VARIABLES IN CERTAIN LINEAR DIFFERENTIAL OPERATORS. 37 
= t ( 66 ) 
r + s ' 
and in like manner write 
= 2 Y"?!-, (67) 
r + s^m 
and 
= X .(68) 
r+sJS^ m '■* 
We may include, if we please, the value zero of m ; but the expansions of r}°, 
consist only of the single terms r)° '’?"• 
The operators to be considered and transformed are the following :— 
in{ix, V, v ; m, n, n]^ — X vr v s) X ^ -s . . . (69) 
m[ju,, V, v'; m, n, n}y = X{(x vr + vs) ^-j . . . (70) 
(^yn + r^n' + s 
V, v; m, n, n}~ = S (/r + *^’ + -— -; . . . (71) 
'’5 ClZji ^ 5 
where p-, v, v are any numerical quantities, 
m a positive integer, 
n, n' positive integers or one or both zero, and 
r, s quantities which take in succession all zero and positive integral values 
subject to r + s <j: m. 
Cases of m negative, and of n, n either or both less than — 1, which have been 
dealt with in the analogous theory of binary operators, will not be here considered. 
The cases of m zero, and of n or n equal to — 1, will not be entirely excluded, but 
will be only dealt with as far as their accordance with the results for m positive and 
n, n' not negative needs no elaboration to make it clear. 
Thus our field of investigation is narrower than in that of the analogous theory 
hitherto considered. Were negative values of n and n admitted, the lower limit of r 
in the operators (69) . . . (71) would be — ?^ + 1 instead of zero, and that of s would be 
m like manner — n + 1. Thus when we admit the value — 1 of n we must exclude 
the value 0 of r, and when the value — 1 of n' we must exclude the value 0 of s. 
Let us now express (69), (70), (71) symbolically as follows :— 
m [p, V, V ; m, n, =■ X {ix vr vs) X'™^ yj + r _ (j^) 
m {p, V, V ; m, n, n'}y = S (p + vr + vs) 
m [p, V, V \ m, n, n]; = ^(p -b d* v‘s)'Z!'"'^ ^n + r y^n>+3. ^ 
