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.

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

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

Linear Ordinary Differential Equations with Constant Coefficients: Identification of Boole's Integral with that of Cauchy

We consider an equation with constant coefficientswhere a≠0 and f(x) is continuous in a suitable interval. Suppose that the symbolic polynomial P(D) has been fully decomposed into its (real or complex) linear factors, so that the equation may be writtenwhere b1, …, bq are distinct, and m1+…+mq = n. The Complementary Function being now known, we may write down a particular integral of (1) by Cauchy's method.