DE. PLtiCKEE ON A NEW GEOMETEY OF SPACE. 
767 
the radius of which is equal to unity. The locus of poles of the wave-fronts, if taken 
with regard to the ellipsoid E, is a new ellipsoid, which, referred to axes of coordinates 
directed along the axes of all auxiliary ellipsoids, is represented by the equation 
! 2 c 2 ~^a 2 6 2 
= 1 , 
aV+% 2 +cV=aW 
(26) 
Its axes are obtained by multiplying the axes of the second auxiliary ellipsoid (8), to 
which it is similar, by abc. 
16. The new fourth auxiliary ellipsoid (26) is fitted to connect the constructions of 
the refracted rays if, the section of the crystal remaining the same, the direction of the 
incident rays vary. Indeed a right line (MM') drawn through any point Y of the fourth 
ellipsoid (26) parallel to OZ', i. e. to the diameter conjugate to xy with regard to the 
third ellipsoid E, meets the wave-surface O, within the crystal, in two points M and M'. 
OM and OM' will be the two refracted rays corresponding to that incident ray which is 
perpendicular to the plane conjugate to OY. 
17. After this digression we resume our subject. 
Let xy be the section of a biaxal crystal and OZ perpendicular to it. Let a ray of any 
direction starting from any point of OZ meet the section of the crystal in a point the 
coordinates of which are 
Let 
x-%, y—a. 
x=pz+q, 1 
y=qz-\-a J 
(27) 
be the equations of the incident ray. 
obtain the following relation, 
Let 
P—t 
q a- 
In order to express that this ray meets OZ we 
(28) 
(29) 
x=rz+g, 1 
y=sz+<r ] 
be the equations of any one of the two corresponding refracted rays. Let us finally 
suppose that, without the crystal, z is negative, within it, positive. Accordingly in the 
equations of the incident ray, positive values of z, in the equations of the refracted rays, 
negative ones are to be rejected. 
Again, let 
0=0 
be the general equation of the wave-surface, and 
E=A^ 2 +2B^-j-C/+2D^+2E^+F2 2 -l=0 
the equation of the third auxiliary ellipsoid ; the position of both being determined by 
the position of the crystal with regard to the axes of coordinates. 
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