The communication on this subject, as originally made to the society, consisted of a series of theorems, giving (1) expressions for the radii of a great many sets of circles, (2) identities connecting several sets of these radii, and (3) miscellaneous identities closely related thereto. As, however, the paper culminated in a general theorem which may be looked upon as fundamental, and the proof of which makes evident the mode of arriving at the said expressions for radii, and as the relations connecting sets of radii are easily found when attention has been directed to their existence, I have thought it best to print little more than the fundamental theorem and a few auxiliary notes.