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THE EESULTANT OF A SET OF HOMOGENEOUS LINEO- 
LINEAE EQUATIONS. 
By Thomas Mum, LL.D., F.E.S. 
(Bead March 20, 1912.) 
1. In several of Sylvester's writings of the year 1863 there crop up 
references to a peculiar entity called a " double determinant." At the 
beginning of a paper " On a Question of Compound Arrangement," he 
says that the question arose in the course of his " successful but as yet 
unpublished researches into the Theory of Double Determinants " ; and, 
having enumerated two results in connection with the said question, he 
appends an observation to the effect that " a double determinant means 
the resultant of a system of m-\-n — l homogeneous equations, each con- 
taining mn terms, and linear in respect to each of tw^o systems of m and n 
variables taken separately but of the second order in respect to the 
variables of these two systems taken collectively." He also states that it 
can be represented by an ordinary determinant of the {m + 7i- l)th order, 
whose elements are sums of simple determinants of the same order. 
Here, however, some error must have crept in, as he says quite correctly 
that the degree of the resultant in respect of the coefficients is 
(m+7^ — 1) ! 
(m-1)! (n-l) !' 
a number which implies that the order-number first mentioned should not 
be m-\-7i — l but 
(m + w — 2)! 
(m-1) ! {71-1) V 
His observation concludes with the statement that the only case previously 
considered is the case where 7n, ?t = 2, 2, and that for this Cayley is 
responsible.! 
* Proceed. R. Soc. of London, xii. pp. 561-563; or Collected Math. Papers, ii. 
pp. 325-326. 
t Cayley's eliminant is of the 2nd order with elements of the 3rd order, in agreement 
with our surmise. See Cambridge and Dub. Math. Journ., ix. p. 171. 
