154 Mr Brill, On Quaternion Functions with especial [Feb. 23, 



where 



f(\ H ,) = A+B\+Cfj, + ^ 1 {D\ 2 + 2EX/J, + F^} 



+ i {GX 3 + SH\y + SKXfj? + Lfi 3 } + &c., 



the coefficients A, B, C, &c, being in general quaternions. The 

 generality of this form will not be affected if after expansion of the 

 exponential symbol and subsequent differentiation we make X and 

 /a both zero. Hence we obtain 



r = A + uB + vC + ^ {u 2 D + (uv + vu) E + v 2 F] 



+ ^ [u*G + {a 2 v + uvu + vu 2 ) H + (uv 2 + vivo + v 2 u) K + v*L) + &c. 



It is important to notice that the quaternion coefficients must 

 always be placed last in the various terms of the series. Graves 1 

 gave a form similar to this as a quaternion solution of Laplace's 

 Equation, but he seems to have only contemplated the possibility 

 of scalar coefficients, and in consequence his result loses in 

 generality. The importance of introducing quaternion coefficients 

 will be at once seen if we attempt to express the elementary 

 solutions mentioned in my former paper in terms of those used in 

 this. We have 



— 2ix + jy + kz = (y + koo)j + (z —joe) k, 

 ix — 2jy + kz = - 2 (y + kso)j + (z —joe) k, 

 ix+ jy — 2kz = (y +kx)j — 2 (z — jx) k, 

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



= (y + kx) (2+i-k) + (z -jx) (2 - i +j). 

 By means of the equation 



df(q)=dq.f'(q) 

 of the preceding article, which holds in the case under con- 

 sideration, we have 



dr = (du -^+ d v.f)e ^ ^f(X, fi) 2 



7 u^ + v^df 7 ^i + v^df 



= du.e dA w^r+dv.e <* ^4~ 

 oX dfi 



= du . U+dv . V, 



1 "On the Solution of the Equation of Laplace's Functions," Proc. B. I. A., 

 vi. 162—171, 186—194. Graves's papers were written with the object of giving 

 the interpretation of a symbolical form obtained by Carmichael, " Laplace's Equation 

 and its Analogues," Cambridge and Dublin Mathematical Journal, vu. 126 — 137. 



2 It is to be understood that after expansion of the exponential symbols and 

 subsequent differentiation, \ and /j. are to be made zero. 



