272 MR. HARGREAVE ON THE RESOLUTION OF LINEAR EQUATIONS IN 
Making use of this relation, we immediately convert our equation, which we assume 
to be in the form 
+ .... + (ao + V+ • • * + 4 -*^+ • .)u-=G, 
into 
a„(D..(D-«+l)>+ J„((D-l)...(D-n))(eeM)+ c„((D-2)...(D-»-l))(.%)+... + /„((D-iB)...(D-n-i,+ l))(/0«)+.. ' 
+«„_i((D-l).. (D-«+l))(/«)+5„_i((D-2).. (D-« ))O20M)+....+^„_j((D-i^).. {\i-n-p+2)){iP»n)+.. 
+a„_2((D-2)...(D-»+l))(A)+... +/i:„_2((D-i?).. (D-«-^+3))(A)+... 
+ +.. . 
or 
\(D -«)+««- 
D 
i)(e««) + (: 
e„(D-Ji)(D-w-l)+*„_i(D-»)+a„_ 2 ' 
D(L)-l) 
D(D-l)(D-2) ^ 
Now if in this equation we change D into d and d into — D, we obtain the equation 
in finite differences (which suppose to be of the wth order), 
or 
f d f d f 6 
M0+ + ^Ug_2-\- ^Ug_3-\- — — [fQ 0 )~^Gg_„—'H.g_„ (suppose) ; 
the solution of which, by section 8, is of the form 
W0=MoH0_„d-MjH0_„_id-M2He_„_2+ .. + ..., 
where Mo=l, 
M,+^M.=0, 
M lyj _Q 
„ /](^ ?w + 1) - ^ . f^{S ?w+ 2) - - I fr{^ m + r) . , 
Restoring the symbols, and thereby converting H 9 _„ into (/o(D))"‘(s”®G), which call 
H, we have 
where Mq, M,... denote a series of operations having the following significations and 
relations; — 
M„=l, 
M,+|§jM.=0, 
