The class of problems treated is one to which considerable attention has been paid of late, as for instance by Glaisher and others in this country, and by Mannheim, &c, abroad.

Two years ago (Proc. 1879, p. 200), in connection with Minding's Theorem, I investigated the space-locus of the feet of perpendiculars from the origin on the rays of a complex. These were found to fill the space bounded by the two sheets of the reciprocal of a Fresnel's wave-surface. The method I employed is capable of very extended application in the same direction, and the following general process is applicable to all the problems I have seen treated by the authors above referred to.