Note on Double Alternants. 
183 
10. In the third place let us look at alternants of the form D„ . 
taking only the simplest case. By proceeding on lines closely analogous 
to the above, we readily find 
6a? 
4ax 
1' 
6a: 
1 
1 
-S/3x 
1 
-2ft 
1 
and at a farther stage 
— (a] + a,a2 + a:) 
-4Sai 
— 4aia2 
-2/3x 
1 \m 
Sax ' 
1 
-S/3x 1 
/5i/32 — S/3t aiacl 
A column of O's is then appended and the additional row 
Sax 4 ... 1 
prefixed, with the result that our next simplification brings us to 
-Sax 
axaa 
/3x/32 
1 
-Sax 
-S/3x 
1 
-S/3x 
(VII.) 
a result which again satisfies the tests regarding invariance. 
11. A general theorem in regard to D,^ . in agreement with 
Garbieri's of 1878 (Giornale di Mat. xvi. pp. 1-17) is thus foreshadowed. 
The fact that in the case of some of the resulting determinants the simple 
symmetric functions of the a's appear in the same element with those of 
the /3's must not be considered an indication to the contrary of this. 
Indeed it is sufficient to point out that the form of every element m 
Garbieri's determinant is a bipartite function, and that an integral power 
of a binomial is a special case of such a function. For example, the 
bipartite 
1 
X 
x^ 
X3 
ftx 
a^ 
a, 
a. 
1 
K 
y 
^3 
d. 
d. 
