Substituting now (10) and (12), in the equations (9), and proceeding precisely as for Sound, we get

by introducing the index of refraction, and the angle of incidence. Thus,

and as e represents half the alteration of phase in passing from the incident to the reflected wave, we see that here also our result agrees precisely with Fresnel's for light polarized in the plane of incidence. (Vide Airy's Tracts, p. 362*.)

Let us now conceive the direction of the transverse vibrations in the incident wave to be perpendicular to the direction in the case just considered; and therefore that the actual motions of the particles are all parallel to the intersection of the plane of incidence (xy) with the front of the wave. Then, as the planes of the incident and refracted waves do not coincide, it is easy to perceive that at the surface of junction there will, in this case, be a resolved part of the disturbance in the direction of the

• [Airy, ubi sup. p. 114, Art. 133.]

normal; and therefore, besides the incident wave, there will, in general, be an accompanying reflected and refracted wave, in which the vibrations are transverse, and another pair of accompanying reflected and refracted waves, in which the directions of the vibrations are normal to the fronts of the waves. In fact, unless the consideration of the two latter waves is also introduced, it is impossible to satisfy all the conditions at the surface of junction; and these are as essential to the complete solution of the problem, as the general equations of motion.

The direction of the disturbance being in plane (xy) w=0, and as the disturbance of every particle in the same front of a wave is the same, u and v are independent of z. Hence, the general equations (4) for the first medium become

These equations might be immediately employed in their present form; but they will take a rather more simple form, by

and being two functions of x, y, and t, to be determined.

By substitution, we readily see that the two preceding equations are equivalent to the system

In like manner, if in the second medium we make

we get to determine 4, and y, the equations

(15),

.(16),

and as we suppose the constants A and B the same for both media, we have

For the complete determination of the motion in question, it will be necessary to satisfy all the conditions due to the surface of junction of the two media. But, since w = 0 and w;= 0, also, since u, v, u,, v, are independent of z, the equations (5) and (6) become

provided x=0. But since x=0 in the last equations, we may differentiate them with regard to any of the independent variables except x, and thus the two latter, in consequence of the two former, will become

Substituting now for u, v, &c., their values (13) and (15), in and, the four resulting conditions relative to the surface of junction of the two media may be written,

or since we may differentiate with respect to y, the first and fourth equations give

which, in consequence of the general equations (14) and (16), become

Hence, the equivalent of the four conditions relative to the surface of junction may be written

If we examine the expressions (13) and (15), we shall see that the disturbances due to 4 and 4, are normal to the front of the wave to which they belong, whilst those which are due to y,

are transverse or wholly in the front of the wave. If the coefficients A and B did not differ greatly in magnitude, waves propagated by both kinds of vibrations must in general exist, as was before observed. In this case, we should have in the upper medium

and

↓ = ƒ (ax+by+ct) + F (− ax + by +

$=x, (−ax+by+ct)

and for the lower one

↓ =ƒ, (a,x+by+ct)

=x(ax+by+ct)

..(19).

The coefficients b and c being the same for all the functions to simplify the results, since the indeterminate coefficients a'aa' will allow the fronts of the waves to which they respectively belong, to take any position that the nature of the problem may require. The coefficient of x in F belonging to that reflected wave, which, like the incident one, is propagated by transverse vibrations would have been determined exactly like a'a, a', as, however, it evidently a, it was for the sake of simplicity introduced immediately into our formulæ.

=

By substituting the values just given in the general equations (14) and (16), there results

c2 = (a2 + b2) y2 = (a,2 + b2) y," = (a'3 + b2) g3 = (a,”2 + b3) g′′,

we have thus the position of the fronts of the reflected and refracted waves.

It now remains to satisfy the conditions due to the surface of junction of the two media. Substituting, therefore, the values (18) and (19) in the equations (17), we get