Professor Mac Cullagh's Lectures on Rotation. 141 



U = qz - ry ; 



V = rx — pz; (1) 



w — py — qx. 



III. — To represent geometrically the Moments of Inertia of a Body with respect to 

 A.res drawn through a fixed Point. 



The moment of inertia of a body -with respect to any axis (o, /3, 7) is 



M = A' cos^ a + B'cos=j3+ C'cos-7- 2Zr'cos/3 cosy - 23/' cos a cosy 



— 2iV'cosa cos^ ; 

 where 



A' = \{if + z"-) dm, L' = \yzdm ; 



B' =\{ar + z-) dm, M' = \xzdm ; 

 C =\{x-^- f) dm, N' = \xydm. . 



Assume M=—,; /i being the mass of the body, and r a distance measured on 



the line (a, /3, 7) and construct the ellipsoid whose equation is 



yl V + B'y' + C'z- - 2L'yz - ni'xz - iN'xy = /x ; (2) 



then it is evident that the moments of inertia of the body with respect to axes 

 passing through the fixed point are represented by the squares of the reciprocals 

 of the radii vectores of this ellipsoid. Assimie A = fia', B — fi¥, C= fju^, and let 

 the axes of co-ordinates be the axes of the ellipsoid; its equation will thus become 



alf^ + ty + cV = l; (3) 



and the equation of the reciprocal ellipsoid will be 



•^+^;+^'=i. (4) 



a^ b- c 



This latter ellipsoid may be called the ellipsoid of gyration, as the perpendi- 

 culars on its tangent planes represent the radii of gyration ; this is evident from 

 the consideration, that these perpendiculars are reciprocal to the radii vectores 

 of the ellipsoid (3). In fact, the moment of inertia with respect to any axis will 

 be represented by the formula 



