318 
[841 
841. 
ON A DIFFERENTIAL OPERATOR. 
[From the Messenger of Mathematics, vol. xiv. (1885), pp. 190, 191.] 
Write X = 1 + bx + cx 2 + ..., = (1 - cue) (1 - /3x) (1 - yx)...; then by Capt. MacMahon’s 
theorem, any non-unitary function of the roots a, ¡3, 7, ... is reduced to zero by the 
operation 
A, = di + bd c + eda + ...; 
for instance, if 
(2), = Sa 2 = b 2 — 2c, 
we have 
A (b 2 -2c) = 2b + b (- 2), = 0. 
We have 
AX = x + bx 2 + cm? 4- ... = xX; 
and writing, moreover, X', = b + 2cx + Mx 2 + &c., for the derived function of X, then 
AX' = 1 + 2 bx + Scx 2 + ...= (xX)'. 
fX' \ 
We can hence shew that A — b) =0; the value is, in fact, 
AX' X'AX A7 + . (xX)' X'xX n 
y X. 2 > that is, 1> 
which is 
X 2 X 
X + xX' xX' 
X 
X 
-1, = 0. 
X' . 
This is right, for Jr is a sum of non-unitary symmetric functions of the roots; in 
fact, 
Z =2^—- = -(l)-(2)x-(3)x 2 -&c., 
X 
or since b = — (1), this is 
1 — ax 
X' 
— — 6 = — (2) x — (3) x 2 — &c., 
a sum of non-unitary functions of the roots.