PROFESSOR CAYLEY’S EIGHTH MEMOIR ON QHANTICS. 
551 
Annex. — Analytical Theorem in relation to a Binary Quantic of any Order. 
The foregoing theory of the superimaginary transformation led me to a somewhat 
remarkable theorem. Take for example the function 
(a, 5, cfx+k, 1 —kz) 2 , 
or, as this may be written, 
k 2 Jc 1 
# 2 
0, 
25, 
a or ( 
; c. 
25, 
a 
X 
25, 
2a— 2c, 
-25 
25, 
2a-2c, 
-25 
1 
a. 
-25, 
c, 
a, 
-25, 
c 
Jk, l)\x , l) 2 , 
then the determinant 
c , 
25, 
a 
25, 
2a— 2c, 
-25 
a, 
-25, 
c 
is a product of linear functions of the coefficients (a, b, c ) ; its value in fact is 
= - 2(a + c)(a+26i + ci*)(a-2bi+ci 2 ), = - 2 (a+ c)[(a - c) 2 + 45 2 ]. 
To prove this directly, I write 
a'=a—2bi-\-ci 2 , 
b'=a — ■ ci 2 , 
c'=a+25«+^ 2 , 
and we then have 
c; 
25, 
a 
1, 2 , 1 
25, 
2a— 2c, 
-25 
i , 0 , — « 
a, 
-25, 
c 
« 2 , — 2i 2 , i 2 
(1, *, * 2 ) 
, (2, 0, - 
-2 *■), (1, * 2 ) 
=( c, 25, a ) 
(25, 2a-2c, -25) 
( a, -25, c) 
i 2 a', 
- 2i 2 5', 
tV 
=a'5'c' 
i 2 , — 2i 2 , i 2 
2ial, 
05',’ 
-2tV 
2t, 0 , —2?; 
a', 
25', 
c' 
1, 2 , 1 
whence observing that the determinants 
1 , 
o 
1 
5 
A2 
v •) 
— 2i 2 , 
. 
*, 
0 , 
—i 
2 1, 
0 , 
-2* 
fc 2 , 
-2i 2 , 
i 2 
1, 
0 
^ 5 
1 
