1916-17.] Operators applied to Solution of Equations. 
Thus from 
23 
De 1 . Dl 1 . D b 2 on 
we have 
C A 0 1x0 
1 0 B 01 
and from 
IV . Dl 2 . IV on - 
we have 
E 
A 1 ^lxOx(-l) 
0 X i x B ' J 
i ‘ 
The development then proceeds as before, 
B 
■Ju — 
A 
B 
A 
C 
B 
B 
A 
or - 
EA 
B 2 
C 
B 
E Y A 
bAb 
a y c 
bAb 
YC\Y A Y _ io /c 
-©©©Ml)’©’ 
B7 \B7 
B 
1Xb)‘ 
I + i«2ViW±Y 
bAb/ \b7 
-XDW 
E\ 4 /AV 
■® W - «©W * 8 ‘©W©‘ - “©©'(A * « 
otc«j etc. 
Here the same properties hold as in the former solution, the only 
exception to this being that the right-hand side, though the work of a 
cubic operator, is not the solution of a cubic trinomial form, as before. 
(c) The third solution, in some respects the most difficult, comes by 
writing 
Bx = -C 
E 
x 
Ax 2 
from which we reach 
C E B A 
x— — — i — x ~ — 
B + B C B 
C x2 
B, 
the operators are thus Djd . EV . D ( -r and DjJ . Dp 1 . D B -, the former having in 
its first action the form jj. These two operators, it will be noticed, are 
both quadratic ones, the two quadratic operators, viz., of the two previous 
solutions. 
Another new feature in this solution is, that many of the elements 
have the value zero. This comes from a zero having appeared in the 
numerator of the coefficient before there is one in the denominator. Till 
