407]
CURVES WHICH SATISFY GIVEN CONDITIONS.
269
106. As regards the case (4Z) (1), taking P an arbitrary point of the given curve
to, and for the curve W the system of the conics (4J£)(1) which pass through the
given point P and besides satisfy the four conditions, then the curve © has with
the given curve (4i/)(l) intersections at P, and the points P' are the remaining
(2to — 1) (4Z) (1) intersections: in the case of a united point (P, P'), some one of the
system of conics becomes a conic (4Z) (1); and the number of the united points is
consequently equal to that of the conics (4^)(1); we have thus the equation
{(4Z) (1) - 2 (2to - 1) (4Z) (I)} + Supp. (4Z) (T) = (4Z) (I). 2D.
107. It is in the present case easy to find a priori the expression for the
Supplement. 1°. The system of conics (4Z) contains 2 (4<Z■) — (4*Z/) point-pairsQ; each
of these, regarded as a line, meets the given curve in to points, and each of these
points is (specially) a united point (P, P'); this gives in the Supplement the term
to (2 (4Z •) — (4Z /)}. 2°. The number of the conics (4Z) which can be drawn through
a cusp of the given curve is =(4Z-); and the cusp is in respect of each of these
conics a united point; we have thus the term /¿(4Z ■), and the Supplement is thus
= to {2 (4iT •) — (4Z/)} + k (4Z•). We have moreover (4Z) (1) = (4Z •), 2D = n— 2to + 2 + k ;
and substituting these values, we find
(4£)(1)= (4to — 2) (4<Z •)
— to {2 (4>Z •) — (4>Z/)} — k (4iZ •)
+ (n — 2m + 2 + k) (4 Z •)
= n(4<Z •) + m(4fZ I),
which is right.
108. It is clear that if, instead of finding as above the expression of the
Supplement, the value of (4J?)(1), =n(4>Z •) +m(4eZ ¡), had been taken as known, then
the equation would have led to
Supp. (4Z) (I) = to {2 (4Z ■) — (4Z /)} + tc (4Z •) ;
and this, as in fact already remarked, is the course of treatment employed in the
remaining cases. It is to be observed also that the equation may for shortness be
written in the form _
(4 Z) {(1)-2(2to-1)(1)}
+ Supp. (I)=(T)2D;
viz. the (4Z) is to be understood as accompanying and forming part of each symbol;
and the like in other cases.
109. We have the series of equations
(4Z) {(1) - (I) (2to - 1) - (I) (2to -1)}
-i- Supp. (1) = (1) ;
1 The expression a point-pair is regarded as equivalent to and standing for that of a coincident line-pair.
see First Memoir, No. 30.
