to the Wave Theory of Light. 33 



Hence, if we draw the right line Oaa' perpendicular to AOA', 

 taking Oa, Oa', always equal to OA, OA', the points a, a, will 

 describe a section of the apsidal surface. This section will evi- 

 dently consist of two circles C'S', C"S", equal to the circle CS, 

 and having their centres C', C", on the opposite sides of in a 

 right line C'OC" perpendicular to 00 ; the distances 00, OC', 

 OC" being equal. The circles C'S', C"S", intersect in two 

 points n, n, on the line OC, and have two common tangents di, 

 d'i', which are bisected at right angles by OC in the points c, c. 



28. Now let the circles C'S', C"S", with their common tan- 

 gents, or only one of the circles with the half tangents, revolve 

 about the axis OC, and we shall have the apsidal surface with 

 nodes at n, n, and with circles of contact described by the 

 radii cd, c'd'. 



The section of the sphere, by a plane passing through at 

 right angles to On, is a circle of which is the centre. If there- 

 fore we suppose that the point n answers to a in Prop. VI., the 

 apsis A corresponding to n will be indeterminate, and the posi- 

 tion of the tangent plane at n will also be indeterminate, which 

 ought to be the case at a node. 



The surface reciprocal to the sphere, the pole being at 0, is 

 evidently a surface of revolution about the axis OC (it is easily 

 shown to be a spheroid having a focus at 0} ; and the section of 

 this reciprocal surface, by a plane perpendicular to the axis at 0, 

 is a circle of which is the centre. This circumstance indicates 

 (15) that on the apsidal surface there is a curve of contact, whose 

 plane is parallel to the plane of circular section ; which agrees 

 with what we have already seen. 



29. When the point is without the sphere, the axis OC 

 will pass between the circles C'S', C"S", without intersecting 

 either of them. The apsidal surface, described by the revolution 

 of one of these circles about OC, will be a circular ring. The 

 nodes have disappeared ; but the circles of contact still exist, as 

 is evident. 



