185 
Beer ie 4) F(a) yore); (32) 
as the part of the mean function Mf, which involves expressly 
wk. 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 (x + tw, y+ jo, 2+ kz) =e" f (&,Y; 2)- (33) 
In like manner, 
Mo (a - iw, y-jw, z—hz) =e4 @ (2, 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*)0; (35) 
that is, the partial differential equation, 
(D? + Di + D} + D3) V=0, (36) 
has its general integral, with two arbitrary functions, f and ¢, 
expressible as follows : 
V= Mf (a+ tw, y + jw, 2+ hw) + Mo (a - iw, y - jw, z— hw; (87) 
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, 
eas D vere dD teh U7, (38) 
In fact, 
D+asersDerst, D-q=esDe4, (39) 
and therefore, 
(D2 ay? (D> dyer ens Des. (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, 
« WituiaM Rowan Hami.ron. 
‘The Rev. Charles Graves, D.D.” 
—_-— 
