VARIABLES IN CERTAIN LINEAR DIFFERENTIAL OPERATORS. 
49 
n,^{z,xy)= t . 
ra + n<t2 L + « —iJ 
= {0, 0, 1 ; 1, 1, - 1], - z,,~ 
”"10 
• ( 116 ) 
Y^{z,xy)= t 
^ Ll -|- R — 2 
S iy '^rs^m — r, n — ^ i 
m + n<j:4 I j’ + s-<2 k 
= C ^ %'*? ^ ~ ^10 ~ ^Ol''?) 
= {0, I, 0 ; 2, - 1, 0], - ^10 (0. 1. 6 ; 1, 0, 0], - Zqi {0, 1, 0 ; 1, 
{ r d 
- z,, {1, 0, 0 ; 1, 0, 0],- 2^0- - - — 
1, 1}, 
= |0, 1, 0 ; 2, - 1, 0}, 
r/r ""Ul 7, 
”~10 ”~01 
10 il, 1, 0; 1, 0, 0], 
- Zoi^l (2, + 
'10 
'10 
(117) 
r r + 5^5* ™ + n — 2 
'V 2 (2, Xy^^^ S'! S ii — s^ 7 
Hi + n-IlL )- + s<2 (^~m, n-l 
d 
= (0,0,1;2,0,~ l],-Zoi{l,0,l; z,,£l,{z,xy)^ (118) 
Thus (114 a) may be written 
d 
d 
diX/, 
01 
= - Og (//, sx) - yoi , = - {0, I, 0; 2, - 1, 0], 
“■yio 
and the result of one cyclical interchange in (114) may be written 
d d 
(^'j 2/^) ^10 7 - J 
^ ^ ^ rfa^oi d^io 
;o, 0, 1; 2, 0, -1],; 
from which two sets of identities, by aid of the facts that 
^lu • ^’oi • 1 — 
MDCCCXC.—7V. 
“ 1 • Vw'- Hi)! — % • 1 • ^10) 
H 
