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

150 HISTORY OF THE THEORY OF DETERMIN.AN-TS

which is found to be
(+ 1 (j~1V a (a2 ~ a2) 19 1a2t02
where a12 + a2 is tecefceto (1112) 3 (t,2 -I- t2 2). Lastly,
Borchardt's derived form *is
a z
at Dt2
z
where Z= (t-ti) /I t2C~tI +C2)(t92 - ct2 -I-c2). Thi s, when the tiresome operations indicated are p erformed, becomes
2t1t2 - C1(t1 + t2) +2C2
(tI2 -celt1 +c2) (t22- C~t2 ---C2)'
which is exactly what is obtained by performing the'simple addition
specified in the original form. Also, since its denominator is
identical with the denominator in Mertens' form, we readily see that
the cofactor of either term of (t.t2) 3 (t2 + t2 2) is
22 Cl1'N
as it ought to be.
GORDAN, P. (1873).
[Ueber den gr6ssten gemeinasamen Factor. Math. Annalen, vii.
pp. 433-448.]
G-ordan here makes an effective application of Brill's theorem of
1871 regarding two arrays so related to one another that the principal
minors of the one are proportional to those of the other. A simple
example or two will make the general procedure clear.
*The following writings concern this. form1856. CAYLEY, A. A memoir on the symmetric functions...Philos.
Tra nsac. B. Soc. (London), cxlvii. pp. 489-499; or Collected Muath. Papers, ii.
pp. 417-439 (see pp. 421-423).
1864. BALTZER, R. Theorie und Anwendung... 2te Aufi. pp. 24:-25'.
1875. Bx-uxo, F. FAk DI. Sur la fonction gyen~ratrice de Borchardt. Crelle's
Journ., lxxxi. pp. 217-219 (cf. Baltzer, 1864).
1876. IKoSTKA, C. Ueber Borchardt's Function. Crelle's Journ., lxxxii. pp.
212-229.