The importance of transformation is due to its effect on singularities. The fixed points Ai, A,, ... in 11 have no correspondents in //', but a point close to in a determinate direction has a determinate correspondent, and if a point describes a small circuit about Ai. the corresponding point in 1r describes a curve as', rational and of order p;; this is a fundamental curve of the second plane. If F passes k times through Ai, F' cuts ai' in k points; if the k directions through Ai are distinct these points are all separate, and thus the multiple point on F is dissipated. If any of the k tangents at Ai coincide, F' has contact or it may have a multiple point on ai' but of a less complex character than the original at Ai; and by repetitions of the process the singularity is made to disappear.
But new singularities may arise, due to fun damental curves in the first plane. Such funda mental curves might be defined by means of the reverted equations of transformation; but as it is not usually convenient actually to form these (at least in the case of a Riemann transforma tion), it is simpler to adopt an independent definition. An irreducible curve /3 that meets the transformation curves P only at their fixed points is a fundamental curve of the system; all points on such a curve, p; correspond to a single point B' of II', or, more precisely, to points close to B' in different directions. If then F meets p in h points, F' has at B' a multiple point of order h.
Finally, new multiple points, in general only simple nodes, arise on F' owing to the passage of F through associated points of the first plane (intersections of two p's). Such passages are indicated by intersections of F with its com panion curve f; hence they are in general in evitable.
By either transformation, Cremona or Rie mann, the multiple points on the given curve can be dissipated, whether the tangents are distinct or coincident. A Cremona transforma tion-system, however, always has fundamental curves, whose number is equal to that of the fundamental points (Bertini,
formation is accomplished by the direct substi tution of y', P2(x', yr, 11),IP3(x',y',11) for x, y, s. (Salmon, (Higher Plane Curves,' chap. 8 in the German translation by Fiedler; also references at end; Scott, (Quart. Jour.,> v. 29, pp. 329-381, 1898; v. 32, pp. 2M09-239,1900).
When a curve is subjected to a birational transformation of either kind, Cremona or Rie mann, the induced points of a group of inter sections. transform into the induced points of the transformed group; this affords one proof (Bertini) of the important theorem that two curves which are birationally connected are of the same genus. The converse is not true; curves of the same genus cannot necessarily be birationally transformed into one another. A curve of genus p depends in general on 3(p-1) characteristic constants, the so-called moduli; if these are equal for two curves, then the two are birationally equivalent. (Clebsch-Linde mann,
Closely connected with this is the trans formation of a curve into itself. This however is more conveniently considered as an inde pendent theory, that of correspondence of points on a curve. (Cayley,
An entirely distinct class of investigations deals with the form of curves (topology, analy sis situs). The method employed is usually variation of the coefficients, by which the curve is derived from a known reducible curve of the same order. In this manner Klein proved (