384] 
1 
384. 
ON THE TRANSFORMATION OF PLANE CURVES. 
[From the Proceedings of the London Mathematical Society, vol. I. (1865—1866), No. ill. 
pp. 1—11. Read Oct. 16, 1865.] 
1. The expression a “ double point,” or, as I shall for shortness call it, a “ dp,” 
is to be throughout understood to include a cusp: thus, if a curve has 8 nodes (or 
double points in the restricted sense of the expression) and k cusps, it is here regarded 
as having 8 + k dps. 
2. It was remarked by Cramer, in his “ Theorie des Lignes Courbes” (1750), 
that a curve of the order n has at most ^ (n — 1) (n — 2), = ^ (n 2 — 3n) + 1, 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 n of a variable parameter 6, is a curve of the order n having this maximum 
number ^ (n — 1) (n — 2) of dps. 
4. The converse theorem is also true, viz.: in a curve of the order n, with 
£ (n — 1) (n — 2) dps, the coordinates (x : y : z) of any point are as rational and integral 
functions of the order n of a variable parameter 6—or, somewhat less precisely, the 
coordinates are expressible rationally in terms of a parameter 6. 
5. The foregoing theorem, as a particular case of Riemann’s general 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,” Grelle, t. lxiv. (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) (n - 2) dps, and through 2n-S other points on the 
given curve of the order n, together (n 2 + n) — 2, (n — l)(?i-f 2) — 1, points, draw 
C. VI. 1