We shall use the notation Σnr= lr + 2r+…+ nr. Mr A. J. Gray suggested to me that n(n+ 1) is always a factor of Σnr, and that, in addition, 2n + 1 is a factor when r is even.

Much has been written on inequalities concerning positive definite matrices, but a new insight may be gained by examining inequalities from the standpoint of the inverse matrix. The standard inequality of Hölder can then be used in a more fruitful manner. This leads to some new results and a rediscovery of some known results.

Lazarsfeld proved a bound for the excess dimension of an intersection of irreducible and reduced schemes. Flenner and Vogel gave another approach for reduced, non-degenerate schemes which are connected in codimension one, using the intersection algorithm of Stückrad and Vogel and defining a new multiplicity k. Renschuch and Vogel considered a condition to ensure that there is no degeneration for more than two schemes. We define an integer which enables us to unify these methods. This allows us to generalize the result of Flenner and Vogel to non-reduced schemes by comparing the multiplicities j and k. Using this point of view we give applications to converses of Bézout's theorem; in particular we investigate the Cohen-Macaulay case.

1. The chief purpose of this paper is to demonstrate the existence of a plane quartic curve with eight undulations, an “undulation” being a point at which the tangent has four-point contact. It is shown that the curve

where (x, y, z) are homogeneous point-coordinates and f a constant, has undulations at the eight points

The curve has, in addition to these undulations, eight inflections which are; in general, distinct. But there are two geometrically different possibilities of their not being distinct, and in either instance they coincide in pairs at four further undulations. Thus two types of curve arise without any ordinary inflections at all, their 24 inflections coinciding in pairs at 12 undulations.

What I intend to put before you first is a graphical view of the solution of indeterminate equations of the third degree.

Not the least interesting portions of the wonderful “Mathematical Collections” of Pappus are those which reproduce parts of the νε⋯σεις, or the two lost books of Apollonius (247–205 B.C.). Pappus (c. 300 A.D.) writes:— “A line is said to verge (using Heath's translation) toward a point if, being produced, it reach the point,” and among other particular cases of the general problem he gives the following as treated by Apollonius:

Problem A: Between two lines, given in position, to place a straight line given in length and verging toward a given point.

We consider rational functions of the form fm(z) = zm/(z – p) which are analytic in |z|<p, p>1, and establish that the asymptotic distribution of the zeros of their Taylor sections and Lagrange interpolants at uniformly distributed nodes is similar. This notion is also illustrated computationally. We conjecture that a similar result can be expected for any function analytic in |z| < p.

[The present paper is a translation of the second part of the third book of Pappus's Mathematical Collection. Pappus's date is uncertain, but 300 a.d. may be taken as an approximation to it.

Throughout the translation I have used the word “progression” as a rendering of the Greek μεσότης, which has no English equivalent. The only other alternative was to employ the term mediety, from the Latin medietas.

The account of the various progressions given by Nicomachus, in his Arithmetical Introduction, differs somewhat from that of Pappus. I hope to have something to say about Nicomachus in a future paper.]

The theorems given in the present section are fundamental in the theory of triangles in multiple perspective. They are all perfectly well known, but are given here because without them the succeeding sections would be unintelligible.

Two triangles A1,A2A3, B1,B2B3, can be in perspective in six different ways, indicated by the following symbols, in which the A's are to be understood as connecting collinearly with the B's standing directly underneath.

The usual methods of proving the existence of regular polyhedra, as given in Wilson and in Todhunter, appear to most students somewhat difficult. It seemed worth while trying, therefore, whether a simpler or more direct proof could not be obtained. The following note shows how this may be done.

The purpose of this paper is to expand upon the results obtained in [4]. We consider the set H of differential equations

where p and r are continuous real-valued functions of period ω (ω being fixed throughout). The equation (1.1) is denoted by P or (p,r), and we regard H as the set of pairs of continuous functions of period ω.On H we define a norm:

In an earlier paper (5) a description was given in set-theoretic terms of the semigroup generated by the idempotents of a full transformation semigroup , one of the results being that if X is finite then every element of that is not bijective is expressible as a product of idempotents. In view of this it was natural to ask whether by analogy every singular square matrix is expressibleas a product of idempotent matrices. This is indeed the case, as was shown by J. A. Erdos (2). Magill (6) has considered products of idempotents in thesemigroup of all continuous self-maps of a topological space X, but a comparable characterization of products of idempotents in this case appears to be extremelydifficult, and no solution is available yet.

Any property of a curve which depends solely upon its form may be styled intrinsic. Thus the circle of curvature at a point, the relation between s and ψ, the envelope of the normals, the envelope of straight lines making a constant angle with the curve, etc., are all intrinsic properties. It is proposed to investigate a few of these properties in this paper.

This paper was based mainly on the results of an investigation which will appear in full in the Transactions of the Royal Society of Edinburgh. Incidentally, however, it led to a discussion of the question:—Find the law of density of a planet's atmosphere, supposing Boyle's law to be true for all pressures, and the temperature to be uniform throughout.

I.—Take the well-known identity

Now if we can transform 4wz into a square we shall have two square numbers whose sum is a square. This will be effected by taking w = p2, z = q2, for then 4wz = 4p2q2 = (2pq)2 and we have

See Mathematical Magazine, Vol. II., No. 5, p.

Theorem. There exist non-Abelian finitely presented lattice-ordered groups which are totally ordered. This disproves a previous conjecture of the author [5]