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

156 HISTORY OF THE THEORY OF DETERMINANTS

BINET, [J. P. M.] (1837).
[Observations sur des theoremes de Geometrie, enoncees page
16() de ce volume et page 222 du volume precedent.
Journ. (de Lioville) de Math., ii. pp. 248-252; or, in
abstract, Nouv. Annales de Math., v. pp. 164, 165.]
The main object of this short paper of Binet's was to draw
attention to the fact that a theorem regarding homofocal surfaces
which Lame had just published was originally given by Binet in
1811. He thus has occasion to say that the form under which he
had considered the equation of homofocal surfaces was
02 b2 C2
K-A K-B ~ K-C- 1,
where a, b, c are the co-ordinates of any point on the surface,
A, B, C are positive constants such that A>B>C, and K is a
quantity which may be of any magnitude greater than C. And
as Lame had obtained expressions for the co-ordinates in terms of
three values given to K, Binet intimates that many years before
he-had not only done the same but had extended the solution to
the case of n equations. It is this purely algebraical problem
which is of interest to us, and fortunately Binet gives it in full.
Taking the set of equations in the form
ra, b c
K-A + K-B + -C +
a h c
K + - +.....1,
K2A K2, -B K2 -C
K-AC
K.-A K1-B * C..
a b c
-A+KB+KC~.....1,
he introduces, for temporary purposes, two functions, F(x), f(x),
the former being
(x-A)(x-B)(x-C)....
and therefore of the nth degree, and the latter being any integral
function of a degree less than A. He then recalls the fact that