ANIMAL MECHANICS. 



305 



This expression might be integrated in terms of the 

 definite integrals, X, Y, Z (p. 282), and the whole theory of 

 work done by plane quadrilateral muscles deduced from it ; 

 but we shall obtain results more in accordance with the actual 

 circumstances of nature, by using the method of differences, 

 as employed in the discussion of the wings of birds. Previous 

 to adopting this method, however, it is worth while to apply 

 equation (69) to the simple case of a triangular muscle, which 

 often occurs in nature. 



Making I = o, and substituting for I its value, 



we find 



SpdO - a sin 0 



b sin /3 

 sin (j3 -T)' 



(a sin $ + I sin 0) dO 



a sin 6 C , 4 9 . . ~ n , T . _ 



j—. — g {asintf> sin ]3cos0 + (b sin/3 - a sin <p cos Q) sin 0j JO 

 0 sin p j _(f 



2a 2 sin* 0 sin 6 

 b J 



(71) 



From this expression for the work done by a triangular 

 muscle, when rotated round an axis parallel to the bisector of 

 its vertical angle, we obtain the following conclusions : 



i°. If the vertical angle and bisector of the muscle be 

 given (0, b), the work done will be independent of the position 

 of the bone. 



2°. The work done varies as the square of the distance of 

 the axis of rotation from the bisector. 



x 



