40 
Transactions of the Boyal Society of South Africa. 
In order to raake a fair start the reader niay assure himself of the 
truth of (Z) by observing that the cofactor of [e,'C] on the right is 
I [y.ffl [y,a] I 
that this is a two-Hne minor of the adjugate of [ay,/3c] ; that, further, it is 
that particular minor which corresponds in position with the minor of 
zero-elements in [ay,/3^] : and that therefore by a theorem of Jacobi's it is 
equal to [ay,/3^] multiplied by the complementary of the said minor of 
zero-elements, namely, by | u^^ \ : and, finally, that this agrees with what 
we see on the left. 
3. The cofactor of | | in (Z) is readily recognized to be a sum of 
products of pairs of determinants, the first factor of each product being 
of the {ii + l)th. order and the second factor of the (7H-2)th order. This 
suggests the possibility of the cofactor being representable as a deter- 
minant of the (272--f3)th order; and without much difiiculty such is found 
to be the case. Thus, taking 7t = 4 merely for shortness' sake in writing, 
and keeping an eye on the first of its three product-terms, namely, 
[£,4] . [ay,/)c] , 
we form a determinant of the 11th order having \e,'C\ for its first five-line 
coaxial minor and having [ay,/33] for its last six-line coaxial minor, the 
only change in form being that the row of /3's and the row of ^'s in the 
latter are transferred from the end to the beginning. Turning next to 
the second product, namely, 
[e,/3] . [ayrc 'C] , 
we see at a glance that not only it but also \_e,c] . [ay,;/3] is provided for 
by merely appending to the five-line coaxial minor a row of /3's and a row 
of o's, prefixing to the six-line coaxial minor a row of 4's, and filling all 
the remaining vacant places with zeros. The result of this is 
