280 ANIMAL MECHANICS. 



If we write 



feos 2 9d0 TT f sin 2 0d0 _ fsin 0 cos i 



feos *0d0 v fsin *0d0 „ (s 



-J— 5— F =J-A- z =y 



A 



equation (48) becomes, writing as before, x = a cos 0, 



= Z sin /3 sin j3' (5 + #) (6' + a?) 

 + F cos j3 cos /3' a: 2 



- Zx { sin |3 cos fi'(b + !c) + sin /3' cos j3 (6' + a?). 



If 



we obtain, for the geometrical representation of the work 

 done, the following conic : 



if = { X sin j3 sin j3' + Feos j3 cos $ - Z sin (j3 + 0') j x* 

 + { (6 + 6') Xsin j3 sin j3' - Z(&sin j3 cos /3' + 5' sin /3'cosj3)J^ 

 + &&' sin j3 sin /3'. (59) 



The centre of the conic, 



if = Lx* + Mx + N, 



is situated at a distance § from the origin, determined by the 

 equation 



2Lx + M - o, 



or 



(b + b f ) Xsinj3 sm(3'-Z(bsin /3cosj3'4 £'sin j3'cos/3) 

 6= " 2 * X sin |3 sin |3' + F cos j3 cos 0' - Z sin (0 + /3') 



(60) 



Conclusions, similar to those obtained from equation (53), 

 may be drawn from equation (59). 



i°. The equilibrium of the plane quadrilateral muscle will 

 be stable when y" is positive. 



2°« The equilibrium will be neutral when y 2 » o. 



