48 
MR. E. B. ELLIOTT ON THE INTERCHANGE OP THE 
A first case of this formula occurs in (96), A second is that the operator on the 
right of (109) is the sum of terms involving 2:^05 ^on of an operator free from those 
derivatives which, when affected with the factor (l/^^io^oi)^; has the persistent property. 
The fact is interesting in its bearing on the theory of the transformation of and o)^. 
In fact (109) may be written 
wi (x, yz) — W 3 {y, zx) = {z, xy) + Zoi^i {^, -^ 2 /) + [— 2 , 1 , 1 ; 2 , 0 , 0 ],. 
From this and its two correlatives a verification of the formula (112) for the case of 
m = 2 is readily obtained. The first few terms of this cyclically persistent operator 
are 
OCcyrT ^ d . _ / • ^ 
~r)T 
+ Sx^qXqq “h 3 {x^qX^i + ^ (^ 20^12 “1“ d" ^ 02 ^ 30 ) 
60 
clx. 
'41 
dx. 
'33 
(L ct d/ 
+ 3 {x.2oXq^ + x^-^x^c^ + Xq^x^^ + 3(a:]ja:o3 + «^03^i2) + 3 0:02^03 
23 
dy. 
14 
dx. 
05 
+ 
■ (IIS) 
With it compare the analogous operator (30) of the binary theory. That, like this, 
is, of course, one of a whole class of persistent operators. 
24. The transformation of the four form annihilators O^, O3, V^, V3 of pure cyclicants* 
will be now considered. They are all operators free from first derivatives, and might 
as such be treated by the methods of the last article. It seems preferable, however, 
to take them in connection with the formulee of transformation 
- (0, 0, 1 ; 2,0, - 1],= - {0, 1, 0; 1, - 1, 1}„ . . (114) 
& = - (0, 0, 1 ; 1,1,- 1}, = - {0, 1, 0 ; 2, - 1, 0}„ . . (114a) 
which are particular cases of (82) and (83), or again of (87) with ni = 0. 
The forms of the four annihilators are 
Hi {z, xy) = S j -1 
m+n<f,2 L n + i] 
dr 
= {0, 1,0; 1,-I, A’. 
Cl~01 
* See my papers in the ‘ London Math. Soc. Proc.,’ vol. 18, pp. 142, &c.; vol. 19, pp. 6, &c., and 
pp. 337, &c.; vol. 20, pp. 131, &c. 
