Account ^Huygens’ Theory of DouUe Refraction. 
through KT, and touching the spheroid, will touch it in a 
point of the ellipse HME. But this point is necessarily the 
point I which we are seeking, since the plane drawn through 
TK cannot touch the spheroid in more than one point. This 
point I is easily determined, since we have only to draw from 
T, which is in the plane of this ellipse, the tangent TI, in the 
manner already shewn. For the ellipse HME is given, of 
which CH and CM are conjugate semidiameters, because a 
straight line drawn from M parallel to HE touches the ellipse 
HME ; for a plane drawn through M, and parallel to the plane 
HDE, touches the spheroid in M, as appears from Vol. iii. 
p. 149, 150. Besides, the position of this ellipse with respect to 
the plane passing through RC and CK is also given, whence it 
will be easy to find the position of the refracted ray Cl with 
respect to RC, 
We may here observe, that the same ellipse HME serves to 
find the refractions of every other ray in the plane^ passing 
through RC and CK. Because every plane parallel to HF or 
TK, which shall touch the spheroid, will touch it in this ellipse 
by the lemma already demonstrated. 
Huygens next proceeds to shew, that by cutting the crystal 
in different directions, he found the unusual refraction to be ex- 
actly the same as the preceding theory indicated. 
In order to explain, says he, what these directions are, let 
ABKF, Fig. 8. be the principal section through the axis of the 
crystal ACK, on which will be the axis SS of a spheroidal 
wave of light, propagated in the crystal from the centre C, and 
the straight line PP, which bisects SS at right angles, will be 
one of its great diameters. 
But as in a natural section of the crystal, made by a plane 
GG, parallel to two of its opposite surfaces, the refraction of the 
surfaces is regulated by the hemispheroids GNG, as has been 
their diameters LD, PP, QQ parallel, their conjugate diameters BK, ON, AR will 
also be parallel : And the centres K, N, R being in the same diameter of the 
spheroid, the parallels B,K, ON, AR will be necessarily in the sani*3 plane v/hich 
passes by this diameter of the spheroid, and consequently the points B, 0, A will be 
in the same ellipse formed by the intersection of this plane, which was to be proved. 
It is manifest that the demonstration would be the same, if beside the points O, A, 
there were others in which the spheroid was touched by planes parallel to BM, 
