28 
MF^ E. B. ELLIOTT OX THE IXTERCHAXGE OF THE 
To discover this interpretation let us reverse the order of investigation and seek the 
operator in x, which is equivalent to the operator in y obtained by putting m = 0 in 
the right hand member of (14). 
By ( 8 ) the symbolical form of 
— (n + 1) {1, 0 ; 71 + 1, — 
is 
_ 1 y^n + 1 
The equivalent operator in x dependent has then for its symbolical form, as in Article 4, 
i.e.. 
Now 
^« + i ^ 
-1 
drj 
*._logp 
+ *2'7 + *3’?® + • ■ •) 
d 
- - g ■>! fe didxi + 2 Xi dldx^ + ZXi djdx^ ^ 
drj ^ 
_ g •>) (X 2 djdxi + 2x^ d/dx-i + Sx^ d/dx^ + . . , 
~ 
ly 'V* /y» 2 ^ Y’ 2/y» ___ Q 'V'* y y 1 y 3 
= H- V +--1-+ 
_.^2 G-Tj G^ajj G%2 , , 
— Td ^ ■ 1 (2 Td ’ 1.2.3 + • • • ’ 
(32) 
where G = x^[ % — + 2 a ;3 — + 3x^ +...)— 373 (Xi + ^3 ; 7 y + 
' dr 
' dr. 
= (2a;ia;3 - x /) - x ^ x .^ + (4a:,0^5 - a:oa:4) + • • • , (32 a) 
so that the numerators Ga; 2 , G"a: 2 , G% 2 , . . . , are a set of seminvariant protomorphs in 
a:,, a: 2 , 2 ! a; 3 , 3 ! a:^, . . . 
Consequently the transformation in x dependent of — (n+l) {IjO: n 1, — l},,is 
d . d ^ G«2 d I G%2 
+ 
+ 
d , G^iCo d 
J- 2 — I 
I O t 4. • 
dXfi x, dxji ^.,2 2! a, dx^ 3 3 ! a, dx^i , 
and it is accordingly this operator which has to be defined as 
{0, 1 ; 0, 71}/. 
(33) 
* Cf. Hammond, ‘ London Matt. Soc. Proc.,’ vol. 18, p. 64, note. 
