102 



Trans. Acad. Sci. of St. Louis. 



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

 ?* , the less the loss. 



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

 P and 7j we have for the approach 



d£7' = (- + -)i¥ 1 sin^^- 

 \r. rj J r sin 



rdq 



if + i\ sin ( 6 — (p ) 



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

 stant, and r = q sin 6 ; hence 



<7, 



■=1 



1 2 h + 

 ^o 



#£?# 



?'j sin (d — <p) 

 sin 6 sin^p 



Letting; -i - ±-1 = h. and integrating we have 



sin a sin <p 



— 



(13). 



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



7i , _ ^sinffl+y) 

 sin d sin <p 



we get by a similar process 



u *={k +: k) M >h- !, 'H i+ £) 



(14). 



12. If as is usual we let q l = q 2 — q, the total energy 

 lost becomes 



t7 l +D, = U t (±.+ ± ; ) 2 q - Alog(l + 2) - 



A'lo g (l+|,) 



