[ 75 .] 
IV. On the Curves which satisfy given Conditions. By Professor Cayley, F.R.S. 
Received April 18, — Read May 2, 1867. 
The present Memoir relates to portions only of the subject of the curves which satisfy 
given conditions ; but any other title would be too narrow : the question chiefly consi- 
dered is that of finding the number of the curves which satisfy given conditions ; the 
curves are either curves of a determinate order r (and in this case the conditions chiefly 
considered are conditions of contact with a given curve), or else the curves are conics ; 
and here (although the conditions chiefly considered are conditions of contact with a 
given curve or curves) it is necessary to consider more than in the former case the theory 
of conditions of any kind whatever. As regards the theory of conics, the Memoir is 
based upon the researches of Chasles and Zeuti-ien, as regards that of the curves of the 
order r, upon the researches of De J onquieres : the notion of the quasi-geometrical 
representation of conditions by means of loci in hyper-space is employed by Salmon in 
his researches relating to the quadric surfaces which satisfy given conditions. The 
papers containing the researches referred to are included in the subjoined list. I reserve 
for a separate Second Memoir the application to the present question, of the Principle of 
Correspondence. 
List of Memoirs and Works relating to the Curves which satisfy given conditions, 
with remarks. 
De Jonquieres: Theoremes generaux concernant les courbes geometriques planes 
d’un ordre quelconque, Liouv. t. vi. (1861) pp. 113-134. In this valuable memoir is 
established the notion of a series of curves of the index N ; viz. considering the curves of 
the order n which satisfy \n(n-\-?>) — 1 conditions, then if N denotes how many there 
are of these curves which pass through a given arbitrary point, the series is said to be of 
the index N. 
In Lemma IV it is stated that all the curves C„ of a series of the index N can be 
analytically represented by an equation F (y, x)=0, which is rational and integral of the 
degree N in regard to a variable parameter A: this is not the case; see Annex No. 1. 
Chasles: Various papers in the Comptes Rendus, t. lviii. et seg. 1864-67. The 
first of them (Feb. 1864), entitled “ Determination du nombre des sections coniques qui 
doivent toucher cinq courbes donnees d’ordre quelconque, ou satisfaire a diverses autres 
conditions,” establishes the notion of the two characteristics (gj, v ) of a system of conics 
which satisfy four conditions ; viz. g> is the number of the seconics which pass through 
a given arbitrary point, and v the number of them which touch a given arbitrary line. 
mdccclxviii. N 
