120 
[186 
186. 
ON THE DETERMINATION OF THE VALUE OF A CERTAIN 
DETERMINANT. 
[From the Quarterly Mathematical Journal, vol. n. (1858), pp. 163—166.] 
Consideeing the determinant 
0,1, 
n, 9, 2 
. n-1, e, 3 
n - 2, 9, 4 
let the successive diagonal minors be U 0y U 1 , U 2 ,...U X it is easy to find 
U 0 = l, 
Vi = 6, 
U 2 = (0*-l)-(n-l), 
U 3 = 6 {6* - 4) - 3 (n - 2) 0, 
u4 = (0 2 - 1) (0 2 -9) - 6 (re - 3) (0 a - 1) + 3 (n - 3) (n - 1), 
which in fact suggests the law, viz. 
U x = (9+ x — 1){9 + x — 3) {9 + x — o)...(9 — x + 5)(9 — x + 3)(9 — x + 1) 
— - ^2 ( n ~ x +1)(0 + x — 3){9 + x — 5)...(0 — x + 5)(0 — x + 3) 
x(x — l)6r — 2) (x — 3) . , w .. 
+ — ^“4 (w — x + 1) (n — x + 3) {0 + x — 5).. .{0 — x + 5) 
— &c. 
, , Se x(x — l)...(x — 2s+1). nw , 
+ (—) s —- 24~2s ~ {n-x+l)(n — x + 3)...(n — x-\-2s — l) 
(9 + x — 2s — 1)(# + x — 2s — 3)...(0 — x + 2s + 1) 
to s — ^x or %(x—l), as x is even or odd.