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In a recent paper (Math. Ann. vol. 114, p. 237), Jonas has given a very interesting new property of W-congruences. After some general remarks, this paper will give here new, easy, and purely projective demonstrations of these theorems. On the last page we state a new problem which seems to be important for the theory of W-congruences. We shall say that two surfaces, whose points are in a given one-to-one reciprocal correspondence, are congruence-transforms of each other if they are the focal surfaces of the congruence generated by the straight lines joining two corresponding points of the two surfaces. We shall denote by x, y, *. the points whose homogeneous projective coordinates are xi, yi, .** (i = 1, 2, 3, 4), by ax + by + * ** the point whose coordinates are axi + byi + * i i . If (p is a function of two parameters u, v, we denote by so", so,,, #uu, Or".,, ... the derivatives 8co/cu, 8cp/8v, l2V/8u2, 8 2V/8U8V, ... .When xi are functions of u, v, we denote by x", X, * the points whose coordinates are 8xi/8u, 8xi/8v, * . By (x, y, z, t) we denote the determinant

In a paper published in these Annals, Mr. V. G. Grove1 studies a relationship, between two congruences r and P related to each other by a transformation T. That is to say, by a correspondence, such that, corresponding lines of the congruences are not coplanar, the developables of the congruences correspond to each other, and there exist at least three transversal surfaces of each congruence whose tangent planes at their points of intersection with the line of that congruence, pass through the corresponding line of the other congruence. He finds that the transformation T is of two types. The asymptotic type is defined by (a) f = f = F = F = gO + gG = O with s = s = S = S and the conjugate type by (b) g = g = G = G = 0 where f, f, F, F, s, s etc. are the coefficients of the differential system of equations (1) of the paper of Mr. Grove, satisfied by the homogeneous projective coordinates of the focal points of the lines of the congruences, and u, v, are the parametric curves on the focal surfaces, which correspond to the developables of the congruences. In the asymptotic case, he finds that if one of the two congruences is a W congruence, the other is also a W congruence. In the conjugate type, he also considers the quadrilaterals formed by the focal points of one of the lines of a congruence, and by two points on the corresponding line of the other congruence, and obtains the following result. If any three of the sides of these quadrilaterals generate a W congruence, so does the fourth. The purpose of the present note is to show, that a transformation T of congruences is only of one type, and in this case, not only the congruences r, I are W congruences, but likewise the congruences of lines formed by the sides of the quadrilaterals above considered. In fact, according to the definition of a couple of stratifiable congruences,2 it results that two congruences related to each other by a transformation T, form a couple of stratifiable congruences with the correspondence of the developables. Let us consider the asymptotic type. According to a theorem of Fubini,2 if the developables of a couple of stratifiable congruences correspond, then the developables cut all the transversal surfaces in conjugate nets. It is easy to deduce from the equations (1), (2a), (2b), (8) of Grove's paper, that the fundamental quadratic form of a transversal surface generated by a point of a line of the congruence r for instance, is proportional to

constitute a fundamental set of simultaneous solutions. These four pairs of solutions may be grouped into two quadruples (yi, Y21 y89 y4) and (z1, z2, z3, z4), and we may interpret these two quadruples as the homogeneous coordinates of two points Py, and P, in space. The coordinates of Py and P2 being functions of x, as x varies P. and P, trace two curves Cy and C, called the integral curves of the system (A). Those points of these curves having the same values of x are said to be the corresponding points. If the corresponding points P. and P, be joined by straight line L., the locus of Ly, will be a ruled surface S, called the integrating ruled surface of the system (A). If the determinant D is not identically equal to zero, this ruled surface is then a non-developable ruled surface. Conversely, for every non-devolopable ruled surface there exists a system of differential equations of the form (A), and their coefficients can be deter

If y(i) *.. y(4) ZM' ... z(4) be interpreted as the homogeneous coordinates of two points P1y and P,, then, as u and v vary, these points will describe two surfaces, Sy and Sz (either or both of which may be degenerate), and the line P1 P, will generate a congruence whose focal surface consists of Sy and S. The ruled surfaces obtained by equating it or v to a constant are its developables. Any congruence whose focal surface consists of two distinct sheets can be studied from the projective point of view by this method. Any transformation of the form

1. In the theory of the transformations of many kinds of surfaces and in the theory of rectilinear W-congruences a fundamental r6le is played by Bianchi's permutability theorem. The conviction that it is possible to state a converse theorem has led me to the following calculations and results. In this paper I state also a problem which is not completely solved. We shall say that two surfaces are congruence-transforms of each other if the points of the two surfaces are in a one-to-one reciprocal correspondence and the two surfaces are the focal sheets of the congruence generated by the lines joining two corresponding points. Let the five points x, y, z, t, v be functions of two parameters u, v (that is, their projective homogeneous coordinates xi, yi, zi, ti, Ii, [i = 1, 2, 3, 4] are functions of u, v). As u, v vary, these points ordinarily generate five surfaces 2z, T, 2z, , 21 ,, We suppose that each of the surfaces I, 2y is a congruence transform of every one of the surfaces 2z, It, 2;,7. It may happen that the curved lines enveloped on 2x by the tangents (xz), (xt), (xq)2 correspond to the lines enveloped on 1, by the lines (yz), (yt), (ye). In this case every line tangent to the surface 2x intersects the corresponding line tangent to 2,, the lines (xy) joining corresponding points of 2x, 2, will pass through a fixed point w (whose coordinates wi are constant). By changing the factors of proportionality we can suppose

In analytic geometry of a fairly elementary sort, we frequently encounter the problem of solving simultaneously a number of linear equations, and a single quadratic equation. In homogeneous form this amounts to solving n 1 linear equations, and one irreducible quadratic in n + 1 homogeneous variables, or, to use the jargon of projective geometry, of finding the intersections of a straight line and a hyperquadric in a space of n dimensions. We all know that the problem can be solved, generally we content ourselves with saying that it is easy, or with writing down the single irrationality that enters into the solution. There do arise cases, however, when we must actually put the work through. We must actually have the coordinates of the intersections of a straight line and a conic, or those of the limiting points of a set of coaxal spheres, or those of the lines that intersect four skew lines in our space. The work of putting through the solution is somewhat repellant, yet when we search in the textbooks we fail to find there any solution in symmetrical form. Such, at least, has been the present writer's experience.* The fact is that every solution must be either unsymmetrical or involve arbitrary quantities dragged in from the outside. Of the two alternatives the latter is certainly preferable, and in the present paper a solution is given which seems to be about as compact and symmetrical as the problem will allow. The final form will be found in formulke (14) and (15). Let a point in n space have n + 1 homogeneous coordinates

1. In the terminology of the Italian School an algebraic variety is called "normal" if its system of hyperplane sections is complete. 0. Zariski applies the term "normal" to an algebraic variety whose associated ring of homogeneous coordinates is integrally closed.' The two concepts are not equivalent. Zariski refers to a variety which satisfies the former condition as "normal in the geometric sense" and to one which satisfies the latter condition as "normal in the arithmetic sense." A variety which is normal in the arithmetic sense is necessarily normal in the geometric sense. Moreover, an r-dimensional variety Vt which is normal in the arithmetic sense has no (r l)-dimehsional singularities. A variety which is normal in the geometric sense need not be normal in the arithmetic sense. For example, a plane quartic of genus two is normal in the geometric sense, but has a double point. A curve may be free from singularities and yet not normal in the arithmetic sense. A rational space quartic illustrates this possibility. The validity of all of these assertions is established in Z. The object of this note is to characterize geometrically those algebraic varieties which are normal in the arithmetic sense. To this end we propose the following theorem: A necessary and sufficient condition that the r-dimnensional algebraic variety Vt be normal in its ambient projective space Pn is that for every integer m the linear system cut out on V, by the hypersurfaces of order m in Pn be complete. In the course of the proof we need the notion of the "character of homogeneity" of an algebraic variety, introduced by Zariski in the paper Z. Let to * n be the homogeneous coordinates of the general point of a variety Vr in the projective space Pn. The underlying field of constants for Pn is assumed to be algebraically closed and of characteristic zero. We denote this field by K. The integral closure, K[o, P, , I '] of the ring of homogeneous coordinates K[t', , .. , in] in its quotient field 2* is denoted by 6*. Each of the quantities to, t, . , may be assumed to be homogeneous of positive degree.2 Zariski proves that there exist integers 6 with the following property.