68] MOMENTS AND PRODUCTS OF INERTIA. 227 



Of these the terms in V are the moments of the vector M V 

 in the direction of the Z-axis applied at the center of mass, while 

 the terms in o are applied elsewhere. The equations of the central 

 axis of momentum are, by 66, 38), x' y ! z' being the running 

 coordinates, 

 H x 



or inserting the values, 



> r\ (o Zmxz y' MV -f z' Miax 



- M.V~x a Emyz + z' Mo>y -f x' MV 



M&HC 



_ co Zm (x* -f y 2 ) x' Max y' May 

 MV 



This does not pass through the center of mass unless, putting 



x ! = x, y' = y, z' =~z, 



*K\ aZmxz -f Mazx _ oEmyz -f M&yz 



M.(oy 

 _ co Zm (x z -f 2/ 2 ) M a) (x 



MV 

 We see that the resultant momentum involves the various sums 



ZmXj Zmy, Zmxz, 2myz, Hmr 2 , 



the axis of Z being the instantaneous axis. These sums are constants 

 for the rigid body, depending on the distribution of mass in it. The 

 first two represent the mass of the body multiplied by the coordinates 

 of the center of mass. The last represents the sum of the mass of 

 each particle multiplied by the square of its distance from the Z-axis, 

 and is what has been called the moment of inertia of the body with 

 respect to that axis. We are thus led to consider the sums 



A = Zm (y 2 4- A B = 2m (z 2 -f # 2 ), C = Zm (x 2 + y 2 ), 

 D = Zmyz, E = Zmzx, F = Hmxy. 



Of these the last three, D, E, F, are termed the products of inertia 

 with respect to the respective pairs of axes. 



In the case of a continuous distribution of mass, we must divide 

 the body up into infinitesimal elements of volume dt, and if the 

 density is 0, the element of mass is dm = gdt and the six sums 

 become the definite integrals 



15 



