182 Transactions of the Royal Society of South Africa. 
in succession it is transformed into 
2«x 
-3 
3 
1 
We thus have finally 
-2/3, 
-3 
1 
-3 -Sa, 
where, be it noted, any two elements situated symmetrically with respect 
to the secondary diagonal do not differ in form, the one being the same 
symmetric function of the one set of variables as the other is of the other. 
As a consequence the invariance referred to in §3 holds here also, as 
it ought. 
In exactly similar fashion there is obtained 
5lw 
and so, generally. 
Sa, 4 
/3,/3./33 
-S/3, 
6 
1 
(VI.) 
9. Doubtless a verificatory proof of this result, similar to that of §7, 
could be devised ; and as a matter of fact in the case of -D^.^-^^i^^ we 
have only got to multiply row-wise the asserted equivalent by - (/32-/3r) 
in the form 
1 /3. /3J 
1 /3. I3l 
. . 1 
. 1 
and then multiply column-wise the product so reached by —a^a2{a2 — aj) in 
the form 
a, 
1 
1 1 
