Arthur Cayley

The Collected Mathematical Papers of Arthur Cayley.vol. 6

From the Proceedings of theL ondon Mathematical Society, vol. I. (186.5 1866), No. ill. pp. 111. Read Oct. 16, 1865.] 1. The expression a double point, or, as I shall for shortness call it, a dp, is to be throughout underetood to include a cusp: thus, if a curve has Snodes (or double points in the restricted sense of the expression) and kcusps, it is here regarded as having +dps. 2. It was remarked by Cramer, in hisT hdorie des Lignes Courbes (1750), that a curve of the order nhas at most (ra l)(w 2), =(w 3w) +l, dps. 3. For several years past it has further been known that a curve such that the coordinates {x :y :z) of any point thereof are as rational and integral functions of the order nof a variable parameter ,is a curve of the order nhaving this maximum number J(n l)(n 2) of dps. 4. The converse theorem is also true, viz.: in a curve of the order n, with J(n l)(n 2) dps, the coordinates {x :y :z) oi any point are as rational and integral functions of the order nof a variable parameter 6or, somewhat less precisely, the coordinates are expressible rationally in terms of a parameter 6. 5. The foregoing theorem, as a particular case of Riemann sgeneral theorem, to be presently referred to, dates from the year 1857 ;but it was first explicitly stated only last year (1864) by Clebsch, in the Paper, Ueber diejenigen ebenen Curven deren Coordinaten rationale Functionen eines Parameters sind, Crelle, t. LX iv. (1864), pp. 43 63. 6. The demonstration is, in fact, very simple; it depends merely on the remark that we may, through the (n 1) (2) dps, and through 2n 3other points on the given curve of the order n, together {n +n)
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