OF ENERGY IN A SYSTEM OF MATERIAL POINTS. 735 



Hence in general, whatever axes we choose, 



(({ i 



J J J [^ A-J ' 1.,'du.dv.dw. = - V - I [ifflJ-^Jf/ AT*/;*- ..... (87). 



For the system consisting of the two particles m, and m 2 the energy of 

 rotation is 



and the energy of internal motion is 



JW-jf*- ( 89 )' 



Hence we may write equation (70) 



f 3 "" 7 / M \~i A 



\ W) 'JW / 



\ I Ill-ill tint 



We have first to express J 2 in terms of quantities having the suffix 3 . 



If we make the plane of yz pass through the three particles m l , m 2 , m 3 , 

 so that the origin coincides with their centre of mass and has the same velocity, 

 and the axis of z passes through m 3 , then a is twice the area of the triangle 

 whose vertices are m lt m 3 and m 3 , 



,u 3 , H, = H 3 (91), 



(92), 



1 , M 3 m 3 z 3 

 ^ 



M.GA v 2 _ _ 4 2 



" h V + ' 



M, 



, 



We have now to integrate 



I \I~*dv 3 dw 3 , 



extending the integration to all values of v, and w t which make 7 3 positive, and 



