16 
Transactions of the Royal Society of South Africa. 
The proof is quite simple. We have only to note in the first place that the 
product of the two arrays is expressible as a single determinant 
Pi Po p. a^ a- 
Qi Qo Qs ^5 
Pj Po P;. C- 
^4 74-1 . 
K 75 • -1 ' 
the latter being transformable into the former by adding to each of the 
first three rows times the 4*'^ row and a-^ times the 5*'' row. Then we 
partition this determinant into four, namely, one representing all the terms 
of it containing both of the elements in its (4,4)^'' and (5,5)*^*^ places, 
one representing all the terms containing one of these elements without 
the other, and one representing all the terms containing neither. 
(2) The general theorem may be formulated thus : 
The ]jrochict of ttvo m-hy-n arrays A, B, is expressible as an aggregate 
of single determinants, the first of which is the product of the first k columns 
of the arrays, and the others are formed from this by bordering, namely, 
bordering first in every way with one of the remaining columns from A and 
the correspo7iding column from B, secondly, with two of the remaining 
columns from A and the corresponding two from B, and so on, those having 
an odd numher of lines in the border being negative and the others 
positive. 
The number of terms in the expansion is evidently 
{n-lc)0 ^- {n-h\-^ . . . + {n - h)n _ jc 
i.e. 2^-^ 
(3) The relation between the old expansion and the new is not at all 
complicated, each term in the latter being the equivalent of a group of 
terms in the former. Thus, in the example with which we started, where 
n, m, k — 5, 3, 3, the apportionment of the 10 product-terms of the old 
expansion among the four terms of the new expansion is 
1+4+4 + 1; 
and when n, m, k = Q, 3, 3 the apportionment of the 20 among the 8 
(that is, 63 among 2*^ - is 
1+3+3+3+3+3+3+1. 
If, further, we group the latter terms according to the number of lines in a 
border, this takes the form 
1 + 3-3 + 3-3 + 1 , 
and we have the verification provided by the known theorem regarding 
