Abridgment of the circle diagram of a transmission network

The first circle diagrams of the character herein described were published almost simultaneously by Thielemans in Europe and Evans and Sels in this country. The diagram of Evans and Sels is not nearly as complete as Thielemans's, but is more easily constructed because of the mathematical methods employed to determine the circle centers and radii. The present paper is an elaboration of the work done by these two investigators. It coordinates the graphical and mathematical methods of construction and extends their application. The principal contributions of this article are incorporated in Tables I and II and in the paragraphs concerning geometrical checks that may be applied to the circle diagram. Tables I and II include formulas for determining the coefficients of a large number of circular loci which have not been heretofore constructed by mathematical methods. Other formulas given in Tables I and II are to be found elsewhere, but generally in a somewhat more complicated, although equivalent, form. The graphical checks that result from the geometrical properties of the circle diagram, as first investigated by Thielemans, have not heretofore been applied to diagrams derived by mathematical computations. By establishing the identity of the Thielemans and Evans and Sels diagrams, it has been possible to utilize the numerous geometrical properties of the diagram that Thielemans has worked out, as well as to make use of other graphical properties. With the aid of the information incorporated in the following paragraphs it is possible to construct a circle diagram on which may be drawn circles representing almost any conceivable locus. This construction is carried out with the aid of computations made from the relatively simple formulas incorporated in Tables I and II. Any errors in the mathematical or graphical work that lead to an incorrect diagram can be simply and quickly uncovered by applying the numerous geometrical checks that are given in the paper. The result is a diagram easy to obtain, almost error proof, and of extreme usefulness.