ANIMAL MECHANICS. 



275 



to zero, which gives the following quadratic, whose roots de- 

 termine the positions of neutral equilibrium : — 



X sin j3 sin j3' (b + x) (// + x) + F cos j3 cos j3,^ 2 = o 

 The condition for real roots in this quadratic is, that 

 X sin j3 sin j3' I (b - b'fX sin|3 sin 0'- 4^' ^cos/3 cos/3'} (55) 

 shall be positive. 



If this condition be fulfilled, and the points corresponding 

 to neutral equilibrium be taken on the bisector of AOB\ for 

 all axes lying outside those limits the equilibrium will be 

 stable, and the work done positive, and represented by the 

 square of the ordinate of an hyperbola constructed with the 

 intercept between the points of neutral equilibrium as its 

 transverse axis; and for all axes lying inside the points of 

 neutral equilibrium, the equilibrium will be unstable, and the 

 work done negative, and represented by the square of the 

 ordinate of an ellipse constructed with the intercept between 

 the points of neutral equilibrium as its major axis. 



Hence a geometrical construction, similar to that shown in 

 Fig. 72, may be made to represent the work done; pro- 

 vided we replace the circle and equilateral hyperbola by an 

 ellipse and hyperbola with unequal axes. 



If the equation of the central conic (53) be referred to its 

 centre, as origin of co-ordinates, it will have the form 



where 



2 = X sin J3 sin ft j (b - b'YX sin sin ft - 466TC03 j3 cos ft) 

 4 { X sin (3 sin ft' + Fcos j3 cos ftj 



* - X sin P sin P 1 sin P sin j3'-4^Fcos (3 cos 



4 { X sin j3 sin ft + Ycas fi cos ftj 2 (771" 



These valuer for m 2 and n l are essentially positive when the 

 T 2 



