230 Transactions of the Uoyal Society of South Africa. 
simple, namely, that factorisation is in general facilitated when one of the 
factors is known d priori. 
Our present purpose is to exemplify and enforce this, the determinants 
taken for factorisation being, not 4-line minors of the 3rd compound as just 
spoken of, but for the sake of brevity 3-line minors of the 4th compound.* 
(3) By the elimination of n, v, w, the derived set of equations in x, y, z is 
I ^i^^3^3/i. 1^+1 ^1^^063/5 \y \ CidoeJc, \z = 0 j 
These are in number 15, /. e. Cg 3, and the number of 3-line determinants 
got therefrom by the elimination of x, y, z is thus 455, i. e. C^j^ 3. 
(4) Probably the best and most expeditious way of dealing with them is 
by means of the Law of Complementaries. Writing for each element of 
the 15-by-3 array its complementary minor in | a^^G^d^e-J\^ \ we obtain the 
companion array 
1 
1 <^J\ 1 
1 1 
dj, 1 
1 dj\ 1 
1 dJ, 1 
1 
1 ^4^6 1 
1 ^A^^ 1 
cj& 1 
1 C4/6 1 
1 ^4/5 1 
^^Q 1 
1 ^'4^6 1 
1 ^^6.3 1 
C3(7g 1 
i Ca^g ! 
1 1 
Kh 1 
! ^4/6 1 
1 hh 1 
^5^6 1 
1 ^4^6 1 
1 ^e- 1 
1 
1 1 
! M5 1 
h'-'G 1 
1 ^y'G 1 
1 ^.i^5 1 
«5/6 1 
1 «-4/g 1 
1 ^4/5 1 
«5^^6 1 
1 «-i^r. 1 
1 ^h^h ! 
agf/g 1 
1 «'4f^6 1 
1 H^o 1 
«5C6 1 
1 «4^6 1 
1 ^''4^5 1 
«5^6 1 
1 «4^6 1 
1 «4^5 1 
and what we have got to do is the comparatively simple task of factorising 
its corresponding 455 3 -line determinants, knowing that when we have 
done so we can pass with ease to the factorisation that is wanted. 
(5) A little examination shows that the auxiliary determinants in ques- 
tion may be conveniently classified according to the number of letters 
involved in them, the smallest possible number being three and the 
greatest six. 
* Probably the mtli compound of | a,„ | is most suitably denoted by 
certainly nothing more helpful has as yet been proposed. In the same way the 
compounds just referred to may be denoted 'bY\\a^h.2C^d^e'^f(^\\-},?a\d\\aih.2C-^d^e-^fi^\\^, 
or even by [Aglg and [AqI^. 
