2GG ANIMAL MECHANICS. 



expanding the root, and remembering that w is moderately 

 small, we obtain 



= P 



i + — ( a cos 0 + ^ cos 0) (a cos <p + /' cos 0)\ 



2 p 2 



') 



and finally, substituting for p, its value I - V, we have 



„/ n - s>„ _ w * ( a co s <t> + l cos 0) (a cos 0 + /' cos 0) , . 

 p - /? - dp - — . r /. ( 47 ) 



This expression denotes the shortening or lengthening of a 

 single fibre aa\ produced by the rotation (w) round the axis 

 KS ; and the work done by the entire muscle, in a single 

 contraction, will be, 



Work done =^pdO = - p cos 0 + / cos 0) (q cos 0 j g cos g) 



^0 



and we are required to investigate the conditions, which shall 

 render this work a maximum. 



Using the same notation, as in Prop. B, we have 



6 sin /3 ;/ b 1 sin fi' 

 = sin (0-0/ = s "in70 7 ^9)' 



Hence, 



^9 = 



w 2 I {a cos 0 sin (|3 - 6) + b sin /3 cos 0} {a cos 0 sin (/3'- 0) + sin/3' cos 9} dO 

 ~z ) b sin |3 . sin (((3' -Q)-V sin /3' . sin (/3 - 0) ' 



If we write the denominator, 



A = 6 sin |3 sin (j3 ; -0)-b' sin |3' sin (|3 - 0) ; 

 we shall find, by expanding the numerator, 



jV*0 = (48) 



• o • n, / ix / 7 a fCOS 2 0^0 



— sin p sin p' (a cos 0 + 0) (a cos 0 + o') J — — 



