ON DOUBLE REFRACTION. 259 
- Aceording to observation, in each of the principal planes the ray polarized 
in that plane obeys the ordinary law of refraction, and therefore if we suppose 
that in polarized light the vibrations, at least when strictly transversal, are 
perpendicular to the plane of polarization, we must assume that R+H=Q-+I, 
P+I=R+G, Q4+G=P-+H, which are equivalent to only two distinct rela- 
tions, namely 
Pi 6=0— BS Rar et oaks UR Sens Seles (3) 
For a wave parallel to one of the principal axes, as that of #, the direction 
of that axis is one of the three rectangular directions of vibration of the waves 
which are propagated independently. For such vibrations the velocity (v) of 
propagation is given by the formula 
v=m (R+H)+n°(Q+1), 
which by (3) is reduced to 
v=R+H=Q+4+1, 
so that on the assumption that the velocity of propagation is the same for a 
wave perpendicular to the axis of y as for one perpendicular to the axis of 
z when the vibrations are parallel to the axis of «, the law of ordinary re- 
fraction in the plane of yz follows from theory. 
For the two remaining waves which can be propagated independently in a 
given direction perpendicular to the axis of w, the vibrations are only approxi- 
mately normal and transversal respectively. In fact, for the three waves 
which can travel independently in any given direction, the directions of vibra- 
tion are not affected by the introduction of the constants expressing equili- 
brium-pressures, but only the velocities of propagation. The squares of the 
yelocities of propagation of the two waves above mentioned are given as be- 
fore by a quadratic; and in order that the velocity of propagation of the 
nearly transversal vibrations may be expressed by the formula 
PaO MOM criss yada a deeb da (4), 
in conformity with the ellipsoidal form of the extraordinary wave surface in 
a uniaxal crystal, and the assumed elliptic form of the section of one sheet of 
the waye-surface in a biaxal crystal by a principal plane, the quadratic in 
question must split into two rational factors, which leads to precisely the 
same condition as before, namely that expressed by the first of equations (2) ; 
and by equating to zero the corresponding factor, we get 
v’=(P+H) m*+(P+4+]) x’, 
which is in fact of the form (4). Applying the same to each of the other 
principal axes, we find again the three relations (2). 
Hence Cauchy’s second theory, in which it is supposed that in polarized 
light the vibrations (in air or in an isotropic medium) are perpendicular to 
the plane of polarization, leads like the first to laws of double refraction, and 
of the accompanying polarization, differing from those of Fresnel only by 
quantities which may be deemed insensible. This result is, however, in the 
present case only attained by the aid of two sets of forced relations, namely 
(2) and (3), that is, relations which there is nothing @ priori to indicate, and 
which are not the expression of any simple physical idea, but are obtained by 
forcing the theory, which in its original state is of a highly plastic nature 
from the number of arbitrary constants which it contains, to agree with 
observation in some particulars, which being done, theory by itself makes 
kmown the rest. As regards the third ray by which this theory like its pre- 
decessor is hampered, there is nearly as much to be urged against the present 
theory as the former. There is, however, this difference, that, as there are 
only five relations, (2) and (3), between nine arbitrary constants, there remains 
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