o 
ON THE TRANSFORMATION OF PLANE CURVES. 
[384 
a series of curves of the order n — 1, given by an equation U+6V = 0, containing an 
arbitrary parameter 0; any such curve intersects the given curve in the dps, each 
counting as two points, in the 2n — 3 points, and in one other point; hence, as there 
is only one variable point of intersection, the coordinates of this point, viz., the coordi 
nates of an arbitrary point on the given curve, are expressible rationally in terms of 
the parameter 6. The demonstration may also be effected in a similar manner by 
means of curves of the order n — 2. 
7. Before going further, it will be convenient to introduce the term “ Deficiency,” 
viz., a curve of the order n with \ {n — l){n—2) — B dps, is said to have a deficiency 
= I): the foregoing theorem is that for curves with a deficiency = 0, the coordinates 
are expressible rationally in terms of a parameter 0. Since in such a curve the 
different points succeed each other in a certain definite order, viz., in the order 
obtained by giving to the parameter its different real values from —oo to co, the 
curve may be termed a unicursal curve. 
8. Riemann’s general theorem, as applied to plane curves, is stated, but not in 
its complete form, by Schwarz, in the Paper, “ De superficiebus in planum explica 
bilibus primorum septem ordinum,” Crelle, t. lxiv. (1864), pp. 1—17 : to complete the 
enunciation it is necessary to refer to page 137 of Riemann’s own Paper, “ Theorie 
der Abelschen Functionen,” Crelle, t. Liv. (1857), pp. 115—155, viz., the enunciation 
will be: 
9. For a curve of any order with a given deficiency B, the coordinates may be 
expressed as follows: 
B = 0, rationally in terms of a parameter 0, or what comes to the same thing, 
rationally in terms of the parameters (£, v), connected by an equation of 
the form (1, £)(1, 77) = 0. 
B > 0, rationally in terms of the parameters (£, 77) connected by an equation of 
a certain form, viz.: 
B — 1, the equation is (1, £) 2 ( 1, 77) 2 =0, or (what comes to the same thing) 
77 is the square root of a quartic function of 
B = 2, the equation is (1, £) 3 (1, t?) 2 = 0, or (what comes to the same thing) 
77 is the square root of a sextic function of £. 
B > 2, viz.: 
B odd, = 2^—3, the equation is (1, 1)^(1, rjY = 0. and is besides such, 
that treating (£, 77) as Cartesian coordinates, the curve thereby 
represented has (7x — 2) 2 dps. 
B even, = 2/j, — 2, the equation is (1, £^(1, 77)^ = 0, and is besides such, 
that treating (£, 77) as Cartesian coordinates, the curve thereby 
represented has (/m — 1) (/x — 3) dps. 
10. To see more clearly the meaning of this, write 
that the coordinates (x : y : z) are expressible rationally 
- , in place of 
V £ 
and homogeneously 
il 
in 
77), so 
terms