262 MR, HARGREAVE ON THE RESOLUTION OF LINEAR EQUATIONS IN 
from which the complete solution would result, so far as known methods extend, in 
the form 
or in an expanded form, 
H-PlGl-l +P«-iP*G^_2 +P^-2P^-iP^G^-3 +*-* + cP,;iP,;j+i...P^ 
. +g: +p:g:_, +p:_,p:g:_2 +p:-2p:-ip:g:_3 +...+c"p;p:,,...p: 
+ . 
+ . 
I I T)(«)/^(^) I T}(^) 'n(n)f^(n.) . 13(71) y^(n) r)(«)^(w) 1 1 ,S^)T)(n)-ry(n) T)( 7 i) 
n ' "1 ^.r— 1 "^r* 1 2 I 3 "1 • • • I ^ -^ 7?^+ 1 • • • -^.r * 
The last column contains, as is well known, the general solution of the original 
equation deprived of the term G^; and the remaining columns contain the particular 
solution of the original equation. 
3. The investigation the results of which are given in this part, although it actu- 
ally succeeds in solving the original equation, will be found to contribute little or 
nothing to the analytical theory as above explained ; and this arises from the circum- 
stance, that the particular solution, instead of being produced in the separated form 
above written, is produced in an aggregated form. 
G.+AG„_,+BG._2+.., 
and the complementary part of the general solution, instead of being produced as 
above, appears in the form of a sum of the complementary functions above written, 
the constants c', c", &c. being the same in all. But as we shall thus obtain a solution 
with an arbitrary constant, capable of solving the original equation deprived of G„ 
we have it in our power to solve the equation completely by reducing the order of 
the equation in successive steps. 
If we take the equation of the first order, 
= P 1 + Gj., 
and arrive at its solution, not properly by any analytical method, but by giving to x 
the successive values 
X, X — 1 , X — 2 w-f- 1 , m, m — 1 , 
and eliminating all values of between the first and the last, we have 
— Px^x—1 "h G^ 
— P^(P^-iW^- 2 +G^_i)- 1-G^ or P^P^_iM^_2+P^G^_,-f-G^ 
P^^PiT— l(Pji;— 2^^ — 3~1“ G^_2) “i" P^G^_ j -(- G^ 01 P^Pj;_lPj;_2W^ — 3 
+ P^Pir- 1 G^_2 + P^G^_ 1 + G^ 
= (P,P._.....P„>_,+(P...P,„.,0G.+(P...P,„+2)G^+,+ ...-hP.P._,G._2+P.G._. + G. 
