270 MR. HARGREAVE ON THE RESOLUTION OF LINEAR EQUATIONS IN 
The particular solution, therefore, is 
+ 4'>^)PrPa;-lGr^-2 + 2a?P^,P ^_iP^_2G^-3 + 2) (P^ . . P^_3) G^_4 
H- (3.r" — 5 - 2 ) (P^ . . P^_4) G^_5 + — 1 U’ -f 6) (P^ . . P^.s) G^_6 + . . . . ; 
and the part of the general solution found by this method is 
{x-\-l)x (a? + l)...(<r— 2) (a?+ l)...(a?— 4) 
«^,= c(P^..Po) 1 
2.4 
2.4.6 
•••)■ 
Ex. 6. Let the equation be 
1 ^2Pj;Pi'— 1^^'— 2~1~ ' • ~i"^n(P.r • • Pa— n+ 1) « Gj. j 
then the equation for determining is 
M,+a,P 
x~~p + 1 M^+agP^. x~-p-^2^^p — 2~\~ • s:~p-¥\* x—p-Vn )M,_„=0; 
or making 
P — P' 
^ x~~p + \ ^ p^ 
Mp + aiPpMp_i + «2P;,Pp-lM^-2~l"-- + ^n(P/)-*Pp-n+l)Mp_„=0 ; 
the solution of which is evidently the solution of 
Mp+aiM^_,+a2Mj,_24---+«nM^-«=0, 
multiplied by (P^..Pi). 
Let (rd-ai#”“*+a2r“^4--*+^*«)~* =711^+^17^ “!“•••? 
then M^=(P;,..P;){c.Aa'’+”+C2B/3^+”+..}; 
and taking the parts affected by each constant separately, it will be seen that the 
original equation reduces itself to a set of equations of the first order, 
— aP^- 1 = ' AG„ 
f3PA-i=|3"‘'BG^, 
so that its complete solution can be adequately represented. 
Solution of D’ff^erential Equations in Series. 
10. I now proceed to point out a method by which the processes above indicated 
may be made to give solutions of certain general forms of linear differential equa- 
tions. 
In a paper on Linear Differential Equations presented by me to the Royal Society, 
and which the Society has done me the honour to publish in the Philosophical Trans- 
actions (Part I. for 1848, p. 31), I have enunciated, and so far as is material to the 
present purpose, demonstrated the following theorem : — 
That if, in a linear differential expression (p{x, D)m=X and its solution u='^p(x, D)X, 
the letter x be changed into the operative symbol D and D into —x, we shall thus 
