302 MOTION OF KIGID BODIES 



We have OL 2 = x 2 + y 2 + z 2 , 



ON 2 = (Ix + my + nz} 2 , 

 so that p 2 =OL 2 -ON 2 



= x 2 +y 2 +z 2 -(lx + my + nz) 2 



= x 2 (m 2 + n 2 ) + y 2 (n 2 + I' 2 ) + z 2 (I 2 + m 2 ) 



2 wm yz 2nl zx 2 Im xy 

 = I 2 (y 2 +z 2 ) + m(a a + ai 8 ) + n 



2 raw -yzZnl-zx 



Hence the moment of inertia, say J, is given by 



2 mn 



= I 2 A + m 2 ^ + 7i 2 (7 - 2 miiD -ZnlE-1 ImF, (124) 



where ^4 = 2^ (y 2 + z 2 ), etc., 



D = mz etc. 



The quantities -4, J?, (7 are seen to be the moments of inertia 

 about the axes of x, y, z respectively. The quantities Z>, E y F are 

 called products of inertia. 



By giving different values to I, m, n in equation (124), we can 

 find the moment of inertia about any line through 0, as soon as 

 we know the values of the six coefficients A,. B y C, D, E, F. 



Ellipsoid of Inertia 

 247. The equation 



Ax*+ Bf+ Cz 2 - 2 Dyz - 2 Ezx - 2 Fxy = K 9 



where K is any constant, being of the second degree, represents a 

 conicoid. If r is the radius vector of direction cosines I, m, n, we have 



r 2 (Al* + Bm 2 + Cn 2 2 Dmn 2 Enl 2 Flm) = K, 

 or, from equation (124), r 2 = (125) 



