24 
MR. E. B. ELLIOTT OR THE IRTERCHANGE OF THE 
considered as the symbolical form of an operator in x dependent, is the expansion in 
terms of ^ of 
considered as an operator in y dependent. 
In other words, by (11) and (9), 
ra{1, 0 ; m, n[ 0, 1 ; 71 + 1, .(13) 
Again, the transformed form of the symbolical expansion in powers of y of 
' ^ (h^ 
is, by Art. 4, tlie symbolical expansion in powers of ^ of 
i.e., of 
^ ' d7) ■ 
ini — m?; + 1 
b 7 > 
since in d^jdy and dyjd^ the derivatives x^, x^, . . and y^_, . , are not regarded as 
variables. In other words, by (12) and (8), 
{0, 1 ; 771, 7i}^ = — (w + 1) {1, 0 ; n + 1, 7)1 — Ijj, .... (14) 
It is to be remarked that (13) and (14) are entirely in accord. Either of them is 
produced from the other by the interchange of x and y and of m and 77 + 1. 
From (13) and (14) by aid of (10) we produce the more general equality of 
operators 
{p., c; 771, 77 }^ = — Jc(77 + 1), ^ ; 77 + 1 , 777. — 1 I .... (15) 
L \ y 
which may be given the more symmetrical form 
{my, y ; 771 , in' — l}x -- — {m y, y ; in, in — 1}^ , 
in which in + m has to be positive. 
In (16) are included two interesting classes of particular ca,ses, viz.; 
^ — 771 , 1 ; 7 ) 7 , in — 1 jj- = {~ 7 ) 7 , 1 ; in, in — 1 , 
ajid 
{ 7 ) 1 , 1 ; 7 ) 1 , 7)1 — 1}^ = — {) 71 , 1 ; 7 ) 7 , 771 — 1 
( 10 ) 
(10 
(18) 
Corresponding to each positive degree in tliere are then two self-reciprocal 
operators.'" The first is of positive character, being entirely unaltered in form by 
* Self reciprocal operator.^, of coarse, generate from absolute reeipirocaiits otiier absolute reciprocauts. 
