ON DOUBLE REFRACTION. 261 
and therefore on the six quantities s,, s,,s,, a, 3, y, which may be expressed 
by means of the nine differential coefficients of uv, v, w with respect to x, y, z, 
of which therefore @ is a function, but not any function, since it involves not 
nine, but only six independent variables. If the disturbance be small, the 
six quantities s,, s,, 8,,a, 3, y will be small likewise, and ¢ may be expressed 
in a convergent series of the form 
$=Pot bi thot Gste: +s 
where ¢,, ¢,, $,, ¢,, ce. are homogeneous functions of the six quantities, of 
the orders 0, 1, 2, 3, &c.; and if the motion be regarded as indefinitely small, 
the functions ¢,, , . . . will be insensible, the left-hand member of equation (5) 
being of the second order as regards u,v, w. ,, being a constant, will not 
appear in equation (5), and g, will be equal to zero in case the medium in its 
undisturbed state be free from internal pressure, but not otherwise. The 
function ¢,, being a homogeneous function of six independent variables of the 
second order, contains in its most general shape twenty-one arbitrary con- 
stants, and ¢, which is of the first order introduces six more, so that the most 
general expression for g contains no less than twenty-seven arbitrary 
constants, all which appear in the expressions for the internal pressures and 
in the partial differential equations of motion*. 
The general expressions for the internal tensions in an elastic medium and 
the general equations of equilibrium or motion which were given by Cauchy, 
and which are written at length in the 4th volume of the ‘ Exercices de Mathé- 
matiques,’ contain twenty-one arbitrary constantswhen the undisturbed state of 
the medium is one of uniform constraint, and fifteen when it is one of freedom 
from pressure. In the latter case, Green’s twenty-one constants are reduced 
to two, and Cauchy’s fifteen to only one, when the medium is isotropic. 
Green’s equations comprise Cauchy’s as a particular case, as will be shown 
more at length further on. It becomes an important question to inquire 
whether Cauchy’s equations involve some restrictive hypothesis as to the 
constitution of the medium, so as to be in fact of insufficient generality, or 
whether, on the other hand, Green’s equations are reducible to Cauchy’s by 
the introduction of some well-ascertained physical principle, and therefore 
contain redundant constants. 
Tn the formation of Cauchy’s equations, not only is the medium supposed 
to consist of material points acting on one another by forces which depend on 
the distance only (a supposition which, at least when coupled with the next, 
excludes the idea of molecular polarity), but it is assumed that the displace- 
ments of the individual molecules vary from molecule to molecule according 
to the variation of some continuous function of the coordinates ; and accordingly 
the displacements w’, v', w' of the molecule whose coordinates in equilibrium 
are w+ Ax, y+Ay, z+ Az are expanded by Taylor’s theorem in powers of. ' 
: l : ' 
Aw, Ay, Sz, and the differential coefficients 7 &e. are put outs‘de the sign of 
summation. The motion, varying from point to point, of the medium taken as 
* The twenty-seven arbitrary constants enter the equations of motion in such a manner 
as to be there equivalent to only twenty-six distinct constants, the physical interpretation 
of which analytical result will be found to be that a uniform pressure alike in all directions, 
in the undisturbed state of the medium, produces the same effect on the internal move- 
ments when the medium is disturbed as a certain internal elasticity, alike in all directions, 
and of a very simple kind, which is possible in a medium unconstrained in its natural state. 
The twenty-one arbitrary constants belonging to a medium unconstrained in its natural 
state are not reducible in the equations of motion, any more than in the expressions for the 
internal tensions, to a smaller number. 
