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V. Second Memoir on the Curves which satisfy given conditions; the Principle of 
Correspondence. By Professor Cayley, F.B.S. 
Received April 18, — Read May 2 , 1867. 
In the present Memoir I reproduce with additional developments the theory established 
in my paper “ On the Correspondence of two points on a Curve ” (London Math. Society, 
No. VII., April 1866); and I endeavour to apply it to the determination of the number 
of the conics which satisfy given conditions ; viz. these are conditions of contact with a 
given curve, or they may include arbitrary conditions Z, 2Z, &c. If, for a moment, we 
consider the more general question where the Principle is to be applied to finding the 
number of the curves O' of the order r, which satisfy given conditions of contact with a 
given curve, there are here two kinds of special solutions ; viz., we may have proper 
curves C' - touching (specially) the given curve at a cusp or cusps thereof, and we may 
have improper curves, that is, curves which break up into two or more curves of inferior 
orders. In the case where the curves C r are lines, there is only the first kind of special 
solution, where the sought for lines touch at a cusp or cusps. But in the case to which 
the Memoir chiefly relates, where the curves C r are conics, we have the two kinds of 
special solutions, viz., proper conics touching at a cusp or cusps, and conics which are 
line-pairs or point-pairs. In the application of the Principle to determining the numbei 
of the conics which satisfy any given conditions, I introduce into the equation a term 
called the “Supplement” (denoted by the abbreviation “ Supp.”), to include the special 
solutions of both kinds. The expression of the Supplement should in every case be fur- 
nished by the theory ; and this being known, we should then have an equation leading 
to the number of the conics which properly satisfy the prescribed conditions ; but in thus 
finding the expression of the Supplements, there are difficulties which I am unable to 
overcome; and I have contented myself with the reverse course, viz., knowing in each 
case the number of the proper solutions, I use these results to determine a, posteriori in 
each case the expression of the Supplement ; the expression so obtained can in some cases 
be accounted for readily enough, and the knowledge of the whole series of them will be 
a convenient basis for ulterior investigations. 
The Principle of Correspondence for points in a line was established by Chasles in 
the paper in the Comptes Rendus, June-July 1864, referred to in my First Memoir; it 
is extended to unicursal curves in a paper of the same series, March 1866, “ Sur les 
courbes planes ou a double courbure dont les points peuvent se determiner individuelle- 
ment — Application du Principe de Correspondance dans la theorie de ces courbes,” but 
not to the case of a curve of given deficiency D considered in my paper of April 1866 
