102 



Trans. Acad. Sci. of St. Louis. 



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

 r Q9 the less the loss. 



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

 P and l x we have for the approach 





9) 



But in the case of involute teeth both 6 and <p are con- 

 stant, and r = q sin ; hence 



1 2 J q + 



qdq 



r x sin (# — <p) 

 sin sin <p 



Letting JL. — \ 12 = h, and integrating we have 



6 sin d sin cp & 8 



(13). 



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



sin sin ^ 

 we get by a similar process 



(14). 



U^^+^M^q-k'lo^l + f) 



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

 lost becomes 



A'log(l+|,)] 



