212 ON THE DETERMINATION OF THE ATTRACTIONS 



we see that the function / will always be two degrees higher 

 than F. But since our formulae become more complicated in 

 proportion as the degree of F is higher, it will be simpler to 

 determine the differentials of V, because for these differentials 

 the degree of F and /is the same. Let us therefore make 



p'(x-xJ}dxdydz 



-n ax J < (x _^ _ y (z _ z 'Y + u ^ 



then this quantity naturally divides itself into two parts, such 

 that 



pdxdydz' * _ 



where A' 



\(x- 



r 

 and A" = 



{(x - xj + (y - y'Y + (s - z'Y + 2 J 2 



By comparing these with the general formula (1), it is clear 

 that n 1 = n + 1, and consequently n = ri + 2. In this way 

 the expression (28) gives 



which coincides with (34) by supposing F=f. 



The simplest case of the present theory is where /(#', y', z) = 1, 

 and then by No. 11, we have $ '= 1 arid 5 = 1, when ^4' is the 

 quantity required, and as the general series (29), No. 11, then 

 reduces itself to its first term, we immediately obtain from the 

 formula (30), the value of A' following, 



A'= -^a'b'c'l'-!^ (35), 



because in the present case H = 1, 5 = 3, and n = ri + 2. 



