( 188 ) 
determinate, we now get what follows. Let from O perpendiculars 
be drawn to all possible wave-fronts belonging to the radius vector 
of a conic point as ray. Let this cone of perpendiculars be intersected by 
a plane normal to the ray, which plane intersects the perpendiculars 
on S, and S, in A and B; let C be a point of that intersection, 
so that OC is the normal to the wave-front, then C B indicates 
the direction of the electric and C A the direction of the magnetic 
force. The intersection is therefore an ellipse, of which AB is 
a diameter and which is similar to the sections of the ray-plane 
with the two reciprocal ellipsoids, and has its axes in the same 
directions as these sections. . (ixternal conic refraction). 
18. We see directly from the wave-surface that the ray for 
which the wave-front is indeterminate, and the wave-front for which 
the ray is indeterminate, do not belong together as ray and wave- 
front. The phenomena of internal and external conic refraction are 
therefore wholly separated from each other. The ellipse of contact 
encloses a conic point of the wave-surface. This ellipse is a spinodal 
curve of the wave-surface; the tangent-plane in one of its points 
intersects the surface in a curve with a double point and two 
coinciding tangents, in such a way however that the curve has 
not a cusp in the point of contact, as is generally the case on a 
spinodal curve, but that it consists of two coinciding curves. ‘The 
surface is everywhere convex-convex, the concave side turned to 0, 
except between the four conic points and the ellipses of contact, 
where the wave-surface is concave-convex. 
19. The existence of tangent planes which touch along an ellipse, 
and which we may call ellipse-tangent planes, can also be directly 
derived from that of the conic points, if we remember that the 
surface is dualistic with itself. As we have four conic points in the 
X Z-plane, so we have four ellipse-tangent planes through the point 
whose point-coordinates are equal to the plane-coordinates of the 
X Z-plane, i.e. through the point at infinite distance on the Y- 
axis; these planes will be parallel to the Y-axis. There are also 
four ellipse-tangent planes parallel to the X- and four parallel to 
the Z-axis, which are, however, unreal. Four unreal ellipse tangent 
planes through O correspond to the four unreal conic points at 
infinite distance. 
