(55) 
call the ray-plane. It is easily seen that the reciprocal principal 
directions are a set of three conjugate diameters of the two reciprocal 
ellipsoids. 
Let the distances in the principal directions from O to the points 
of intersection with the electric ellipsoid be «, ¢, and 7, and to the 
points of intersection with the magnetic ellipsoid em, Em and mn. 
We use the same symbols but distinguished by accents, for the dis- 
tances along the reciprocal principal directions, cut off by the reei- 
procal ellipsoids. 
8 
Let us now put S= bi = by and iar bz and let us take 
Em m Nm 
bi > by > bs. If bj, bo and 53 all differ, which we shall assume, 
then the electric and the magnetic ellipsoid have only one set of 
three conjugate diameters. 
De Pads eae 
how Ue — oa) if D is a radius vector of the electric ellipsoid, 
7 
so that ee cos (1, 1) = 1 and in the same way &m &'m cos (1, 1) = 1. 
€ 1 17: WG 1 me 1 
Therefore é¢ ce = &mém and —-—— and likewise =p — = = 
Em ] 6 m by Hm bs 
So, if the bs are all different, it is also seen that the reciprocal ellip- 
soids have only one set of three conjugate diameters. 
We have seen that, for a given wave-front, 8 and D are found as 
common conjugate diameters of the sections of the wave-front with 
the electric and the magnetic ellipsoid. These directions are always 
determinate, except when the two sections are similar and have as 
centre of similitude. If this be the case, we shall call the section a 
section of similitude; there are two of these sections, both passing 
through the principal direction 2, which we shall call the middle 
principal direction, and their position is such that the planes through 
2 and 1 and through 2 and 3 are diametral planes for both, resp 
with 3 and 1 as direction of chords. If the wave-front is such a 
section of similitude, we can choose D arbitrarily, but then D is 
determined, being doubly conjugate to D. 
The same thing holds good for € and in the ray-plane; 
if this is a section of similitude of the two reciprocal ellipsoids, passing 
through the middle reciprocal principal direction 2', then € en £ 
are indeterminate in the plane; however, if one is given, the other 
may be found. 
It is easy to find: 
A principal direction is normal to the plane of the two not cor- 
responding reciprocal principal directions and vice versa. 
