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X. Investigation of an Extensive Class of Partial Differential Equations of the 
Second Order, in which the Equation of Laplace’s Functions is included. 
By G. W. Hearn, Esq., of the Royal Military College, Sandhurst. Communicated 
by Sir John F. W. Herschel, Bart., F.R.S., tyc. 
Received February 19, — Read April 2, 1846. 
Theorem. If u be a function of x and y satisfying the equation 
where 
then the solution will be 
where 
where 
d^u 
dxdy 
dff 
dxdy 
+a n eHi — 0, 
-j-ce^O, 
u = D n v n 
D = e~V, 
dy 
v n=J e A ft, xydy+^r, 
X y and 4*x arbitrary functions of y and x, 
. A 2 
and 
where 
ft 
(Aa„_!— c)(A«„_ 2 — c) ' 
Act r a r+1 ci r , 
and a n is a function of n vanishing for n — 0 and for n= — I. 
I will proceed to demonstrate this curious theorem as briefly as possible. 
According to the notation, we may write the given equation 
D 
du 
dx +u H u- o. 
v n =D n u 
v n+l =T> n+ 'u=Dv n . 
D^+/3,^,=D-{d £+«„«]. . . . 
where z is a function of x and y to be determined, also (3 n a function of u. 
Writing n-f-1 for n in equation (a.), we ought to have 
^ +& + ,zD*„ +1 =D* + '{D £+«.*,«,} . . . 
Then if 
we have 
Suppose that 
(a.) 
D 
m 
