34 
MR. E. B. ELLIOTT ON THE INTERCHANGE OE THE 
14. Operators for which m + n is negative still remain to be considered. In 
particular, those of the type {0, 1 ; 0, — n} have still to be defined. About the 
right definition of them there can, however, after articles 8 and 12, be no doubt. 
The general principle by means of which if {/x, v; m, n]y is known, the form of 
{/X, v; m, n — r}y is deduced is expressed by the rule — “Write [/x, v, m, n}y 
symbolically, by putting for each d/dyp, divide through by reject all terms, if 
any now occur, which do not contain a positive power of f as factor, and then for 
each write dldi/p.” 
Thus to accord with (33) and (50) the right operator to be defined as 
:0, 1 ; 0, — n'} 
where — n is a negative integer, is 
d 
+ 
+ 
G'‘'+V2 
d 
dy^ (7h + 1)! dy„ (A + 2) ! dy. 
+ 
(53) 
We now proceed to the transformation of {/x, v ■, m, —m — where r is a 
positive integer. 
The symbolical form of m{l, 0 ; m, — m — is as in Arts. 6 and 11 
•“ Tfh - 't' . 7 
7] \C — JL 7) — jL 7} 
' m ' m + l ' 
. . — A 71 j. 
m + r ' ^ 
(54) 
The symbolical form of its transformation is, therefore. 
^ni — III — r dy — r drj 
(y rfi + A’? 
+ X‘"’ 
' m+l ' 
■ + 1 dy 
cij 
+ 
m+ rcl^ 
(55) 
This when expanded in terms of ^ can involve no zero or negative powers. For it 
is a sum of multiples of 7j dirjjd^, 7f d7]ld^, . . . only, since (1, 0 ; w, — m — r}a; is a 
sum of multiples of t), t^, . . . only, and these when expressed in terms of ^ are 
all free from . . . Thus the coefficients of ... 1, which 
would appear to occur in the above symbolical form of the transformation of 
(1, 0 ; m, — m — r] are in reality absent; and, consequently. 
m{l, 0; m, — m — r],, = — {0, 1; 1 — in — r, 
+ X'"^ {0,1; 2-7’,-1] +...+X'”‘^ 
‘ )h+G ’ ’ ’ ' ' m+r-l 
m-l] +X^™'{0, 1; ] - r, - 1] 
y la ^ ^ y 
(0,1;0,-1} +X'"'JO.l;!,-!} , (56) 
the various terms on the right consisting of the parts with positive indices of ^ from 
the corresponding terms of (55). 
