396 
Transactions of the Royal Society of South Africa. 
More proofs than one suggest themselves. That which lends itself 
most readily to generalization consists in changing the last column into 
— ^ma, —^mb, —^mc, 0, 0, 
and then expanding in terms of the minors formed from the last two rows. 
Doing this we see that the cofactor of Ic^i^al 
= — a. a^ m^a^^ + m^a^ 
b^ m^bj^ + mjj^ 
C3 C4 m^c^ + m^c^ 
= —mT\a^b^c^\—m2\aJ)^c^\= —mJ)^-\-mJ)^ 
and similarly with the cofactors of \d^e.\, etc. 
7. Applying this result to Clebsch's determinant we see that the latter 
is equal to 
\a^bj^ . Iz^jWal + I^Xi^jl • \Ut10.^\-\- ... ; 
and its equivalent being known to vanish for all values of the a's and 6's, 
it follows that the second factor of every term of it must vanish, and that 
therefore 
U2 
8. The pecuhar identity estabhshed in §6 is, however, only the first of 
a series. Thus taking the same basic determinant as before, namely, 
{a^b^c/l^l, and repeating the process of bordering, we obtain for investiga- 
tion the determinant 
Olearly it is equal to 
a^ 
a. 
«4 
K 
bl 
K 
C2 
c. 
C4 
dz 
I 
%md 
%nd 
S^e 
/. 
/a 
fl 
tmf 
tnf 
a-, 
a^ 
a. 
a. 
%ma 
Sna 
K 
bl 
\ 
^mb 
%nb 
C2 
C3 
C4 
'%mc 
%71C 
d. 
^3 
h 
.fl 
