37 
1916-17.] Operators applied to Solution of Equations, 
both increasing or both decreasing. We choose the latter, and introduce 
as many changes of sign from the indeterminate factor on the right as 
are needed to change, first, the sign of the element taken from the 
denominator and as many others as survive. 
Thus to get the coefficient of 
B 5 Eys IF J_ + 
5!' ' AV or "5! ' AV El 
11 11 
we change — — to -j- and make it decrease till it meets 
3 3 
of E, stopping, of course, one short of this. Thus we have 
or 
and the term is 
1 8 5 2 1 IF i 
+ 3 3 3 3 3 5! AV Ej • 
4 
3’ 
the index 
( — introduces another — etc. sign here. 
L 5! & 
11 4 
In the latter case, where — — 5 and - have the same sign, there is 
o o 
no difficulty in seeing why the infinite factor to the right should be +1. 
But the same holds also for the above case, where the first factors that 
cancel, ( — — + 5) and have unlike signs. In this case, both above 
\ 3 / 5 
and below, we pass through 0, and thus have the form . 
Having now ascertained the general form of operator for the solution 
of the trinomial form AaF — Baf 1 + C = 0, we can complete our proof of 
the validity of such solutions for the general equation, left over from the 
beginning of the section. For clearly, if there be solutions of the trinomial 
form Ax n — Bx m + C = 0, got from the operator D A 1 .D^ ) Vw! 7 . D B ™ and the 
operand — (g)”] and if the same operand remain available for the form 
-i ) - 
A^ n + Rrc r — Bcr m + C = 0, then the operator I) R . D c ] . D B m must take 
part in the solution of the latter, so as to recover the particular solution 
(A = 0) from the general, and so for the inclusion of Rptf 1 , etc., in the form 
of the general equation. 
