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

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

combining these with the system (23), there results

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, sin e and sin e, are small, and cos e, cos e, do not differ sensibly from unity; we have, therefore, as a first approximation,

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 B2 approximation: but the formula (26) gives the correct value of or 29 a2

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

which agrees with experiment, and this minimum value is, since (27)

4 (u2 + 1)3 u* + (u3 − 1)*

4

=

3

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 refracting 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 ß 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.)

γ

Lastly, if the velocity y, of transmission of a wave in the lower exceed 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 =

μ

Y

e, e, and as before, 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.

II.

On Molecular Equilibrium. Part I. By the Rev. PHILIP Kelland, M.A., Queens' College, Cambridge; Professor of Mathematics in the University of Edinburgh.

1.

[Read March 26, 1838.]

INTRODUCTION.

WHATEVER ideas may have been entertained of the nature of forces at a distance from the centre of action, there appear to have been no very definite notions current respecting molecular forces, till within a few years from the present time. The obvious change in the attractions of the different parts of a solid body, produced by separating the particles by ever so small an interval; the fact that the attraction of cohesion when destroyed cannot be restored by any ordinary pressure, indicated that the force which the particles exert on each other in their positions of equilibrium, is of a nature totally distinct from the appreciable attractions and repulsions at finite distances. Newton only threw out hints respecting the nature of forces of this kind, never applying them, except in a popular manner in his Optics. One kind of molecular force which he conjectures is that of the particles of air and the magnetic ones, Newton applies to calculation, but he by no means supposes his hypothesis the correct one; on the contrary he appears to entertain great doubts on the subject, for he concludes his scholium by observing: "Whether elastic fluids do really consist of particles so repelling each other, is a physical question. We have demonstrated the properties of fluids consisting of particles of this kind, that hence philosophers may take occasion to discuss that question,"