to the Wave Theory of Light. 33 



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

 taking 0#, Oa', always equal to OA, OA f , 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 C8 9 

 and having their centres (7, (7", on the opposite sides of in a 

 right line C' OC" perpendicular to 0(7; the distances 0(7, 00', 

 00" being equal. The circles <7'S', 0"S", intersect in two 

 points n 9 n, on the line 0(7, and have two common tangents di, 

 d'i' 9 which are bisected at right angles by OC in the points c, c. 



28. Now let the circles C'S', (7"$", with their common tan- 

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

 about the axis 0(7, and we shall have the apsidal surface with 

 nodes at n 9 n' 9 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 9 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 0(7 (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 0(7 

 will pass between the circles (7'$', C"S" 9 without intersecting 

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

 of one of these circles about 0(7, will be a circular ring. The 

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

 is evident. 



