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

456 HISTORY OF THE THEORY OF DETERMINANTS
where, be it observed, it is the rows 1, /3, ' and 1, 8, (' that are
diminished on both occasions. There is thus obtained
=_ (a-13)(a'- 8) {(3-y)(I-y') -(8- y)(/3'-y )}
- (y-/3)(7')0{(/3-a)('-a') - (3'-a')(8-a)}
= (a-/3)(a'- 8')(y- )(/3'-y') + (y^-/3)(y'-8')('3-a')(8-a),
= -(a- )(^y -)(a'- ')(y' - ) + (a'-/3')(y'- 8')(a-)(a - )
(72) SYLVESTER'S UNISIGNANT.
SYLVESTER, J. J. (1855, April).
[On the change of systems of independent variables. Quart.
Jozrn. of Math., i. pp. 42-56; or Collected Math. Papers,
ii. pp. 65-85.]
In the course of Sylvester's investigations a peculiar three-line
determinant turns up, which he considers deserving of attention
on its own account, namely, the determinant
a1 + a2 +a3 -a2 -a3
-bl b+ b2+ b3 - b3
- C - C2 C1 -+ 2 3,
the final expansion of which consists of 16 terms, all positive.
To obtain this expansion a "simple rule" is laid down, namely,
to substitute
a ab a, a} a% a3
ba b bc for { b1 b2 b3
Ca Cb C c1 C2 C3
* It will be seen that merely by accident the three ways in which v can be
expressed as the difference of two products have turned up in succession, and
that they may be written
IPQ', I QR' I, IPR' I
if we put
(a-/P)(7- ) = P,
(a - o) ( - 6) = Q,
(a - ) ( -) = R.