RIGID BODY ABOUT A FIXED POINT. 291 



In the initial position of the body, therefore, this vector passes through the in- 

 tersection of the ellipsoid 



S . (V + ^)~ V a (V + } f)~^ = S&i = - h 2 , 



with a second ellipsoid 



T(V+^ *=T5 = T 7 . 



It therefore lies on the cone 



7 *s . <v +py%-i(V + py 1 * + ** . (*-' + ^y 1 , <v + Ky* = o , 



or 



[We might have saved the last seven lines by noticing that 



Sye = o 

 in the present position of the body, involves 



S£<r = 



in the initial state, which, with the value of £ in terms of <r above, gives the 

 result at once.] 



44. This cone is seen at once to be normal to the £-cone in the initial body, 

 viz., by § 26, 



or 



s.e(W'p>=o. 



The vector o- constantly changes so as to be perpendicular to £. Hence in the 



* In fact any equation such as 



Sg-vJ/g = , 



where -^ is a constant self-conjugate linear and vector function, gives 



s-4^g = o , 



whence 



where v represents the normal-vector. For its locus, we have 



g = -\>-h, 

 and by substitution for g and -vj/g in the given equation, we have 



S^-^v = . 



