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

BIGRADIETNTS (MANSION, 1879) 6

561

where the X's, MS's I/'s stand for
a~b1a~b2a~b a~b+a~2+a b, a~b3+,+aab1 -a~b a~3+b. - 0 a3b3-ab,
a-Ob2 a0b3 + a, b. a, b + aob. a~b + a~b, + a b a~b. -aqub0+ a b. - a~b1 ab-a b,
a0b, a1b3 ajb3 4~3-aob0 a4b.3- a(bl a~b3- a~bt
are only four of the ten noted by Trudi, the others being excluded,
so to speak, by drawing three vertical dotted. lines on the right of
each determinant instead of continuing the horizontal dotted lines
all the way towards the right.
From the foregoing there is probably no serious omission of
papers dealing directly with bigradients, and in particular with the
bigradlient eliminant. But, as it is possible to study the subject
of the common roots of two integral equations without direct
reference to bigradients, and as the other determinants that may then
be used can generally be transformed into bigradlients, it w ill doubtless be of service to the student of determinants to give the following
list of titles of papers on elimination. When taken together with
the preceding papers on the same subject they will also be helpful
to the student of the theory of equations:
1870. GORDAN, P. Ueber die Bildung der Resultante zweier
Gleichungen. Math. Annalen, iii. pp. 355-414.
1872. NAEGELSBACH, H. Ueber die Resultante zweier ganzen
Functionen. Zeitschrift f. Math. u. Phys., xvii. pp. 333 -346.
1876. DARBOUX, G. Sur la th~orie de l'61irination entre deux
equations 'a une variable. Bull. des Sci. Math., x. pp.
56-64; (2) i. pp. 54-64.
1877. ROUCH~k, E. Sur Pelimination. Nouv. Annales de Math.,
(2) xvi. pp. 105-113.
1877. IGEL, B. Einige Siitze und Beweise zur Theorie der Resultante. Sitzungsb.... Akad. d. Wiss. (Wien), lxxvi.
pp. 145-168.
1877. FORESTIR, C. Exposition succincte de quelques m6thodes
d'e'limination entre deux equations. Me'mn. de l'Acad. des
Sci. (Toulouse), (7) ix. pp. 142-163.