VARIABLES IN CERTAIN LINEAR DIFFERENTIAL OPERATORS. 
25 
interchange of x and y ; and the second of negative character, persisting in form but 
for a change of sign. (A coraplex self reciprocal operator can of course be found by 
taking the sum or difference of any two correlative operators; e.g., 
w{l, 0 ; m, {0, 1 ; « + 1, m — 
At greater length (17) and (18) are 
(m) (m) ^ (m) J 
%m + 1 
d,r.i 
“im + 2 
and 
(m) (m) ^ (m) V 
(m) J (m) g (m) g 
2m X™. ^ h (2m + 1) -1“ "i" ^™ + 2 3l + • • • 
(^jCn- 
tm 
■'2m +1 
(m) ^ (m) ^ (m) ^ 
= -n-«Y. ^ + (2m + l) Y.,.^ + (2m + 2)Y,.,,^ 
2m — 1 ^ytm ^y%in +1 
(17a) 
(18a) 
In particular for m = 1 we have 
and 
+ 2a?3 — + 
^ d , „ d , ^ d , 
d.C;^ 
fteo 
Again m = 2 gives us that 
^^ 1^2 + 2 (2(riX3 + a;/) + 3 (2a:ia?4 + 2(r5a;3) £- 
+ 4 (2a?iX5 + 2a;2a;4 + a;32) ^ + . . . , .... (21) 
and 
^ ^ + 6 ( 2 aJia :3 A xi) ^ + 7 {2x^x^ + 2 ^ 2 X 3 ) (22) 
are self reciprocal operators of positive and negative characters respectively. 
7. As other examples of the important formulae of transformation (13) and (14), let 
us write down cases corresponding to m = 1, ri <|: 0. 
For m = 1, n = 0 we obtain 
a^i 5“ + ^^2 
dx 
d d 
+ *»* + 
dx, 
d . , d . ^ d , 
2/1X7+“3/2 77+ %3^ + 
ddi 
dy2 
(23) 
x.—-\- 2x, 
dx^ 
MDCCCXC.-A. 
+ 3a;3 
^ dvo dx.y 
•f . . • = 
r d , d , <i 1 
“ “ A </7. + rfS-S ‘ ■••f 
E 
(24) 
