FINITE DIFFERENCES AND LINEAR DIFFERENTIAL EQUATIONS. 
267 
is a solution of the original equation, if 
M„=l, 
M. = P,Mo, 
M, = Pi— 1 M , 
M3 = Pi.gMa + Qi_ iM 1 + RiMo, 
M,= ■^.r— ^ + 1 1 + I R.y— + 3 1 • • 1 ^ + n— w-Hl I ^^x—p-\-7^^p—n 9 
which last is a general relation determining the coefficient of any term from the 
coefficients of the n preceding terms ; or from the coefficients of all the preceding terms 
when the number of these preceding terms is less than n. This relation is the equa- 
tion of formation, and may be regarded as universal, bearing in mind that when 
is less than n some of the terms of this relation vanish. 
The solution of the original equation is therefore reduced to that of a similar 
equation without second member : and, by what has preceded, this solution is 
it being understood that Mo=l ; and this is the value of Mj, before given. 
But the value of may be found otherwise, thus : 
Make p + 1 ■” Pp5 jO + 2 — Qpj &c. &c. 
Then the solution is evidently 
M^=£XPp.-P'i), 
wffiere the operation v has the same meaning as before, except that it is applied to 
the accented letters. 
Consequently 
M^ = (Pp .. Fi ){ 1 -p Ap -h -f- , 
where 
Ap= 2i§'^+i H- ^ar^+i -f . .+ 
RjO “h Aj,_ 2) ~1“ • • • “k 22^—1 (2j5 + lAp_^+l), 
and generally the terms are formed as stated in section 5, using p for x. 
9. I shall conclude this part of the subject with a few simple examples for the pur- 
pose of illustrating the processes here given. 
Ex. 1. Let the equation be 
-f &X-2 + 
That part of the solution which is independent of G^, is 
u^-=c{a'’-\-(x—\)a''~^h^-\-^{x—2){x—3)a‘‘~‘^b^-\-^{x—d){x—A){x—b)a^~^b^^.,.}-, 
2 M 2 
