203 
1907-8.] Dr Muir on Compound Determinants. 
hand members are identical, and that the equating of the two left-hand 
members which is thus legitimised gives Cauchy’s extended multiplication- 
theorem. 
The former remark, one regrets to note, is another instance of inaccuracy. 
The specialisation given therein is only one of two which are needed, the 
other being that every element of the determinant 
<h a 2 a B . . . a m i 
^1 ^2 ^3 4 * * ' 
be made 0. To make the matter clear, let us take the case of the general 
theorem where m = 4, n — 5, r = 2, and perform the requisite specialisations. 
The general theorem then is 
“l 
a 2 
«5 
a 6 . . 
. «9 
“l 
a 2 
a 5 
• s 
“l 
«2 
«5 
°6 ' ' 
■ «» 
h 
h 
• h 
h 
h 
b 5 
*6 •• 
• *9 
*3 
*4 
*5 
be- 
b. 
«1 
<h 
«5 
«6 ■ • 
. « 9 
1 a \ 
«■ s 
(X 5 
«6 • ' ' 
. a g 
a i 
«3 
«5 
a e ■ ■ 
. . a 9 
h 
h 
K ■■ 
• b. 
l \ 
h 
h 
*9 
h 
*4 
h 
be- 
a 3 
«4 
a a 
% ■ ■ 
■ «9 
a 3 
a i 
«6 
«6 ' • • 
a 9 1 
a 3 
°4 
«5 
a e . . 
. . s 
h 
h 
be .. 
• \ 
h 
h 
b 5 
*9 1 
" ' 
h 
»4 
• &9 
II 
«s 
a Q • • • 
®9 1 3 
“l 
«2 • • 
• “9 
h 
• • • 
&9 1 ’ 
*2 ••• 
. />, 
Changing now the matrix of the determinant 
a b a 6 
^5 ^6 
a , c 
b 
into matrix unity, and the matrix of the determinant 
«2 
b 2 b 3 & 4 
into matrix zero, the first element of the compound determinant becomes, 
if we write a r b s in place of 
a r 
V 
. 
«A 
«A • • • 
«2 6 5 
a 2 & 6 . . . 
°2 6 9 
“564 
«5 6 2 
1 
• 
a ^ > \ 
• 
1 ... 
* 
a A 
1 
