162 
Monpay, JANUARY 22ND, 1855. 
LIEUT.-COL. LARCOM, F.R.S., Vicse-PresipEnt, 
in the Chair. 
Rey. Proressor Graves, D.D., read a Paper on the solution 
of the equation of Laplace’s functions. 
‘«¢ Mr. Carmichael was the first to observe that the partial 
differential equation of the second order, 
PV PV @V 
de dy? + de =U, 
or (1) 
D}V + DV + DjV=0, 
known as the equation of Laplace’s functions, may be reduced, 
by means of Sir William Hamilton’s imaginaries, to the sym- 
bolic form, 
(D,+jD2+hD,) (Di-jD.-kD;) V=0. 
Its complete solution is, therefore, the sum of those of the 
two equations of the first order, 
(D, +jD, Se kD;) V= 0, 
(D, -jD, ae kDs) V= 0 3 
and these latter solutions have been presented to us by Mr. 
Carmichael in the symbolic forms, 
ae alata oh (Y, Z), (2) 
Vu Fail eet oF 7 (Y, z)s (3) 
in which f; and f, stand for quite arbitrary functions. Follow- 
ing, however, too closely the analogies of ordinary algebra, 
Mr. Carmichael has fallen into an error in interpreting the 
right-hand members of these formule. He has made 
