Woodward — The Efficiency of Gearing under Friction. 103 

 The energy exerted by the driver during this action is 



17 = ** 



and the ratio of energy lost to energy exerted is 



p - % + U> _ ( 1 , l\r, 

 U [r^ r 2 ) 2q 



22-/Uog(l+!)-A'log(l + f,) 



(15), 



Let Wj and n 2 represent the numbers of teeth in the driver 

 and follower. Then, but one pair of teeth being in action at 



a time, 



irr, 



q = — K 



When the obliquity is as small as possible, 



max. r = q sin 6 = r, cos 6 



hence 



tan 6 = - = -» and 



q 7T 



, sin Id — <p) , Q , ft, — 7rf 



h — r,- \ . r/ = ?\ (cot (p — cot 0) — r x 



1 sin 6 sin <p 



n J 



h' = r. . \ . r/ = r, (cot cc + cot 0) = r. -* — ~ 

 1 sm o sm <p v n 2 / 



\r, rj 2q i \n, nJ 2mr l 



Substituting the above values in (15) and removing r, from 

 the brackets, we have 



Vftj nJ 27, 



27T Wj — irf 





log (l + 



7T 



•/ 



n, 



irf) 



• ( 



log 1 + 



w, + tt/V 



(16), 



