276 



ANIMAL MECHANICS. 



quantity (55) is positive, which is the condition necessary for 

 the existence of real positions of unstable equilibrium. 



When the equilibrium is stable, y 2 is positive, and the 

 equation becomes 



x 2 y 2 



m l n 2 1 



which represents an hyperbola, whose axes are m and the 

 real axis being m, and situated on the bisector of the angle 

 A OB. 



When the equilibrium is unstable, y 2 is negative, and 

 the equation becomes 



x 2 y 2 

 m 11 1 



which represents an ellipse, whose axes are m and n\ the axis 

 m being situated on the bisector of the angle AOB; and 

 the maximum value of the negative work, corresponding with 

 the position of maximum instability of the muscle, determined 

 by equation (54), will be represented by n 2 . 

 From the preceding equations it appears that 



n 2 



— = X sin |3 sin j3' + Fcos j3 cos j3' ; 



or, that the conjugate axis will be greater or less than the 

 transverse axis, according as this function is greater or less 

 than unity. 



There are three distinct cases to be considered according 

 as the expression (55) is positive, zero, or negative. 

 i°. If the expression (55) 



(/; - b') 2 X sin /3 sin fi' - ^bb'Y cos /3 cos /3' 



be positive, then m 2 and w 2 , in the equations (56, 57), are both 



