ON OPTICAL THEORIES. 215. 
the value of U in terms of wu, etc., it is shown that these approximate laws 
become, respectively, 
Ve=Vot(1+ S454 2... ) deuten oay oL(&) 
rie T 
V being the velocity, and Vo, «, etc. constants, while for w, the rotation 
produced by a length z of the substance, he finds 
ged Aelia Hx thi ae 
For the explanation of double refraction Boussinesq supposes that the 
constants in the above formula giving U, V, W in terms of u, v, w may 
be functions of the direction of displacement; but, arguing from the 
relative importance of A, C, and D in the ordinary theory of refraction 
(refraction is due to the existence of A, dispersion only to that of D), he 
supposes that we may toa first approximation treat C and D as constants, 
while we consider A as a function of the direction, and write for the 
three axes of symmetry, the existence of which is assumed, the values 
A(1 +a), A(1 + 6), and A(1 + y). 
This leads to the equations— 
d?u dé 
Le Lg 4 a6 2 
7B (1 + a) qe + L(1 + a) 77u 
2 
a = R(T on) el + 8) 4 ° (6) 
dt? da 
a dé 2 
Toy = K(1 + c) Fat Ut +c)V7w 
K, L, a, b, c being functions of the other constants. It is clear that 
these are the same equations as were given by Lord Rayleigh,! and 
which have been already considered. The wave surface they lead to 
is not Fresnel’s, at least if we suppose the vibrations to be necessarily 
transversal. 
By retaining the terms involving the coefficient B, the elliptic polari- 
sation produced by quartz in directions oblique to the axis is explained. 
The formula for the difference in velocity in the two elliptically polarised 
waves traversing the crystal in any given direction agrees closely with 
that given by MacCullagh. In this case the squares of the velocities parallel 
; i 4 2 
to the axis are given by the expression N(1 a aah while the ve- 
locities in a direction making an angle 6 with the axis depend on the 
equation 
Qh? 
72 
w=N + 
eae ily + 
+ 3a/ [Qt - sint 9 +87" {2N + QM — N)sinto t ] @ 
i 
1 See p. 179. 
