28 Transactions of the Royal Society of South Africa. 
method, but by a quite different and more rapidly effective procedure. 
Before entering on this, however, we shall, in introducing four fresh results, 
establish them by the old method, in order that a knowledge of both may 
be available. 
(3) Increasing the last row of E,„, namely, 
(2?i — — 3)„_2, , 5.,, 3i, 1, 
by ' 
row„_i'2 + row„_2'2 + row„_3-4 + . . . , 
where the r^^ of the multipliers 
2, 2, 4, 10, 28, .... 
4 
is ~ C2r— 3, /•— 2» we find that the new row is divisible by A^n- 3, and that 
this factor being removed the row becomes 
2 2 2 2 
^(2?'i— 3)„_2, ;,(2?^ — 5)„_3, . . . ., --31, ^"S,,, 1. 
n n — i o 2 
It thus follows that, if we denote the resulting determinant by U„, we have 
IJ. = = (- l)"-i (2/1 + 1) {2n + 3) ... (49^ - 5) ; . . (I) 
'±n — o 
for example, 
12. 
U4 = 
o 
o 1 4 . 
10 3 1 6 
- 911, 
5 2 11 
Note should l^e taken of the relation between the the series of multipliers 
and the row which ultimately comes of using them, namely, that the multi- 
pliers wheti halved give the elements of the said n*'^ row in reverse order. 
(4) Again, by adding to 3 times the row of R„ 
l-row„_i + 2-row„_2 + 5-row„_3 + . . . . 
we obtain a new row which is divisible by 27i + 1 and which on 
removal of the said factor becomes 
■4t(2«-1)»-i' -(2«-3)»-2' , |-5,, |8i, |l„; 
n+i n 4 6 Z 
so that, if the resulting determinant be denoted by y„, we have 
Y„= (-iy~' S (2n^S) (2n + b) (4?^ - 3) . . (II) 
Here the multipliers are 
1, 2, 5, 14, .... ,| (2/^-3)„.2 
it 
and the last row of is 
— (2/i- l)„_i, .... ,14, 5, 2, 1. 
