85, 86, 87, 88] 



PROJECTIONS OF POLHODE. 



265 



a constant, so that all the projections are similar. The motion about 

 the axis A is most stable when the small polhode is a circle, that 

 is when the above ratio is unity, or B = C. 

 Eliminating g we obtain 



46) d 2 {A(A C')x 2 + B(BC)y 2 }= 1 <7tf 2 , 



an ellipse the ratio of whose axes is 



-i/B(B-C) 



V A(A-C)' 



and for maximum stability this is unity, or A = B. These projections 

 are shown in Fig. 79. 



Fig. 79. 



Fig. 80. 



V, 



Eliminating y, we have 

 47) 6 2 {A(A B)x* - 



an hyperbola the ratio of whose axes is 



C(B-C) 



A(A-B)' 



All the hyperbolas have the separating polhode projections as 

 asymptotes (Fig. 80). 



88. Invariable Line. The invariable line describes a cone in 

 the body. Its equation may be simply found from consideration of 

 the reciprocal ellipsoid 



4g ) Z + F+Z?- 1 ' 



whose radius in the direction of d is -y and therefore constant. The 



cone of the invariable axis is accordingly the cone passing through 

 the intersection of the ellipsoid 48) with the sphere 



