102 



Trans. Acad. Sci. of St. Louis. 



The fourth and fifth terms show that the greater the value of 

 r^j, the less the loss. 



10. Involute Teeth. Keturning to (3) and substituting for 

 P and ?i we have for the approach 



dU = {--{--] M. sin w — ^ 

 \»', rj '^ r sin 



— rdq 



ip + r^ &m{d — <f) 



But in the case of involute teeth both d and (p are con- 

 stant, and r = q sin d ; hence 



U, 





' = (?;+r. 



qdq 



rj sin (d — (f) 

 sin 6 sin <p 



Letting J— ^ ^^ ^ = h, and integrating we have 



sin d sin cp 



U^={^ + -)m^ ..-/'log(n-l) 



(13), 



11. During the recess, we use Z/ from (2), and putting 



sin d sin (p 

 we get by a similar process 



K = {^+~) ^1^ [22- '''i"g(i + f )] (I*)- 



12. If as is usual we let q^ = q^_ = q, the total energy 

 lost becomes 



C/.+ P, = i)i.(i+^) 2«- /.log (1 + 1)- 



