, 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

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,2 = (a3 + ba) 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

where to abridge, the characteristics only of the functions are written.

By means of the last four equations, we shall readily get the values of F"xf"x" in terms of f", and thus obtain the intensities of the two reflected and two refracted waves, when the coefficients A and B do not differ greatly in magnitude, and the angle which the incident wave makes with the plane surface of junction is contained within certain limits. But in the introductory remarks, it was shewn that

Α

B

= a very great quantity.

which may be regarded as infinite, and therefore 9 and g, may be regarded as infinite compared with y and y,. Hence, for all angles of incidence except such as are infinitely small, the waves dependent on 4 and 4, cease to be transmitted in the regular way. We shall therefore, as before, restrain the generality of our functions by supposing the law of the motion to be similar to that of a cycloidal pendulum, and as two of the waves cease to be transmitted in the regular way, we must suppose in the upper medium

y= = a sin (ax+by+ct + e) + ß sin (— ax+by+ct +

ct+e) } (20) ;

These substituted in the general equations (14) and (15), give

2

c2 = y2 (a2 + b2) = y,” (a,” + b2) = g3 (− a'a + b2) = g,” (— a‚a + b2), or, since g and 9, are both infinite,

b=a' = a,.

It only remains to substitute the values (20), (21) in the equations (17), which belong to the surface of junction, and thus we get

bA sin+bB cos y + ba cos (y,+e) + bẞ cos (+e)

=

0

-b4, sin -bB, cos

+ ba, cos Y.,

(y,+e) + aß cos (Y, +e,)

ba cos - bB sin y. — aa cos

=bA, cosy.-bB, sin

-a,a, cos y

........(22).

(4 sin ↓ ̧ + B cos ¥.) = —, (4, sin,+B ̧ cos ↓.),

1

— (a sin (y, + e) +ẞ sin (✨, + c)} = a, sin v

Expanding the two last equations, comparing separately the coefficients of cosy, and sin y,, and observing that

In like manner the two first equations of (22) will give

0=A+A,- a sin e-ẞ sin e,

α

0 = B− B, + 1⁄2 (8 sin e, — a sin e) ;

combining these with the system (23), there results

0 = A + A,

0 =B + B,+ (μ3 − 1) a,

(23).

Again, the systems (23) and (24) readily give

When the refractive power in passing from the upper to the lower medium is not very great, μ does not differ much from 1. Hence, sine and sine, are small, and cos e, cose, do not differ sensibly from unity; we have, therefore, as a first approxima

which agrees with the formula in Airy's Tracts, p. 358, for light polarized perpendicular to the plane of reflexion. This result is only a near approximation: but the formula (26) gives the correct, B*

value of , or the ratio of the intensity of the reflected to the incident light; supposing, with all optical writers, that the intensity of light is properly measured by the square of the actual velocity of the molecules of the luminiferous ether.

From the rigorous value (26), we see that the intensity of the reflected light never becomes absolutely null, but attains a minimum value nearly when

* [Airy, ubi sup. p. 110.]

which agrees with experiment, and this minimum value is, since

as when the two media are air and water, we get

It is evident from the formula (28), that the magnitude of this minimum value increases very rapidly as the index of refraction increases, so that for highly retracting substances, the intensity of the light reflected at the polarizing angle becomes very sensible, agreeably to what has been long since observed by experimental philosophers. Moreover, an inspection of the equations (25) will shew, that when we gradually increase the angle of incidence so as to pass through the polarizing angle, the change which takes place in the reflected wave is not due to an alteration of the sign of the coefficient ß, but to a change of phase in the wave, which for ordinary refracting substances is very nearly equal to 180°; the minimum value of B being so small as to cause the reflected wave sensibly to disappear. But in strongly refracting substances like diamond, the coefficient ẞ remains so large that the reflected wave does not seem to vanish, and the change of phase is considerably less than 180°. These results of our theory appear to agree with the observations of Professor Airy. (Camb. Phil. Trans. Vol. IV. p. 418, &c.)

Lastly, if the velocity y, of transmission of a wave in the lower exceed y that in the upper medium, we may, by suf ficiently augmenting the angle of incidence, cause the refracted wave to disappear, and the change of phase thus produced in the