crystalline Reflexion and Refraction. 33 
we get, by subtracting the one from the other, 
225402 440-320 “00 
for the equation of the cone A which has its vertex at O, and passes through the 
curve of intersection. If OQ be equal to 7, it will be a side of this cone; and a 
plane touching the cone along OQ will make in the ellipsoid a section of which 
OQ will be a semiaxis; so that OT will be perpendicular to that plane, and 
equal in length tor. Therefore, as OQ describes the cone A, the right line OT 
describes another cone B reciprocal to A, and the point T describes the curve in 
which the wave-surface is intersected by the sphere above-mentioned ; this curve 
being a spherical ellipse, reciprocal to that which the point Q describes on the 
surface of the ellipsoid. The equation of the cone B is found from that of A, 
by changing the coefficients of the squares of the variables into their reciprocals, 
and is therefore 
Se 0 (11) 
which, of course, is also the equation of the wave-surface, if 7 be supposed to be 
the radius drawn from O to the point whose coordinates are x, y, z. Combining 
this equation with that of the sphere, we have 
Wi 
po = ap 
bl Bae sie es 12 
at maple eee cas (12) 
which represents a hyperboloid passing through the common intersection of the 
sphere, the cone B, and the wave-surface. 
Since the differences between the coefficients of the squares of the variables 
in the equation (10) are the same as the corresponding differences in the equation 
of the ellipsoid, the cone A has its planes of circular section coincident with those 
of the ellipsoid. The cone B, being reciprocal to A, has therefore its focal lines 
perpendicular to the circular sections of the ellipsoid. These focal lines are con- 
sequently the nodal diameters* of the wave-surface, that is, the diameters which 
pass through the points where the two sheets of that surface intersect each other. 
If the direction of OT cut the other sheet of the wave-surface in T’, and if 
* Tbid, vol. xvii. p. 247. 
VOL. XXI. F 
