184 Transactions of the Royal Society of Soutli Africa. 
degenerates into {x + y)^ when a^, 63, c^, = 1,3,3,1 and all the other 
fl^'s, 6's, c's, d'^ vanish. 
12. Lastly, a momentary glance may be taken at Cauchy's double 
alternant, that is to say, the alternant which in our temporary notation 
is denoted by 
This when the two sets of variables are identical is axisymmetric and 
admits of special treatment. Thus, using D^ ^, to stand for 
1 
1 
1 
a, + 
a, + ttj 
1 
1 
1 
02 + ai 
Sa^ 
+ a3 
1 
1 
1 
a3 + ai 
a3 + 
2a3 
and for (a^ - a^) (a,. + a,) we have 
2ax.2a3.2a3.D;^_, - 1+0 1+/,, 1 +/,3 
1-/,, 1 + 0 l+/,3 
1-/13 1-/.3 1+0 
But for this latter determinant may be substituted the four-line de- 
terminant 
1111 
|-1 • /.a /.3 
-1 -/-, ■ /.3 
-1 -/.3 ■ 
which on account of the cofactor of the (l,l)th element being equal to 0 
may itself be replaced by 
1 1 1 
-1 • /.3 
-1 -/.» ■ fn 
-1 -/.3 -U 
We thus have finally 
/.3 
'-7-2Via2a3 
(VIII.l) 
