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1. Statement of the problem. It has been shown by Enriquest that if an algebraic surface contains a pencil of elliptic curves and also two curves cutting each curve of the pencil in two groups of points which are not equivalent and which do not possess equivalent multiples, the surface is invariant under a discontinuous infinite series of birational transformations. In a more recent memoir: the same author states that, with the exception of the case in which

It was shown by SCHWARZ t that no algehraic curve of genus greater than unity can remain invariatlt under a continuous group of birational transforma-tions. Later HURWITZ t showed that no such curve could belong to any birational group of infinite order. The corresponding theory for surfaces is by no means complete. WhileA those belonging to continuous groups have been determined, only a few isolated examples are known of surfaces having an infinite discontinuous group. § All the groups which have been discussed are illustrations of two principles, the first of which refers to quartic surfaces and will be considered in two parts, the latter including the former as a particular case; the second principle is appli-cable to a much wider category. I propose to discuss these principles and apply the second to the determination of an extended family of new surfaces having an infinite group.

Cubic varieties of 003 poillts in space of four dimensions have been extensively studied by CASTELNUOVO t and by SEGRE, t showing the close connection between the section of the enveloping cone by ordinary space (apparent contour) and the well knowll congruences of the second and third orders, with their focal surfaces. The treatment is exclusively synthetic and the methods there employed do not lend themselves readily to the discussion of particular cases. In the following paper I wish to call attention to two particular projections of the variety having nine double points. This variety is given a three page discussion in SEGRE'S first paper, containing most of the results of Articles 1, 2 3, 4, 6 of the present paper. The result of Art. 5 is stated by CASTELNUOVO in a footnote without proof, but no use is there made of it. The results in the remaining articles are new. I have made extensive use of all the papers cited, but as I shall proceed analytically, the development will be along diflerellt lines, and no knowledge of their contents will be assumed. @ Let r 1Z2z3 + \z4z5z6 ° define a cubic variety in space of four dimensions 4, where Ewt O. § Every plane defined by wi ° ( i 1, 2, 3 ) and wk-O (k 4, 5, 6 ) will lie entirely on r. These nine planes may be designated by av. Any two of these having neither subscript in common will intersect in just one point which is a double point on r. Since two values of i and of k have been used to define each point, it ulay be denoted by A with the remaining pair as subscripts. It follows immediately that r contains nine double ?oints; through every double point pass four planes and ill every plane lie four double points.

In a recent paper Dr. ROSENBLATT gives two interesting examples of algebraic surfaces which are invariant under an infinite discontinuous group of birational transformations and at the same time are not envelopes of quadric surfaces.t In an earlier paper I mentioned 1: that all surfaces belonging to such groups which have thus far been noticed were examples of surfaces defining an ordinaxy elliptic ( 2, 2 ) correspondence. Practically all the memoirs bearing on the problem are mentioned in the articles just named. The surfaces discussed by DR. ROSENBLATT have a pencil of elliptic curves, and the transformations are expressed by a linear transformation of the parameters u, v of elliptic functionsn in terms of which the coordinates of a point on either surface can be rationally expressed. The treatment is transcendental, the transformation is defined only for points on the surface and no explanation of the geometric meaning of the transformation is given. It is a curious fact that these transformations are birational for all space, that they have a simple geometric interpretation in terms of the (2 X 2 ) correspondence, and that surfaces of any order or of as high a geometric genus as may be desired can be constructed which are invariant under this group, or a larger group under which that discussed by Dr. Rosenblatt is a subgroup of infinite index. Consider the surface whose equation is of the form

1. Statement of problem. Although there are many isolated examples of involutorial space transformations, the only type which has been systematically invest'gated is the monoidal one.t A (2, 1) correspondence between two spaces (x) and (x') may be expressed by three equations, algebraic and homogeneous in x1, x2, X3, X4 and all linear and homogeneous in xl, x2, x3, x4. With a point P1 in (x) a unique point P' in (x') is associated, but to P' correspond two points P1 and P2. Thus, when either P1 or P2 is given the other is uniquely defined. The pairs of points in (x) thus determine an involutorial transformation I in the space (x) . By this method it is possible to discuss various types of involutions and to develop certain properties common to all. Every space involution previously mentioned and having a surface of invariant points is included as a particular case in the present scheme. 2. Equations of the transformation. Let the defining equations be

1. Statement of the problem. Given two algebraic curves, C (xl, x2, x3) C C(x) = 0 of genus p in the plane (x), and C'(x') = 0 of genus p' in the plane (x'). Suppose that to a point (y) on C correspond n' points (y') on C', and that to a point (y') on C' correspond n points (y) on C. The two curves C, C' are then said to be in (in, n') correspondence. It is the purpose of this paper to give some methods of constructing curves having such correspondences, and of obtaining the equations which define them. For certain positions of the point (y), two of the n' images on C' may coincide. Such a point is called a branch-point, and the image point that is counted twice is called a coincidence. If the number of branch-points on C is denoted by 'j, and on C' by -q', then we have by Zeuthen's formula