In her paper (Gazette, May 1950, p. 94) on the roots of two types of determinantal equation, M. J. Moore obtained various results by elementary but somewhat complicated methods . In this note, these results are obtained by a simpler method, which moreover helps to exhibit their real significance. The essential facts are expressed by the following theorem:

THEOREM I. If a rational function of x has poles and zeros (including a possible pole or zero at infinity) which are all real, simple and strictly alternate, then its partial fraction expansion

has coefficients kr (including k∞ if non-zero) all of one sign, while its reciprocal has expansion coefficients all of the opposite sign.

In this note I give the important trigonometric relations between the Brocard and Steiner angles of the cross-section of a triangular prism and the angle which the plane of this cross-section makes with the plane cutting the prism in an equilateral triangle. To facilitate a comprehension of the whole problem, I also give a complete and brief solution of the known problem for the equilateral section of the prism, deducing the equations of the equilateral case from the equations for the general case in which the section is a triangle similar to a given triangle.

The solution of linear differential equations under given boundary and initial conditions has been greatly expedited in recent years by the use of the method of integral transforms. It is the purpose of this note to show that an entirely parallel method may be developed for the solution of linear difference equations and that this leads to a new view of the generating function of a sequence as a transform of the sequence, analogous to the Laplace transform of a function.