586 
Proceedings of Eoycd Society of Edinhurgh. 
and the other similar determinants each into an aggregate of ten 
products, the two factors of any product in the expansion of 
I ] tti 02 «3 O 4 «5 \ 
I ^1 ^2 ^2 ^3 / 
being, as we should nowadays say, a minor formed from the first 
two rows and the complementary minor. In this way would 
arise 20 rows of 10 terms each, and these being combined by 
a second use of Laplace’s expansion-theorem in columns of 20 terms 
each, the outcome would be an aggregate of 10 products, viz., the 
aggregate 
1 
1 
«2 \ 
II "3 
«5 
!-i 
^3 \ 
II 
«3 \ II 
«2 
«4 
«5 
&2 
h 
1 
aJ' 
•|b, 
B2 
B3 
B4 
Bs 
Be)- 
-|A 4 
As) -I 
Bx 
Bs 
Bs 
B4 
Bs 
Be 
«4 \ 
II “2 
«3 
«5 
h 
II 
\ |1 
do 
«3 
^1 
h 
Ad' 
|Bi 
Bs 
B4 
Bs 
Be)- 
-|Ai 
As) -11 
Bx 
Bs 
Bs 
B4 
Bs 
Be 
-1 
«3 \ 
II **1 
«5 
61 
&2 
h\ 
II «2 
«4\ II 
«3 
«5 
h 
&2 
^3 
Ad' 
■||Bi 
B. 
B3 
B4 
Bs 
Be)- 
-|ki 
As) -I 
Bx 
Bs 
Bs 
B4 
Bs 
Be 
1 Oo 
«5 \ 
II "^1 
O3 
«4 
!>■ 
^3 \ 
II "3 
«2 
«5 
h 
h 
|a; 
Aj- 
'll Bi 
B2 
B3 
B4 
Bs 
Be)- 
-|1Ai 
As) -II 
Bx 
Bs 
Bs 
B4 
Bs 
Be 
, 1 
1 
«5 \ 
I! 
«2 
«4 
h 
^3 \ 
li 
«2 
«3 
h 
62 
h 
^1 
|Ai 
Aj' 
I|Bx 
B^ 
Bs 
B 4 
Bs 
Be)- 
-|1Ax 
As)-| 
Bx 
Bs 
Bs 
B4 
Bs 
Be 
The following is Schweins’ statement of this most general 
theorem: — (l. 2) 
B' 
1 
n • • • ^ m — q \ 
B. ;• 
The only points about it requiring explanation are the exact effect 
to be given to the symbol 5, and the meaning of the dashes affixed 
to certain of the letters. The two symbols are connected with each 
other, the dashes not being permanently attached to the letters, but 
merely put in to assist in explaining the duty of the On the 
left-hand member of the identity, the two symbols indicate that the 
first term is got by dropping the dashes, and that from this first 
term another term is got, if we substitute for B^ ... . B^, some 
other set of q B’s chosen from B^ . . . . B„^ , and take the remain- 
ing B’s to form the B’s of the second determinant, — the two sets of 
