185 
a (ert —e"4) f(x, ys 2), (32) 
as the part of the mean function Mf, which involves expressly 
yk. Adding the two parts, (28) and (32), we are conducted 
_ finally to the very simple and remarkable transformation of 
the mean Function Mf of which the discovery is due to 
you: 
Mf («+ tw, y+ jw, 2+ kz) =e4 f (&; Ys 2) (33) 
_ In like manner, 
Mo (a - tw, y—jw, 2—hz) =e" (a, y, 2). (34) 
Each of these two means of arbitrary functions, and therefore 
also their sum, is thus a value of the expression 
(D?- 4*)70; (35) 
that is, the partial differential equation, 
(D? + Di +D} + D3) V=0, (36) 
has its general integral, with two arbitrary functions, fand ¢, 
- expressible as follows : 
a 
V= Mf (2+ iw, y+ jw, 2 + kw) + Mp (2 - iw, y - jw, 2 - kw; (37) 
_ which is another of your important results. You remarked 
that if the second member of the equation (36) had been U, the 
expression for V would contain the additional term, 
ews D1 ew 4 D1 evs U7, (38) 
Tn fact, 
D+qz=ersrDerst, D-q=ersDers, (39) 
and therefore, 
CDS ayo Aye = eu? errs Dane (40) 
“‘ Most of this letter is merely a repetition of your remarks, 
but the analysis employed may perhaps not be in all respects 
identical with yours: a point on which I shall be glad to be 
informed. 
‘«‘T remain faithfully yours, 
«© Witt1aM Rowan Hamitton. 
-“ The Rev. Charles Graves, D.D.” 
— 
