140 



KINETICS OF A RIGID BODY. 



[256, 



mass ; and let us denote by A, B, C the moments of inertia of 

 M for the axes of x y y, z\ by A\ B\ O those for the planes yz r 

 zx, xy\\>yD ) E y F the products of inertia (Art. 240) for the 

 co-ordinate planes ; i.e. let us put : 



B' = 2mjP, E=^mzx, (6) 



C = 2m (x* +y 2 ), C = ^mz*, F= ^mxy. 



256. These nine quantities are not independent of each other. 

 We have evidently 



A=B' + C', B= 



hence, solving for A', B\ C', 



The moment of inertia for the origin O is 



). (7) 



257. The moment of inertia 7 for any line through O can be 

 expressed by means of the six quantities A, B, C, D, E, F\ and 

 the moment of inertia /' for any plane through O can be 

 expressed by means of A', B\ C, D, E, F. 



Let TT (Fig. 31) be any plane passing through O ; /its normal ; 

 a, j3, 7 the direction cosines of /; and, as before (Art. 245), p, 



q, r the distances of any point (x,y, s) 

 of the given mass from TT, /, and O, 

 respectively. Then, projecting the 

 closed polygon formed by r, x, y, z 

 on the line /, we have 



Fig. 31. 



hence, squaring, multiplying by 

 m, and summing over the whole 

 mass, we find 



or, with the notations (6), 



r = A 'a 2 + '/3 2 + C V 4- 2 D/3y + 2 Eja + 2 Fa/3. 



(8) 



