Woodward — The Efficiency of Gearing under Friction. 105 

 V + t 2 / 2 1 3tt 2 / 2 



(,^2—^/2)2- n ? -i n 

 1 Girl 



n* + Zn*ir>P _ 1 67r 2 / 2 



( Wl 2_^ /2) 3 ^ 



These values put in (17) give 



*- U; + 1?) t[ x - "k; + -h?— -jjt + &c ' 



(18) 



This gives the ratio of energy lost to energy exerted on 

 Involute Teeth. 



15. A comparison of the efficiencies of the two kinds 



of teeth is easily made by means of (12) and (18). The 



agreement is surprising. For two terms they are identical. 



For the purpose of seeing which is the greater, the third and 



fourth and fifth terms in (12) should be compared with 



the third in (18), for the terms containing higher powers of 



/ in the numerator, and of w, in the denominator, may well 



1 f 2 f 2 

 be omitted. That is to say, how does— + J —-L J — forepi- 



6e 2 ■■ Ir S 



3 f 2 

 cycloidal teeth, compare with -±— for involute teeth? If e 



z 



is as large as possible, that is unity, we have — + f 2 for 



6 



the first; and the second expression, ~ f 2 can equal it only 



on the supposition that f 2 = - = 0.33 or/= 0.575, which 



3 



is a rather violent supposition. For common values of e and 



f, it is evident that the loss is greater for epicycloidal teeth, 



hence their efficiency is less; but the difference is too small 



to be of practical value. 



Issued May 10, 1898. 



