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

112 HISTORY OF THE THEORY OF DETERMINANTS

CAYLEY, A. (1846).
[Note on the maxima and minima of functions of three variables.
Cambridge and Dub. Math. Journ., i. pp. 74-75; or Collected
Math. Papers, i. pp. 228-229.]
Using A to stand for the determinant
a h g
h b f
hbf
g f c
we may formulate Cayley's first theorem by saying that if
(a+b+c)A and A+B+C be positive, then aA, bA, cA, A, B, C
are all positive. By way of proof it is noted that the equation
A-x H G
H B-x F =0,
G F C-x
i.e. x3 - (A+B+C)X2 + (a+b+c)Ax - A2 = 0,
has by reason of the data all its roots positive: that therefore
the roots of the equations
A-x H B-x F C-x G
=0, =0, 0,
H B-x F C-x G A-x
i.e. X2 -(A+B)x + cA = 0, 2-(B+C)x + aA=0,
x2 -(C+A)x+bA = 0,
on account of Cauchy's localisation of them in relation to the
roots of the previous equation, must also be positive; and
consequently that
A+B, B+C, C+A, cA, aA, bA
must be positive. This last is essentially what was to be
proved; because, for example, to say that A+B and cA are
positive implies that A+B and AB are positive, and therefore
that A and B are positive.
Cayley also puts on record the theorem that the equation in x