ments in the two media are in general denoted by , 77, ; f j, rjj, Ci > but in the present case , t], fj, rj 1 all vanish. Moreover f, are independent of ~. The equations to be satisfied in the interior of the media are accordingly ( 24) _ dt" V dx-dtj- cPCi_^(d^ d^ dP V ^da? + dy 2 J At the boundary the conditions to be satisfied are the continuity of displacement and of stress ; so that, when x=0, "dx-^idx The incident waves may be represented by (3). where Dc; 2 = N(a 2 + 5 2 ) ....... (4); and ox + by const, gives the equation of the wave-fronts. The reflected and refracted waves may be represented by The coefficient of t is necessarily the same in all three waves on account of the periodicity, and the coefficient of y must be the same since the traces of all the waves upon the plane of separation must move together. With regard to the coefficient of x, it appears by substitution in the differential equations that its sign is changed in passing from the incident to the reflected wave ; in fact c 2 =V 2 {(a) 2 + & s }=V 1 2 {a 1 s + & 2 } .... (7), where V, V x are the velocities of propagation in the two media given by V =N/D, V^^/Di ..... (8). Now 5//(^ 2 + ^ 2 ) is the sine of the angle included between the axis of x and the normal to the plane of waves in optical language, the sine of the angle of incidence and b//(a l 2 + b z ) is in like manner the sine of the angle of refraction. If these angles be de noted (as before) by 0, 0^ (7) asserts that sin : sin 6 1 is equal to the constant ratio V : V a , the well-known law of sines. The laws of reflexion and refraction follow simply from the fact that the velocity of propagation normal to the wave-fronts is constant in each medium, that is to say, independent of the direction of the wave-front, taken in connexion with the equal velocities of the traces of all the waves on the plane of separation (V/sin0 = . . . (9), The boundary conditions (3) now give whence no) i Realized a formula giving the reflected wave in terms of the incident wave expres- (supposed to be unity). This completes the symbolical solution. sions. If a l (and 0J be real, we see that, if the incident wave be or in terms of V, X, and 0, = cos (x cos + y sin + V<) .... (11), X the reflected wave is ._Ncot0-N 1 cot0 1 27r The formula for intensity of the reflected wave is here obtained on the supposition that the waves are of harmonic type ; but, since it does not involve X and there is no change of phase, it may be ex tended by Fourier s theorem to waves of any type whatever. It may be remarked that when the first and second media are inter changed the coefficient in (12) simply changes sign, retaining its numerical value. Alter- The amplitude of the reflected wave, given in general by (12), native assumes special forms when we introduce more particular supposi- supposi- tions as to the nature of the difference between media of diverse tioas. refracting power. According to Fresnel and Green the rigidity does not vary, or N = N 1 . In this case N cot 9 - N] cot 9 1 _ cot 9 - cot 9 l sin (0, - 9) Ncot + NjCotfl ""cote + cot^^sin (9 L + e) If, on the other hand, the density is the same in various media, N 1 : N = Vj 2 : V 2 =- sin 2 ^ : siu 2 , and then N cot 9 - N cot 0! _ tan (6 l - 9) N cot + N! cot 0j "" tan (9 l + 9) If we assume the complete accuracy of Fresnel s expressions, either alternative agrees with observation; only, if N = N 1 , light must be 457 supposed to vibrate normally to the plane of polarization ; while, if D = D 1; the vibrations are parallel to that plane. An intermediate supposition, according to which the refraction- is regarded as due partly to a difference of density and partly to a difference of rigidity, could scarcely be reconciled with observation, unless one variation were very subordinate to the other. But the most satisfactory argument against the joint variation is that derived from the theory of the diffraction of light by small particles ( 25). We will now, limiting ourselves for simplicity to Fresnel s sup- Total position (Nj = ]S T ), inquire into the character of the solution when reflexior total reflexion sets in. The symbolical expressions for the reflected and refracted waves are 2 ^ and so long as a^ is real they may be intrepreted to indicate .. -, (13), (14), (15), corresponding to the incident wave ...... (17). In this case there is a refracted wave of the ordinary kind, con veying away a part of the original energy. "When, however, the second medium is the rarer (V 1 >V), and the angle of incidence exceeds the so-called critical angle [sin" 1 (V/V 1 )] ) there can be no refracted wave of the ordinary kind. In whatever direction it may be supposed to lie, its trace must necessarily outrun the trace of the incident wave upon the separating surface. The quantity i, as defined by our equations, is then imaginary, so that (13) and (14) no longer express the real parts of the symbolical expressions (5) and (6). If-m/ be written in place of a l} the symbolical equations are from which, by discarding the imaginary parts, we obtain (18), where tan = // ....... (20). Since a: is supposed to be negative in the second medium, we see that the disturbance is there confined to a small distance (a few wave-lengths) from the surface, and no energy is propagated into the interior. The whole of the energy of the incident waves is to be found in the reflected waves, or the reflexion is total. There is, however, a change of phase of 2e, given by (20), or in terms of V, V 1} and 9 tan = /{tan 2 0-sec 2 0(V 2 /V 1 2 )} .... (21). The principal application of the formulae being to reflexions when Fresnel the second medium is air, it will be convenient to denote by /x the interpre index of tlie^rs^ medium relatively to the second, so that /JL= Vj/V. tation. Thus tane=V{ tan20 - scc20 /A i2 } .... (22). The above interpretation of his formula sin (0 : - 0)/sin (0 1 + 0), in the case where : becomes imaginary, is due to the sagacity of Fresnel. His argument was perhaps not set forth with full rigour, but of its substantial validity there can be no question. By a similar process Fresnel deduced from his tangent-formula for the change of phase (2e ) accompanying total reflexion when the vibra tions arc executed in the plane of incidence, tane = v/{/j?ta.u-e-SGC 2 e} .... (23). The phase-differences represented by 2e and 2e cannot be investi gated experimentally, but the difference (2e - 2e) is rendered evident when the incident light is polarized obliquely so as to contribute components in both the principal planes. If in the act of reflexion one component is retarded more or less than the other, the resultant light is no longer plane but elliptically polarized. , From (22) and (23) we have tan ( -) = cos 0/{l ~ M~ whence The most interesting case occurs when the difference of phase Different amounts to a quarter of a period, corresponding to light circularly of phase: polarized. If, however, we put cos (2e - 2) = 0, we find from which it appears that, in order that sin may be real, /j~ must
XXIV/ $B