On Systems of Rays. 135 
Ac=0, Ar=0, (Q”) 
when we take for the plane of x y the tangent plane to the refiecting or refracting 
surface ; they show therefore that the components of normal slowness parallel to this 
tangent plane are not changed, which is a new and general form for the laws of 
reflexion and refraction. It is easy to combine this general theorem with F'Rresne1’s 
law of velocity, and-so to deduce new consequences from that law with respect to 
biaxal crystals. 
For this deduction, our theorem may be expressed as follows, 
ov ov a 
a fee ee nes 19 
o=A (a, s+ bag tox)» (R") 
in which v is the undulatory slowness of a ray considered as a homogeneous function 
of the first dimension of the cosines a B y of its inclinations to any three rectangular 
semiaxes a b c, while A refers to the changes produced by reflexion or refraction, the 
unaltered trinomial to which it is prefixed being the component of normal slowness in 
the direction of any line ¢ on the tangent plane of the reflecting or refracting surface, 
and a, b, c, being the cosines of the inclinations of this line to the semiaxes a bc: and 
in order to combine this theorem with the principles of 'rrsneL, we have only to 
suppose that the rectangular semiaxes a 6 c in each medium are the semiaxes of elas- 
ticity of that medium, and that the form of the function v is determined as in the 
twenty-seventh number. 
Thus, to calculate the refraction of light on entering from a vacuum into a biaxal 
crystal a 6 c bounded by a plane face F, we may denote by a, B, y, the cosines of 
the inclinations of the external or incident ray to two rectangular lines s, ¢ upon the 
face F, and to the inward normal, and we shall have the two equations following, 
q5=0, a shel gig (=ca, +1), +vc,), 
oa (3 oy 
(S") 
ov év én 
po=a > + bse + C, 3y (=ca,+7b, +v0,), 
which contain the required connexions between a, 3, y, anda y, that is, between 
the external and internal directions. In this manner we find in general two incident 
rays for one refracted, and two refracted for one incident ; because a given system of 
values of a B y, that is, a given direction of the internal ray, corresponds in general 
to two systems of values of the internal components of normal slowness o 7 v, and 
therefore to two systems of values of a, 3, y,, that is, to two external directions ; 
while, reciprocally, a given system of two linear relations between o, tr, v, deduced by 
(S") from a given external direction, corresponds in general to two directions of the 
internal ray. But there are two remarkable exceptions, connected with the two sets 
of lines of single velocity, and with the conoidal cusps and circles of contact on Fres- 
NEL’s wave. 
