The theory of relativity leads to a relation between matter and radiant energy such that a mass m is equivalent to a quantity of energy E given by

in which c is the velocity of light. According to the quantum theory a photon of wavelength λ has energy E given by

The general problem may be stated as : Given a plane polygon A1A2 … An, to find the polygon inside A1 … An which minimises the perimeter S of , subject to a condition

Prologue. This article began in an attempt, for purposes which shall be nameless to find the result of inverting a triangle and associated lines and circles with respect to its nine-points circle. The work outran its aim, and I now present it from a quite different starting-point.

It is proposed to consider all triangles congruent with a given triangle and having their vertices on the sides (or sides produced) of the given one with the various alternative allocations of vertices and sides; and, conversely all triangles congruent to a given one, but whose sides (or sides produced pass through the vertices of the given one, again with the various allocations

To introduce the proposed rule we make use of it to settle the signs of terms in the expansion of determinants such as :

where the symbol for the element has superscript r denoting rows and subscript s denoting columns.