VARIABLES IN CERTAIN LINEAR DIFFERENTIAL OPERATORS. 
31- 
In like manner 
1 I dP 
standing for their expansions in terms of y], are the symbolical forms of 
{1, 0 ; m, — m]x and {0, 1 ; m, — m}^. 
As in Art. (6) the result of transforming {1, 0 ; to, — m]j; to its form in 
dent is, then, symbolically. 
i.e., 
i.e., 
_ i tZni _ /y. m ^m\ 
m ^ ^ 
1 o dr] , 1 dr] 
m 
d^ m d^ 
m 
-rjh 
- -J "'+ - 2/r'M:;i - ni ]; 
d^ 
I ^ (dr] 
m 
whence, by aid of (40), 
TO {1, 0 ; TO, — to}^ = — {0, 1 ; 1 — TO, TO — 1 [0, 1; 1, — i 
Once more the result of transforming {0, 1 ; to., — is, in like manner. 
i.e.. 
i.e. 
5 
d^ 
dr] 
'Im — 1 rq—m + \ 
+ Vi— 
dr] 
w 
t-H-n 
— M+\ 
Ui 
“W + 1 d—/fl + 1 
/ dr] 
— )}) I _^ 
d^ 
SO that, by (39) and (40), 
{0, 1 ; TO, — w]^ = — (I — to) {1, 0 ; I — to, to — 1 {0 1 ; 1, — 
From (43) and (44) follows the more general identity, 
TO {y, z/; TO, — m]^— — {vm (1 — m), p, ; 1 — m, m — 1 \y 
+ (/. + TOi.)/yi-“{0, 1 ; I, - I] 
(41, 42) 
y depen* 
ir («) 
]r (44) 
