of a rigid Body round affixed Point. 439 



Let the equations of the right line, passing through the 



centre and the point x' j/ z' , be x — — r a, y = -^- z : by 



the help of these four equations, eliminating x'y' z\ we get 



■£ (a*-u*)+£ (P-u^+^-^-u*) = . . (12.) 



Hence during the rotation of the body, the axis u of the im- 

 pressed moment, is found on a cone of the second degree, 

 whose circular sections are parallel to the circular sections 

 of the ellipsoid, as we now proceed to show. 



In an ellipsoid, let <p be the angle between the plane of a 

 circular section, and the principal plane containing the greatest 

 and mean axe of the ellipsoid, >j and ■ the eccentricities of the 

 two principal sections perpendicular to this plane, whose 

 semiaxes are b, c and a, c respectively, 



x. 1 ■ « 2 (& 2 -c 2 ) ;, A% 



then cos <p = — , or cos 2 <f> = gjT CTzA ( 13 -) 



If now in this formula, instead of a 2 , 6 2 , c 2 , the squares of the 



a 2 b 2 c 2 



semiaxes of the ellipsoid, we substitute -5- — 5, ^ $» -5 s- 



r a 1 — u z b l —u l c 2 — w 2 » 



which are proportional to the squares of the corresponding 

 semiaxes of the cone, we shall have, calling <p' the angle be- 

 tween a circular section of the cone and the plane of a b, 

 . a? (6 2 -c 2 ) 



cos ¥ = w (^ry 



a result independent of 11; hence <p' ss <p. 



XVI. The instantaneous axis of rotation moves on a cone of 

 the second degree. 



Assume a point on the perpendicular P, (whose coordinates 

 let be xy z) at the distance I from the centre. 



Then cos \ = —3-, or 

 a z 



a 4 cos 2 a = x'* {a? cos 2 A + 6 2 cos 2 p + c 2 cos 2 v } . 

 Now, I cos \ = x, I cosjk, = y, and / cos v = z, substituting 



a 4 a? 2 

 these values, we find x 12 = * » . t* - • - . a v 



a 1 x l + 6T y l 4. c l z* 



Finding analogous values for y' and z 1 , introducing the rela- 

 tion 



d* + y 2 + z n — u% we obtain for the equation of this cone, 

 a 2 (a 2 - u 2 ) .r 2 + 6 2 (b 2 -u*)y 2 + c 2 (c 2 -m 2 ) z* - 0. (1 *.>* 



* It may be worth while, and it is not difficult to show, that the equa- 

 tions of these cones, the loci of the axis of the impressed moment and of 



