1904 - 5 .] 
Dr Muir on General Determinants. 
933 
l a 1 a 2 . . . a n ) | aj a 2 . . , a n j 
t \ b 2 . . . b„ f X 1 ft & . . . ft ) 
a 1 a 2 a n ) j a x a 2 a n ) 
... bp foop+i fidp+'Z ••• fion ' ^ $6\ ft 02 ••• @0p ^p + 1 ^jp+2 • • • ' 
where 0 1 , 0 2 . . . , 0 P are any p different integers taken from 1 , 
2, . . n, or where, as Sylvester puts it, 6 lt 0 2 , . . 0 n are 
* disjunctively ’ equal to 1 , 2 , . . . , n. 
A proof of a verificatory character is given at some length, the 
aim being to show that the terms arising from the expansion of 
the products occurring on the right-hand side are classifiable under 
two heads : (1) those which appear twice, viz., once with a positive 
sign and once with a negative sign ; (2) those which appear only 
once, and which in their aggregate agree with those arising from 
the expansion of the single product on the left. An improvement 
of it by Faa di Bruno will be given later. 
By removing to the right the solitary product at present on the 
left, the theorem is seen to belong to the class of vanishing aggre- 
gates of products of pairs of determinants, and ■ therefore to be not 
entirely new, the first instances of it, viz., 
\ab'\-\cd'\ - | ac |-| bd! | + \ ad'\\bc'\ — 0, 
| ab'c” |*| def" | - | ab r d" |*| ce'f" | + | acd" |-j bef" \ - \ he'd" |-| acf" \ = 0, 
that is to say, the cases where n = 2, p — 1 , and n = 3 ,p = 1 , being found 
in Bezout (1779). Nothing, however, done by Bezout, Cauchy, or 
Schweins ought to dissociate Sylvester’s name from the theorem. 
His claim, too, is all the stronger from the fact that he it was who 
in 1839 first formulated the case n = n, p= 1, — a fact which seems 
to have dropped entirely out of remembrance, writers giving the date 
1851 when referring to the case enunciated a dozen years earlier. 
Sylvester, J. J. (1851, Oct.). 
[On a remarkable discovery in the theory of canonical forms 
and of hyperdeterminants. Philos. Magazine , ii. pp. 
391-410 : Collected Math. Papers , i. pp. 265-283.] 
Here, amid matter of great algebraical importance, there is 
incidentally suggested the discarding of the use of the term 
