FINITE DIFFERENCES AND LINEAR DIFFERENTIAL EQUATIONS. 
263 
= ( P,. . . P.) f«._. + 1 ^ + 
I ^ m * m + 1 
Gr^— 2 
• P^-2 
Gr^-1 I Gjt 1 
which is the solution ordinarily given, though it is arrived at by a process somewhat 
less coarse than the above. 
By applying the same process of successive elimination to the general linear equa- 
tion ( 1 .), and carefully observing the law by which the entrance of the factors 
P^, Q^, R^, &c. is governed, it will be found that an exactly similar solution may be 
found in the form 
«.=.*{p....P.(2(p^) + c)], 
or 
where v denotes the sum of a set of distributive operations Vj, V^, V 3 , . . . V„_j not of 
a strictly algebraical character, which are capable of being performed only upon fac- 
torial expressions containing consecutive values of P^, and which have the following 
significations. Vj denotes one operation of this character, signifying that the factor 
Pm-i Pm is changed into Q,„ as often as it occurs, any term in which it does not occur 
disappearing, and the sum of the terms thus obtained being the result of the opera- 
tion ; so that, for example, 
V,(P._,P.)-Q., VXP._.P.)= 0 , 
V.(P,_3P._.P.) =Q.-,P.+P.-2Q« V?(P,_2P,_,Pj =0, 
Vl(P^- 3 P.r- 2 P*-lP*) — Q^- 2 Pi— lP 4 :“l“P^- 3 Q^-lPa’"f'P«- 3 Pa?- 2 Q.« (P^_ 3 P^_ 2 P^^- iP*) = 2 Q^_ 2 Q,r, 
&c. &c. 
Again, V 2 denotes another operation of a similar character, signifying that the 
factor P,„_ 2 P^_iP,;i is changed into as often as it occurs, the result of the operation 
being as before the sum of the terms ; so that, for example, we have 
V ''P P P ^ — P 
^ 2^-^ a— 2^ I-*- 
V2(P^-3P*-2P^-iP^) = I^^-iP^r+P^-sR^, 
V2(P^-4P^-3P^-2P^-lP^)=I^^-2P^-lP^+P^-4R^-lP^+P^-4Pa^-3Rw 
V2(P4:-5- -P4^) =I^^-3P4^-2P4^-lP4:+P^-5l^^-2P^-lP*+P^;-5Px-4R^-lP«+P^-5P4^-4Px-3R^; 
V2(P^_5...P^)=2R^_3R^, &c. &c. ; 
V 3 denotes the change in a similar manner of Pm- 3 Pm- 2 Pm-iPm info ; 
V „_2 denotes the change in a similar manner of Pm_„+ 2 ..-Pm into ; 
and 
V„_, denotes the change in a similar manner of P^_„+i...P,„ into Z«. 
