199 
1907-8.] Dr Muir on Compound Determinants. 
compound determinant, namely, the theorem which Sylvester writes in the 
form 
( a 2 ... a r a r+l a 2 . . . a r a r+2 .... a x a 2 ... a r a r+s ^ 
' ®1 ' ’ * ®r ®r+ 1 ®]_ a '2 " • • ®r+2 «... ttj • • • ®r ®r+s i 
l a l 
a 2 . . 
( «1 
«2 • ■ 
, . a r 
CL r +\ 
a r+ 2 • • 
• ®r+s ) 
\ a x 
a 2 • • 
• «J 
! a x 
a 2 * * 
. a r 
a r+ 1 
a r-f2 • ' 
■ • a r+s • 
but which would now be better understood in the slightly modified form 
^1 ^2 * ' * 
#2 • • • a r+s 
b x b 2 . . . b T b r+ 1 
6 2 . . . b r b r+ 2 
b 1 b 2 . . . b r b r+s 
| 
«2 
. . . 
S-l 
«i 
« 2 • < 
, . a r+s 1 
1 ^ 
* 2 
\ ■■ 
■ • &r+ S 
No proof of it is given. At a later date it would have been viewed as the 
“ extensional ” of the manifest identity 
05 r+ i 
^r+l 
®r+l 
b r +2 
a r+ 1 
b r + s 
a r+2 
7 
a r + 2 
7 
a r+2 
7 
®r+l 
a r+2 
^r+1 
b r + 1 
b r +2 
^r+s 
®r+s 
^r+1 
b r +2 
&r+ f 
«r +S 
br+s 
Later on in the same paper Sylvester gives for a particular purpose 
what he calls an “ important generalisation.” His words are (p. 304) : 
“ Suppose two sets of umbrae 
a 2 • • • a m+n 
& 2 • • • b m+n ) 
and let r be any number less than n, and let any r-ary combination of the 
m numbers 1 , 2, 3, . . . , m be expressed by q 0 v 0 2 , ... , 6 m , where q 
goes through all the values intermediate between 1 and fx , /x being 
m(m — 1) ... (m-r+ 1) # 
1 • 2 77d r ; 
then I say that the compound determinant 
a i 9l a i0 2 * * * a i g m a m + 1 a m + 2 • • • a m+n CI "2 q 1 a 2# 2 k * * a 2 9m a m+\ a 'm + 2 • • • a m+n 
h 9l b m+ i b m+ 2 . . . b m+n b 2e ^ b 2& g . . . b 2Q ^ b m+1 b m+2 . . . b m+n 
^^ 0 ., * • * Q'm + 1 ^vi+2 • • • ® m+n 
^ ‘ ‘ ^ 0 m ^ m+1 ^ m + 2 * * * ^ m+n 
