tage gained in the spectra of even order is not in dispersion, nor in resolving-power, but simply in brilliancy, which is increased to four times. If we now suppose half the grating to be cut away, so as to leave 3000 lines in half an inch, the dispersion will evidently not be altered, while the brightness and resolvingpower are halved. If, therefore, resolving-power be our object, we should aim at covering a considerable breadth with very accurately placed lines, rather than at extreme closeness of the lines themselves. On the other hand, for experiment on dark heat, or whenever a narrow slit is not available, resolving-power is of less importance, and the best grating will be one that covers the largest space with the finest lines.

I have already mentioned that my 6000-to-the-inch Nobert defines not only not better, but decidedly worse, than the one with 3000 lines in the inch. This inferior definition is probably an accident; for there seems to be no theoretical reason for it. In brightness the closer-ruled grating has greatly the advantage.

The preceding investigations are founded on the principles ordinarily adopted in explaining diffraction-phenomena, and not on a strict dynamical theory. In the present state of our knowledge with respect to the nature of light and its relations to ponderable matter, vagueness in the fundamental hypotheses is rather an advantage than otherwise; a precise theory is almost sure to be wrong. Nevertheless it is often instructive to examine optical questions from a more special point of view; and therefore I hope that an investigation of an ideal grating on dynamical principles will not be out of place, though not very closely connected with the preceding portion of the paper.

In actual gratings the lines or grooves occur at the boundary of two media of different refrangibilities; but, for the sake of simplicity, we shall here suppose the medium on both sides to be the same. The grating will thus consist simply of bars (infinitely long) whose optical properties differ from those of the rest of the medium; and we further suppose

(1) that the variation of optical properties depends upon a difference of inertia, and is small in amount;

(2) that the diameter of the bars is very small in relation to the wave-length of light.

The supposition that refraction depends upon a difference of inertia is that of Fresnel and Green, and has been shown by the latter to lead to Fresnel's laws. In several papers in this Magazine*, I have shown that, if the analogy with an elastic solid holds good at all, no other supposition is reconcilable with the facts of the reflection of light from surfaces and its diffraction

*Phil. Mag. 1871, February, April, June, August.

by small particles. Whether true or not, it is at any rate mechanically possible.

Since the bars are very small, the effect of each is quite independent of the rest; and so the dynamical theory need only concern itself with one. In my paper "On the Light from the Sky "*, I proved that the effect of a body in disturbing the waves of light incident upon it may be calculated by ordinary integration from those of its parts, provided that the square of the alteration of mechanical properties may be neglected. This proposition, though true as stated, requires some caution in use, and is practically inapplicable when the body is elongated in the direction of original propagation, because the dimension of the body in this direction divided by λ may occur as a factor in the terms omitted. In the present case, however, where the light is incident normally to the plane of the grating, this difficulty

does not arise.

P

Let the bar under consideration be taken for axis of z, and let the axis of x be parallel to the direction of propagation of the original light. The original vibration is thus, according to the polarization, parallel to either z or y. We will take first the former case, where the disturbance due to the bar

R

must be symmetrical in all directions 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 corresponding to the distance r. In calculating the effect of the whole bar, we have to consider the integral

Now the denominator =√r-R√r+R, a product of which the second factor may be treated as constant in the integration in view of the fact that the parts for which r differs much from R destroy each other's effects. After this simplification the integral may be evaluated by means of the formula

λ

8

showing that the total effect is retarded behind that due to the

element at O. This result is analogous to, though different from, that of the ordinary integration by Huyghens's zones. In that case the effect of each zone is very nearly the same, and therefore the whole is the half of that of the first zone. If the first zone be divided into rings by circles drawn so that r increases in arithmetical progression, the rings will be of equal area, and therefore the phase of the resultant vibration will be halfway between that corresponding to the centre and circumference-that is, will be retarded relatively to the centre by one fourth of λ. In the present case, if the bar be divided on the same principle so that each piece gives a result retarded behind its predecessor, the lengths will rapidly diminish from the centre outwards, and therefore the same argument does not apply. The retardation of the resultant relatively to the central element is less than before, on account of the preponderance of the central parts.

λ

2

By the result investigated in my paper previously referred to, if T be the volume of the element PQ, D and D' the original and altered densities, the disturbance at A due to the element is D'-DT.T D ηλε

2π

sin a cos

(bt-r)*,

λ

the original vibration at P Q being denoted by cos

the angle between the ray PA and the direction of original vibration OZ; but in the present application we may put sin a=1, since only the central parts are really operative. If we replace T by Adz, A being the sectional area of the bar, and use the integral just investigated, we find for the effect of the whole bar D'-D AT λ D X2

COS bt-R
λ

bt for the incident light.

When the original vibration is parallel to y, the disturbance due to the bar will no longer be symmetrical round O Z. If a be the angle between Ox and the direction of the scattered ray, it is only necessary to introduce the factor cos a in order to make the preceding expression applicable.

* The factor was inadvertently omitted in the original memoir.

If the direction of the original vibration be inclined at an angle to O Z, and that of the scattered ray at an angle o, we have on resolution

tan = cos a tan 0,

which expresses the important law enunciated by Stokes*.

The preceding investigation depends upon the smallness of D'-D as well as of A; but where the original vibrations are parallel to the bar, the result is correct to all powers of D'-D. I find that, if the bar be circular and composed of material for which the density is D' and rigidity n' (the corresponding quantities for the rest of the medium being D and n), the expression for the scattered vibration is

for the incident light at the centre of the bar. If we suppose that n'=n, this agrees with the result already found; and it is correct if the bar be small enough, whatever may be the magniD'-D

tude of

D

XXVI. On some Phenomena of Polarization by Diffusion of Light. By J. L. SORETT.

IN Na recent memoir of great interest‡, M. G.-A. Hirn has occupied himself with the optical properties of flames; and, for the explanation of the whole of the phenomena, he suggests the hypothesis that the solid incandescent particles to which the flame owes its brightness become transparent at that high temperature and have no longer any sensible reflecting-power. A section of the memoir is devoted to the phenomena of polarization in, or rather to their absence from, the light of flames, and contains the account of some experiments on the effect obtained by causing the light of the sun to fall on a flame and on the smoke which may escape from it §.

• "Dynamical Theory of Diffraction," Camb. Trans. vol. ix.

+ Translated from the Bibliothèque Universelle, Archives des Sciences for November 1873, vol. xlviii. pp. 231-241.

"Mémoire sur les propriétés optiques de la flamme des corps en combustion et sur la température du Soleil," Annales de Chimie et de Physique, November 1873.

§ We will quote the following passage :

"One of Arago's first observations in optics demonstrated that the light

This publication of M. Hirn's induces me to make known a few of the results I have obtained in observing the polarization

of flame in general presents no trace of polarization. As is known, it was this observation that served to place one of the first and most important landmarks of the theory of the sun, by informing us that the solar light emanates from a gas, and not from a liquid or a solid.

"When we have to do with a homogeneous flame, such as that of strongly compressed hydrogen burning in compressed oxygen, when the light emanates from all parts of the incandescent gas itself, the fact discovered by Arago exhibits nothing unusual, nothing inconsistent with other known facts. It is not so when the flame is heterogeneous, formed of a real mixture of a gas with the dust of a solid body-dust which is far from being, as has often been erroneously advanced, in a state of infinitesimal division; in a word, it is not so with nine tenths of ordinary flames, the brightness of which can only be explained by Davy's theory. Here, in fact, not only does each of the solid particles emit light of its own, but it must reflect light from other sources; for it is illumined by the other particles. Then how is the total absence of polarization to be explained? The interpretation, it seems, is not difficult in the case of carburetted gases.

In an

"Of all known substances, carbon (at least in its most usual state) is that which reflects light the least. Lampblack, for example, reflects very little or none at all; now, in flame, it is precisely in the form of lampblack that carbon is found, although incandescent. The absence of light polarized by reflection seems therefore very natural in this particular case. experiment, however, of which I shall speak subsequently, I have ascertained that the smoke of carbon, when produced in a very hot atmosphere and strongly illuminated, appears not black, but of dazzling whiteness. The above interpretation is therefore not so correct as it seems; and we shall see besides that it cannot be applied in its entirety in a great number of other cases. I cite as striking examples the two following

"1. I have examined with the Arago-lune polariscope the flame of phosphorus burning in shade or in bright sunshine, and I have not been able to perceive the least appearance of coloration in either of the two images. When, on the contrary, I directed the instrument towards the vapour of phosphoric acid strongly illuminated by the sun, the coloration became manifest.

"2. I have examined in the same manner the lofty and bright flame that issued from the top of a cupola coke-furnace, supplied with air by an energetic blast, which served for melting cast iron. Neither at night, nor in bright sunshine have I been able to perceive in it the slightest trace of polarization. Now, although here also the combustible was carbon, the light of the flame (a very bright light yellow) was certainly not due solely to particles of carbon precipitated in the burning gas; instead of not solely, it would be perhaps more correct to say not at all. When, at the end of the operation, the furnace-door was opened for drawing out the scoriæ, the calcareous flux, &c., a thick bluish smoke (various metallic oxides or salts) escaped from the top instead of the flame. This smoke, strongly illuminated by the still-glowing interior, gave to the polarimeter two images as much coloured as those obtained on looking at any light whatever, reflected under the most favourable angle, through an unsilvered glass plate. It is evidently to this vapour of metallic oxides or salts that the flame owed the greater part or even the totality of its brightness while the furnace was in action.

How is it to be accounted for that phosphoric acid and the vapour of