338 Mr MURPHY'S THIRD MEMOIR ON THE 



Laplace's equation occurs when we put m = 0, and therefore 



the first term alone of which is the type of Laplace's functions, the 

 equation is therefore more general than the functions it was used to 

 designate. 



The term ^"+' («')-<"+"'+') gives n + 1 constants of integration which 



enter as coefficients of the appendage which is a rational function of 



n dimensions, but this must be rejected, since the constants must be 



determined so that the rational function of n dimensions may satisfy 



the given equation, and this only identifies the appendage with the 



d' ift'Y^"' 

 other term in u, viz. aitf)"" — , / — . 



17. To find explicitly the omitted part of the complete integral in 

 Laplace's equation. 



The general equation of Art. 16. becomes in this instance 



and the complete solution is 



u = a^^^^+bfr'{tt')-^'-'\ 



the first term being Laplace's function, and the second the transcendant, 

 it is required to find explicitly. 



Let a, /3 be any arbitrary quantities, then we have 



dar\t-a' fi-a) ~ ^-a da^\t-a)^'^da\fi-a] da'-'\t-al 



n{n-l) d^ / 1 X d'-^ / 1 N 



^ 1.2 Ma'[(i-a) da"''[t-a)' ^' 



