407]
263
407.
SECOND MEMOIR ON THE CURVES WHICH SATISFY GIVEN
CONDITIONS; THE PRINCIPLE OF CORRESPONDENCE.
[From the Philosophical Transactions of the Royal Society of London, vol. clviii. (for
the year 1868), pp. 145—172. 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), [385] ; and I endeavour to apply it to the deter
mination 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 r 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 G r 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 G 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 number 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 furnished 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
