These curves are instances of unicursal bicircular quartics.
[In particular a curve and its reciprocal have this rational or (I, r) correspond ence, and it has been already seen that a curve and its reciprocal have the same deficiency.] A curve of a given order can in general be rationally transformed into a curve of a lower order; thus a curve of any order for which D=o, that is, a unicursal curve, can be transformed into a line; a curve of any order having the deficiency r or 2 can be rationally transformed into a curve of the order D+2, deficiency D; and a curve of any order deficience = or> 3 can be rationally transformed into a curve of the order D+3, deficiency D.
In particular if D =o, that is, if the given curve be unicursal, the transformed curve is a line, 4 is a mere linear function of 0, and the theorem is that the co-ordinates x, y, z of a point of the unicursal curve can be expressed as proportional to rational and integral functions of 0; it is easy to see that for a given curve of the order m, these functions of 0 must be of the same order m.
The theorem of united points in regard to points in a right line was given in a paper, June-July 1864, and it was extended to unicursal curves in a paper of the same series (March 1866), " Sur les courbes planes ou a double courbure dont les points peuvent se determiner individuellement - application du principe de correspondance dans la theorie de ces courbes."
The theorem is as follows: if in a unicursal curve two points have an (a, 0) correspondence, then the number of united points (or points each corresponding to itself) is=a+ (3.
In fact in a unicursal curve the co-ordinates of a point are given as proportional to rational and integral functions of a parameter, so that any point of the curve is determined uniquely by means of this parameter; that is, to each point of the curve corresponds one value of the parameter, and to each value of the parameter one point on the curve; and the (a, t3) correspondence between the two points is given by an equation of the form MO, I) u (4), 01 3 =0 between their parameters 0 and 4); at a united point 4)=0, and the value of 0 is given by an equation of the order a+ 0.
When D = o, the curve is said to be unicursal, when = i, bicursal, and so on.