44 
MR. E. B. ELLIOTT ON THE INTERCHANGE OE THE 
If the operation be on a homogeneous and doubly isobaric function we are thus 
told that 
i Wi-\- 
i + oiiv^ + 
2j 
— i coiv^ . 
(99) 
are characteristics which persist after one cyclical transformation but for the 
multipliers 1, oj, respectively, and after a second but for 1, (o. 
21. The quadro-linear operators (linear operators with coefficients quadratic in the 
derivatives) both whose cyclic transformations are also quadro-linear, are obtained by 
giving to every one of m, n l,n -{■ 1 the value 2 in the formulse of Art. 19. Their 
agofregate is involved in 
oo o 
2 [ 1 , 0 , 0 ; 2 , 1 , 1 }.. = - [ 0 , 0 , 1 ; 2 , 1 , 1 ], = - { 0 , 1 , 0 ; 2 , 1 , 1 }„ . ( 100 ) 
and its two correlatives in y, z, x and z, x, y. 
I'he same system is expressed by the three cyclically persistent quadro-linear 
operators 
{ — 2, 1, 1 ; 2, 1, 1], of character 1,.(lOl) 
{- 2, a,,co3; 2, 1, 1], 
{-2, c.; 2, 1, 1], 
oi. 
<y®. 
. ^ (102) 
. . (103) 
Of these the first expanded to a few terms is 
where 
-y(2) A j A -y-(2) A A /„(2) A 
Ax^„^ Ax^^ + Ao3 + 2 (^^40 + 
+ 
(104) 
^90 - 
“-20 
-A-qo — 
^ 2 Y^^^ — 9^ -r ^ 2 
^10 5 -^n — -^02 — '^01 5 
'-30 
-A.in — 
(•2) (2) -r (2) 
2i^iQi^oQj X 21 2XiqXi-i "4" 2xQ-iXor\^ X 22 ^*^10^02 ^03 ■“ 2xQ■^x^ 
v(2) _ 
Aso — 
(*^) 
^iX^yc^Q -j- 3 ^ 20 ^ X31 = 2cCj^qX2i “b 'ix^yX-^:^ ‘lX(^fy]CYi, X2- — -b “b 
( 2 ) (2 
"b 2 ir 2 QCCQ 2 , Xi 3 = 2 x^qXq^ -b ‘ 2 iXQyic-^<^ -b Xof = 2 xq^Xq^ -b ^^02^5 
Q /yj /yt I V '■y* 
01"^02> 
and generally 
Y^^^ _ 
^mn 
'4-s}>m-l-«—1 
r -1- 5 1 
(Xj-gXpi _ ^) • 
