FINITE DIFFERENCES AND LINEAR DIFFERENTIAL EQUATIONS. 
265 
^ V’(P,...P,„)=P.-P,« 2 ; 4 r.., 2 ;- 4 (r.., 2 ;;>„.)). and so on, 
|vL(P...Pp.,) = (P...P,.,)2;„„.(n„,2;;>,„), 
2 ^Vl(P...P,.,) = (P,..P,*,) 24 ,.+ 5 (n,+, 2 ;;r,+,(t>„, 2 ;;>..,)), and so on, 
being that term of the series qjr^s„ See, which corresponds to the operation V^, as 
q_, corresponds to Vj, to Vg, &c. 
V^V^(P^...P^+i) = P^...Pp+i2^+^+^+i(4+,2p;Lv^+i)5 (4 corresponding to V,), 
- • 'Pp+l) “Par* • • Pp + l^p+m + 2l+2(^x+l^p+m+l+l(^a:+l^p + m‘^.r+l))} ^C. &C. 
Finally, if Nl,,p+N" •• represent a series in which 
/) 1 5 p J q^+i ”1“ ^p+2^j;+l “H • • I 1^*+15 
and generally 
■VT(a + l)_'2^ / -|yT(a) / |^(a) \ | Va' / -|yj-(a) v . 
p "" ^/?+2c— 1 \ 1, p/ ^p+3a— 1 \' .r+l-^^.r— 2, p) ^^^+40— 1 ,r— 3, p/ • • • • 
H~ + + p) "f" 2p + «a— 1 + n + 1, p) 5 
then, if this series be called N^,^, we shall find that s'’(P^...Pp+i) = (P^...Pp+i)N^_p. 
It may be here observed that the number of the terms of the series N^_^+N" ^4-.. 
cannot exceed ^{x—p)-\-\ when x—p is even, and ^{x—p-\-\) when x—p is odd ; and 
since p has the successive values x—2, x— 3, &c., it is always known whether x—p be 
odd or even. The number of terms may be less ; for if Q^, be zero, the number of 
terms would be the next whole number above ^{x—p), &c. 
That the equation 
w^=c(P^...P^+i)N,,,p 
is a complementary solution of the original equation, or in other words, a solution of 
the original equation wanting the term G^, may be directly verified ; for we have 
u, - P^M^_i = c(P^ . . Pp+ ,) p - N^_,, p) 
— c(Pj,..P p+i)AN^_,^^. 
Now AN^_i,p=0, 
AN;_,, p= p +r^N"_3, ;,+<s,.NL 4, p + . . . 4- , 
and generally 
ANr,;i=?,Nl«„+r,N<“>,,,+s,N?i,,,+ . . . 
,P’ 
whence 
and 
AN^_i,p — ^4rN,j.-2,p4"^4;N^-3,p4"'^^Nj,_4_j„4" •• 4“*c^N,_„+,_p4~^4'N,i._„, 
.p’ 
p’ 
MDCCCL. 2 M 
