VARIABLES IN CERTAIN LINEAR DIEEERENTIAL OPERATORS. 
23 
5. Now the symbolical form of any MacMahon operator for which m -\- n 1 is 
very simple. By (4) that of, 
{ix,v, m, n]y 
is 
i.e. 
i.e. 
£ t + 2/2^" + i/3^® + • • •)" •+ £ ^ (^1^ + 2/2^^ + 2/3^® + • • •)" 
^Uy^m _p + l y^M-l .(7) 
Thus in particular 
{1, 0 ; in, n,] y = - t r] 
m 
and 
;0, 1 ; m, n}y = + ^ , 
( 8 ) 
( 9 ) 
the right hand members being supposed to be expanded in terms of ^ by aid of (1), 
and then to have each power of ^ which occurs replaced by the corresponding d/di/s. 
We may of dourse write (7) 
{fx, p; in, n]y = ix{l, 0 ; in, n}y + v{Q, 1 ; in, n]y, .... (10) 
so that in (8) and (9) we have involved all MacMahon operators in y for which 
m + n is not less than unity. A reservation must for the moment be made of the 
case m = 0. 
Exactly corresponding to (8) and (9) we have the symbolical forms 
[\,Q-, = , .. (11) 
{0, 1-, in, = + ■ ..( 12 ) 
where the expansions on the right are to be in ascending powers of y by (2), and 
where in an expansion each is to be replaced by the corresponding dldx^. 
6. The transformation of {y, v; in, n}x for the cases at present under consideration 
of m + n not less than unity is now immediate. By Art. 4 the transformed form of 
the expansion in terms of y of 
