Dr T. Muir on the Theory of Determinants. 
535 
The further generalisation of which this is possible, and whicli 
Schweins effects, depends on the fact, that the law of formation 
twice used in proving the identity, is but the simplest case of 
Laplace’s expansion- theorem, and that the said theorem can be 
similarly used in all its generality. In other words, instead of 
taking only one of the B’s at a time to go along with the A’s to 
form the first factors of the left-hand aggregate, we may take any 
fixed number of them. For example, out of six B’s we may take 
every set of three to go along with two A’s, and we shall have the 
aggregate 
II «1 «2 «3 «4 «5 \ 
II ^1 ^2 
^3 \ 
II Oi O2 O3 O4 Og \ 
II 61 62 ^3 \ 
II -^1 ^2 ^1 ^2 ® 3 / 
IIB4B3 
bJ- 
1 -^1 ^2 ^1 ^2 ^ 4 / * 
IIB3B3BJ 
II ai 02 «3 «4 «5 \ 
II h 
h \ 
II Oi O2 O3 O4 Og \ 
+ 11 AiA,B,B,Bj- 
11B3B4 
BJ- 
II ^1 ^2 ^1 ^2 ^6/ 
IIB3B4BJ 
II Oi 02 03 04 Og \ 
II ^>2 
^3 \ 
II O4 O2 O3 O4 Og \ 
II ^2 ^3 \ 
+ |AiA,b, B3BJ' 
•IIB3B3 
BJ- 
II ^1 ^2 ^1 ^3 ^ 5 / 
1 ^2 ^4 ^6 ) 
|{ Oi 02 03 O4 Og \ 
II &2 
^3 \ 
|| O4 O2 «3 O4 Og \ 
+||a,a,b,B3bJ’ 
IB3B4 
Ba)- 
|l -^1 -^2 ^4 ^ 5 / 
1 ^2 ®3 ^6/ 
II Oj 02 03 O4 Og \ 
II h h 
h \ 
|| Oj Oo O3 O4 Og \ 
II ^2 *3 \ 
+||a,a,b,b,bJ' 
1 ^2 ^3 
B5)- 
-IIajA^BiB^bJ- 
IIB3B3BJ 
II Oi 02 03 04 Og\ 
II &2 
h \ 
|| Oj 02 03 O4 Og \ 
II h ^2 h \ 
+|a,a,b,B3bJ- 
|BiB, 
bJ- 
II Aj A2 B2 B3 Bg j ' 
•IIB4B4B3) 
II Oi O2 «3 «4 «5 \ 
II h h 
h \ 
II h\ 
+||a,a,B3B3bJ' 
BJ- 
1 ■'^1 -^2 ^2 ^4 / 
I1B1B3B3) 
II 0 | O9 «3 «4 Og \ 
II ^2 
h \ 
|| Oi O2 «3 «4 Og \ 
II h h\ 
+ 1 A^ A2 Bg B4 Bq y ' 
IIB4B3 
Ba)- 
ij -^1 -^2 ^2 ^5 ^6 / 
IIB1B3BJ 
II «l «2 «3 «4 «5 \ 
h \ 
II Oi O2 03 O4 Og \ 
II ^1 &2 &3 \ 
+ 1 |a,A 3 B 3 B,bJ. 
IIB1B3 
Baj- 
II ^1 -^2 ®3 ^4 ^6 / 
IIB1B3B3} 
II Oi Oo O3 O4 Og \ 
II h h 
h \ 
II Oi O2 O3 O4 Og \ 
II ^2 &3 \ 
+ ||AiA 3 B 3 B 3 BJ' 
'IIB4B3 
Bj- 
II ^1 "^2 ^4 ^5 ^6/ ’ 
1 ^1 ^2 ^ 3 / 
— the sign of any term being determined by the number of inver- 
sions of order among the suffixes of all the B’s of the term. In this 
particular case the first use of Laplace’s expansion-theorem is to 
transform 
