770 
DR. PLUCKER ON A NEW GEOMETRY OE SPACE. 
that the cone in question of double refracted rays is not at all altered if its centre moves 
along a right line passing through the origin O. 
22. In the peculiar case where M lies within the section of the crystal xy all corre- 
sponding incident rays likewise meet in that same point, constituting the plane of inci- 
dence passing through OZ, and represented by 
y'x=a?y. 
Here the cone of refracted rays degenerates into a system of two planes, which after 
putting z' 0 =0., are represented by 
z'=0, I 
L (42) 
Xo{y-yo)=PX{x-oco)'\ 
The second of these equations represents the plane of refraction corresponding to the 
plane of incidence *. 
23. If M fall within one of both the other planes of coordinates XZ and YZ, the cone 
of double refracted rays likewise degenerates into two planes. 
24. Either by putting s' = 0 in (40), or, after having eliminated r and s between the three 
equations (37) and (38), byreplacing the remaining variables § and a by x and y, we obtain 
y 0 x-p 2 x°y=(l-(3*)xy (43) 
This equation represents, within xy, the trace of the cone of refracted rays which meet 
in M. It is an equilateral hyperbola, having its asymptotes parallel to OX and OY, and 
passing through the projection of M. The coordinates of its centre are 
whence 
V— l— /3 2 x ~ 
y_ _JL Vo, 
X /3 2 Xq 
P a *b 
1-/3 2 ’ 
As the equation (43) does not involve the constant z' 0 , we conclude that 
The cone of double refracted rays continually changes if its centre be moved along a 
right line parallel to OZ', but its trace within the section of the crystal always remains 
the same hyperbola. 
25. Secondly, we may determine the curve enveloped by refracted rays confined 
within any given plane. If the plane be 
tx-\- uy-\-vz-\-w=0. 
* In the present researches, the auxiliary ellipsoid E, which may he considered as described round any point 
of the section of the crystal, as well as the wave-surface itself, has no other signification than to indicate by 
its constants the molecular constitution of the crystal so far as the transmission of luminous vibrations is con- 
cerned. Our equations only containing the ratio of these constants, the ellipsoid E and its elliptical trace (41) 
may be supposed here to have any dimensions whatever. 
The last equation (42) represents the plane of refraction as it represents its trace within xy. It likewise 
represents, if the point M falls within the circumference of the ellipse (41), the normal to that curve in the 
point M. Hence is derived an elegant construction of the plane of refraction. 
If within xy round any point of incidence as centre the ellipse (41) be described, the traces of the planes, 
both of incidence and of refraction, are such two diameters of that ellipse, the second of which is parallel to the 
normal to it at the point where the first intersects it. 
