264 MR. HARGREAVE ON THE RESOLUTION OF LINEAR EQUATIONS IN 
It might at first sight be supposed that this process, if successful at all, would give 
the complete complementary solution in the form 
w«=CiS'’(Pi,.P^) + C22'’(P2..PJ + .. ; 
but it will be found that in reality this introduces only one arbitrary constant. 
It now remains to place these expressions under a properly algebraic form, and to 
verify the result ; and in order to do this we must express the general term £'’(P^...Pp+,) 
in terms of the factors of the original equation; and afterwards give top, which in 
the first instance is regarded as a constant, the successive values required for form- 
ing the several terms of the solution. 
This may be done as follows : — 
j Gtx 
■L<Gt p p §'i5 P P P f'xy P P p p 
and p — p- 
n+2***^x i’— n + 1 • • • 
= 2 .. 
Then it is easily seen that 
V,(P^,...Pp+l) = P^... Pp+l{^p+2-{-9’p + 3+... + 9^} =P^...Pp + l2p + l</^+I, 
V2(P^...Pp+i) = P^,. ..Pp+1 {rp+3-l-rp+4 + =P^... Pp+i2p+2^^.+i? 
and generally 
• • Pp + l) • • Pp + l { '^p + l^Jx + l ~l~ 2p + 2^j;H T 2p + 3.S^ + i T" • • •'\~'^p+n--2^x+\ H” '^p + n-\^x+\} • 
To find Vj (P^-.P^+i), it will be convenient to proceed by steps, beginning with a 
small number of terms. 
Thus 
Vi(P^...P«_4) (P«- ••P^-4) 9^-1 + (7^-2 + S'a^-s)} 2 
’V i(P^. ..P^-s) = (P^ - ••P,r-5)(5'ifH~ 9’4^-1 + §'4-2 + 94-3 + 94-4)5 
.. . . P^_4) = P.r • • .P^-4(94 (9^-3 + 9^-3) + 
2 Vj(P^...P^_5) — (P4;-”P4^-5)(94;(94-2 + 94;-3 + 94’-4)+94 ;-i( 94-3 + 94;-4)+9‘^-294?-4) ? 
and generally, V,(P^. ..P^+J being P^...P^+i(2;+,9^+i), we have 
y ^i(P4 - ••Pyj+i) = P,r---Pp+i(942p+i94.+i+94-i2p+i94’+i + ••+9 p+ 4 ^p+i 9 ^+i) 
= P.,,...P^+i2p4,3(9^+i2^4^jy^+i). 
Similarly, it will be seen that 
^ V'(P^...P,,+,) = P^...P^+,2;+3(5r^+,2;;3(9,+,2;~|7,,+0)5 ami so on. 
In like manner we sliail find 
