The theory of determinants in the historical order of development, by Sir Thomas Muir.

COMPOUND DETERMINANTS (SYLVESTER, 1863) 179
equations each containing mn terms and linear in respect to each
of two systems of m and n variables taken separately, but of the
second order in respect to the variables of these two systems taken
collectively." It is also stated to be representable by an ordinary
determinant of the (m+-n-l)t"l order* whose elements are themselves "sums of simple determinants of the (m+n-l)tl' order."
Cayley's resultant of the year 1854 is referred to as the particular
case where m, n = 2, 2.
The theorem of the next paper is spoken of as the auxiliary by
means of which he obtains " the resultant of a lineo-linear system
of equations in its most perfect form. It is easy," he adds, "to
obtain two different solutions, each of them unsymmetrical in
respect of the data of the question," but this theorem effects " a
conversion and fusion of each of these into one and the same determinant, symmetrical in all its relations to the data."
An important part of the third paper concerns the elimination
of u, v, w and x, y from the equations
(bu+b'v+b"w)y - (au+a'v+-a"w)x = 0
(cu+c'v+-c"w)y - (bu+b'v+b"w)x = 0
(duZ+d'v+d"w)y - (cu +c'v +c"w)x = 0
(eu+e'v —e"w)y - (du+d'v +d"w)x =
the eliminant found being a compound determinant of the third
order, which we may write in the form
114561 112461-113451 112341
124561 112561-123451 112351
i34561 113561-123461 112361,
if we use [rstu\ to stand for the determinant whose columns are the
rtll, stl, ttll, uYt columns of the 4-by-6 array of given coefficients.
We may add for ourselves in passing that this would still be the
eliminant if the 24 coefficients were all different. Further, it may
be noted that though Sylvester in this last paper viewed the discovery of double determinants as " the dawn of a new epoch in the
history of modern algebra," he never afterwards, returned to the
subject.
This cannot be correct, because the "resultant is of the degree
(e+ n- 1)!/(n- 1)!(n- )!
in respect of the given coefficients."