1891.] Reference to the Discussion of Laplace's Equation. 155 



where U and V are formed from of/d\ and df/djj, in a similar 

 manner to that in which r is formed from f(\,, //.). Thus we see 

 that if we take u and v as our fundamental solutions of Vr = 0, the 

 formula discussed in my former paper is very closely analogous to 

 the formula 



7 dw lf . dw , 

 dw = yp d% + ^— d-q, 



which would hold if w were a scalar function of the two variables 

 £ and y. 



Instead of the pair of special solutions that we have here made 

 use of, we might have chosen either of the pairs 



z + ixj and x — ky, 



x +jz and y — iz. 



I have investigated the matter and find that, adhering to real 

 quaternions, the most general form that we can take for our pair 

 of special solutions, in order that a third solution may be expressed 

 in terms of them in the simple form of the present article, can by 

 a suitable choice of axes be expressed by 



y + lex, y cos a + z sin a — x (j sin a — k cos a). 



5. The expression for r given in the preceding article is 

 obviously not perfectly general, as it is derived from a series con- 

 taining only positive integral powers. In the second part of the 

 paper referred to above, Graves gave a method of deriving solutions 

 from scalar series containing negative and fractional powers, but 

 he did not succeed in expressing his results in terms of the two 

 special solutions he made use of. I think that it is highly probable 

 that if we take any two independent special solutions, any other 

 solution can be derived from them ; but at the same time it is 

 possible that the said other solution may not be expressible in 

 terms of the two special solutions in the ordinary functional 

 form. 



[There is one remark to be made in completion of my former 

 paper. It is there proved that if p and q be any two inde- 

 pendent solutions of the equation Vr=0, and r any other solu- 

 tion, then there exists a relation of the form 



dr — dp.P + dq.Q, 



where P and Q do not involve the ratios dx : dy : dz. The converse 

 of this is also true : for since the above relation is to be satisfied 



