By a finite lattice† is meant any finite class having the following properties:

This paper is a sequel to one recently published, in which the regularity of surfaces was investigated by means of certain criteria suggested by Comessatti. We suppose throughout that the surface F under consideration is of general type in Sr, and that it has curve sections C of order n and genus π. It may be recalled that in the previous work the two most useful criteria were the following theorems:

If π < ½ (n + 2), then F is regular or referable. (I)

If πr (n) ≤ π ≤ n, where πr (n) is the maximum genus of a Cn in Sr, then F is regular or referable. (II)

The main object of this paper is to study the quadrics which have simultaneously certain poristic relations with a rational norm curve. We shall begin with a résumé of the work done in this direction in the ordinary space [3].

Let SsTt denote a space of s + t dimensions with the property that the square of the distance between any two points

is of the form

where s of the ε's are equal to + 1 and t of them to — 1.

Castelnuovo has shown that the maximum freedom of a linear system of curves of genus p is 3p + 5, and that a system with this maximum freedom consists of hyperelliptic curves which can be transformed into a system of curves of order n with an (n − 2)-ple base point and a certain number of double base points; the only exceptions being that when p = 3 the system may be transformable into that of all quartics, and when p = 1 the system may be transformable into that of all cubics. Further, since, in the transformed system, the characteristic series is a it is non-special and hence the redundancy of the base points is zero, therefore each of the double points of this system reduces the freedom by exactly three; and hence if we remove all the double points we get a system of curves of genus p′ and freedom r′ = 3p′ + 5 with only one base point. If we take this point for origin the system of curves can be represented by a single Newton polygon containing in its interior exactly p points, collinear since the curves are hyperelliptic (p ≠ 3), and containing on its boundary 2p + 6 points. From this we can deduce immediately a theorem concerning convex polygons drawn on squared paper; I shall now give an a priori proof of this theorem.

The general theory of surfaces has been developed without much attention to the simpler aspects of the subject and, in particular, to the problem of determining some elementary criteria of regularity or referability; there is, in fact, only one well-known criterion of this kind, due to Castelnuovo and Enriques. (Theorem 2 of the present work.) It seems desirable, therefore, to establish some tests which require no more knowledge of a surface than its order, sectional genus and normal space. A number of such tests is given in the present paper. Before it was written the author had the advantage of consulting an unpublished manuscript by Professor Comessatti on the classification of surfaces in S4; in this work the regularity of surfaces of order eight is considered and, in part, those of order nine; the four-dimensional case of Theorem 11 and a particular case of Theorem 6 and Theorem 14 are given. Theorems 4 and 5 are also proved by Comessatti. For permission to publish these results and for the help derived from them the author's grateful thanks are due.

In a previous paper the author has examined the various types of non-singular surfaces of sectional genus four; in the present work the same method is applied to non-singular surfaces of sectional genus five. The examination of this case completes the classification of non-singular surfaces in higher space as far as those of the seventh order; for a septimic surface of sectional genus six, necessarily normal in S4, must lie on a quadric, and its characters may be determined from this fact.

L. M. Milne-Thomson has recently described a method of solving linear finite difference equations by using processes some-what analogous to those employed by Heaviside.

In recent seismological work several questions have arisen concerning the construction of tables from data containing a certain amount of accidental error, and it seems that some of the methods may be of more general interest. The first is the problem of smoothing. A method suggested by Dr L. J. Comrie was as follows. Suppose that we have values of y for five equally spaced values of x, and that we wish to make the second differences vary as smoothly as possible. We try to find a cubic polynomial such that the sum of the squares of the deviations of y from it will be a minimum. It is found that the polynomial is less than the observed value of y at the centre of the range by of the central fourth difference; or approximately by of the fourth difference. I have since found that the method had previously been suggested by Sir G. H. Darwin. It gives a great improvement in the steadiness of the differences and thereby makes interpolation much easier.

1. If the canonical series of an algebraic curve of genus p is compounded of an involution of sets of points on the curve then the involution must be rational and of order two, and the canonical model is a repeated rational normal curve of order p − 1 with 2p + 2 branch points. An analogous question suggests itself for algebraic surfaces. Under what conditions is the canonical model of a surface, whose geometric genus is not less than two, a multiple surface, and what in this case are the properties of the simple surface on which the multiple surface is based? In this paper we give particular examples of multiple canonical surfaces and attempt to go some way towards the solution of the general problem.

The problem of the torsion of beams of ⊥- and L-cross-sections has received attention from very few authors despite its important technical applications. The first mathematical solution in this connection was obtained by F. Kötter in 1908 for an L-section both of whose arms are infinite. He attacked the problem by the use of the known solution of the rectangle and by application of the scheme of conformal transformation. Kötter's method, however, does not lend itself readily to the solution of the problem involving more than one re-entrant angle. The first solution for the torsion of a beam whose cross-section is a rectilinear polygon of n sides was published in 1921 by E. Trefftz who also applied his method to an infinite L-section. Recently I. S. Sokolnikoff has suggested a more general method depending upon the fundamental theorem of potential theory that a harmonic function is uniquely determined by the values assigned along the boundary of the region within which the harmonic function is sought, the boundary condition and the region being subject to certain well-known assumptions of continuity, connectivity, etc. As an illustration of his method he has given an approximate solution for a ⊥-section whose flange and web are both infinite.

1. Saks calls an integral complete if it satisfies two conditions, of which one, the condition (C), is as follows.

If f(x) is defined in (a, b) and is integrable in each interval (a + ε, b − η), where a < a + ε < b − η < b, and

exists, then f(x) is integrable in (a, b) and the integral over (a, b) is equal to the above limit.

Several papers have recently been published which include proofs of the following:

Theorem 1. Those trisecant planes of a rational normal quartic curve Cu which meet a second rational normal quartic curve Cv having six points in common with Cu also meet a third rational normal quartic curve Cw through these same six points. The three suffixes may be permuted in any way, the three curves forming a symmetrical set.

E. Trefftz has discussed the problem of the torsion of a beam whose cross-section is bounded by a polygon with the help of the Schwarz-Christoffel transformation given by

where a1, a2, …, an are external angles of the polygon in the w-plane, and ξ1, ξ2, …, ξn are the points on the real ξ-axis in the t-plane that correspond to the angular points of the polygon in the w-plane. In the case of regular polygons a further transformation of the upper half of the t-plane into the interior of a circle in the z-plane with the help of the transformation

greatly simplifies the problem, and some definite results can be obtained.

It is familiar that a “double plane”, i.e. an algebraic surface conceived as consisting of each point of a plane counted twice, is unambiguously determined when its “branch curve”, or locus of points in the plane which count as two coincident points of the surface (the general point of the plane counting as two distinct points), is assigned; the only condition which must be satisfied by this curve being that it is of even order. In fact, if f(x, y) = 0 is the Cartesian equation of the curve, the double plane can be regarded as the limit of the surface

where A is a large constant, or as the projection of this surface, for finite A, from the point at infinity on the z-axis. If, however, f = 0 is of odd order, it is easily seen that the line at infinity also forms part of the branch curve of the double plane. A repeated portion of the branch curve is ineffective, for if φ (x, y) = 0 is any other curve, the surface

is birationally identical with the former merely by putting

1. In two previous papers the regularity of non-singular surfaces in higher space was investigated by geometrical methods based on theorems which are themselves deducible by algebraic-geometrical methods. Among the results obtained, one of the most important was a consequence of the following theorem, due to Severi:

The general conception underlying the following analysis is that of a field of potential which is invariant under a group of geometrical transformations. The boundary is to consist of a number of parts which transform into each other and the boundary values also transform into each other. To satisfy the conditions we build up functions which are invariant under the same group of transformations and combine them to give the prescribed boundary values over one section of the boundary. The boundary conditions on the other sections are then automatically satisfied. When the groups are simply translations or rotations we get functions periodic in either a Cartesian or an angular coordinate.

The quantum theory of the electron allows states of negative kinetic energy as well as the usual states of positive kinetic energy and also allows transitions from one kind of state to the other. Now particles in states of negative kinetic energy are never observed in practice. We can get over this discrepancy between theory and observation by assuming that, in the world as we know it, nearly all the states of negative kinetic energy are occupied, with one electron in each state in accordance with Pauli's exclusion principle, and that the distribution of negative-energy electrons is unobservable to us on account of its uniformity. Any unoccupied negative-energy states would be observable to us, as holes in the distribution of negative-energy electrons, but these holes would appear as particles with positive kinetic energy and thus not as things foreign to all our experience. It seems reasonable and in agreement with all the facts known at present to identify these holes with the recently discovered positrons and thus to obtain a theory of the positron.

1. In a paper by Castelnuovo and Enriques the following fundamental theorem is proved:.

“An algebraic surface containing a linear system of curves of freedom r ≥ 0, genus π > 0, and grade n > 2π − 2 is necessarily scrollar.”

The aim of the following exposition is to give some idea of the scope of modern algebra in the light of the theory of ideals. It consists for the most part of definitions, examples, and the statement of theorems without proof; but the selection is severely restricted even in idealtheory. Thus no reference is made to the highly important matter of homomorphic, or multiply isomorphic, correspondence and its consequences.