768 
DR. PLtiCKER ON A NEW GEOMETRY OF SPACE. 
18. According to the footnote of [7], we have between the four constants j?, q, r, s, upon 
which the direction of the incident and the refracted ray depends, the following relation, 
(Aq-Bp)r+(Bq-Cj))s+(Dq-Ep)=0 (30) 
By means of (28) this equation may be transformed into the following one, 
(Ac — B%)r-\- (B<r — Q>)s + (D<r — E^>) = 0, (31) 
and then represents a complex of refracted rays. As no supposition is made regarding 
the position of the luminous point on OZ, the corresponding incident rays may start in 
every direction from all its points. They ’constitute therefore a complex of rays emanating 
from OZ, perpendicular to the section of the crystal, and considered as a luminous right 
line. This complex of incident rays, after entering the crystal, passes into the complex 
of double refracted rays represented by the last equation. 
19. By admitting that OX and OY, within the section of the crystal, were directed 
along the axes of the ellipse, along which xy is intersected by the ellipsoid E, the constant 
B disappears from the equation of the complex, which then may be written thus, 
(Ar+D)<r=(Cs+E)g> (32) 
We have hitherto supposed OZ to be perpendicular to xy , and will continue to do 
so for incident rays without the crystal ; but for the refracted rays entering it (the axes 
OX, OY, perpendicular to each other, remaining the same) the direction of OZ may 
be changed by replacing it by the diameter OZ' of the ellipsoid E, conjugate to xy. Then 
the constants D and E likewise disappear, and the equation of the complex assumes the 
most simple form, 
Arc=Csg. 
20. On denoting by a 0 and b 0 the two semiaxesof the ellipse along which xy is 
intersected by the ellipsoid E, we get 
A=- 2 > b=4- 
a o K 
We may suppose, too, that a 0 falling within OX, is greater than b Q falling within OY, 
a?—b 2 
whence the square of the excentricity of the ellipse e\ becomes 0 2 0 • 
a o 
After having introduced the new constants, the last equation may be written in the 
following ways, 
(34) 
sg — r<r 
s 
Besides, on observing that 
f _£ 5 
g p ’ 
blq 
a l P 
(35) 
(36) 
