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

138 HISTORY OF THE THEORY OF DETERMINANTS
and similar non-determninant forms are given for the whole of
the thirty-six primary minors and for the first fifteen of the
secondary minors of the second determinant. Thus, the primary
minors which are the cofactors of the elements in the places
(1, 1), (1, 3), (1, 5) are
1-b 2 - b22 -b32 - el2 -c22 _ c3 2 + (b2c3 - b3c2 )2 + (b3c1 - blc3 )2 + ( blc2 - b2c1)2,
(ale, + a2c2 + a3C3) (1 - b12 - b2 2- b3 2) ~ (b lei+ b2c2 + b3c3) (atbl ~ a2b2 + a3b3),
alb2C3 bc.1- aljl _-b12 -b2 2-_b3 2-_c12 -c2 2-_ c32) - b1 (a~b1+ a2b2 +a.b3)
- c1 (aec1 + a2C2 +a3(3.,),
all the others being similar in f orm. to one or other of these
three; and in like manner the secondary minors are exemplified
by -e 22_C
-(bic1 + b2c2 + bNCO),
- b1(l -c12- c22 -c3 2) + c1(blc1 + b2C2 +bVe3),
c,1(blc1 + b2c2 + bVc3 - b1 (cla, + C2a2 + C3a.),
- el Ialb2C. + a2b3l.
CAYLEY, A. (1853*).
[Note sur la me'thode d'6limination de Bezout. Crelte's Jomrn.,
liii. pp. 366-367; or Collectedl Math. Papers, iv. pp. 38-39.]
T he determinant, to which Bezont's so-called 'abridged method'
leads, we have already seen dealt with by 'Jacobi (1835) and
Cauchy (1840).t Cayley's noteworthy contribution to it is contained in a single sentence, namely, " Pour 6fliminer les variables
x, y entre deux equations du nimme degre
on 'aqnitformer l'~quation identique
* The author's date is April 1855; but the rule was published as Cayley's by
Sylvester in 1853. See ~ 62 of the memnoir 'On a theory of the syzygetic relations...'; and note also that Sylvester there first introduced the closely related
idea of the Bezoutiant. (See footnote on p. 137 above.)
t See History, i. pp. 214, 243, 485-7.