266 MR. HARGREAVE ON THE RESOLUTION OF LINEAR EQUATIONS IN 
In this part of the solution, I apprehend that it will not generally be necessary to 
have regard to the lower limits, since p may have any constant value, and that value 
may be taken which is most conv^enient in each case; and if we make p= — qc, all 
the lower limits will have this value. 
This part of the solution would then be 
where 
and generally each term of the part within the brackets is formed from the preceding 
term thus : first change x into ^—1, multiply by and effect a summation ; then 
change (in such preceding term) x into x—‘2, multiply by r^+i, and effect a summation; 
and so on until lastly we change x into x—n-\-\, multiply by 2^+1, and effect a sum- 
mation ; and the sum of the parts thus obtained is the next term. 
The verification of the particular solution is easily derived from the above ; but it 
rests on the assumption that the algebraic value above given for £"(P^..Pp+i) is correct; 
to which therefore particular attention is directed. 
The equation 
considered as a solution of the original equation wanting the term G^, implies that 
£^(P...P^^0=P.^”(P.-i-.P^-..)+Q.2XP.-2..P.+i)+R/(P.-3..P.+0+..+Z/(P.-„..Pp-..). 
Now in order to verify the particular solution. 
+ P^G^_ 1 -b s” ( P^P^- 1 ) ^ 1^-2 + P;rP^- 1 P^-a) G ^-3 (P^- •• P^-p + 1 ) + ••• 5 
it is only requisite that 
£«(p,..p._,,0=p.^*(p.-i-P-^+i)+Q^^“(P-2..p.-.+i)+R.£“(P.-3..p.-,+0+.-+z.£'’(p.-„..p.. 
and this is true, for it is identical in form with the equation last above given, not- 
withstanding the occurrence of x in the lowest value of P, for this lowest value re- 
mains the same throughout the expression. 
7. If P^=0, the expressions £'’(P^...P^_ot) reduce themselves to those terms which 
do not contain any value of P. It would not be difficult to determine generally 
what terms these are; but probably the most convenient general method of arriving 
at the solution in an algebraical form would be to make P^ equal to a constant 5, and 
to make h equal to zero in the result finally obtained. 
8. The above particular solution of the original equation is in such a form that the 
general term of the indefinite series representing the solution is given in explicit 
terms ; but that general term may be represented in an implicit form, which perhaps 
is more convenient for practical use. 
By attending to the formation of the successive expressions £'’(P^...P^), it will readily 
be seen that the series 
MoG^-f MiG^.i -f MgG^.a + . . + MpG^_^-l- . . 
