98 



Trans. Acad. Sci. of St. Louis. 



by simplifying and putting tan ^ = /. This formula gives 

 the energy lost during the "approach," the initial value of 

 d being d^. 



Let (l — — )/= ^" in the above. Integrating as indicated 

 — loge (sin ^^ — A; cos 6^) — Jc l-^ — ^J 



1+Jc'- 



(5), 



4. The above assumes that but a single pair of teeth are in 

 action. It does not appear that the introduction of more 

 pairs would affect the ratio I am aiming to get provided the 

 pairs are in all respects alike, and the moment of the action 

 of each driving tooth be constant. 



5. During the " recess " dq \s positive and we have 



and (3) becomes 



/I 1 \ / cos ddd \ 

 dU,= 4r,W, sin ^ (^- + — j ( j^j 



Substituting the value ot l\, simplifying and integrating from 



- to d^ we get 



2 ^ ° 



4?- 2 / 1 1 \ 



X 



^■' C^- — e^ — loge (sin d, + k' cos d,) 



1 + A;'2 



(6). 



2r 



^^•'=(1 +-/)/• 



in which 



This formula gives the energy lost during the recess. 



