2 The Double Refraction of Light in a Crystallized 



parts of his theory. For this purpose it will be convenient to 

 premise the following Geometrical Lemmas : 



1. If a, b, c, be the semiaxes of an ellipsoid, and a, /3, 7, 

 the angles which they make with a perpendicular from the centre 

 on a tangent plane, the square of the perpendicular will be 



equal to 



a 1 cos 2 a + b 2 cos* |3 + c z cos 2 y. 



Let a plane through the point of contact Q, and one of the 

 semiaxes OA, intersect the ellipsoid in 

 the ellipse A ON, and the tangent plane 

 in the tangent QL, and draw QM per- 

 pendicular to OA ; then OA is a semi- 

 axis of the ellipse A ON, and therefore is 



a mean proportional between OM and 



2 



OL ; whence OL = , denoting OM by Fig. i. 



*c 



x. But if p denote the length of the perpendicular from the 



centre on the tangent plane at Q, the cosine of the angle 



*j 

 which it makes with OA will be equal to -^., and therefore 



Similarly, 

 Hence 



px px 



cos o = ^-r, and a cos a = . 



o py pz 



o cos p = -^-. and c cos y = . 

 o c 



fy yZ g2\ 



2 cos 1 a + b z cos 2 Q + c 2 cos 2 y = 2 + J + } = p*. 



\a* b 2 c z j 



Cor. Since x, y, z, are as the cosines of the angles which OQ 

 makes with the semiaxes, it appears from the demonstration that 

 the cosines of the angles which the perpendicular to a tangent 

 plane makes with the semiaxes are, with respect to each other, 

 directly as the cosines of the angles which the semidiameter 

 through the point of contact makes with the semiaxes, and in- 

 versely as the squares of the semiaxes themselves. 



2. If the semiaxes or, 6, c, and a', b', c', of two concentric 



