470 ANIMAL MECHANICS, 



suppose that the arm in rising from the vertical to the hori- 

 zontal position rests at each division of the quadrant, for the 

 ftith. part of the time t. The angle 9 in our last equation will 

 have, successively, the following value i — 



ff 27T 37T Mlt 



%m %iri am* ' 2M 

 Let us write 



nr . 

 * %m 



then the total work done will be 



b) (to + a) xn— (sin <b + sin 26 + &c. 4- sin wrf, 

 m 



Comparing this with equation (136), we find 



K = ~ (sin 6 + sin 26 + &c< + sin m<j>) f (137) 

 where m is to be made indefinitely large ; but 



sin & + sin 26 + &c< + sin m6 = sin 



• 0 



sin - 



. (m +1)0 . m6 

 sin ' sin — 



K~ - -1 _L. 



m sm — 

 2 



Substituting, in this equation, — for fa we obtain 



. (m + l) 7T . 7T 



sin — — sm - 



4 m 4 



" . i 



m sm — 

 4^ 



K. 



and, finally, making m very large, we find 



