455 To obtain the total force which must be supposed to act, the factor T (representing the volume of the particle) must be introduced. The opposite of this, conceived to act at 0, would give the same disturbance as is actually caused by the presence of the particle. Thus by (18) ( 24) the secondary disturbance is expressed by <r= D - D 7t 2 T sin </> sin (nt - kr) D 4irZr r D -D irTsin* , , , > The preceding investigation is based upon the assumption that in passing from one medium to another the rigidity of the tether does not change. If we forego this assumption, the question is necessarily more complicated ; but, on the supposition that the changes of rigidity (AN) and of density (Al)) are relatively small, the results are fairly simple. If the primary wave be represented b y t=e- te (4), General the component rotations in the secondary wave are sxpres- / A N y ~>
- ions. ^3 =1 ^~~N~^
AD y AN a-?/ where . AD* AN s*. ^Fr + TT 7 The expression for the resultant rotation in the general case would be rather complicated, and is not needed for our purpose. It is easily seen to be about an axis perpendicular to the scattered ray (x, y, z), inasmuch as x-a l + y-sfo + z-a. A = . Let us consider the more special case of a ray scattered normally to the incident ray, so that x=0. We have -a i + - 2_p2 | AJN - Z" | p2| AU V 2T
N / r 2 D / r 2
ANmust vanish. If AN, AD be both finite, we learn from (7) that there is no direction perpendicular to the primary (polarized) ray in which the secondary light vanishes. Now experiment tells us plainly that there is such a direction, and therefore we are driven to the conclu sion that either AN or AD must vanish. The consequences of supposing AN to be zero have already been traced. They agree very well with experiment, and require us to suppose that the vibrations are perpendicular to the plane of polarization. So far as (7) is concerned the alternative supposition that AD vanishes would answer equally well, if we suppose the vibrations to be executed in the plane of polarization ; but let us now revert to (5), which gives PAN yz PAN xy PAN z--y?
- lTp> i~ - N r 2 Wa ~ N ~^~
According to these equations there would be, in all, six directions from along which there is no scattered light, two alongthe axis of y normal to the original ray, and four (y = 0, a=cc) at angles of 45 with that ray. So long as the particles are small no such vanishing of light in oblique directions is observed, and we are thus led to the conclusion that the hypothesis of a finite AN and of vibrations in the plane of polarization cannot be reconciled with the facts. No form of the elastic solid theory is admissible except that in which the vibrations are supposed to be perpendicular to the plane of polarization, and the dif ference between one medium and another to be a difference of density only. 2 .- Before leaving this subject it may be instructive to show the applica tion of a method, similar to that used for small particles, to the case of an obstructing cylinder, whose axis is parallel to the fronts of the primary waves. We will suppose (1) that the variation of optical properties depends upon a difference of density (D D), and is small in amount; (2) that the diameter of the cylinder is very small in comparison with the wave-length of light. Let the axis of the cylinder be the axis of z (fig. 26), and (as before) let the incident light be parallel to x. The original vibration is thus, in the principal cases, parallel to either z or y. We will take 1 In strictness the force must be supposed to act upon the medium in its actual condition, whereas in (18) the medium is supposed to be absolutely uniform. It is not difficult to prove that (3) remains unaltered, when this circumstance is taken into account; and it is evident in any case that a correction would depend upon the square of (D D). 2 See a paper, "On the Scattering of Light by Small Particles," Phil. Mac., June 1871. Fig. 26. first the former case, where the disturbance due to the cylinder must evidently be symmetrical round OZ and parallel to it. The element of the disturbance at A, due to PQ (dz), will be proportional to dz in amplitude, and will be retarded in phase by an amount corre sponding to the distance r. In calculating the effect of the whole bar we have to consider the integral f^dz f" dr sin (nt - kr) I sm (nt - kr) = / 77-- ^.-, - Jo r _/R v v ~ **) The integral on the left may be treated as in 15, and we find showing that the total effect is retarded | behind that due to the central element at 0. We have seen (3) that, if a- be the sectional area, the effect of the element PQ is D - D Tro-ffesin 4> . ^=r 5 - sin (nt - kr) , L) ~r where (/> is the angle OPA. In strictness this should be reckoned perpendicular to PA, and therefore, considered as a contribution to the resultant at A, should be multiplied by sin 0. But the factor sin 2 <, being sensibly equal to unity for the only parts which are really operative, may bo omitted without influencing the result. In this way we find, for the disturbance at A, .. D X !tfY corresponding to the incident wave sin (nt-kx). When the original vibration is parallel to y, the disturbance due to the cylinder will no longer be symmetrical about OZ. If a be the angle between OX and the scattered ray, which is of course. always perpendicular to OZ, it is only necessary to introduce the factor cos a in order to make the previous expression (9) applicable. The investigation shows that the light diffracted by an ideal wire- grating would, according to the principles of Fresnel, follow the law of polarization enunciated by Stokes. On the other hand, this law would be departed from were we to suppose that there is any difference of rigidity between the cylinder and the surrounding medium. 26. Reflexion and Refraction. So far as the directions of the rays are concerned, the laws of reflexion and refraction were satisfactorily explained by Huygens on the principles of the wave theory. The question of the amount of light reflected, as dependent upon the characters of the media and upon the angle of incidence, is a much more difficult one, and cannot be dealt with a priori without special hypotheses as to the nature of the luminous vibrations, and as to the cause of the differ ence between various media. By a train of reasoning, not strictly dynamical, but of great ingenuity, Fresnel was led to certain Fresnel s formula?, since known by his name, expressing the ratio of the re- formula!. fleeted to the incident vibration in terms of one constant (/a.). If 6 be the angle of incidence and 0j the angle of refraction, Fresnel s expression for light polarized in the plane of incidence is where the relation between the angles 6, O i} and fj. (the relative re fractive index) is, as usual, sin6 = ^sin0 1 ....... (2). In like manner, for light polarized perpendicularly to the plane of incidence, Fresnel found tan (fl- tan (0 + 0!) In the particular case of perpendicular incidence, both formulae coincide with one previously given by Young, viz., ( M -1)/( M + 1). / ...... (4). Since these formulae agree fairly well with observation, and are at any rate the simplest that can at all represent the facts, it may be advisable to consider their significance a little in detail. As 9 increases from to %ir, the sine-formula increases from Young s value to unity. We may see this most easily with the aid of a slight transformation : sin(0-0i) 1 - tan 0J tang = p - cos 0/ cos } ~ 1 + tan Oj tan 6 ~ yU + COS 0/ COS X Now, writing cos6/cos9 1 in the form l-sin 2 1 -yK- 2 sin 2 we recognize that, as increases from to |TT, cos0/cos0 x dimin ishes continuously from 1 to 0, and therefore (1) increases from (/i-iyOi+1) to unity. It is quite otherwise with the tangent-formula. Commencing at Young s value, it diminishes, as increases, until it attains zero, when d + O^^^ir, or sin 0j = cos0 ; or by (2) tan# = ju. This is the
polarizing angle defined by Brewster. It presents itself here as the
