384] 
ON THE TRANSFORMATION OF PLANE CURVES. 
3 
°f (£ V> 0> connected by an equation of the form (£ (£, vY~ 0. Such an equation, 
treating therein (£, rj, £j as coordinates, belongs to a curve of the order 2/a, with a 
/¿-tuple point at (£ = 0, £=0), a /¿-tuple point at (?? = 0, £=0), and which has besides 
(/a — 2) 2 or (/a - 1) (/a — 3) dps, according as D = 2/a — 3, or 2/a — 2. The coordinates (x:y:z) 
of a point of the given curve are expressible rationally in terms of the coordinates 
: V '• 0 °f a, point on the new curve ; and we may say that the original curve is 
by means of the equations which give (x : y : z) in terms of (£:??: £) transformed 
into the new curve. 
11. A curve of the order 2/a may have £ (2/a—1) (2/a — 2), = 2/a 2 — 3/t + 1 dps; 
hence in the new curve, observing that the /¿-tuple points each count for § (/¿ 2 — /a) dps, 
we have 
In the case D = 2/a — 3, 
Deficiency = 2/a 2 — 3/a + 1 
-/¿ 2 + /A 
— /a 2 + 4/a — 4 
= 2/a — 3, = D 
In the case D = 
Deficiency = 
2/a — 2, 
2/a- — 3/A -f-1 
-/a 2 + /a 
— /A 2 + 4/A — 3 
2/a —2, =D 
Moreover for D = 0, the transformed curve is a conic, with 0 dps, and therefore with 
deficiency = 0; in the case D = 1, it is a quartic with 2 dps, and therefore deficiency 
= 2; in the case D = 2 it is a quintic with a triple point = 3, and a double point 
= 1, together 4 dps, and therefore deficiency =2. Hence in every case the new curve 
has the same deficiency as the original curve. 
12. The theorem thus is that the given curve of the order n, with deficiency D, 
may be rationally transformed into a curve of an order depending only on the 
deficiency, and having the same deficiency with the given curve, viz.: D = 0, the new 
curve is of the order 2(=D-f2); D = l, it is of the order 4(=D + 3); D — 2, it is 
of the order 5 (= D + 3); and D> 2, it is for JD odd, of the order D + 3; and for _D 
even, of the order D + 2. It will presently appear that these are not the lowest values 
which it is possible to give to the order of the new curve. Riemann’s object was, 
not that the order of the transformed curve might be as low as possible, but that the 
equation in (£, rj) might be in each of these parameters separately of the lowest 
possible order; and this he effected by giving to the transformed curve the two /A-tuple 
points. 
13. It is to be noticed that the theorem that for any rational transformation of 
one curve into another the two curves have the same deficiency is in effect given (as 
a consequence of Riemann’s theory) by Clebsch in the Paper, “Ueber die Singularitäten 
algebraischer Curven,” Grelle, t. lxiv., pp. 98—100. I have, by the assistance of a 
formula communicated to me by Dr Salmon, obtained a direct analytical demonstration 
of this theorem. 
14. I remark that (x, y, z) being connected by an equation, if (x : y : z) are 
given rationally in terms of (£ : v : 0> th en if follows that (£ : V '• 0 are also 
1—2