250 Geometrical Propositions 
two apsides 4, 4’, with respect to the point O, except when the plane is perpendicular 
to OC. Hence if we draw the right line Oaa’ perpendicular to 404’, taking Ga, Oa’, 
always equal to OA, OA’, the points a, a’, will describe a section of the apsidal surface. 
This section will evidently consist of two circles C’S'; C" S”, equal to the circle CS, 
and having their centres C’, C”, on the opposite sides of O in aright line C'OC" per- 
pendicular to OC; the distances OC, OC’, OC" being equal. The circles C'S’, C" 8”, 
intersect in two points m, ’, on the line OC and have two common tangents di, d’7’, 
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 tangents, 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 7,n’, and with circles of contact described by the radii 
ed, cd. 
The section of the sphere by a plane passing through O at right angles to On, is a 
circle of which Ois the centre. If therefore we suppose that the point » answers to 
ain Prop. VL, the apsis A corresponding to 7 will be indeterminate, and the position 
of the tangent plane at 7 will also be indeterminate, which ought to be the case at a 
node. 2 
The surface reciprocal to the sphere, the pole being at O, is evidently a surface of 
revolution about the axis OC (it is easily shown to be a spheroid haying a focus at O) ; 
and the section of this reciprocal surface, by a plane perpendicular to the axis at Q, is 
a circle of which Ois 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 cir- 
cular section; which agrees with what we have already seen. 
29. When the point O is without the sphere, the axis OC will pass between the cir- 
cles C'S’, CO" S", without intersecting either of them. The apsidal surface, described 
by the revolution of one of these circles about OC, will be a circular rng. The nodes 
have disappeared ; but the circles of contact still exist, as is evident. 
Part Il.—On tHe Wave Tueory or Licurt. 
30. Some of the foregoing propositions lead to a simple transformation of the 
wave theory of light. 
In this theory, the surface of waves, or the wave surface, is a geometrical surface 
used to determine the directions and velocities of refracted or reflected rays; being 
the surface of a sphere in a singly refracting medium ; a double surface, or a surface 
of two sheets, in a doubly refracting medium; a surface of three sheets on the sup- 
position of triple refraction 5 and having always a centre O round which it is sym- 
metrical. The radii of the wave surface, drawn from its centre O in different di- 
rections, represent the velocities of rays to which they are parallel. 
