Toroidal-Coordinate Integration Limits

In support of our accompanying discussion of the gravitational potential of a uniform-density circular torus, here we explain in detail what limits of integration must be specified in order to accurately determine the volume — and, hence also the total mass — of such a torus using toroidal coordinates.

Preamble

Apollonian Circles (schematic) (see also Wikipedia's Apollonian Circles)

Quantitative Illustration of Employed Toroidal Coordinate System

Schematic Zones
Zone I <math>~Z_0 > r_t</math> for any <math>~a</math>	Zone II <math>~r_t > Z_0 > 0</math> and <math>~a < (\varpi_t-r_t)</math>	Zone II <math>~r_t > Z_0 > 0</math> and <math>~a < (\varpi_t-r_t)</math>