6 
ON THE TRANSFORMATION OF PLANE CURVES. 
[384 
the order D + 2; and if I) > 2, then the given curve can be by a transformation of 
the order n — 3 transformed to a curve of the order D +1 : the transformed curve 
having in each case the same deficiency D as the original curve. 
23. In particular, if D = 1, a curve of the order n with deficiency 1, or with 
^ (n 2 - on) dps, can be transformed into a cubic curve with the same deficiency, that 
is with 0 dps; or the given curve can be transformed into a cubic. This case is 
discussed by Clebsch in the Memoir “ Ueber diejenigen Curven deren Coordinaten 
elliptische Functionen eines Parameters sind,” Crelle, t. lxiv., pp. 210—271. And he 
has there given in relation to it a theorem which I establish as follows: 
24. Using the transformation of the order n — 1, if besides the 2n + D — 4 (= 2n — 3) 
points on the given curve U = 0, we consider another point 0 on the curve, then we 
may, through the ^ (n 2 — 3n) dps, the 2n — 3 points and the point 0, draw a series of 
curves of the order n — 1, viz., if P 0 , Q 0 , P 0 , are what the functions P, Q, R, become 
on substituting therein for (as, y, z), the coordinates (x 0 , y 0 , z 0 ) of the given point 0, 
then the equation of any such curve will be aP + bQ + cR — 0, with the relation 
oP 0 + bQ 0 + cR 0 between the parameters a, b, c ; or (what is the same thing) eliminating 
c, the equation will be a (PP 0 — PoP) + b (QR 0 — Q n R) = 0, which contains the single 
arbitrary parameter a : b. In the cubic which is the transformation of the given curve 
we have a point 0' corresponding to 0 and if (£ 0 , rj 0 , £"„) be the coordinates of this 
point, then corresponding to the series of curves of the order n — 1, we have a series 
of lines through the point 0' of the cubic, viz., the lines a^ + br]+c^= 0 with the 
relation a£ 0 + br% + c£ 0 = 0 between the parameters; or, what is the same thing, we 
have the series of lines a — ££ 0 ) + b (77^, — £V/ 0 ) = 0, containing the same single 
parameter a : b. By determining this parameter, the curves of the order n — 1, will 
be the curves of this order through the dps, the 2n — 3 points, and the point 0, 
which touch the given curve U — 0; and the lines will be the tangents to the cubic 
from the point 0'; as the number of tangents to a cubic from a point on the cubic 
is = 4, it is clear that the values of the parameter a : b will be determined by a 
certain quartic equation; and there will of course be 4 tangent curves of the order 
«—1 corresponding to the 4 tangents to the cubic. Now by Dr Salmon’s anharmonic 
property of the tangents of a cubic, if on the cubic we vary the position of the 
point O', the absolute invariant P J 2 of the quartic in (a : b) remains unaltered; 
that is the absolute invariant P J 2 of the quartic which determines the 4 tangent 
curves of the order n— 1 is independent of the position of the point 0 on the given 
curve U = 0, and since the tangent curves in question have the same relation to 
each of the 2n — 3 points and to the point O, it follows that the invariant is also inde 
pendent of the position of each of the 2n — 3 points; that is, we have the following 
theorem, viz. : 
25. Considering a curve of the order n with deficiency = 1; we may, through the 
(11 2 — 3n) dps, and through any 2n — 2 points on the curve, draw so as to touch the 
curve, four curves of the order n — 1; viz., these are given by an equation aP' + bQ' = 0, 
where the ratio a : b is determined by a certain quartic equation (*$a, 6) 4 = 0; then 
theorem, the absolute invariant P-j-P of the quartic function, is independent of the