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

DETERMINANTS IN GENERAL (SALMON, 1859)

107

by row-by-row multiplication. This is shown to vanish, not by
pointing out that it contains at least one zero determinant of
the third order as a factor, but by partitioning it into eight
determinants with monomial elements, and showing that all the
eight vanish.*
Unfortunately, for terms of a determinant the word " elements"
is used, and for adjugate the word "reciprocal," although the
elements of the adjugate are spoken of as the "inverse constituents."
SPERLING, J. F. DE (1860, April).
[Note sur un theoreme de M. Sylvester relatif a la transformation du produit de determinants du meme ordre. Joursn.
(de Liouville) de Math.... (2), v. pp. 121-126.]
This is a carefully formulated proof of Sylvester's theorem
of 1839 and the extended theorem of 1851, the lines followed
being those suggested and illustrated by Cayley in 1843. Unfortunately, however, instead of extending Cayley's method to
prove directly and at once the generalisation of 1851, Sperling
repeats Cayley's proof of the simpler theorem, and then uses
the method of so-called mathematical induction to arrive at the
generalisation.
The two determinants whose product is the subject of discus* In using the notation!I I he is not more explicit than its author, Cayley.
If it were explained that
a1 a2 a3
b b2 b3
stands for a, b, | a 13 I a2 h
it would readily follow that the statement
a, a2 3a. I
b1 b2 b3
was short for la, b2, 1abalb 2, b = 0, 0, 0;
and that a1 al aU. a, a a |
b1 b2 9 2 b 1
was short for
( alb, a l b 1, a., b3 | aP, |l, |al03 I aI. ),