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

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

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 equation

where

where A is an arbitrary constant. It is the object of this note to prove that this is the only continuous solution.

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, then

x* = Ax

where A is a non-singular 3×3 matrix, the coordinates and the elements of A being complex numbers.

We consider an equation with constant coefficients

where 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 written

where 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.

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