188 
AN EIGHTH MEMOIR ON QU AN TICS. 
[405 
whence observing that the determinants 
12 1 
J. , j -L 
y 
i 2 , — 2 i 2 , i 2 
i, 0 , — i 
2 i, 0 , —2 i 
i 2 , — 2 i 2 , i 2 
1 , 2 , 1 
are as 1 : — 2, we have the required relation, 
c, 
26, 
26, 
2a — 2c, 
a, 
-26, 
a 
26 
c 
= — 2 a'b'c, 
= — 2 (a + c) {(a — c) 2 + 46 2 }. 
It is to be remarked that the determinant 
1, 
2 , 
1 
, taken as the multiplier of 
c, 
26, 
a 
i, 
0 , 
— i 
26, 
2a — 2c, 
- 26 
i 2 , 
- 2 i 2 , 
i 2 
a, 
-26, 
c 
is obtained by writing therein a = 6 = 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, 6, c, cT^k + x, 1 — kx) 3 , 
k 3 
k 2 
k 
1 
= X 3 
- d, 
3c, 
-36, 
a 
X 2 
3c, 
— 66 + 3d, 
3a — 6c, 
36 
X 
-36, 
3 a — 6c, 
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, ^ii 3 , viz. the multiplier is 
-1, 
3, 
- 3, 
1 
i, 
- i, 
- », 
- 
¿ 2 , 
i' 2 
i 3 , 
3 i 3 , 
3i 3 , 
i 3 
and the value of the determinant is found to be 
9 (a — 3bi + 3a 2 — di 3 ) (a — bi — ci 2 + di 3 ) (a + bi — ci 2 — di 3 ) (a + 3bi + 3a 2 + di 3 ), 
= 9 ((a - 3c) 2 + (3b - d) 2 ) ((a + c) 2 + {b + d) 2 ).