.56 



OF THE VEIOCITV AND FUICTIO-!f OF WHEEL WORK. 



40 EF, EFG becoming a right line parallel to CD, and 

 the angle CFG:iDCF=DBC ; but the angle ECFrrBCH, 

 and in the triangle ECF, sin. CFE : sin. ECF : : CE : EF 

 (139) : : sin. CBD : sin. BCH : : CD : BH. For the sub- 

 stance of this demonstration I am indebted to Mr. Ca- 

 vendish. In the epicycloidal tooth, CD coincides with CB, 

 and CBD is a right angle, so that the friction is to the mo- 

 tion CE of the tooth of A, as the sine of BCH, or of ACB, 

 to the radius. In the involute, CD is constant, and the fric- 

 tion varies as BH, supposing the motion in CE constant. 

 If the pinion acted within the concave surface of a cylin- 

 drical wheel, the friction would be as the sine of the differ- 

 ence, instead of the sura, of the angular distances from the 

 line of junction. 



Scholium. The immediate quantity of the force of 

 friction does not appear to be materially altered by the re- 

 lative velocity of the surfaces : but its mechSnical effect in 

 resisting the motion of a machine is so much the greater as 

 the relative velocity is greater. 



360. Theokem. If a given number of 

 "iceih are to be disposed on an unlimited 



number of wheels and pinions, so as to in- 

 crease or diminish the velocity of the last 

 wheels of the series as much as possible, the 

 proportion of each pinion to its wheel must 

 be nearly that of 1 to 3,59. 



In order to increase the angular velocity in the ratio 

 of 1 to a, with the least possible number y of wheels, 

 each having to its pinion the ratio of a to i, we 

 must make x the number of which the hyperbolical 

 logarithm exceeds the reciprocal by unity. For x'lza 



I 

 :c=n'; and y.{x+i), the number of teeth, must be st 



minimum: but y.(j;+l)=a'j/-f-y, and orjr — y{h.l. a)a> 



Z.+i/=.0, a'+ 1 =a'_!li2, or x+ l=Zx(h.\.x] since h.l.a= 

 yv y 



yii>.\^); therefore 1-f— =h.l.i, and x is found =3.iB» 



This is therefore the most advantageous proportion for pro- 

 ducing the greatest velocity with a given number of tcicth. 



