202 Proceedings of the Koyal Society of Edinburgh. [Sess. 
imagination could we see comprised in it “ all the heretofore existing 
doctrine of determinants.” His last words thereanent are : “ This very 
general theorem is itself several degrees removed from my still Unpublished 
Fundamental Theorem, which is a theorem for the expansion of products of 
determinants.” 
Sylvester (1852 Dec.). 
[On a theorem concerning the combination of determinants. Cam- 
bridge and Dub. Math. Journ., viii. pp. 60-62 : or Collected Math . 
Papers, i. pp. 399-401 .] 
The statement of the theorem referred to in the title unfortunately 
shows want of proper care,* with the result that it is unnecessarily lengthy. 
It may be recast as follows : — 
If from the array 
a \\ a i2 • • 
• <hn 
a 2 1 a 22 • • 
• a 2 n 
or A, say, 
a ml a m2 • • 
• ®mn , 
we form every possible array of r rows 
Aj, A 2 , . . . , A, where of course /uL = C mt 
formed from 
b n b 12 . . . 
^21 ^22 * * * 
(r^>m<n), calling the said arrays 
r ; and if the corresponding arrays 
b ln 
or B, say , 
bn il b m 2 . . 
• b mn , 
be denoted by B 1? B 2 , . 
. • ., B„ ; then 
A^Bj A r B 2 . . 
A 2 *B] A 2 *B 2 . . 
. A r B^ 
A 2 *B /a 
= (A • B) Cm-1 • r_l . 
A jX A 1 A |U /B 2 . . 
• Aju,*B^, 
By way of proof, Sylvester merely states that it is obtainable from his 
general theorem of March 1851, “by making 
a m + 1 ®m+2 • • • a m+n ) 
bm + 1 ^m + 2 • • • b m+n J 
represent a determinant all whose terms (i.e. elements) are zeros except 
those which lie in one of the diagonals, these latter being all units. ” 
His only other remark is that when r— 1 and when r — m the right- 
* See especially line 8 from bottom of p. 61, where in every case m should be m - 1. 
