1. Ratio tests for the convergence or divergence of infinite series of positive terms are well known; they are used in and out of season. On the other hand, ratio tests for infinite integrals are never used. What is the reason for this disparity between series and integrals?

Since the publication of my article on “The advantage of differentials in the technique of differentiation” both Dr H. A. Hayden and Prof. A. Oppenheim have kindly pointed out to me that there is a much shorter solution of the problem by partial derivatives than the one which I gave as Solution 2. The solution is as follows:—

If y and z are the independent variables we have, since

which gives the result required.

1. In a tetrahedron ABCD with its opposite edges perpendicular there are two tetrahedra which can be described as pedal tetrahedra. (1) the tetrahedron A0B0C0D0 where these points are the feet of the perpendiculars from ABCD on to the faces BCD.

In a recent paper, I showed how the properties of algebraic systems of circles in the (x, y) plane could be investigated by means of a representation in which to the circle x2 + y2 − 2px − 2qy + r = 0 there corresponds the point (p, q, r) in space of three dimensions. The plane of (x, y) may be considered to lie in the space (x, y, z), so that the centre of the mapped circle is the orthogonal projection of the representative point.

1. It is known that

(i) the line of striction of a ruled surface is the locus of points at which the geodesic curvatures of the orthogonal trajectories of the generators vanish,

(ii) if at each point of a curve C on a surface, a tangent to the surface is drawn, and these tangents generate a ruled surface of which C is the line of striction, then, if each tangent is turned through a constant angle α about its point of contact in the tangent plane, the new set of tangents also form a ruled surface with C as a line of striction.

1. Teachers tell me that in the final proof of the Remainder Theorem, where lurks the possible fallacy of dividing by x – h which cannot be zero, and then putting x equal to h, they adopt Hyslop's (Math. Notes 25, 1930) redefinition. I here suggest an approach and treatment which I do not find in the textbooks.

§ 1. The importance of proving inequalities of an essentially algebraic nature by “elementary” methods has been emphasised by Hardy (Prolegomena to a Chapter on Inequalities), and by Hardy, Littlewood and Polya (Inequalities). The object of this Note is to show how some of the results in the early stages of Number Theory can be obtained by making a minimum appeal to irrational numbers and the notion of a limit. We use the elementary notion of a logarithm to a base “a” > 1, and make no appeal to the exponential function. The Binomial Theorem is only used for a positive integer index. Our minimum appeal rests in the assumption that a bounded monotone sequence tends to a limit. We adopt throughout the usual notation. Finally, it need scarcely be added that the methods employed are not claimed to be new.

Given an infinity of polygons which form the boundary of a finite number of polyhedra, we shall consider the complex K consisting of the polyhedra, and of the faces, edges and vertices of the polygons. We consider only those cases in which the Eulerian Characteristic N of K is finite. Then if the mean number of sides meeting at a vertex is p, and the mean number of sides of a polygon is q, then