552 
PROFESSOR CAYLEY’S EIGHTH MEMOIR ON QUANTICS. 
are as 1 : —2, we have the required relation, 
c, 2 6, a 
2 6, 2a-2c, -2b 
a , —2b, c 
= -2a!b'c'=-2(a+c){(a-cy+ib 2 }. 
It is to be remarked that the determinant 
1, 2 , 
1 
, taken as the multiplier of 
c, 
2b, 
a 
*, o , 
—i 
26, 
2a— 2c, 
-2b 
i\ -2i\ 
i 2 
a. 
-2b, 
c 
is obtained by writing therein a=b=c, =1; and multiplying the successive lines 
thereof by 1, \i, i 2 ( 1, 1 are the reciprocals of the binomial coefficients 1, 2, 1), the 
proof is the same, and the multiplier is obtained in the like manner for a function of 
any order ; thus for the cubic 
(a, b, 
c, d~Jk-\-x, 1 
—kxf, 
k 3 
¥ 
k 
1 
■■X s 
- d, 
Sc, 
— 36, 
a 
X 2 
Sc, 
— 66 +3^, 
Sa— Sc, 
36,' 
X 
-36, 
3 a— Sc, 
66 -3d, 
3c 
1 
a, 
36, 
3c, 
d 
the multiplier is obtained from the determinant by writing therein a = b=c=d= 1, and 
multiplying the successive lines by 1, ^i, i \i 2 , i 3 , viz. the multiplier is 
-1 3-3 1 
i —i — i i 
—i 2 —i 2 , i 2 i 2 
i 3 3 i 3 , Si 3 , i 3 
and the value of the determinant is found to be 
9 (a— Sbi + Sci 2 — di 3 )(a—bi — ci 2 + di 3 )(a-\- bi —ci 2 — di 3 ){a + 3 bi 4- 3 ci 2 -f di 3 ), 
= 9((a-Scy+(3b-dy)((a+cy+(b+‘iy)- 
But the theory may be presented under a better form ; take for instance the cubic, 
x k 
viz. writing - and -j in place of x and Jc respectively, we then have 
(a, b, c, dJky-\-lx, ly ■ 
—kxf 
k 3 
k 2 l 
kl 2 
r 
= X 3 
- d, 
Sc, 
-36, 
a 
x 2 y 
Sc, 
• — 66 *-| -Sd, 
Sa— 6c, 
36 
xy 2 
-36, 
Sa— 6c, 
66 -3d, 
3c 
f 
a, 
36, 
3c, 
d 
