Dr Muir on General Determinants. 
701 
1907-8.] 
but by partitioning it into eight determinants with monomial elements, and 
showing that all the eight vanish.* 
Unfortunately, for terms of a determinant the word “ elements ” is used, 
and for adjugate the word “ reciprocal,” although the elements of the adjugate 
are spoken of as the “ inverse constituents.” 
Sperling, J. F. de (1860, April). 
[Note sur un theorem e de M. Sylvester relatif a la transformation du 
produit de determinants du meme ordre. Journ. {de Liouville) 
de Math. . . . (2), v. pp. 121-126.] 
This is a carefully formulated proof of Sylvester’s theorem of 1839 and 
the extended theorem of 1851, the lines followed being those suggested and 
illustrated by Cayley in 1843. Unfortunately, however, instead of extend- 
ing Cayley’s method to prove directly and at once the generalisation of 1851, 
Sperling repeats Cayley’s proof of the simpler theorem, and then uses the 
method of so-called mathematical induction to arrive at the generalisation. 
The two determinants whose product is the subject of discussion 
being | a n a 22 . . . a nn \ and | b n b 22 . . . b nn | , or, say, A and B, he forms 
the determinant 
Si 
S 2 ’ * ' 
S n—m 
. . . Sn 
^12 • • 
• h n 
Si 
a 22 . . . 
O' 2n-m 
. . . a 2n 
t>2i 
^22 ’ " 
• b‘2 n 
a n i 
Si2 . . . 
St , n—m 
• • • Sin 
b n i 
b n 2 . . 
. b nn 
n—m+1 
S. n—m+2 ’ ■ 
• • S,n— 1 S n 
K 
b l2 . . 
■ • b ln 
S , n—m+1 
S , n—m+2 • • 
^21 
^22 * ' 
. b 2n 
Si, n—m+1 
Si,n— ra+2 • • 
• • Si,n— 1 
b n , .. 
• m b nn 
* In using the notation || || he is not more explicit than its author, Cayley, 
explained that 
1 S 
a 2 
I \ 
b, b 3 
stands for 
I I S h 1 3 
it would readily follow that the statement 
a l 
Ct 2 Qj 3 
bi 
J)c) J) 3 j J 
was short for 
I «2 & 3 U 
i 0 
and that 
S 
b, 1 ) 
j S h , 
j 1 a 2 bs \ 
= 
0 
0 
0 
I S 
(X 2 ^3 
a i 
a 2 
“ 3 1 
1 b x 
^2 b 3 j 
' II *1 
*2 
s 1 
I «i 6 3 1 
, I <%2 b 3 
m 
1 a i @3 1 
If it were: 
was short for 
