Plane Collineations

If in a complex projective plane a point P, with coordinate vector x, corresponds to a point p*, with coordinate vector x*, under a non-singular collineation, thenx* = Axwhere A is a non-singular 3×3 matrix, the coordinates and the elements of A being complex numbers.

Linkages for the Trisection of an Angle and Duplication of the Cube

In this note some linkage systems for trisecting an angle and for finding the cube root of a number are described. The models are easily made and are of considerable pedagogic value

An Alternative Proof of a Theorem on the Lebesgue Integral

The theorem concerned is the following: iff is continuous in [a, b], and f exists and is finite except at an enumerable set of points and Lebesgue integrable in [a, b], then

Some Properties of the Zeros of Bessel Functions

Let jnm be the mth positive zero of Jn(x) (n not necessarily integral). Then Relton (1), p. 59, has conjectured from numerical considerations that

On a Functional Equation

We have recently discussed in (1) the general solution of a certain functional equation arising in statistical thermodynamics, and we propose in this note to deal with another functional equation arising from the same source(2).The problem is to obtain the most general function f(x) which, for all positive integral values of m, n, satifies the functional equationwherewhere A is an arbitrary constant. It is the object of this note to prove that this is the only continuous solution.