46 
MR. E. B. ELLIOTT ON THE INTERCHANGE OF THE 
or in present notation (—1, 1, 1; 1, 0, 1} and { — 1, 1, 1; 1, 1, 0]. For the 
transformation of these we have, by putting 1 for each of m and n' in (106), 
coi(a:, yz) = zx) = 2, 1, 1 ; 2, 0, 0},, .... (109) 
of which right hand operator the expansion is 
,(2) d 
'*30 (f- 
+ z: 
'30 
( 2 ) 
21 dZoi 
^ 03 ^ 
,( 2 )_ 
"'40 dz 
-t 
'40 
+ 3 Z' 
^( 2 ) d 
^50 dz,, 
+ 
+ 
where the coefficients have meanings, as in Art. 21. 
Closely resembling, but distinct from oj^ and Wg, are Mr. Forsytpi’s and i.e., 
A 2 {x, yz) = S 
»■ + s <1; 1 
Ai {x, yz) = t 
r + 5 1 
= [ 0 , 1 , 1 ; 1 , 0 , 1 }., 
= { 0 , 1 , 1 ; 1 , 1 , 0 ],. 
These are also transformed by means of the present article, but have not the property 
of companionship belonging to and In fact, by (105), 
Aja;, 2/^)= [-1, 1, 0; 1, 1, 0},= {-2, 0, 1 ; 2, 0, 0],, . . (110) 
and, by (105), with z, x, y put for x, y, z, 
Ai(x,2/z)= (-2, 1, 0; 2, 0, 0},= {- 1, 0, 1 ; 1, 0, 1}.. . . (Ill) 
23. The special importance of many operators in which the first derivatives do not 
occur is well known. The form of such operators (in z dependent) is symbolically 
(a + + cy {C — z^q^ - z^^y)^. 
As every such operator is a sum of multiples of complete operators {/a, v, v ; m, n, n']: 
so that them theory is implicitly involved in that above discussed, no attempt will be 
made here to develope it independently. In the present article, however, an interesting 
class of cyclically persistent operators will be obtained, and a method of procedure in 
a much wider class of cases will be thus exemplified. 
It is required to prove that the result of replacing each first derivative by zero in 
{— w, 1, 1 ; m, 0, 0} 
* See Lis Memoii’ “ A Class of Functional Invariants,” ‘ Phil. Trans.,’ A., vol. 180 (1889), pp. 71-118, 
