The differential geometry of circle systems has received a large amount of attention of late. The subject has been approached from two quite distinct sides. On the one hand we have Koenigs, Cosserat, Moore, Bompiani and others who fix their attention on what we may call the descriptive differential properties of circles. The totality of circles in three-dimensional space may be represented by an S6, Iying in an Ss, and this variety may be studied by the now familiar methods of projective diBerential geometry. The theorems so reached are invariant under the twenty-four parameter group of sphere transformations. The other class of writers, wherein we may include Bianchi, Tzitzeica, Guichard and Eisenhart, have confined themselves largely to congruences (two-parameter systems) of circles, frequently to normal congruences. The methods employed have been the general ones of differential geometry and the center of interest has been rather more in the axes of the circles than the circles themselves. It has seemed for some time to the present writer that the last word on these subjects had not by any means been written, and that by a different method of approach not only might we obtain simpler proofs of known theorems but discover a number of new theorems as well. The most interesting properties of circles are those which are invariant under inversion, or under the tenparameter group of conformal transformations of space; the best approach to this group is through the use of pentaspherical coordinates. It is true that some of the writers mentioned above have made use of these coordinates, and still more has been done by Darboux in his Theorie des Surfaces, yet the possibilities of these coordinates have been by no means exhausted, and in the present paper they are more systematically applied to problems in differential circle geometry than has been the case in the past. A circle may be regarded in two different aspects, either as a locus of points, or an envelop of spheres. The two points of view are, of course, closely related, but the change of emphasis leads naturally to rather different sets of theorems. The first section of the present article is devoted to preliminary formulse for points and spheres in pentaspherical coordinates, and certain