58 Proceedings of the Royal Society of Edinburgh. [Sess. 
nant 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 bigradient 
eliminant. But, as it is possible to study the subject of the common roots 
of two intregral equations without direct reference to bigradients, and as 
the other determinants that may then be used can generally be transformed 
into bigradients, it will 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. 
187 2. Naegelsbach, H. Ueber die Resultante zweier ganzen Functionen. 
Zeitschrift f. Math. u. Phys., xvii. pp. 333-346. 
1876. Darboux, G. Sur la theorie de lelimination entre deux equations 
a une variable. Bull, des Sci. Math., x. pp. 56-64 : (2) i. pp. 
54-64. 
1877. Rouche, E. Sur l’elimination. Nouv. Annates de Math. (2), xvi. 
pp. 105-113. 
1877. Igel, B. Einige Satze und Beweise zur Theorie der Resultante. 
Sitzungsb. . . . Akad. d. TFiss. (Wien), lxxvi. pp. 145-168. 
1877. Forestier, C. Exposition succincte de quelques methodes d’elimi- 
nation entre deux equations. Mem. de V Acad, des Sci. 
(Toulouse) (7), ix. pp. 142-163. 
1879. Biehler, C. Sur la theorie des equations. Dissert. 60 pp. 
Paris. 
1879. Falk, M. Sur la methode de l’elimination de Bezout et Cauchy. 
Nova Acta Reg. Soc. (Upsala), x. No. 15, 36 pp. 
1879. Hioux, V. Note sur la methode d’elimination Bezout-Cauchy. 
Nouv. Annates de Math. (2) xviii. pp. 289-295. 
1879. Mansion, P. Sur l’elimination. Bull. . . . Acad. . . . Belgique 
(2), xlvi. pp. 899-903: xlvii. pp. 532-541: xlviii. pp. 463-472, 
473-490, 491-526. 
The papers of Falk and Manson devote some little space to reviewing 
the work of their predecessors, and are therefore additionally helpful. They 
do not, however, mention Trudi, nor indeed does any one of the other 
writers of the period. 
