 
 
 880 
 
 2 4 xoxll 
 
 910 
 
 2X5X7X13 
 
 940 
 
 2 2 x5x47 
 
PROVIDENCE, R. I. 
 
 57 
 
 941 
 
 
 956 
 
 2 2 x239 
 
 971 
 
 
 986 
 
 2x17x29 
 
 942 
 
 2x3x157 
 
 957 
 
 3x11x29 
 
 972 
 
 2 2 x3 5 
 
 987 
 
 3x7x47 
 
 943 
 
 23x41 
 
 958 
 
 2x479 
 
 973 
 
 7x139 
 
 988 
 
 2 2 x 13x19 
 
 944 
 
 2 4 X59 
 
 959 
 
 7x137 
 
 974 
 
 2x487 
 
 989 
 
 23x43 
 
 945 
 
 3 3 xox7 
 
 960 
 
 2x3xo 
 
 975 
 
 3x5 2 xl3 
 
 990 
 
 2x3 2 X5Xll 
 
 946 
 
 2x11 X43 
 
 961 
 
 31 2 
 
 976 
 
 2 4 x61 
 
 991 
 
 
 947 
 
 
 962 
 
 2x13x37 
 
 977 
 
 
 992 
 
 2 5 x31 
 
 948 
 
 2 2 x3x79 
 
 963 
 
 3 2 xl07 
 
 978 
 
 2x3x163 
 
 993 
 
 3x331 
 
 949 
 
 13x73 
 
 964 
 
 2 2 x241 
 
 979 
 
 11X89 
 
 994 
 
 2x7x71 
 
 950 
 
 2x5 2 xl9 
 
 965 
 
 5x193 
 
 980 
 
 2 2 x5x7 2 
 
 995 
 
 5x199 
 
 951 
 
 3x317 
 
 966 
 
 2X3X7X23 
 
 981 
 
 3 2 xl09 
 
 996 
 
 2 2 x3x83 
 
 952 
 
 2 3 x7xl7 
 
 967 
 
 
 982 
 
 2x491 
 
 997 
 
 
 953 
 
 
 968 
 
 2 3 xll 2 
 
 983 
 
 
 998 
 
 2x499 
 
 954 
 
 2x3 2 x53 
 
 969 
 
 3x17x19 
 
 984 
 
 2 3 x3x41 
 
 999 
 
 3 3 x37 
 
 955 
 
 5x191 
 
 970 
 
 2x5x97 
 
 985 
 
 5x197 
 
 1000 
 
 2 3 x5 3 
 
58 BROWN & SHARPE MFG. CO. 
 
 CHAPTER VI. 
 
 INTERNAL GEARING. 
 
 PART A. INTERNAL SPUR GEARING. 
 (Figs. 12, 13, 14, 15, 16.) 
 
 A little consideration will show that a tooth of an internal 
 or annular gear is the same as the space of a spur external 
 gear. 
 
 We prefer the epicycloidal form of tooth in this class of 
 gearing to the involute form, for the reason that the difficulties 
 in overcoming the interference of gear teeth in the involute 
 system are considerable. Special constructions are required 
 when the difference between the number of teeth in gear and 
 pinion is small. 
 
 In using the system of epicycloidal form of tooth in which 
 the gear of 15 teeth has radial flanks, this difference must be 
 at least 15 teeth, if the teeth have both faces and flanks. Gears 
 fulfilling this condition present no difficulties. Their pitch 
 diameters are found as in regular spur gears, and the inside 
 diameter is equal to the pitch diameter, less twice the adden- 
 dum. 
 
 If, however, this difference is less than 15, say 6, or 2, or i, 
 then we may construct the tooth outline (based on the epicy- 
 cloidal system) in two different ways. 
 
 First Method. To explain this method better, let us sup- 
 pose the case as in Fig. T2, in which the difference between 
 gear and pinion is more than 15 teeth. Here the point o of 
 the describing circle B (the diameter of which in the best 
 practice of the present day is equal to the pitch radius of a 15 
 tooth gear, of the same pitch as the gears in question) gene- 
 rates the cycloid o, o 1 , o 2 , o 3 , etc., when rolling on pitch circle 
 L L of gear, forming the face of tooth ; and when rolling on 
 the outside of L L the flank of the tooth. In like manner is the 
 face and flank of the pinion tooth produced by B rolling out- 
 side and inside of E E (pitch circle of pinion). A little study 
 
PROVIDENCE, R. I. 
 
 59 
 
 of Fig. 12 (in which the face and flank of a gear tooth are 
 produced) will show the describing circle B divided into 12 
 
 equal parts and circles laid through these points (i, 2, 3, etc.), 
 concentric with L L. We now lay off on L L the distances 
 o i, 1-2, 2-3, etc., of the circumference of B, and obtain points 
 
6o 
 
 BROWN & SHARPK MFG. CO. 
 
 jl > 2 \ 3\ et c. [Ordinarily it is sufficient to use the chord.] It 
 will now readily be seen that B in rolling on L L will success- 
 ively come in contact with i 1 , 2', 3', etc., c meanwhile moving 
 to c\ <r a , <r 3 , etc. (points on radii through i 1 , 2', 3*, etc.), and the 
 generating point o advancing to o 1 , o 2 , o 3 , etc., being the inter- 
 sections of B with f l t <r 2 , <r*, etc., as centers and the circles laid 
 through i, 2, 3, etc. Points o, o 1 , o a , o 3 , etc., connected with a 
 curve give the face of the tooth ; in like manner the flank is 
 obtained. 
 
 In this manner the form of tooth is obtained, when the 
 difference of teeth in gear and pinion is less than 15, with the 
 exception that the diameter of describing circle B 
 
 where P = diametral pitch, N and n number of teeth in gears. 
 The distances of the tooth above and below the pitch line 
 as well as the thickness / are determined as in regular spur 
 gears by the pitch, except when the difference in gear and 
 pinion is very small, where we obtain a short tooth, as in Figs. 
 13 and 14. In such a case the height of tooth is arbitrary and 
 only conditioned by the curve. In internal gears it is best to 
 allow more clearance at bottom of tooth than in ordinary spur 
 gears. 
 
 42 T. 
 
 8 P. 
 
 30 Teeth 
 
 Fig. 13. 
 
 In a construction of this kind it is suggested to draw the 
 tooth outline many times full size and reduce by photography. 
 An equally multiplied line A B will help in reducing. 
 
PROVIDENCE, R. I. 
 
 61 
 
62 BROWN & SHARPK MFG. CO. 
 
 Second Method. The difference between gear and pinion 
 being very small, it is sometimes desirable to obtain a smooth 
 action by avoiding what is termed the " friction of approach- 
 ing action."* This is done, the pinion driving, by giving gear 
 only flanks, Fig. 15, and the gear driving, by giving gear only 
 faces, Fig. 16. In both these cases we have but one describ- 
 ing circle, whose diameter is equal to the difference of the two 
 pitch diameters. The construction of the curve is precisely 
 the same as described under A. The describing circle has 
 been divided into 24 parts simply for the sake of greater 
 accuracy. 
 
 PART B. INTERNAL BEVEL GEARS. 
 
 The pitch surfaces of bevel gears are cones whose apexes 
 are at a common point, rolling upon each other. The tooth 
 forms for any given pair of bevel gears are the same as for a 
 pair of spur gears (of same pitch) whose pitch radii are equal to 
 the respective apex distances of the normal cones (/. <?., cones 
 whose elements are perpendicular upon the elements of the 
 bevel gear pitch cones). (Compare Fig 19, page 68-) 
 
 The same is true of internal bevel gears, with the modifica- 
 tion that here one of the pitch cones rolls inside of the other. 
 The spur gears to whose tooth forms the forms of the bevel 
 gear teeth correspond, resolve themselves into internal spur 
 gears (Fig. 17). The problem is now to be solved as indicated 
 in the first part of this chapter. 
 
 McCord, Kinematics, pages 107, 108. 
 
PROVIDENCE, R. I. 
 
 8 P. 
 
 Gear 4O Teeth 
 Pinion 20 Teeth 
 
 Fig. 
 
64 BROWN & SHARPS MFG. CO. 
 
 CHAPTER vn. 
 
 V 
 
 GEAR PATTERNS. 
 
 (Fig. 18.) 
 
 To place in bevel gears the best iron where it belongs, the 
 tooth side of the pattern should always be in the nowel, no 
 matter of what shape the hubs are. 
 
 Hubs, if short, may be left solid on web ; if long they should 
 be made loose. A long hub should go on a tapering arbor, to 
 prevent tipping in the sand. i taper for draft on hubs when 
 loose, and 3 when solid is considered sufficient. 
 
 Coreprints as a rule are made separate, partly to allow the 
 pattern to be turned on an arbor, partly for convenience, 
 should it be desirable to use different sizes. 
 
 Put rap- and draw-holes as near to center as possible. 
 Referring to Fig. 18, make L D for D from %" to i/^", or 
 even more, should hubs be very long. Otherwise if D is more 
 than 1%" leave L = i%". 
 
 Iron pattern before using should be marked, rusted and 
 waxed. 
 
 Shrinkage For cast-iron, ^6" per foot. 
 For brass, -f$" " 
 
 Cast-iron gears, especially arm gears, do not always shrink 
 H ff per foot. In making iron patterns the following allow- 
 ances have been found useful : 
 
 Up to 12" diameter allow no shrink. 
 From 12" to 18" " " ^ regular shrink. 
 
 " I8"t02 4 " " % " 
 
 " 24" to 48" " " ft " 
 
 Above 48" " " .10" " 
 
 for cast-iron. 
 
PROVIDENCE, R. I. 
 
66 
 
 BROWN & SHARPK MFG. CO. 
 
 If in gears the teeth are to be cast, the tooth thickness t in 
 the pattern is made smaller than called for by the pitch, to avoid 
 binding of the teeth when cast. No definite rule can be given, 
 as the practice varies on this point. For the different diam- 
 etral pitches we would advise making / smaller by an amount 
 expressed in inches, as given in the following table : 
 
 DIAM. PITCH. 
 
 AMOUNT t 
 is SMALLER. 
 
 DIAM. PITCH. 
 
 AMOUNT t 
 is SMALLER. 
 
 16 
 
 .010" 
 
 5 
 
 .020" 
 
 12 
 
 .012" 
 
 4 
 
 .022" 
 
 10 
 
 .014" 
 
 3 
 
 .026" 
 
 8 
 
 .016" 
 
 2 
 
 .030" 
 
 6 
 
 .018" 
 
 I 
 
 .040" 
 
PROVIDENCE, R. I. 67 
 
 vui. 
 
 DIMENSIONS AND FORM FOR BEVEL GEAR 
 
 CUTTERS. 
 
 (Fig. 19.) 
 
 The data needed to determine the form and thickness of a 
 bevel gear cutter are the following : 
 P = pitch. 
 
 N = number of teeth in large gear. 
 n= number of teeth in small gear. 
 F = length of face of tooth, measured on pitch line. 
 After having laid out a diagram of the pitch cones a b c and 
 a b f, and laid off the width of face, the problem resolves itself 
 into two parts : 
 
 PART I. DETERMINE PROPER CURVE FOR CUTTER. 
 It will be remembered that in the involute system of cutters 
 (the only one used for bevel gears that are cut with rotary 
 cutter), a set of eight different cutters is made for each 
 pitch, numbering from No. i to No. 8, and cutting from 
 a rack to 12 teeth. Each number represents the form of 
 a cutter suitable to cut the indicated number of teeth. For 
 instance, No. 4 cutter (No. 4 curve) will cut 26 to 34 teeth. 
 In order to find the curve to be used for gear and pinion 
 we simply construct the normal pitch cones by erecting 
 the perpendicular p q through b, Fig. 19. We now measure the 
 lines b q and b p, and taking them as radii, multiplying each by 
 2 and P we obtain a number of teeth for which cutters of 
 proper curves may be selected. From example we have : 
 
 Gear : b q - 9^" ; 2 X P X 9.75 = 97 T No. 2 curve. 
 
 Pinion: bp = 3^" ; 2 X P X 3-5 = 35 T No - 3 curve. 
 The eight cutters which are made in the involute system 
 for each pitch are as follows : 
 
 No. i will cut wheels from 135 teeth to a rack. 
 
 2 " " " 55 " " 134 teeth. 
 
 3 " " " 35 " " 54 " 
 
 4 " 26 " " 34 " 
 
 c- " " '* 21 " " 2< " 
 
 6 " " " 17 " " 20 " 
 
 7 << I4 i I( j << 
 
 8 " 12 " " 13 " 
 
68 
 
 BROWN & SHARPE MFG. CO. 
 
PROVIDENCE, R. I. 69 
 
 PART II. DETERMINE THICKNESS OF CUTTER. 
 
 It is very evident that a bevel gear cutter cannot be thicker 
 than the width of the space at small end of tooth ; the practice 
 is to make cutter .005" thinner. Theoretically the cutting angle 
 (h} is equal to pitch angle less angle of bottom (or h = a ft'). 
 Practically, however, better results are obtained by making 
 h = a ft (substituting angle of top for angle of bottom), and 
 in calculating the depth at small end, to add the full clearance 
 (/) to the obtained working depth, giving equal amount of 
 clearance at large and small end. This is done to obtain a 
 tooth thinner at the top and more curved. As the small end 
 of tooth determines the thickness of cutter, we shall have to 
 find the tooth part values at small end. From the diagram it 
 will be seen that the values at large end are to those at small 
 end as their respective apex distances (a b and a I). The 
 numerical values of these can be taken from the diagram and 
 the quotient of the larger in the smaller is the constant where- 
 with to multiply the tooth values at large end, to obtain those 
 at small end. In our example we find : 
 
 a 7 = = .655 = constant T? 
 
 a b = .8 For 5 P we have : 
 
 ^ = .2057 . 
 
 S = .2000 / = .I3IO 
 
 /=-03 I 4 /=.Q3i4 
 
 ^+/=.23i4 /+/=.i624 
 
 D" + / = .4314. J' = .1310 
 
 D'"+/=.293 4 
 
 From the foregoing it is evident that a spur gear cutter 
 could not be used, since a bevel gear cutter must be thinner, 
 
 If in gears of more than 30 teeth the faces are proportion- 
 ately long, we select a cutter whose curve corresponds to the 
 midway section of the tooth. The curve of the cutter is found 
 by the method explained in Part I. of this Chapter. 
 
BROWN & SHARPK MFG. CO. 
 
 IX. 
 
 DIRECTIONS FOR CUTTING BEVEL GEARS 
 WITH ROTARY CUTTER. 
 
 (Fig. 20.) 
 
 In order to obtain good results, the gear blanks must be of 
 the right size and form. The following sizes for each end of 
 the tooth must be given the workman : 
 Total depth of tooth. 
 Thickness of tooth at pitch line. 
 Height of tooth above pitch line. 
 
 These sizes are obtained as explained in Chapter VIII. 
 The workman must further know the cutting angle (see 
 formula on page 13 and compare Chapter VIII.), and be pro- 
 vided with the proper tools with which to measure teeth, etc. 
 In cutting a gear on a universal milling machine the opera- 
 tions and adjustments of the machine are as follows : 
 
 1. Set spiral bed to zero line. 
 
 2. Set cutter central with spiral head spindle. 
 
 3. Set spiral head to the proper cutting angle. 
 
 4. Set the index on head for the number of teeth to be cut, 
 leaving the sector on the straight or numbered row of holes, 
 and set the pointer (or in some machines the dial) on cross-feed 
 screw of milling machine to zero line. 
 
 5. As a matter of precaution, mark the depth to be cut for 
 large and small end of tooth on their respective places. 
 
 6. Cut two or three teeth in blank to conform with these 
 marks in depth. The teeth will now be too thick on both their 
 pitch circles. 
 
 7. Set the cutter off the center by moving the saddle to or 
 from the frame of the machine by means of the cross-feed 
 screw, measuring the advance on dial of same. The saddle 
 must not be moved further than what to good judgment 
 
PROVIDENCE, R. I. 
 
 Fig. 20. 
 
72 BROWN & SHARPE MFG. CO. 
 
 appears as not excessive ; at the same time bearing in mind 
 that an equal amount of stock is to be taken off each side of 
 tooth. 
 
 8. Rotate the gear in the opposite direction from which the 
 saddle is moved off the center, and trim the sides of teeth (A) 
 (Fig. 20'.) 
 
 9. Then move the saddle the same distance on the opposite 
 side of center and rotate the gear an equal amount in the 
 opposite direction and trim the other sides of teeth (C). 
 
 10. If the teeth are still too thick at large end E, move the 
 saddle further off the center and repeat the operation, bearing 
 in mind that the gear must be rotated and the saddle moved 
 an equal amount each way from their respective zero settings. 
 
 It is generally necessary to file the sides of teeth above the 
 pitch line more or less on the small ends of teeth, as indicated 
 by dotted lines F F. This applies to pinions of less than 30 
 teeth. 
 
 For gears of coarser pitch than 5 diametral it is best to 
 make one cut around before attempting to obtain the tooth 
 thickness. 
 
 The formulas for obtaining the dimensions and angles of 
 gear blanks are given in Chapter III. 
 
PROVIDENCE, R. I. 
 
 73 
 
 THE INDEXING OF ANY WHOLE OR FRAC- 
 TIONAL NUMBER. 
 
 (Fig. 21.) 
 
 Change Gear 
 Fig. 21. 
 
 In indexing on a machine the question simply is : How 
 many divisions of the machine index have to be advanced to 
 advance a unit division of the number required. To which 
 is the 
 
 divisions of machine index 
 answer = 
 
 number to be indexed 
 
 Suppose the number of divisions in index wheel of machine 
 to be 216. 
 
 EXAMPLE I. Index 72. 
 
 Answer: 
 
 216 
 - = 3 
 
 / 
 
 (3 turns of worm). 
 
74 BROWN & SHARPS MFG. CO. 
 
 EXAMPLE II. Index 123. 
 
 = i + .93 
 123 123 
 
 If now we should put on worm shaft a change gear having 
 123 teeth, give the worm shaft, Fig. 21, one turn, and in addi- 
 tion thereto advance 93 teeth of the change gear (to give the 
 fractional turn), we would have indexed correctly one unit of 
 the given number, and so solved the problem. Should we not 
 have change gear 123 we may try those on hand. The ques- 
 tion then is : How many teeth (x) of the gear on hand (for 
 instance 82) must we advance to obtain a result equal to the 
 one when advancing 93 teeth of the 123 tooth gear? We have : 
 
 -93. ^ _X_ where y = 62 
 123 82 
 
 EXAMPLE III. Index 365, change gear 147. 
 
 == -%- where j = 87 JL 
 
 365 147 365 
 
 Here 147 is the change gear on hand. In indexing for a unit 
 of 365 we advance87teeth of our 147 tooth gear. It is evident 
 that in so doing we advance too fast and will have indexed 
 three teeth of our change gear too many when the circle is 
 completed. To avoid having this error show in its total amount 
 between the last and the first division, we can distribute the 
 error by dropping one tooth at a time at three even intervals. 
 
 EXAMPLE IV. Index 190. 
 
 216 __ 26 
 
 ^, ~ ^90 Change gear on hand 88 T 
 
 26 y , 8 
 
 = A where j = 12 + 
 
 190 88 190 
 
 To distribute the error in this case we advance one additional 
 tooth ot a time of the change gear at eight even intervals. 
 
 EXAMPLE V. Index 117.3913. 
 
 216 _ 986087 
 
 "739 I 3 
 
 This example is in nowise different from the preceding 
 ones, except that the fraction is expressed in large numbers 
 This fraction we can reduce to lower approximate values, 
 which for practical purposes are accurate enough. This is 
 done by the method of continued fractions. [For an explana- 
 
PROVIDENCE, R. I. 75 
 
 tion of this method we refer to our " Practical Treatise on 
 Gearing."] 
 
 986087 
 1173913 
 
 986087) 1173913 (i 
 986087 
 
 187826) 986087 (5 
 939130 
 
 46957) 187826 (3 
 140871 
 
 46955) 46957 Ci 
 46955 
 
 2) 46955 (23477 
 46954 
 
 1)2(2 
 2 
 
 986087 =T 
 "73913 
 
 i + i 
 
 23477 + 1 
 
 2 
 
 <r=3 i 23477 
 
 {r=i b = 5 d '= 16 21 493033 986087 
 a l =i b l 6 d l = ig 25 586944 1173913 
 
 NOTE. Find the first two fractions by reduction = - and - = - ; the 
 
 others are then found by the rule \ b c + a ~ d 
 
 \ t> 1 c -j- a 1 = 
 
 i + i 6 
 5 
 
 The fraction |J- is a good approximation; putting therefore 
 a change gear of 25 teeth on worm shaft, we advance (beside 
 the one full turn) 21 teeth to index our unit. 
 
 Of course, in using any but the correct fraction we have an 
 error every time we index a division ; so that when indexed 
 around the whole circle, we have multiplied this error by the 
 number of divisions. 
 
 In the present example this error is evidently equal to the 
 difference between the correct and the approximate fraction 
 used. Reducing both common fractions to decimal fractions 
 we have : 
 
 = .84000006 
 
 1*739*3 
 
 21 
 
 ~ 
 
 .00000006 = error in each division. 
 
76 BROWN & SHARPE MFG. CO. 
 
 .00000006 X 117.3913 .00000704348 total error in complete 
 circle. This error is expressed in parts of a unit division. (To 
 find this error expressed in inches, multiply it by the distance 
 between two divisions, measured on the circle.) In this case 
 the approximate fraction being smaller than the correct one, 
 in indexing the whole circle we fall short .00000704348 of a 
 division. 
 
 EXAMPLE VI. Index 15.708 
 
 216 _ 11796 
 
 15.708 15708 
 
 11796 _ 983 
 15708 1309 
 983) 1309 (i 
 983 
 
 326) 983 (3 
 978 
 
 5) 326 (65 
 30 
 26 
 
 I 5 
 
 D5(S 
 5 
 o 
 
 983 =I 
 '309 T^i__ 
 
 3+ x _ 
 65 + 1 
 5 
 
 i 3 65 5 
 
 i 3 *9 6 9 8 3 
 i 4 261 1309 
 
 In using the approximation j-J the error for each division 
 (found as above) will be .000002927, for the whole circle 
 .0000460. In this case, the approximation being larger than 
 the correct fraction, we overreach the circle by the error. 
 
PROVIDENCE, R. I. 
 
 77 
 
 CHAPTER XI. 
 
 THE GEARING OF LATHES FOR SCREW 
 CUTTING. 
 
 (Figs. 22, 23.) 
 
 The problem of cutting a screw on a lathe resolves itself into 
 connecting the lathe spindle with the lead screw by a train of 
 gears in such a manner that the carriage (which is actuated by 
 
 Simple Gearing. 
 
 Fig. 22. 
 
78 BROWN & SHARPK MFG. CO. 
 
 the lead screw) advances just one inch, or some definite dis- 
 tance, while the lathe spindle makes a number of revolutions 
 equal to the number of threads to be cut per inch. 
 
 The lead screw has, with the exception of a very few cases, 
 always a single thread, and to advance the carriage one inch it 
 therefore makes a number of revolutions equal to its number 
 
 Compound Gearing. 
 
 Fig. 23. 
 
 of threads per inch. Should the lead screw have double 
 thread, it will, to accomplish the same result, make a number 
 of revolutions equal to half its number of threads per inch. It 
 follows that we must know in the first place the number of 
 threads per inch on lead screw. 
 
PROVIDENCE, R. I. 79 
 
 It ought to be clearly understood that one or more inter- 
 mediate gears, which simply transmit the motion received from 
 one gear to another, in no wise alter the ultimate ratio of a 
 train of gearing. An even number of intermediate gears 
 simply change the direction of rotation, an odd number do not 
 alter it. 
 
 The gearing of a lathe to solve a problem in screw cutting 
 can be accomplished by 
 
 A. Simple gearing. 
 
 B. Compound gearing. 
 
 Referring to the diagrams, Figs. 22 and 23, we have in Fig. 
 22 a case of simple, and in Fig. 23 a case of compound gear- 
 ing. 
 
 In simple gearing the motion from gear E is transmitted 
 either directly to gear Ron lead screw or through the interme- 
 diate F. In compound gearing the motion of E is transmitted 
 through two gears (G and H) keyed together, revolving on the 
 same stud , by which we can change the velocity ratio of the 
 motion while transmitting it from E to R. With these four 
 variables E, G, H, R, we are enabled to have a wider range of 
 changes than in simple gearing. 
 
 B and C, being intermediate gears, are not to be considered. 
 If, as is generally the case, gear A equals gear D, we disregard 
 them both, simply remembering that gear E (being fast on 
 same shaft with D) makes as many revolutions as the spindle. 
 Sometimes gear D is twice as large as gear A, then, still con- 
 sidering gear E as making as many revolutions as the spindle, 
 we deal with the lead screw as having twice as many threads 
 per inch as it measures. 
 
 SIMPLE GEARING. 
 
 Let there be : the number of teeth in the different gears 
 expressed by their respective letters, as per Fig. 22, and 
 
 s = threads per inch to be cut, 
 L threads per inch on lead screw ; then 
 
 i. s ^R 
 
 L E 
 
80 BROWN & SHARPE MFG. CO. 
 
 If now one of the two gears E and R is selected, the other 
 will be : 
 
 R = lE E = LR 
 L s 
 
 2. The two gears may be found by making 
 
 ~^ ^ > where/ may 'be any number. 
 
 3. The above holds good when a fractional thread is to be 
 cut, but if the fraction is expressed in large numbers, as, for 
 instance, s = 2.833 ( 2 T 8 77W)> we fi rst reduce this fraction (yWoO to 
 lower approximate values by the process of continued fraction 
 (see pages 73 and 74). 
 
 833) icoo (i 
 
 833 
 
 i6 S ) 167 (i 
 165 
 
 2) l6 5 (82 
 
 16 
 
 I 
 
 4 
 
 i 
 
 
 
 82 
 
 2 
 
 I 
 
 I 
 
 S 
 
 5 
 6 
 
 497 
 
 833 
 1000 
 
 L = .833 (nearly) and s = 2?- 
 o 6 
 
 If in this case L = 4, and we select E = 48, then, since 
 
 COMPOUND GEARING. 
 
 4. In a lathe geared compound for cutting a screw the 
 product of the drivers (E and H, Fig. 23) multiplied by the num- 
 ber of threads per inch to be cut must equal the product of the 
 driven (G and R) multiplied by the number of threads on lead 
 screw. This is expressed by 
 
 . L, 
 
PROVIDENCE, R. I. 8l 
 
 If three of the gears E, H, G, R have been selected, the 
 fourth one would be either 
 
 ..i o, 
 
 TT G R L 
 
 H = or 
 
 G = n " J or 
 
 = R_G_L ^_ L / R.G \ 
 EH VL.E.H/ 
 
 If a fractional thread is to be cut, as under " 3," we reduce 
 the fraction to lower approximate values. 
 
 EXAMPLE. Gear for 5.2327 threads per inch, lead screw is 
 6 threads. 
 
 IOOOO 
 
 2327) looco (4 
 9308 
 
 692) 2327 (3 
 2076 
 
 "251)692(2 
 502 
 
 190) 251 (i 
 190 
 
 61) 190 (3 
 183 
 
 7) 61 (8 
 S 6 
 
 5)7(i 
 5_ 
 
 4 
 
 1)2(2 
 
 a 
 o 
 
 43213 8 I 2 2 
 
 L A _L 15. 37 3 6 343 ^92 2327 
 4 13 30 43 159 1315 J 474 4263 loooo 
 
 .2327 (nearly) and 5.2327 = 5 15 
 43 "43 
 
 Selecting E 43, H = 52, R = 50, and 
 
 * we have G = 4 3 ' 5* 5 = 39 
 
 R . L 50 . 6 
 
82 BROWN & SHARPK MFG. CO. 
 
 5. The examples so far given all deal with single thread. 
 The pitch of a screw is the distance from center of one thread to 
 the center of the next. The lead of a screw is the advance for 
 each complete revolution. In a single thread screw the pitch 
 is equal to the lead, while in a double thread screw the pitch 
 is equal to one-half the lead ; in a triple thread screw equal to 
 one-third the lead, etc. 
 
 If we have to gear a lathe for a many-threaded screw 
 (double, triple, quadruple, etc.), we simply ascertain the lead, 
 and deal with the lead as we would with the pitch in a single 
 thread screw, *'. e., we divide one inch by it, to obtain the num- 
 ber of threads for which we have to gear our lathe. 
 
 EXAMPLE. Gear for double thread screw, lead = .4654. 
 Number of threads per inch to be geared for is : 
 
 Lead -4654 
 
 Lead screw is four threads per inch. 
 
 As in previous examples, we reduce the fraction .i4%7=itffo 
 to lower approximate values by the process of continued frac- 
 tion. 
 
 From the different values received in the usual way we 
 select : 
 
 ^J = .1487 (nearly) and 2.1487 = 2ij-J 
 
 We have therefore : 
 
 =74 
 Selecting -j G = 30 
 
 R _ E . H . s _ 74 . 4Q . Hi - -- 
 G . L 30 . 4 
 
 NOTE. In using any but the original fraction we commit an error. This error 
 can be found by reducing the approximate fraction used to a decimal fraction, and 
 comparing it with the original fraction. In the above example the original fraction is 
 
 .1487 and 
 H = . 14864 
 Error = .00006 inch in lead. 
 
 In cutting a multiple screw, after having cut one 
 thread, the question arises how to move the thread tool the 
 correct amount for cutting the next thread. 
 
PROVIDENCE, R. I. 83 
 
 In cutting double, triple, etc., threads, if in simple or com- 
 pound gearing the number of teeth in gear E is divisible by 
 2, 3, etc., we so divide the teeth ; then leaving the carriage 
 at rest we bring gear E out of mesh and move it forward one 
 division, whereby the spindle will assume the correct position. 
 
 When E is not divisible we find how many turns (V) of 
 gear R are made to each full turn of the spindle. Dividing 
 this number by 2 for double, by 3 for triple thread, etc., we 
 advance R so many turns and fractions of a turn, being careful 
 to leave the spindle at rest. 
 
 For compound gearing : 
 
 V _E.H 
 ~G~R 
 
 When the gear D is twice as large as the gear A (as ex- 
 plained in fifth paragraph, page 78.) the formula would be 
 
 y= E. H. 
 
 2 G. R. 
 
 If in simple gearing both E and R are not divisible, one 
 remedy would be to gear the lathe compound ; or the face- 
 plate may be accurately divided in two, three or more slots, 
 and all that is then necessary is to move the dog from one slot 
 to another, the carriage remaining stationary. 
 
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