Several formulas related to the zonal calculation system

Abstract

In the article “Extreme simplification of zonal calculations and crystallographic calculations in general,” I noted the extreme simplicity that calculations of spherical bipolar coordinates obtain if we take Miller’s formula as a basis. Now I will supplement the formulas derived then with those that relate to the calculation of the cotangents of the angles formed with any initial face of the belt (see note). We can conclude that the system of zonal calculations with bipolar coordinates, in its special simplicity, is applicable without the use of a sequential, recursive course of calculation, and directly to faces with arbitrary complex indices.