It remains to prove how A and B are connected with the rigidity and the bulk modulus. If e, f, g are equal

P=Q=R=e (A + 2B).

Hence the stress is uniform.

But the cubical dilatation being 3e and the bulk modulus being equal to the ratio of the uniform stress P to the cubical dilatation, it follows that

As a second special case take R = 0, and Q = − P, which conditions indicate a shearing stress in planes equally inclined to the axis of X and Y, and these will cause a shearing strain equal in amount to Pn. This shearing strain is equivalent by Art. 127 to an elongation in the direction of P of P/2n, and an equal contraction in the direction of Q. Substituting e-f=P/2n into the first of the equations (4), we find if g=0

2n = A - B

Combining (5) and (6), it follows that

A =k+n, B = k - n.

..(6).

In place of the components of strain, we may introduce their equivalents in terms of the displacement (Arts. 126 and 127). Equations (3) and (4) then become.

dy+da),

dz

dy dx

132. Equations of Motion in a disturbed medium. Returning to the stresses acting on the cube in Art. 129, we consider the case where these stresses are not constant through the volume, but alter slowly from place to place. If the distance between the two faces of the cube which are at right angles to the axis of X is dr, there will be a force

Xdydz

acting in the negative direction on the face which is coincident with the coordinate plane and a force on the opposite face equal to

dz

Similarly the force Xdady acting on the plane ay in the direction of a together with the force

combine to a resultant

dX

(X+ d) drdy

dz dx

of the two forces which act on the faces which are normal to the axis of y. If p be the density of the substance, so that pdx dydz be the mass of the volume considered, and if a be the displacement in the direction, the equations of motion may be written down by the laws of dynamics, leaving out the factor dadydz on both sides,

Re-introducing the notation of Thomson and Tait, the equations become

dy dT

Pat2

=

dx

+

+

ds

+

dy

dz

ds

dR

+

dy dz

To eliminate the stresses use equations (7) and (8).

Substituting the values of A and B from Art. 131, and rearranging the terms, we obtain

d3y d3y + d2 y) + (k + }n) dz dx dy dz

+

dx dy

These equations govern wave propagation in all elastic media. We may obtain from them the characteristic equations for the longitudinal waves of sound by putting the rigidity n of the medium equal to zero. When applied to light, the medium is taken as incompressible, so that

but k at the same time becomes infinitely large. Writing

These equations, together with certain relations which must hold at the surfaces of the elastic body, constitute the elastic solid theory of light.

For plane waves, the displacements are the same at all points of the wave-front, which we may imagine to be at right angles to the axis of . The differential coefficient of a, ß, y with respect to x and y must therefore vanish. The equations (9) then reduce to

The last equation represents a longitudinal wave propagated with infinite velocity and having no relation to any observed phenomenon of light. Each of the first two equations represents a rectilinear wave propagated with velocity √np, a result already deduced by the simpler but less general methods of Art. 12.

The investigation of wave propagation in crystalline media presents great difficulties. The simplest hypothesis from a mathematical point of view is that of assuming that the inertia of the medium may differ for displacements in different directions. By substituting P1, P2, P3, respectively, for p on the left-hand side of equations (9), we obtain equations which lead to a wave surface which is similar to, but not identical with, Fresnel's wave surface. A theory of double refraction

based on this hypothesis was brought forward by Lord Rayleigh, but abandoned because observations made by Stokes, and afterwards by Glazebrook, decided in favour of Fresnel's surface. Instead of taking the inertia as variable, we may adopt the very plausible hypothesis that the rigidity is different in different directions. Thus different values of n in the first two equations (11) would give two plane waves propagated with different velocities, along the axis of z. A general theory cannot however be formed by a simple modification of the equations holding for isotropic media. According to Greent, there may be twenty-one different coefficients defining the properties of crystalline media, which shows the complication we might be led into if we wished to attack the problem in its most general form.

133. Equations of the Electromagnetic Field. The line integral of the magnetic force round a closed curve is numerically equal to the electric current through the curve multiplied by 4′′. It is shown in treatises on Electricity that the mathematical expression of this law is contained in the three equations:

where a, B, y are the components of magnetic force, and u, v, w the components of current density. The factor 4 depends on the units chosen, which are those of the electromagnetic system.

Another proposition which embodies Faraday's laws of electromagnetic induction states that if a closed curve encloses lines of magnetic induction which vary in intensity, an electromotive force acts round the curve, and the line integral of the electric force round the closed curve is equal to the rate of diminution of the total magnetic induction through the circuit. This leads to the equations

where μa, uß, uy are the components of magnetic induction, μ being the permeability, and P, Q, R those of electric force.

The two sets of equations may be taken to represent experimental facts and to be quite independent of any theory, although equations (13) may be deduced from (12) with the help of the principle of the conservation of energy. Both sets of equations would be equally true if we considered electric and magnetic forces to be due to action at a distance.

There are some additional equations to be considered.

Differentiating equations (12) with respect to x, y, z respectively and adding, we find

Similarly we derive from (13), if u be constant and a, B, y periodic,

μ

134. Maxwell's Theory. The fundamental principle of Maxwell lies in his conception of an electric current in dielectrics and the way in which this current is made to depend on electric force. His views are best explained by an analogy taken from the theory of stress and strain. A stress in an elastic solid produces certain displacements which are proportional to the stress. If the stresses increase, the displacements increase, and the change of displacements constitutes a transference of matter. This flow or current of matter is proportional to the rate of change of the elastic stress. Taking this as a guide we may imagine the medium to yield in some unknown manner to the application of electric force, and if so, the rate of change of that force will be proportional to a "flow" which according to Maxwell is identical in all its effects with an electric current.

dE

If the electric force is E, the electric current is proportional to

and if the law that the total flow is the same across all crossdt sections of a circuit holds good for these so-called "displacement currents" or "polarization currents," it can be shown that the current

is equal to K 4, where K is the specific inductive capacity of

dt

the medium. In a conductor, the current would, according to Ohm's law, be CE, where C is the conductivity. If we imagine a medium to possess both specific inductive capacity and conductivity, we must introduce an expression which includes both cases and put the current equal to