192 
ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 
[40G 
be analytically represented by an equation F(y, &■) = (), which is rational and integral 
of the degree iV in regard to a variable parameter A, : this is not the case ; see 
Annex No. 1. 
Chasles : Various papers in the Comptes Rendus, t. Lvm. et seq. 1864—67. The 
first of them (Feb. 1864), entitled “ Determination du nombre des sections coniques 
qui doivent toucher cinq courbes données d’ordre quelconque, ou satisfaire à diverses 
autres conditions,” establishes the notion of the two characteristics (p, v) of a system 
of conics which satisfy four conditions ; viz. p is the number of these conics which 
pass through a given arbitrary point, and v the number of them which touch a given 
arbitrary line. The Principle of Correspondence for points on a line is established in 
the paper of June—July 1864. Many of the leading points of the theory are repro 
duced in the present Memoir. The series of papers includes one on the conics in space 
which satisfy seven conditions (Sept. 1865), and another on the surfaces of the second 
order which satisfy eight conditions (Feb. 1866). 
Salmon: “On some Points in the Theory of Elimination,” Quart. Math. Joumi. 
t. vu. pp. 327—337 (Feb. 1866) ; “ On the Number of Surfaces of the Second Degree 
which can be described to satisfy nine Conditions,” Ibid. t. vm. pp. 1—7 (June 1866),— 
which two papers are here referred to on account of the notion which they establish 
of the quasi-geometrical representation of conditions by means of loci in hyper-space. 
Zeuthen: Nyt Bidrag... Contribution to the Theory of Systems of Conics which 
satisfy four conditions, 8°. pp. 1—97 (Copenhagen, Cohen, 1865), translated, with an 
addition, in the Nouvelles Annales. 
The method employed depends on the determination of the line-pairs and point- 
pairs, and of the numerical coefficients by which these have to be multiplied, in the 
several systems of conics which satisfy four conditions of contact with a given curve 
or curves. It is reproduced in detail, with the enumeration called “ Zeuthen’s Capitals,” 
in the present Memoir. 
Cayley : “ Sur les coniques déterminées par cinq conditions d’intersection avec une 
courbe donnée,” Comptes Rendus, t. lxiii. pp. 9—12, July 1866. Results reproduced in 
the present Memoir. 
De Jonquières : Two papers, Comptes Rendus, t. lxiii. Sept. 1866, reproduced and 
further developed in the “ Mémoire sur les contacts multiples d’ordre quelconque des 
courbes du degré r qui satisfont à des conditions données de contact avec une courbe 
tixe du degré m ; suivi de quelques réflexions sur la solution d’un grand nombre de 
questions concernant les propriétés projectives des courbes et des surfaces algébriques,” 
Crelle, t. lxvi. (1866), pp. 289—322,—contain a general formula for the number of curves 
C having contacts of given orders a, b, c, . . with a given curve U m , which formula 
is referred to and considered in the present Memoir. 
De J onquières : Recherches sur les séries ou systèmes de courbes et de surfaces 
algébriques d’ordre quelconque ; suivies d’une réponse &c. 4°. Paris, Gauthier Villars, 
1866 C). 
1 The foregoing list is not complete, and the remarks are not intended to give even a sketch of the con 
tents of the works comprised therein, but only to show their bearing on the present Memoir.