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

20 HISTORY OF THE THEORY OF DETERMINANTS
which is not expressible in the form of a determinant. So far,
therefore, as this section of the memoir is concerned, it is evident
that the title is somewhat misleading, and it is unnecessary to
enter into detail regarding the properties in question.
In the course of the section, however, having occasion to use
Jacobi's theorem regarding a coaxial minor of the adjugate,
Cayley gives at the outset a formal proof which it is most
important to note, as it is the natural generalisation of Cauchy's
proof for the ultimate case, and consequently has since become
the standard proof given in text-books. The passage is
"Let A, B...., A', B',.... be given by the equations
A = y'... B =+ ' '...
A' = p~lY 1" B' y" 3"
t A' = / or 3" l y"... B" a "...
the upper or lower signs being taken according as n is odd or even.
"These quantities satisfy the double series of equations
Aa + B/ +....= K
Aa' + B/' +....= 0
A'a + B'i +.... 0 6)
Aa''+ B'I'+....=/K
Aa + A'a' +.... = K
A/ + A'P'+... = 0
Ba + B'' +.. = 0
BPD + B'P' +....= K
the second side of each equation being 0, except for the rth equation of
the,th set of equations in the systems.
"Let X, /L,... represent the rth, ( +l)th,... terms of the series
a, /P,...; L, M,... the corresponding terms of the series A, B,...,
where r is any number less than n, and consider the determinant