76 
PROFESSOR CAYLEY ON THE CURVES 
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 reproduced 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. Journ. t. vii. pp. 
327-337 (Feb. 1866) ; On the Number of Surfaces of the Second Degree which can be 
described to satisfy nine Conditions, Ibid. t. viii. 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 determinees par cinq conditions d’intersection avec une 
courbe donnee. — Comptes Bendus, t. lxiii. pp. 9-12, July 1866. Besults reproduced 
in the present Memoir. 
De Jonquieres : Two papers, Comptes Bendus, t. lxiii. Sept. 1866, reproduced and 
further developed in the “ Memoire sur les contacts multiples d’ordre quelconque des 
courbes du degre r qui satisfont a des conditions donnees de contact avec une courbe 
fixe du degre m ; suivi de quelques reflexions sur la solution d’un grand nombre de 
questions concernant les proprietes projectives des courbes et des surfaces algebriques,” 
Crelle, t. lxvi. (1866), pp. 289-322, — contain a general formula for the number of curves 
C r having contacts of given orders a, b, c, . . with a given curve U m See., which formula 
is referred to and considered in the present Memoir. 
De Jonquieres : Becherches sur les series ou systemes de courbes et de surfaces alge- 
briques d’ordre quelconque ; suivi d’une reponse &c. 4°. Paris, Gauthier Yillars, 1866 *. 
On the quad-geometrical representation of Conditions. — Article Nos. 1 to 23. 
1. A condition imposed upon a subject gives rise to a relation between the parameters 
of the subject; for instance, the subject may be, as in the present Memoir, a plane curve 
of a given order, and the parameters be any arbitrary parameters contained in the 
equation of the curve. The condition may be onefold, twofold, ... or, generally, #-fold, 
and the corresponding relation is onefold, twofold, ... or &-fold accordingly. Two or 
more conditions, each of a given manifoldness, may be regarded as forming together a 
* The foregoing list is not complete, and the remarks are not intended to give even a sketch of the contents 
of the works comprised therein, but only to show their bearing on the present Memoir. 
