ON OPTICAL THEORIES. 173: 
and if Green’s arguments as to the relative magnitude of the constants be 
still supposed to hold, the quasi-normal wave will disappear, and the 
vibrations will be very nearly indeed transversal. The theory, however, 
interesting as it is, does not enable us to overcome the difficulty of 
reconciling the theories of double refraction and reflexion so long as we 
adopt the view of Fresnel and Green, that the latter depends on difference 
of density, not of rigidity, in the two media. It is also open to the 
objection that if the medium be incompressible the displacements must 
be in the wave front, and we must get in this case Green’s conditions, 
not the above ; while if the medium be not incompressible an appreciable: 
amount of energy must exist in the form of longitudinal vibrations. 
§ 6. The question of the propagation of waves through an isotropic 
medium, which is rendered anisotropic by the production of three elonga- 
tions, a, b, c, in three rectangular directions, has been studied by 
Boussinesq.! The elastic constants are taken to be linear functions of 
these permanent strains, and the number of constants involved in their 
expression is reduced from the considerations involved in the symmetry 
of the medium and the principle of the conservation of energy, 
The equations of motion may be written 
d*u dé 
aa Q+Na)7 + (+ pa) 7 2u 
rere ine du d*u du \ 
Peg apd 
with the condition implied by the principle of the conservation of energy 
that \’/= v, while if the normal stresses in the equilibrium condition 
vanish =p. These may be deduced from Green’s equations by putting 
A=(c—p)a, B=(o-p)b, C=(e —p)e, 
G=A+p+2(p+y)a, 
L=p+p(b+e), 
P=d—-p+(y—p)(b+c), 
with similar expressions for the other constants. \ and p are the two 
elastic constants of the unstrained medium in the form in which they 
are written by Lamé, /A and (A+ ,) being the velocities of trans- 
verse and normal waves respectively, the density being taken as unity. 
It is thus shown that on the assumption that a, b, c are small quan- 
tities, such that their squares and products may be neglected, Fresnel’s 
wave surface is given ifeithero =0 ors =p. In fact, the condition « = 0 
leads to Fresnel’s surface without any assumption as to the value of 
a, b, c, for then the theory becomes identical with Green’s second theory ; 
while if c=p we have either St. Venant’s ellipsoidal condition or his 
suggested modification of Cauchy, for to this degree of approximation the 
two theories are identical. 
(7) 
1 Boussinesq, Liowville’s Journal, S. ii. t. viii. 
