crystalline Reflerion and Refraction. 31 
we must come to a different conclusion, and say that the vibrations of a polarised 
ray are parallel to its plane of polarisation. 
Conceive an ellipsoid with its centre at O, the common origin of the coordi- 
nates 7, y, 2, 2’, 7’, 2’; and let its semiaxes be parallel to x, y, 2; their lengths 
being equal to 4 > i respectively. From the identity of the condition (7) 
abc 
with that marked (¢’) in Lemma IIL, it is evident that the directions of 2’ and y/, 
when they are the two directions of vibration, coincide with the axes of the 
ellipse in which the plane of «’y' intersects the ellipsoid; yy 
and if the right line OR, meeting the ellipsoid in R, be the 
direction of 2’, we have 
Ss 
ae = a’ cos’a + Lb? cos’B +c? cosy, 
or, by (8), 
i 
OR 
so that OR is the reciprocal of the velocity with which the vibrations parallel to 
y are propagated. Thus we see that the vibrations parallel to either semiaxis of 
the elliptic section are propagated with a velocity which is measured by the 
reciprocal of the other semiaxis. 
Again, conceive an ellipsoid with its centre at O, and its semiaxes parallel to 
L, Ys 2, as before, but equal to a, b, ¢ respectively. Let this ellipsoid be touched 
in the point Q by a plane which cuts OR perpendicularly in P, and draw the 
right lines OQ, PQ. Then as the condition (7) is identical with that marked 
(1) in the corollary to Lemma IV., it follows that Oy’ (if we so call the direction 
of y’) is perpendicular to OQ, and also that Oy’ and OQ coincide with the axes 
of the elliptic section made in this ellipsoid by the plane QOy’, just as Oy’ and 
OR coincide with the axes of the section ROv/ in the first ellipsoid. The plane 
QOR is therefore perpendicular to Or/ and to the plane of the wave. More- 
over, we have 
= Ss 
(OP) = @ cos*a + 6’ cos’B + ¢? cos’y = 8’, 
so that OP is the reciprocal of OR, and is equal to the velocity s with which 
the wave is propagated when its vibrations are parallel to Oy/. 
