VARIABLES IN CERTAIN LINEAR DIFFERENTIAL OPERATORS. 
33 
Now, proceeding exactly as in Art. 8, it is seen that the symbolical form of an 
X operator equal to this is t) dldrj log and that its expanded form is obtained by 
omitting the first term, and then putting n = 0 in the general value (33). Thus, the 
operator which has to be defined as [0, 1 ; 0, 0}^ is 
d Gr”^2 
d 
xd dxo 2 ! d®. 
-j- . . ., 
(50) 
where G is the generator defined in (32a). 
13, Examples of important operators which occur among those transformed in the last 
two articles are before us in (48) and in the ec[uality of (49) and (50), It is unnecessary 
to multiply particular instances as they can be deduced without number by giving 
m particular integral values in (43) , , , (46), It is to be noticed that, excluding the 
special cases of wi = 0 and m = I, one or other of the two equivalent operators will 
involve as coefficients those in a multinomial expansion of negative index. Thus, 
for instance, 
2{1, 0; 2, - 2}, = - (0, 1; - 1, 1),+ 2/r'{0, 1 ; E - 1},, 
a particular case of (43), is at more length 
d d d 
dx\ + ^ -+• Sa^s^Tg) • 
_ _ ! yj - d _ 2 - ‘^vdj^h + Viyu) , 
y\ " y^^ dy,^ 
y\ 
d 
d 
d 
+ ,7^ 2^3 — + 32/3— + 42/4vy + • • • - 
dyi 
dyg 
(51) 
The transformation of the operator G of (32a) is an application of (49), Thus 
d 
d 
d 
y> 
_ -ijy2 d d G2//2 d 
— iJi i r A, "T 777 XT + + • • • 
Vi dyi yd d //2 2 ! yd dyg 
— y \ ®2/2 i yiir + 23/2 7 , + ^2/3 X- + . . . 
d 
d 
d 
dyi 
dys 
_ 2yiy3 - m d_ 3yd"y^ - Qy^y^y^ + yj 
3 1- i- • • • 
yi 
Vi 
dys 
dys 
MDCCCXC,—A. 
dx.i 
G {X, y) = X, y- + 2*3 - + 3*3- + , , .j - *3 p. - + *3- + *3 - 
= 1 , 1 ; 1 , - 0 ; 1 , 0 }, 
= 2/rHo, 1 ; 0, 0}^ - yC^iJz{0, 1 ; 1, 0} 
by ( 49 ) and (23), 
(52) 
