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

14 HISTORY OF THE THEORY OF DETERMINANTS

interest attaching to the little exposition is connected with the
"proof." The first essential paragraph is:"The result of the elimination of the variables from the equations
alx1 + a2, +... + a,;,, = 0,
blz1 + b2 +... + bzx = 0,
^11 ' + 2 X2 +... +,X, = 0,
is an equation of which the second member is 0, and of which the first
member is formed from the coefficient of xx2... xI in the product of
the given equations, by assuming a particular term, as alb2... r,
positive, and applying to every other term a change of sign for every
binary permutation which it may exhibit, when compared with the
proposed term a b2...?,."
The curious point worth noting here is that we are directed
first to form the terms of the expression afterwards denoted by
alb2... r,, I and called a "permanent," and then to alter the
signs of certain terms of it. Boole then proceeds:"The truth of the above theorem is shown by the following considerations. The elimination of x1 from the first and second equation
of the system introduces terms of the form alb - a2b a, abs - abt, etc.,
in which the law of binary permutation is apparent, and as we may
begin the process of elimination with any variable and with any pair of
equations, the law is universal. From the same instance it is evident
that no proposed suffix can occur twice in a given term, which condition is also characteristic of the coefficient of xx2... x, in the product
of the equations of the system, whence the theorem is manifest."
It will be observed that neither the word "determinant" nor
the word "resultant" occurs: indeed, throughout the paper,
instead of resultant he uses "final derivative," a term which
probably may be traced to Sylvester.*
CAYLEY, A. (1843).
[Chapters in the analytical geometry of X dimensions. Cambridge Math. Jourr., iv. pp. 119-127; or Collected Math.
Papers, i. pp. 55-62.]
Of the four short chapters which compose this paper, the only
one which concerns us is the first, although in the others deter* See Sylvester's paper of 1840.