406] 
191 
406. 
ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 
[From the Philosophical Transactions of the Royal Society of London, voi. clviii. (for the 
year 1868), pp. 75—143. 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 
considered 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 Zeuthen, as regards that of 
the curves of the order r, upon the researches of De Jonquières : 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 Jonquières: “ Théorèmes généraux concernant les courbes géométriques planes 
dun 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+ 3)-l 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 JS T . 
In Lemma IV it is stated that all the curves C n of a series of the index N can