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

200 HISTORY OF THE THEORY OF DETERMINANTS

At this stage considerable space (pp. 222-233) is devoted to
examples illustrating the theorems thus far reached and leading to
another theorem (no. 12) in which the mtl compound of I a(,,, is
made to produce a mixed mtl compound by replacements from the
mtl compound of f b,, [. There then follows Sylvester's theorem of
August 1851 (Hist., ii. pp. 61-62): and the memoir ends with a
theorem regarding a special minor of the mixed n"t compound Vabove referred to.
RUBINI, R. (1878).
[Formole di trasformazioni nella teorica dei determinanti. Giornale
di Mat., xvi. pp. 198-208, 344.]
Rubini's main theorem is that extreme case of Bazin's which had
already been rediscovered by Zehfuss in 1862. His mode of
establishing it is to multiply a1,,, by I x, 1, equate the elements
of the resulting product to the elements of [ bl,, I in order, solve for
the x's in rows, and then substitute their values in I x, \. The
outcome being a compound determinant equal to AB"-1, the companion determinant equal to Al-1B is noted, and easy deductions
made. Attention is also drawn to special cases; in particular, how
by taking one of the given determinants equal to unity, say equal to
(k)o (k+l)o (k+2)0..
(k)1 (k+1)1 (k+2)1...
(k)2 (k+1)2 (k+2)2..
the adjugate of the other determinant can be obtained in any
number of forms.
The really fresh result of the paper is contained in ~ 5, where he
shows, merely by making a column of A identical with a column of
B, that his supposed new compound determinant equal to AB'"has every one of its primary minors exactly divisible by B~-2, and
that as a consequence of dividing by this factor there is obtained
as an expression for AB an aggregate of products of pairs of determinants,-in other words, Sylvester's expansion of the years 1839,
1851. From this, again, easy specializations and deductions are
made.