190 Transactions of the Royal Society of South Africa. 
As each of the a's in the last form occurs in only one place, all that 
remains necessary in view of (IV.) is to develop the cofactor of one 
of them, say the cofactor of a^, which 
-b, -b. 
c, — c. 
d,, -d. 
d, 
= Cj^{b^d^ -f- b^d^) + dj^{b^c^ + b^c^) ; 
the whole result thus is 
^4 = a,^ (Ci + c J (b^d, + b^d.) + {d^ + ^^3) {b^c, + bx^) | 
+ ^3 I {d, + d.) {cbj, + cJd^ + {br + c J {cd^ + c^d,) | 
+ a, I (61 + 63) (d^c^ + d,c^) + (Cx + <^,) {djb^ + | 
(VL) 
so that ^4 is a determinant whose final development consists of 24 
positive terms. 
7. An alternative and equally important result is obtained by observ- 
ing that in (V.) solitary positions are also occupied by b^, c^, d^, namely, 
= Cidia^b^^-i-bi{a/lx^^-\-ax,d,,^ ) 
-\-d^b,a^c^,-]-c^(aJ)^d^,-Ya\dfy [ + a,^^{b^c,d^^bx^d,) (VII.) 
+ bxia^d,^ + di(a^c^b^^ + a.b^c^,) ] 
where the aggregate of the terms free of b^, c^, d^ is most readily got by 
going back to the original form of ^4 in § 4.''' 
8. Before proceeding to the consideration of ^5 it is necessary to 
point out that there are variant determinant forms of which, though 
of little value for development purposes, are well deserving of notice on 
other grounds. For example, there is the form 
a,. 
a.. 
c,. 
* Further, taking from (VII.) the portion free of the suffix 1, and using (III.) on it, 
we have 
€4= (a^ + a^ + a^) {h^c^d^+h^c^d^U 
+ (^>3+&4+&i) (M3a4 + C4^i«'3) I (YU1\ 
+ (c4+c, + c,) {d,a,b,-]-d,aJ),) j ^^^^^-^ 
This form, however, has no interest in connection with the general argument, as it is 
peculiar to ^4- 
