DR. PLUCKER ON A NEW GEOMETRY OE SPACE. 
769 
In order to get a geometrical interpretation of these equations, let any refracted ray 
of the complex be projected in the ordinary way on the three planes of coordinates XY, 
XZ and YZ ; each axis of coordinates will be met by two of the three projections. The 
intercepts on OZ' are j and^; on OY, <r and on OX, g and — - 1 cr . Hence 
With regard to all rays of the complex , the two intercepts on each axis of coordinates 
are in the same ratio. 
For OZ', i. e. for the diameter of the ellipsoid E conjugate to the section of the 
crystal, this ratio is the ratio of the squares of the axes of the ellipse within this plane. 
For OY, i. e. for the shorter axis of this ellipse, it is equal to the square of its excentri- 
city ; for OX the greater axis equal to 
Finally, if any incident ray, without, he projected on the section xy of the crystal 
along OZ, i. e. perpendicularly, and one of the two corresponding refracted rays, within 
the crystal, along OZ', the projections thus obtained are the traces of the planes of inci- 
dence and of refraction, - and - indicatin 
g the trigonometrical tangents of the angles, 
between the two traces and the greater axis of the ellipse within the section xy. The 
ratio of the tangents is egual to the ratio of the squares of the axes of the ellipse. 
21. In order to get a general idea of the distribution of the refracted rays constituting 
the complex, we may determine first the cone formed by rays passing through any given 
point within the crystal. If M he this point and x 0 , y 0 , z' 0 its coordinates, the equations 
x Q =rz’ a +gf 
y 0 —sz' 0 -\-tr,\ 
are to be combined with the equation of the complex, which, on putting 
written thus, 
sg=j3V<r 
By eliminating g> and <r, we get 
x 0 s — (3 2 y 0 r = ( 1 — / 3 2 )z' 0 rs 
. . . (37) 
~=i 3, may be 
. . . (38) 
. . . (39) 
This equation shows that the locus of rays of the complex which pass through the point 
M is a cone of the second degree. Its equation in ordinary coordinates x, y, z' (z 1 being 
referred to OZ') is 
®«(y-yoX^-^)-^-^-^)=(i-W(*-*o)(y-y^ • • • ( 40 ) 
From this equation we immediately derive that, whatever may be the position of M 
within the crystal, the cone always contains three rays parallel to OX, OY, OZ', as well 
as a fourth ray passing through the origin O. Besides, the cone depends upon the only 
constant (3, the ratio of the two axes of the ellipse, here represented by 
f! , £ =1 
«n ^ bl ’ 
(41) 
along which xy is intersected by the third auxiliary ellipsoid E. 
The equation (39), only depending upon the ratio of the constants x 0 , y 0 , z 0 , shows 
