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"x"f"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 g 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 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 and = a sin (ax+by+ct+e) + ẞ sin (— ax + by - and in the lower one = a, sin (a,x+by+ct) Þ, = e ̃a; (A, sin ¥ +B, cos where to abridge = by + ct. (20); These substituted in the general equations (14) and (15), give c2 = y2 (a2 + b2) = y," (a," + b2) = g3 (— a'" + b2) = g," (— a," + b3), or, since g and g, 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) =-bA, siny-bB, cos + ba, cos y, 0 ba cos-bB sin - aa cos (y,+e)+aß cos (y,+e,) (A sin ↓ + B cos ¥.) = —, (4, sin v. + B, cos ↓.), — (a sin (y, + e) + ẞ sin (y,+ e)} = a, sin 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, 0=B-B,+(8 sine, — a sin e); combining these with the system (23), there results (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, Hence, sine and sine, are small, does not differ much from 1. and cos e, cos e, 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 B3 value of or the ratio of the intensity of the reflected to the a2 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 (27) gives 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 B 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 sufficiently augmenting the angle of incidence, cause the refracted wave to disappear, and the change of phase thus produced in the reflected wave may readily be found. As the calculation is extremely easy after what precedes, it seems sufficient to give the result. Let therefore, here, μ= ช , also e, e, and as be ช 1 fore, then e,e, and the accurate value of e is given by The first term of this expression agrees with the formula of page 362, Airy's Tracts*, and the second will be scarcely sensible except for highly refracting substances. * [Airy, ubi sup. p. 114, Art. 133.] |