162 
Monpay, JANUARY 22ND, 1855. 
LIEUT.-COL. LARCOM, F.R.S., Vicze-PReEsipENT, 
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, 
dV av a&V 
da dy? d@’ 
or (1) 
DiV+ D}V + D3V =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,+jD,+kD;) (D:-jD:-kD;)V=0. 
Its complete solution is, therefore, the sum of those of the 
two equations of the first order, 
(D,+jD,+ kD: V=0, 
(D, —jD, os kD;) V= 0 3 
and these latter solutions have been presented to us by Mr. 
Carmichael in the symbolic forms, 
We ETE D fF (Y, Z), (2) 
Va at Df, (Ys 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 formula. He has made 
