440 Prof. Booth on the Rotation 



XVII. Now as the axe of the impressed moment is al- 

 ways on the surface of this cone whose circular sections coin- 

 cide with those of the ellipsoid, the plane of the impressed 

 moment will envelope during the rotation a cone supple- 

 mental* to the former; whose Jbcals will be therefore per- 

 pendicular to the circular sections of the first cone, that is, to 

 the circular sections of the ellipsoid; hence the optic axes\ of 

 the ellipsoid of moments will be the focals of the cone en- 

 veloped by the plane of the impressed moment ; thus the 

 whole motion of the body consists in the uniform rotation 

 of the plane of the impressed moment round its axis, while 

 this plane rolls on the surface of the latter cone. 



XVIII. From these considerations it follows, that we may 

 dispense altogether with the Ellipsoid of Moments, and say, that 

 if two right lines are drawn from the fixed point of the body, in 

 the plane of the axes of the greatest and least moments of inertia, 

 making an angle with the axe of greatest moment whose cosine 



squared may be equal to w-W — ~ci i an ^ a cone ^ e conce ^° e ^ 



having these lines as focals and touching the plane of the im- 

 pressed moment, the whole motion of the body will consist in the 

 uniform rotation of this plane section of the body in its owti 

 plane, while this plane envelopes the cone. 



Let A C B be the mean section of the ellipsoid, O N', 

 O N the optic axes, then if the plane of the impressed mo- 

 ment coincides with any of the principal planes, the cones 

 round the optic axes as focals become also planes, and the 

 axes of rotation coincide with the axes of the figure. 



the axis of instantaneous rotation, are equivalent to the equations of the 

 same cones given by Poisson, Traite de Mecaniquc, torn. ii. pp. 151, 152. 

 For this purpose then, assuming the equation given by Poisson, torn. ii. 

 page 140, A p"- + B q 1 + C r 2 = h ; and putting for A, B, C, p, q, r t their 

 values given in the preceding pages, we find h = rfif 2 . 



Eliminating then from (12.) and (14.) the quantities a, b, c, u, and intro- 

 ducing A, B, C, k, h instead, equation (12.) is changed to 



(£*— Ah) . , (Jc*-Bh) , , (tf — Ch) , n ,_... . 

 - — A — '- x> + v — g — i-y"*- + v g — '- z J = 0, and (14.) is changed to 



(P-A£)*2 + (* a — BA)y + (k*-CK) z 2 = 0. 



* Two cones are said to be supplemental when their corresponding 

 vertical angles are supplemental. 



The focals of a cone are two right lines drawn through the vertex of 

 the cone, making with the axis through the vertex equal angles, such that 

 the cosine of one of these angles is equal to the ratio of the cosines of the 

 semiangles of the cone. 



•f- I have ventured to use the appellation optic axes, a term borrowed 

 from the wave theory of light, to denote the diameters perpendicular to 

 the circular sections of an ellipsoid. 



