324 Transactions of the Royal Society of South Africa. 
As thus written, however, much greater complexity is suggested than in 
reality exists. A first simplification results from reversing the order of 
the rows and thereafter the order of the columns, the first row then 
becoming simple in appearance and the same for all orders, and the other 
rows flowing readily from it. Further, without loss of generality we may 
write X for B//3. With these changes and the change of t^ into the 
cases for the 4th and 5th orders are 
X Va 2(1-2) 
3(a + 2) X 2(<x + 3) 2(2-3) 
2(a + 3) X 3(a + 6) 
l(a-i-4) X 
= {x^ 3a-hl2)(a; + (x)(a3 — 3a-6) 
X 
4(a-f 3) 
X 
3(a + 4) 
2(1-2) 
2(a + 3) 
X 
2(a + 5) 
2(2-3) 
3(a + 6) 
X 
1(^1 + 6) 
2(3-4) 
4(^ + 9) 
X 
= (ii;+4a + 24) {x^2a+^) {x-Q) (a: ~ 2a -12) (a; -4a -12). 
It is thus seen that the determinant in each case is a function of only 
two variables, and that on account of the absence of both of them from 
the shortest diagonal the determinant cannot degenerate into a continuant, 
and the theorem therefore cannot include any of those established in 
1903-04. 
(3) In bringing forward our new result in extension of this the method 
of the former papers need not be again explained; it will suffice to make 
clear what the result really is, and to show its relation to others. In the 
case of the fifth order it is : 
X 
4(7-3c 
/3 
d(y-2c-d) 
c-h^U 
2(^-6) 
2(7 -c) 
'^{c-^h + U) 
3(^_26 + 3cZ) 
a? + E 
7 
&{c-h + bd) 
4(^-36 + 9cZ) 
^ + S 
^ (ar_4y + 12c + \Uy{x + /3 - 37+ 6c + 3c?)-(^+2/3 - 27 - 6 + 3c + 3^^) 
.(a, + 3/3-7 - 3?> + 3c+ Myix + 4/3 - 66 + 6c + 18c?), 
where P = /:? - 7 + 6c -f 9(^, 
Q = 2/3 - 27 - 6 + 9c + 9(Z, 
E ^ 3/3 - 3y - 3& + 9c + 9c?, 
S = 4/3 - 47 - 6& + 6c + 18cZ. 
The factors of the determinant are brought into evidence by means of the 
row-multipliers 
