98 
Transactions of the Royal Society of 8outh Africa. 
have the peculiarity that the combinatory numbers involved in them are all 
of the type 
(2r+l),: 
they thus have a link of connection with the recurrents studied in a previous 
paper.* 
(8) In seeking to verify any one of the three for a determinant of definite 
order the fact that the (1, 1)^^^ element is 1 naturally suggests condensation 
into a determinant of the next lower order ; for example, 
1 3 10 35 126 
3 10 35 126 462 
10 35 126 462 1716 
35 126 462 1716 6435 
120 462 1716 6435 24310 
, 1 7 36 
=r j 7 50 261 
! 36 261 1378 
1 5 21 84 
5 26 112 456 
21 112 491 2025 
84 456 2025 8434 
1 9 
9 82 
= 1, 
where, be it noted, the one process suffices to show that the determinants 
of orders 5, 4, 3 are equal to 
1 9 
9 82 
1 7 
7 50 
1 5 
5 26 
respectively, and consequently are all equal to 1. 
(9) The most instructive general proof consists in multiplication by 
unity in the form— 
1 
-3 
1 
o 
-5 
1 
-7 
14 
-7 
1 
9 
-30 
27 
-9 
the row of the multiplier being — 
This leads to the product — 
3i 
1 
1 
7. 
7i 
1 
9* 
% 
9i 1 . . . . 
* ' Trans. E. Soc. S. Africa/ viii, pp. 27-32. 
