228 Transactions of the Boyal Society of Soiith Africa. 
Since we know 
n^-\-n^_^ = {n+l)^, 
ni + 27Zi_, ^ (n + 2)^, 
and in general 
we can, by grouping rows, express in the form 
{n-^v-1)^, (n + v-l)i„ 
Which 
_ {n-{-v-l)^.{n-\-v-l), 
"(n-^v- iy-\ {n + v - 2y- 
{n- a + l)(n-a + 2)...{n + v -1-a), do. in 6, ... 
{n- a + 2)... {n + v -1 -a), 
This determinant is (i) symmetrical in {n — a), {n — b), ... ; (ii) of order 
(v — — 2) + ...+0. By (i) it is divisible by the product of differences 
(b — a). {c — a)(c — b) This product is of order l + 2 + ... + ('u — 1), there- 
fore any other factor is independent of (n — a), (71 — b), ... and can be 
obtained by taking special values, n-a = 0, n-b=-l, 7^-c=-2, ... 
n- last = -v + 1. All the elements above the leading diagonal become 0, 
and the diagonal is /v — 1 /v — 2 ... /I. 
The product of differences becomes /v — 1 /v — 2 ... 1, therefore the 
numerical factor is 1, and we obtain 
(n + v - l)a{f^ + V - 1)6 
'{n + v - lY~\n + v - 2y- 
prod. of diff. 
Now, the original D,.= 
can be reduced to the more general form by bringing the last row to the 
top, the last row but one to the third, and the last but two fifth, etc., and 
putting a for 1, b for 3, c for 5, ... 
