230 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION. 



any axis. Let p be the perpendicular distance of a point P from 

 the axis, r its distance from 0, and q the distance from of the 

 foot of the perpendicular. Now since q is the projection of r on 

 the axis, 



61) q = ax + fty + yz, 

 and we have 



tf = r * - <? = x* + f + z* - (ax 



62) = ^(i_^ ) + 2/2(1 _^ ) + ^2 ( 



- 2(fiyyz + yazx + afixy). 

 Now since we have 



- = + y, -/ = r +, 

 and replacing in 62), 



4- 

 -f 



-f 



Thus the moment of inertia K about any axis whose direction cosines 

 are a, /3, y, is given by 



64) K=Aa* + Bp+Cf-2Dfly-ZEra-2Fap = F(a,p,y), 



as a homogeneous quadratic function of the direction cosines of 

 the axis. 



The sum of products of the mass of each particle multiplied by 

 the square of its distance from a given plane is called the moment 

 of inertia of the system with respect to the plane. Although it has 

 no physical significance it will be convenient to consider it. For a 

 plane normal to the preceding axis we have 



65) Q = Zmq 2 = a?Zmx* + (P2my* + fZmz 2 



+ SfiyZmyz -f 2yaZmzx + 2a(lZmxy, 

 and if we put 



A' = Zmx 2 , B' = 2my*, C' = 2mz*, 

 we have 



66) Q = A ! a 2 + B'p*+C ! f + 2I>p r +2E r a + 2Fap = F'(a,p,>y*). 



The six quantities, A, B, C, A', B\ C r , being sums of squares, are all 

 positive. We have evidently 



67) 



