322 ANIMAL MECHANICS. 



Hence (since w is a small angle), 



p2 _ _ 2 A A' (cos y - cos 

 (p + p) {p-p') = sin /. sin a sin a': 



or, 



£>S/> = AA'u> . sin / . sin a sin a. 



Hence we obtain, for the shortening of a single fibre, by a 

 rotation w, 



« . -r AA' sin a sin a 

 bp = w sm 1 . . 



P 



If, from the ends of the fibre p, we let fall two perpen- 

 diculars, 7r and 7r', upon the common intersection of the planes, 

 it is easy to see that 



7r = A sin a, 



it' = A' sin a' ; 



from which we find 



ho = u) sin I . — , 



P 



obtain 



2 ($/>) = o> sin / 2 ( — ). (78) 



It is evident that the work done by the muscle will be a 

 maximum when 



S ($p) = maximum ; 



and this quantity will be a maximum independent of the 

 sum, 2 (which depends on the special arrangement of 



the muscular fibres), when /= 90 0 . 



