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

142 HISTORY OF THE THEORY OF DETERMINANTS

We may note for ourselves (1) that if, n be the number of variables,
the connecting sign-factor is (- 1 )t' 1(- 1), the said determinant being
equal to
1 a a2 ao... 1
1/I2f. P " ~ a~ii~ -1
1 y y2 y3...l Ilaf -a 1.
1 6 62... Iac8y -ICa8 a I -1
and (2) that, save for this question of sign, Cauchy's reasoning
about I a0/31y2... might with equal effectiveness be applied~here
(Hist., i. pp. 309-310).
COTTERILL, T. (1867).
[Question 2409 or 4830. Educ. Times, xx. p. 42; xxviii. p. 194
solution by J. Hammond in Math. from Educ. Times, xxviii.
pp. 26-27.]
The theorem set for proof is that if (pxy) be an alternating function
of p, x, y, and the function be of such a nature that
(pxy)(pzt) + (pyz)(pxt) + (pzx)(pyt) 0,
or, say, 0(p,xyzt)= 0,
whatever p, x, y, z, t may be, then
(acd) (aef)(bde) (bcf) - (bcd)(bef)(ade) (acf)
is an alternating function of a, b, c, d, e, f.
Neglecting Hammond's procedure and taking one of the fifteen
pairs of letters that may be interchanged, say b and e, we see that the
new function will be alternating with respect to the pair if
(acd) (aef) (bde) (bcf) - (becd) (bef) (ade) (aef)
-(aecd)(abf)(edb)(ecf) + (ecd)(ebf)(adb)(acf),
that is, if
(acd)(bde){ (aef') (bcf) + (cabf)(cef)} = (bef)(acf){(bcd)(ade) + (cde)(adb)},
and therefore if
(acf) (bcf) +c(abf)(cef) = (bef)(acf) '
and (bcd)(ade) + (cde) (adb) = (acd)(bde)