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 ).