VARIABLES IN' CERTAIN LINEAR DIFFERENTIAL OPERATORS. 45 
The two imaginary cyclically persistent quadro-linear operators (102) and (103) are 
easily written out in like manner. They commence with terms in djdx^,^, 
d/dx-y^,, which it is to be observed are wanting from the above. 
Once more by giving to each of m, w + l,n 1 the value 3 in Art. 19, an aggregate 
is obtained of linear operators with coefficients of the third degree, whose transforms 
have both of them coefficients of the third degree also. The aggregate may, as before, 
be considered involved in three cyclically persistent operators of the type, one of each 
character. Similarly as to operators with coefficients of any higher degree. 
22. Some of the most important linear operators which have been used in recent 
theories of functional invariants, cyclicants, &c., have the property of persistence 
of degree in the derivatives after one cyclical transformation, but not after a second. 
Such operators occur among those obtained by putting t-z. + 1 = m in (87), viz., 
{jx, V, V ; m, m — 1, = \ ~ pm, v, — —m, n, m — 1 ^ 
L J y 
= I — V (A + 1), — p; n'-{■ 1, TO — 1, m — l| . . . (105) 
In particular, there are three classes of operators which have a property closely akin 
to that of persisting in form after a first cyclical transformation, being, in fact, only 
altered by the interchange of first and second suffixes ; they are 
{— TO, 1, 1 ; TO, TO. — 1, n}a; = { — m, 1, 1 ; to, n, to — 1}^, 
= {-(w'+1), 1, 1 ; 7^'+l,m - 1, TO - 1], .... (106) 
{— TO, w, w®; TO, TO — 1, n}x = 0 ) { — m, (o, ; to, n, m l},j 
= 0 )^ { — [n + 1), oi, cxx^; n + 1, m — 1 , m — 1}~ . . . (107) 
[— m, (ji^, co; m, m — I, n}x = { — to, oj®, oj ; to, n, to — 1 }^ 
= w (— (A+ 1), <y^, (y ; 1, TO — 1, TO — l}s . . . (108) 
It is to be noticed, in the case of the first of these, that the second cyclical 
transform.ation, which is of different degree from the first, is quite symmetrical in first 
and second suffixes. 
Among the operators comprised in (106) occur the two, which I have called coy and 
Wg,* two of the six form annihilators of projective cyclicants, viz., 
(Oy{x,yz)= t j(r + 5 
?- + s 1 L 
co^ix, yz)= t J (n + s 
5' + S<l L 
— 1) a?: 
5 - 1-1 
* ‘ London Math. Soc. Proc.,’ vol. 20, pp. 131-160. 
