geneous." Here follows the well-known argument as to the effects of want of homogeneousness in the interfering rays.

Foucault and Fizeau* more definitely put it as a question for experimental determination, whether interference phenomena are ultimately stopped by want of homogeneousness or want of "regularity." Experimenting with white light, they came to the conclusion that their experiments "reveal in the emission of successive waves a persistent regularity which no phenomenon has hitherto suggested."

The opinion put forward by Fizeau and Foucault that observations on interference of light with large difference of path could teach us something as to the "regularity" of the vibrations of a particle was generally accepted, until more closely examined by Gouy and Rayleigh in the papers already alluded to. Their conclusions did not, however, receive the attention they deserve. Thus, in the last part of the Encyclopædia der Naturwissenschaften, it is stated that "if natural light really consists in a succession of different states of vibration, these take place so slowly, that in a time which is greater than 50,000 period, light must still be considered as essentially of the same type of vibration."

3. Compare with this the following passage taken out of Gouy's paper :

"Ainsi l'existence de franges d'interférences à grande différence de marche, dans le cas des sources de spectres continus et de la lumière blanche n'implique nullement la régularité du mouvement lumineux incident. Cette regularité existe dans le spectre, mais c'est l'appareil spectral qui la produit, en séparant plus ou moins complètement les divers mouvements simples, qui jusque là n'avaient qu'une existence purement analytique."

Lord Rayleigh, in his article on the "Wave Theory," expresses the same idea as follows:

"Or, following Foucault and Fizeau, we may allow the white light to pass, and subsequently analyse the mixture transmitted by a narrow slit in the screen upon which the interference bands are thrown. In the latter case we observe a channelled spectrum, with maxima of brightness corresponding to the wave-lengths bu/(nD). In either case the number of bands observable is limited solely by the resolving power of the spectroscope, and proves nothing with respect to the regularity or otherwise of the vibrations of the original light."

Put shortly, the argument in favour of the above view is this:-Any disturbance, however irregular, may by Fourier's * Académie des Sciences, 24th Nov. 1845 and 9th March 1846.

theorem be resolved into a series of disturbances each of which corresponds to homogeneous light. It is the object of a spectroscope to separate laterally these homogeneous vibrations, and no matter what the nature of the original light, the greater the resolving power of the spectroscope, the more nearly will it give us homogeneous light.

4. The argument is at once seen to be conclusive in one direction. However irregular the original light, interference bands with large difference of path may always be produced, if a spectroscope of sufficiently high resolving power is used. But there is another side to the question. In order that Fourier's series should correctly represent a given disturbance, there must be a definite phase relation between the different terms of the series. If a spectroscope of small resolving power is used, and if it is required to calculate the disturbance at the focus, we are not allowed to consider the different components of Fourier's series as independent, but must take account of this relation of phase. An example will show that in special cases altogether incorrect results will be obtained if this is neglected. Thus, let the light falling on the slit of a spectroscope be screened off until a given time, when the screen is removed but replaced subsequently after a short interval. The disturbance may by Fourier's series be decomposed into a series of simple * vibrations lasting through an infinity of time; and if each component could be treated independently of another, it would follow that an eye examining the spectrum would continue to see light for any length of time after the incident beam has been cut off. This of course is absurd. analysis by Fourier's series leads to results which are perfectly correct, if, when finite resolving powers are used, we take account of the phase relations between the component vibrations.

The

It seems therefore advisable to consider the effects of finite resolving powers a little more in detail, although it will appear that even then the effects produced can give us no

*Is it too late to abolish the term "harmonic " vibration to express the projection of a uniform circular motion? The term "harmonic" seems to me to imply a relation between two things, and is very useful when we want to distinguish between, say, harmonic and inharmonic overtones. It is quite correct to speak of the expansion in a Fourier's series as an harmonic expansion, because the different terms are harmonically related, but each of them taken separately is not harmonic. Much confusion has been caused by a series of lines in a spectrum being called harmonics, because their distribution suggested some definite connexion between their periods, independently of the question whether that connexion was of the harmonic character. The term "simple" vibration seems to me to be well adapted to express a sine vibration.

information concerning "regularity" of vibration. But in the first place we must ask ourselves whether we can attach any meaning to the expression "regularity of vibration when applied to white light.

5. To quote Lord Rayleigh once more:-" It would be instructive if some one of the contrary opinion would explain what he means by regular white light. The phrase certainly appears to me to be without meaning-what Clifford would have called nonsense"

If we

I believe that those who speak of the regularity of vibration in a continuous spectrum have in their mind the state of motion of the original vibrating system, and not that of the medium through which the light is transmitted. imagine the light to be produced by a number of different vibrating systems, each sending out homogeneous vibrations, prismatic decomposition would give what we should call a line spectrum. We may imagine the lines of the spectrum to be so close together that any resolving power which we can apply fails to separate them.

There is nothing to prevent our imagining for the sake of argument that all continuous spectra if examined by resolving powers, say a million times larger than we can apply at present, would ultimately appear to be line spectra, each line representing perfectly homogeneous light; and if each line is due to a separate vibrating system, we may reasonably say that that system remains perfectly regular for an indefinite time. Let us then imagine a practically continuous spectrum of this nature in which, for the sake of simplicity, there is no difference in intensity, all wave-lengths being equally represented. The spectrum might be made completely continuous by imagining each vibrating system to be split up into several, moving with uniform velocities in different directions, and that velocity may be made indefinitely small, as the difference in wave-length between the vibrating systems is indefinitely reduced. But for my purpose it will be sufficient to take the spectrum as ultimately discontinuous, and I shall take such a system of vibrations as the representative of what may be called "regular" white light. On the other hand, imagine a set of molecules setting up a luminous disturbance by perfectly irregular and indefinitely short impulses. When examined by a spectroscope we should see a continuous spectrum ranging over all wave-lengths with equal intensities; of such we may imagine "irregular" white light

to consist.

The question I shall discuss in detail is this :-Can we by * Phil. Mag. xxvii. p. 463, note (1889).

any interference experiments distinguish between the light sent out on the one hand by a regularly vibrating system of molecules, and on the other hand by irregular short impulses?

6. One of the fundamental facts of our subject is the impossibility of two different sources producing interference phenomena. This fact is generally taken as a proof that molecules must suffer frequent disturbances. But two bodies sending out what we have called regular white light would have the same property. For imagine each of the two sources to send out a series of waves of lengths A1, A2, &c., differing by quantities which are so small that no instrumental power at our command can resolve them. The illumination at any point of a screen will depend on the manner in which the trains of waves coming from the two sources will combine. The difference of phase between the two sets of waves of length A depends not only on the relative distance of the sources from the screen, but also on the difference of phase at the two sources. As there is no connexion between that difference of phase for the different wave-lengths, the wave-lengths in close proximity to λ will partly destroy each other, partly have fourfold or any intermediate intensity, the result being the same at all points of the screen as if no interference had taken place. This leads to an important result. Imagine a body, the molecules of which vibrate in a perfectly definite period, but have a translatory motion such as we imagine the molecules of a gas to possess. Such a body would give a spectrum of one line, which, however, would have a certain width owing to the motion of the molecules. The want of homogeneousness produced by this translatory motion would be quite sufficient to prevent the possibility of interference between two similar sources, and we need not take refuge in the impacts between molecules to explain the non-existence of interference.

Fig. 1.

H1

G

7. In fig. 1 let L1 and L2 be two lenses, H, H, and K1 K, two screens in their focal plane having small apertures at the foci G and F. If ACB is a grating, a disturbance passing through G will produce a certain effect at F. Unless the relative positions of what for convenience' sake we may call collimator and telescope are adjusted so that a direct image of G is formed at F, the optical lengths of rays such as GBF, GCF, GAF, will not be the same, consequently an instantaneous impulse at G will not remain an instantaneous

K,

F

H2

impulse at F, but the light reflected from each line of the grating will reach F at certain intervals of time, and a disturbance will be set up at F which will last during a finite time. The time it takes for an impulse at G to pass completely through F will be that required by light to go over a space equal to the difference in optical length of the extreme rays GBF and GAF. The fact that in every dispersive system an instantaneous impulse entering the system is changed into a disturbance lasting through a finite time is of fundamental importance. It is easily seen that this is brought about in the case of a grating, but the fact is equally true if the dispersion is due to refraction, as will be pointed out further on.

Imagine that in fig. 1 two impulses of the same type succeed each other. If the second impulse begins to reach F before the effects of the first impulse have passed away interference may occur, but if the disturbance due to the first impulse has completely passed through F before the second impulse reaches it, no interference can take place, and this will happen whenever the distance between the impulses before they reach G is greater than the difference in the optical length between the extreme rays GAF and GBF. It is thus seen that if light were to consist of separate instantaneous impulses, interference at the focus of a grating-spectroscope of finite resolving powers could only take place if the retardation were smaller than a certain amount; if the retardation were greater, no interference could possibly occur.

Now consider the light to be what we have called regular, that is to say, to consist of the superposition of a large number of homogeneous vibrations. If the beam of light is split into two, one being retarded with reference to the other, the spectrum formed in the focal plane of L, will be crossed by bright and dark bands; if the retardation is gradually increased these bands will gradually get nearer and nearer together, and at the same time become less distinct as their distance approaches the limits which are still resolvable by the grating. Looking at it in this general manner, there seems to be no reason why there should be a definite limit when interference ceases, but it would seem as if there ought to be always a fluctuating intensity along the spectrum, although the difference in intensity between the bright and dark bands would with increasing retardation of the interfering beams become so small, that they could no longer be traced. This is the test case previously alluded to, by means of which I originally thought we might distinguish between regular" and irregular white light. It was found, however, that on actually calculating the difference in intensity between