90 
Transactions of the Royal Society of South Africa. 
an alternative form for the preceding theorem is 
d^ d 2 d^ 
a Y a 2 % 
h i h h 
C-i Co Co 
■ Pi P 9 Dg 
l & 2 C 3l — l^l^ 1 IV2I a '4 a l a 2 H A* 
— k 2 c 3 | Ifl^Cgl — kxCgl b 4 b 1 b 2 63 B 4 
\a 2 \\ — \Oyb 3 \ \a Y b 2 \ c 4 c x c 2 c 3 C 4 
where it is curious to note that the two bilinear functions are such that the 
elements in the square array of the first are the cof actors, in | a Y b 2 c s |, of 
the elements in the square array of the second, and the elements in the 
laterals of the second are the cofactors, in the bordered determinant, of the 
elements in the laterals of the first. 
(3) The theorem thus reached recalls another* in which occur two of the 
same component parts, namely, 
1^ ii>.j 11/3 
&i b, 6, 
Do D3 
<Xj a>2 &3 
b, K h 
A, 
B 4 
C 4 
\b 2 c 3 | 
-\a 2 c 3 | 
1*2 ^3 1 
D, 
-l&i c 3 | 
\a Y c 3 | 
D 2 
I 6 1 C s' A 4 
- loj c 2 | B 4 
1*1 & 2l C 4 
D, . 
and, being thus able to combine the two, we deduce the still more interesting 
equality, 
b l b, b. 
1 a 1 a 2 a A a 4 1 
^1 hh b i l 
\d\dldl ?\ 
b 2 c 3 l —\b 1 c s \ 1 6 X c. 3 | A 4 
-\a 2 c 3 | \a Y c 3 | — loj c 3 | B 4 
k 2 b s \ — \a Y bo) \a Y b 2 \ C 4 
Pi P2 P3 
where the two bordered determinants are related in the matter of their 
elements quite similarly to the two bilinears in §2. 
As, however, the first factor on the left-hand side of the theorem which 
we have quoted is, when general, raised to the power n — 2, our deduced 
result in its general form is that the product of any bordered determinant 
of the (n + \y h order by the (n — l) th power of the mibordered determinant is 
expressible as a bordered determinant of the (n + 1)'* order also. 
(4) We have next to note that the extent of the border in the foregoing 
need not be restricted to one line : for example, 
I «i h c 3 I 
a Y a 2 « 3 a 4 a b 
— |a 2 & 4 c 5 | la-file^ 
|a x 6o| 
l^&oj \^ b s\ 
\c x d 2 e^ 
h h h h 4 h 
-\b x d^\ 
C \ c 2 c 8 C i C 5 
\ a l C 2\ 
klC S | l«2 C 3l 
di d 2 c2 3 . . 
'Vsl | 6 a c sl 
\a x d 2 e ? j 
e l e 2 e 3 * • 
' Trans. K. Soc. Edinburgh., 
xxxii (1885), § 35. 
