326 
Transactions of the Royal Society of South Africa. 
= {x-4^y + 126) {x + IS - + Qh) +213 - ^y ^- 2h) (x + 8 + y) (x + 4/5) 
and on examination it will be found that the class to which this belongs is 
that dealt with in the first of the papers mentioned above. The width of 
the new extension can thus be appreciated. 
(6) In order to make clear the law of formation of the elements of the 
determinant it is best to view them as diagonally arranged. When the 
order is the n*'' the shortest diagonal is 
l'\c - h + U^Sn - 9)d}, 3.{c - h + U^n - 7)d}, .... 
where the inner co-efficients of d decrerse by 1, and the outside multipliers 
are 1, 3, 6, 10, 15, ... . The next diagonal is 
2(13 - h), 3(/:^ - 26 + I'M), 4^(13 - 36 + S-Sd), .... 
where the co-efficients of Sd are again 1, 3, 6, 10, 15, ... . The main 
diagonal has for the co-efficients of c 
l'(2n - 4), 2'(2n - 5^), 3(2/^ - 7), 4(2ti - 8i), .... 
for the co-efficients of h 
0, -1, -3, -6, -10, -15, .... 
and for the co-efficients of d 
11 (,,2 _ 5^ + 5 + 12)^ 3(7i2_6in + 6^ + 22), 4^^(n^ - 8n + ^ + S^), . . . 
The lower minor diagonal is 
(n- S^y - (n-2)c-i(n-2)(n-S)dy 
(n^2)fy-(n-^d)c-h(n-S)(n-4:)d,\, 
where the inner co-efficients of d are 
0, 0, 1, 3, e, 10, 15, ... . 
in reverse order. 
It is worth noting, too, that the elements of each diagonal form what 
used sometimes to be called an arithmetical progression of the third order. 
Cape Town, S.A., 
December 2Qth, 1916. 
