20 
MR. E. B. ELLIOTT OR" THE INTERCHANGE OF THE 
The only previous contribution, of which I am aware, to the subject of the reversion 
of MacMahon operators, is a paper by Professor L. J. Rogers,"^ in which he obtains 
the operator reciprocal to {p,, v •, 1,1}, and alludes to the self reciprocal property of V 
which is expressed with more precision in (38) below. 
I. Binary Operators. 
• 1 cJ/^ If 
2. Let X and y be two variables connected by any relation. Let y-, denote —, and 
. , 1 d’-x 
jy ■ cIgiiolg ~ • 
rl dif 
Let ^ and r\ be corresponding increments of x and y, so that by Taylor’s theorem 
’I = 2/1^+ +2/3^^ +.(1) 
and 
^=X^y + x.pf + + . . . ,.(2) 
the one expansion being a reversion of the other. 
Let denote the coefficient of ^ in the expansion of i.e., of ^ + 
fi- . . .)® in ascending integral powders of and X/®* the coefficient of in the 
expansion of i.e., of + xpf + . . .)® in ascending integral powers of y. 
It is supposed that m is not fiuctional. It need not, however, be positive. Nor is it 
necessary to exclude the value zero, which, though somewhat special, will be seen to 
be of importance later. 
Let 71 be a positive or negative integer or zero, and let p and v be any numerical 
quantities. 
Denote 
..... (3) 
and 
m ^ by [p, z/; w, ? 2 }^ ..(4) 
the summations being, with regard to s, which assumes in turn all integral values not 
less than the greater of the two w and — n 1, so that, if m + n > 1, only symbols 
of differentiation with regard to all derivatives from ym + n onwards may occur, while, if 
m + ^ 1, symbols of differentiation with regard to all deiivatives may be present. 
It is the operators (p, r; ???, ?i}^ and [p, v •, m, n}^ of which I propose to speak as 
MacMahon operators in x and y, respectively, dependent. It will be seen upon reference 
to the first paper referred to above that they are the results of substitution in Major 
MacMaiion’s operator (p, v ; on, n) for 
* “Note on Conjugate Anniliilators of Homogeneous and Isobario Diiferential Equations,’’* Messenger 
of Matbematics,’ vol. 18, pp. 153-158. 
