VARIABLES IN CERTAIN LINEAR DIFFERENTIAL OPERATORS. 
The only lineo-linear operators, of the classes with which we are dealing, both of 
whose cyclical transformations are also hneo-linear, are found by putting m, n -h 1 
and n' -j- 1 all equal to unity in the results of the last article. Thus 
[1, 0, 0 ; 1, 0, 0]. = - (0, 0, 1 ; 1, 0, 0], = - [0, 1, 0 ; 1, 0, 0},, (91) 
with the correlative equalities obtained by writing y, z, x and 2 , x, y respectively for 
X, y, z, involve the aggregate of all such operators. At length the equalities (91) are 
A 
^^0 dx 
X-ic\ + ^01 ^ “T •^20 y. 
10 
d d 
^-20 ,7.. ' “ 1 ~ ^ 
01 
d d 
'20 
11 
7 I ^10 7 "h 7 
dXQ„ diCgy dllgi 
d , d 
+ ^12 + • 
12 
dx, 
03 
d 
+ . . . 
03 
= — ^ 2: 
^10 7 ^ “h 1-h ^11 J “h SZon - -(- 2^.27 -p .i/io , 
+ 
12 
+ ■•■ . (92) 
We thus learn that, if a function of the derivatives of x with regard to y and 2 is 
homogeneous, the equivalent function of the derivatives of y with regard to 2 and x is 
isobaric in second suffixes, while the equivalent function of the derivatives of 2 with 
regard to x and y is isobaric in first suffixes ; and that 
i {x, yz) = — {y, zx) — — ( 2 , xy), . (93) 
where the notation explains itself. The correlative facts are 
and 
— iv^ {x, yz) = i {y, zx) = — {z, xy), . (94) 
— iv.2 {x, yz) = —w^ {y, zx) = i ( 2 , xij) . (95) 
The same aggregate as is involved in (91) and its correlatives is also expressed by 
the facts that 
{ - 1 , 1 , 1 ; 1 , 0 , 0 }, 
{ — 1, CO, 0)^; 1, 0, 0}, 
{ — 1, co^ CO; 1, 0, Oj, 
(96) 
(97) 
(98) 
obtained by giving m the value 1 in (88) to (90), are cyclically persistent lineo-linear 
operators of characters 1, co, co^ respectively. 
G 2 
