134 
MR. HEARN ON A CLASS OF PARTIAL DIFFERENTIAL EQUATIONS. 
This circumstance will serve to determine 2 and j3„ as follows: we have identically 
du 
' l U 
D r 
D 2 { zDti.l + A*. . Di>„ by (cc) 
=d*%- +&D |«,. +1 J+a«,.». +1 . 
But 
dvn+1 _d _ d / _ <t dv n \ 
dx dx V ' n dx\ dy ) 
dx 
d<$ -pv . -p.^: 
=-s D!, -+ D 
, - 0 d dv n 
dy ' dy dx 
dx 
Hence 
u dx — U \ dx ^ dx Vn+1 J‘ 
D %T + D (S^ +1 ) +P" D (zv n+l ) + Au n v n+i 
dv 
ought to be identical with -\-(3 n+1 zDv n+1 , and hence the conditions 
+&* = &.+ !* 
[dx == ~ 
Eliminating z, we have 
j}d<p ^ A«„A/3« 
dx (3 n+ i C ' 
or 
and 
or 
'«/ 5 
^+ ce ^— 0, 
Cfin + 1 — (3 
@ n+1= Aa„ — c ' @ n ’ 
n A«„_1 . Aoe„_2 .... 
P n ~ ( A«ft_ ! — c) ( A«„_ 2 — c) 
by these determinations we establish the formula ((3.) as a consequence of (a.), and 
therefore if the formula (a.) be true for any value of n, it will be (subject to the above 
conditions) true for the next superior value. 
Now, when n — 0, v 0 = D°u = u, and provided « 0 and u_ 1 are each =0, a 0 and A «_ l 
dv 
will be each 0, and .\ a 0 and (3 0 each =0, and the equation (a.) reduces to D ^ 
