VARIABLES IN CERTAIN JRNEAR DIFFERENTIAL OPERATORS. 
27 
%.e. 
or 
cl cl (1 
% : 7 V + 2a;3 A- + -- 
dx. 
clx^ 
dx,- 
= 4 |Ys®t- + 2Y*® A+ 8Y®A + ,.. \ 
I i 
= i { 21/11/3 £- + 2 {2y, y, + y^^) + 3 {2iy y_, + 2//3 y,) + 
+ 4 
yi d _d 
v” 1 —1“ 34/2 2/3 ^ 
2 (hji 
- yl\ A 
« c. ?/) = ?/i« fe *) + 2 V £■; + ^y-i A ■ 
1 ^ 2 /. y* 4 tjrfy^ + 
(29) 
Since acj = 1 we infer from this conclusion and its correlative that 
x. 
■* i, + ** + v) A, + •■•}=»>*" 2/) 
is a self reciprocal operator of negative character. The operator is one of considerable 
interest in connection with the theories of invariants and reciprocants. (See 
MacMahon, ‘London Math. Soc. Proc.,’ vol. 18, p. 75). 
It also follows that the sum of 2xy Cl [x, y) and the operator on the left of (30) is 
a self reciprocal operator of positive character. 
8. To complete our theory of the reversion of MacMahon operators, for which m n 
is not less than unity, we must consider the somewhat special and exactly correlative 
cases m = 0, n 1, and n= — 1, m <]: 2. 
The operator 0 {1, 0 ; 0, is dldx„ in accordance with the general definition of 
m {y, Vm, 74 }^ in (3). Thus the identity (6) may be written, by aid of (9), 
0{1, 0; 0, p},,= - [0, 1; r+ 1,-1},,.(31) 
for any positive integral value, not excluding unity, of r : which is strictly in agree¬ 
ment with the general formula of transformation (13). 
On tlie other hand the general definition (3) gives to 0[0, 1 ; 0, 11} 3 : no other 
meaning than zero. So far then the operator {0, f ; 0, n]^ is indeterminate in form. 
An interpretation of it is now sought which shall make the case not exceptional to 
the general formula of transformation (14). 
E 2 
