766 
DR. PLUCKER ON A NEW GEOMETRY OE SPACE. 
If especially the crystal be cut in such a way that xy become a circular section of the 
ellipsoid E, each ray grazing the surface of the crystal will be contained within the cor- 
responding plane of refraction. This plane therefore is easily obtained by means of the 
trace of the plane of incidence and the diameter OZ' of the ellipsoid E conjugate to its 
circular section xy. 
14. In the preceding numbers the plane of refraction has been determined without 
determining SS confined within it. This right line, passing through the infinitely distant 
pole of xy, is parallel to the diameter OZ' conjugate to xy and represented by the equa- 
tions (16), which by eliminating successively y and x may be replaced by the following 
ones, 
(B 2 — AC> + (BE— CD>=0,1 
• (B 2 -AC)y+(BD-AE>=0.j [ } 
The direction of SS being known, any one of its points, i. e. the pole of any plane passing 
through RR, will be sufficient to construct it. If the plane be parallel to the diameter 
just determined, its pole will fall within the plane xy, and may be also regarded as the 
pole of RR, with regard to the ellipse (22) along which this plane is intersected by E. 
The trace RR being represented by 
qy-\-jpx=w, 
where 
the two lines, the equations of which are 
(A^+By) ~ = 1, 
(Bx+Cy) ^=1, 
will meet in the pole mentioned. Hence, on denoting its coordinates by x° and y°, 
By-Cp 1 , 
X ~ B 2 -AC w 
PrAg.l. [ 
y B 2 — AC w J 
(24) 
Finally, the equations of SS thus obtained are 
x—x° y—y° z 
CD— BE = AE — BD B 2 — AC 
(25) 
In order to complete the construction of the two refracted rays, the points (M, M') 
in which SS meets the wave-surface O within the crystal are to be joined with O by 
means of two right lines OM and OM 7 . 
15. If rays of every direction meet the crystal in O, the corresponding wave-fronts in 
that moment when, within the crystal, the wave-surface O is formed, will envelope a 
sphere, 
