444 Mr. R. Har greaves on 



lead through (114) to the conclusion stated above. Thus for 

 example the part involving a 2 is ^a 2 x i (ax-\-c'y + b'z)dr i 



= 3 ^i (3A 2 a + 3AOV + 3AB'6') = ^ (3A>) , 



the term in the bracket being the same as that in the square 

 bracket of (114). 



With a view to other linear functions of xyz as density, 

 we note that the energy-integral when Xi * s multiplied by 



p(e\v + by + a'z) as density ( i. e. £ ~r instead of ■- - T -\. may 



r v u ' J \ 2 dy 2 dx) ' 



be found by the same method to be -^ I rf\ or w ^ n 



£ - 7 - as density, to be ~~- I — 7= - The transformation 

 2d* J? 70 Jo v/j 



corresponding to (114) is got by using fydx/^/j, and quoting 



the second relation of (87) as written for 7'. 



In dealing with a density px, we see that x and Xx are 



,i ,. « .. P l <#m 1 dw 1 <:/« 



the same linear functions of 5 - - 7 , ~ -, ;--, s^r-? and of the 



2 o# 2 ay 2 ffe 



potentials due to densities ^ y ,... say %i % 2 %s ; then 



%x = (Axi + ^'%2 + B'% 3 )/A a . The energy-term belonging to 



X\ with its proper density is ^~ I — -,- , but when it is 



'^ Jo v J 

 associated with the densities proper to ^ 2 and ^ 3 , a is re- 

 placed by 7' and /3' respectively. Thus the energy belonging 

 to px is got by writing the square of ( A^i + C^ 2 + B'^/A^ 

 and putting a for X i 2 > for ^ 2 2 , 7' for %i % 2 - . •; and is 



&± f ^(A 2 a + G^ + B' 2 7 + 2B'CV + 2AB^' + 2ACV). . (116) 



For py the bracket is replaced by 



C' 2 a + B 2 y g + A' 2 7 + 2A'B«' + 2A / C'/3'-r2BCV ; 



and for the integral in which either px is associated with 

 Xjf or py with ^ it is replaced by 



C'(A* + B/9) + A'BV + (BB' + C'A')*' 



+ (A A' + B'C')£' 4- (AB + C /2 )y. 



