538 
Proceedings of Royal Society of Edinburgh. 
A, ... A 
B. ) 
■Uv 
k) 
^i>+s/ 
^'p+S+l 
B,.. 
^p\ 
B,A, Aj 
This may be viewed as an extension of Laplace’s expansion-theorem 
to which it degenerates when p is put equal to 0. Though a com- 
paratively very special identity it is considerably beyond anything 
attained by Schweins’ predecessors. In fact, we only come upon 
something like known ground, when in descending further, we put 
in it g = 1. The result thus obtained is 
ai . 
Ai 
a'l 
A. 
V+s+l 
A^-i-s 
)• 
aJ 
)■ 
a'p+s+l ^p\ 
Bi Aj ..... j, (xLvi. 2). 
which closely resembles a theorem of Desnanot’s. The difference 
between them consists in the fact that here the second factor on the 
left-hand side is any determinant of a lower order than the cofactor, 
whereas in Desnanot the second factor is a minor of the cofactor. 
A further specialisation, 
result 
viz. putting Bj 
brings 
|| a' , &i . . . 
'•Ui 
. . b N 
2{-hu: . . 
• > 
. A^+i^ 
)=0.'| 
or 
1 
2(-HU,... 
&p+l\ 
II hi ... . 
* 1 B 2 ) B 3 . . . 
• • &i>+2 '' 
• B^+3y 
)=0.) 
(xxiii. 7). 
The form here is that of a vanishing aggregate of products of pairs 
of determinants, and identities of this form we have already had to 
consider in dealing with B6zout, Monge, Cauchy, and Desnanot. 
To the last of these only does Schweins refer. His words are 
(p. 352)— 
“ Wird in dieser Gleichung s = 2 gesetzt, so entsteht 
folgende: — 
6i hi ... . 
2(-)*l 
A..B' 
B'J- 
B 2 B 3 . 
. B' 
iJ+3 
)=0 
wovon Desnanot einige ganz specielle Dalle gefunden hat, oder 
vielmehr der ganze Inhalt seiner Untersuchung ist in folgendeii 
dreien Cleichungen begriffen 
