1890.] to the Discussion of Laplace's Equation. 121 



and therefore p, a, t are vector solutions of equation (1); but, 

 since p + <r + t = 0, 



they only supply us with two independent solutions. This, how- 

 ever, will be sufficient for our purposes, since for the development 

 of the theory we only require to know two independent special 

 solutions. There is a certain disadvantage about these forms as 

 the selection of two of them renders the work unsymmetrical. 

 There is a symmetrical quaternion solution of (1), which involves 

 x, y, z linearly, viz. 



2 (x + y + z) +i(y - z) +j (z - x) +k (x -y); 



but 1 have not been able to hit upon a second symmetrical 

 solution involving x, y, z linearly. 



2. We are now in a position to shew that there exists a 

 relation of the form 



dr = dp.R + da.S (2), 



where R and S are quaternions whose expressions involve x, y, z, 

 but not dx, dy, dz. To verify this, assume 



R = u + if+jg + kh, 



S = v + il +jm + hi ; 



and then equation (2) becomes 



— dS + ida +jd/3 + kdy = (— 2idx +jdy + kdz)(u + if+jg + kh) 



+ (idx — 2jdy + kdz) (v + il +jm + kn). 



This involves the existence of the four relations 



-dB = (2/- I) dx + (2m -g)dy- (h + n) dz, 



da = (v — 2u) dx + (h — 2n) dy — (g + m) dz, 



d/3 = (2h -n)dx + (u -2v)dy + (/+ I) dz, 



dy = (m - 2g) dx + (21 -/) dy + (u + v) dz; 



and since these are to be satisfied independently of the values of 

 the ratios dx : dy : dz, we obtain the following twelve equations : 



