ANIMAL MECHANICS. 



339 



It is evident, from inspection, that equation (97) has a 

 very small root and a very large root, both real ; because the 

 coefficients of the first and last terms are small, as compared 

 with the intermediate coefficients ; and the smallness of these 

 coefficients depends on the small value of K, which is a factor 

 of the first and last coefficients. 



These small and large roots of equation (92) would 

 become 



X = o, 



, 1 (98) 



in one or other of two cases, viz. : — 

 m= i, 



K=o* (99) 



Having determined the value of X from equation (97), we 

 must introduce this value into equation (92), in order to 

 calculate the actuaj value of the maximum work or moment 

 of the muscle. This equation may be thus written, by 

 elimination of X' : — 



cmXX 2 + K\ + cmX' (m 2 - 1) 

 \p)~ 4 V\ 2 + m 2 + 1 \/(m 2 + i)X 2 +0 2 - i) 2 ' (l ° 0) 



Since the denominator of this fraction does not change sign 

 with the change of sign in X, it is plain that the numerator 

 equated to zero determines the position of the two neutral 

 generators of the hyperboloid, with respect to which the skew 

 muscle does no work, and that these generators divide all the 

 generators of the hyperboloid into two groups, for one of 



* It may be interesting to note that the geometrical meaning of K = o is, 

 that the line joining X and X' is a tangent to the hyperboloid conjugate to the 

 locus hyperboloid (84). 



z 2 



