VARIABLES IN CERTAIN LINEAR DIFFERENTIAL OPERATORS. 
29 
that (14) may be regarded as holding for the value m = 0 as well as for non-vanishing 
values of m. 
It affords an instructive verification to conduct the investigation of the same trans¬ 
formation in the order of Article 6. 
9. For the case n = 1 the two formulae of transformation, 
0 {1, 0; 0, n}^= — {0, 1 ; 1, — 1}^, 
[0, 1; Q, n}x— — (u+l) [1,0; n 1, — 1}^, 
of the last article become respectively 
and 
A __ 1 J 9 Y ®) 4^ _ 4 _ qv (2) dL 4 _ lY (2) i 
dx,- 2 dy,^"\ 
= - i +?//) + 
d d, 2x-^x^ — x,^ d ^ ?>x-^x^ — Sx^x^x.^ x^^ d 
Ll/'t/t/g tv 2 
dXr, 
dx^ 
= - A 
dy- 
■ (34) 
By combination of these we have the equivalence, free from cljclx^ and djdy-^, 
x^d 2x1^3 — x^ d “^yx^x^ — A- X:^ d 
Xj dx^ x-^ dx^ x^ dx^ 
d 
f 
= yyy^ ¥ ^ + tj ¥ + ^ ¥■'■•■■ 
/ d d d \ , d „ d 
= y^ b^¥ + ^^■‘¥ + ®^*¥ +■••) + ^ V ¥ + ¥ 
+ 4(„, + f)A +. 
(36) 
In like manner, for any positive integral value of 'll, 
[0, 1 ; 0, 71 }X — 0(1, 0 ; 0, 7^}x — “ (vi + 1) [1, 0; ^^ + 1, — 1}^ 
-= [0, 1; 1, - 1}, . . ..(37) 
is an equivalence of operators which do not involve any lower symbols of operation 
than djdxn+i and dld'y^ + i respectively. 
10. From (34) is easily derived in exact form the known fact that the anaihUator T 
