DE. PLUCKEE ON A NEW GEOMETEY OE SPACE. 
771 
the equation of this curve will result from the combination of the equation of the 
complex 
sg=@ 2 rcr (38) 
with the two equations 
tr-\-us-\-v =0, 
tg-\-ua-\-w=b, 
expressing that a ray ( r , s, g, <r) falls within that plane. By eliminating r and g, we 
obtain 
ws— (3 2 v<r-\-(l — /3 2 )us(r=0, (44) 
^ and ^ ^ being the coordinates of the projection, within xy', of the refracted ray. 
The projection envelopes an hyperbola ; so does the ray itself within the given plane. 
The last equation (44) does not contain t, and therefore will not be altered if the given 
plane turns round its trace within YZ', represented by 
uy+vz' +w = 0 (45) 
Hence it follows that the projections of all refracted rays which meet that trace are 
tangents to the same hyperbola (44), the asymptotes of which are parallel to OY and 
OZ', and which especially is touched by the trace itself, with regard to which 
W <7 W 
U S V 
The refracted rays themselves are tangents to a hyperbolic cylinder having as base the 
hyperbola (44) and OX as axis. 
26. In order to particularize, let us, in the first instance, suppose that the trace (45) 
is parallel to OZ' and intersects OY in any point Q, OQ being equal to Then 
v being equal to zero, the equation (44) becomes 
(w+(l-/3>)s=0, 
indicating that the hyperbola of the general case degenerates into two points, falling 
within OY, one at an infinite distance, while the distance of the other (Q') from O is 
OQ'=,= - tz «- = i ^OQ. 
(46) 
Accordingly the hyperbolic cylinder degenerates into two right lines, met by all 
refracted rays. One of the two lines within the plane xy along which the crystal is cut 
is parallel to OX, and intersects OY in Q ', the other is infinitely distant. Hence all 
rays within a plane intersecting xz' along a trace (QZ' 0 ) parallel to OZ' are divided into 
two sets. The rays of one set being parallel to the plane xy may be here omitted. The 
rays of the other set meet in a fixed point of that same plane along which the crystal is 
cut. If the plane turns round its trace QZ„ the fixed point moves, within xy, parallel 
to OX, describing a right line Q'X 0 . Each ray meeting both right lines QZ' 0 and Q'X 0 
is a ray of the complex. 
