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1 millim. (on which the electrostatic measure is founded) is presupposed, and in (II.), on the contrary, the acceleration in a free fall (which forms the basis of the calculation with the usual weights).

e

n

If the conductor has a cross section of the unit of surface,

is simply the density of the fluid reckoned in ordinary weights.

Let it be put equal to D. Then would

✓M=

2e2R 9811.C.D

(III.)

Now this relation would attain a very much further-reaching practical signification if, in connexion with hypotheses which, especially recently, have been repeatedly mooted, by M were to be understood the number of æther molecules contained in a cubic millimetre of the conductor, and by C the velocity of propagation of light in the free æther. Then would D be at all events much greater than the density of the æther in free space. Sir 1 W. Thomson* has calculated that the latter is greater than

1022;

1

Bellit, indeed, has subsequently stated it to be at least 21.1012 As, unfortunately, I have not been able to examine Belli's calculation, while the value given by Thomson is certainly very much

1

too small, may perhaps be taken, as lying between the two.

1018

We should then have, understanding by M and C the abovestated quantities, the inequality

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and the inequality hereby expressed would indeed be considerable.

Now R is known for a definite conductor; and so a maximum value of M is given by the preceding relation, in case e is known -or, conversely, a minimum value of e for a known M. It is true, indeed, that both M and e are unknown to us; but we have more closely approximate ideas concerning e than concerning M, and can therefore make use of the relation we have obtained in order to gain similar notions with respect to M.

If, e. g., for mercury we assume e=1020 nearly, remembering that, on the hypothesis of the equality of the velocity of elec

* Phil. Mag. S. 4. vol. ix. p. 36.
↑ Cf. Fortschritte der Physik, 1859.

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trolytic motion to our electrical v, e has been found, according to Weber*, to be about 3.1018 for 1 cubic millim. of water, even this value will perhaps be sufficiently great.

For Siemens's mercury unit, according to Kohlrausch + the resistance is, in electromagnetic measure, 9717. 106; according in mechanical mea

to Webert, therefore, it is about =

1 24.1011

1

24.1014

sure. Consequently for 1 cubic millim. of mercury, R= Accordingly, in this volume of mercury we should have

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If the number above admitted for the density of the free æther be divided by this value, we get a lower limit for the weight (in milligr.) of each molecule of æther. It would thus amount to considerably more than

1

10100

These values have of course only a widely approximate signification, since we have no very exact knowledge of the numerical values of e; nevertheless it would be of high importance to have, only so far, at least an idea of the relations of the æther.

Aachen, October 26, 1873.

XXV. On the Manufacture and Theory of Diffraction-gratings. By LORD RAYLEIGH, M.A., F.R.S.

[Concluded from p. 93.]

THE HE remainder of this paper is principally occupied with theoretical considerations relating to the performance of gratings considered as light-analyzing apparatus. The more popular works on the theory of light give only the main outlines of the subject, and pass over almost in silence the important questions of illumination and definition. On the other hand, the mathematical treatises, such as Airy's Tracts' and Verdet's Leçons, though they give analytical results involving most of the required information, are occupied rather with explaining the production of spectra as a diffraction-phenomenon than with investigating on what conditions their perfection depends. On * Electrodynamische Maassbestimmungen, 1856, p. 281. + Pogg. Ann. Ergänz. vol. vi. p. 1.

Elektrodynamische Maassbestimmungen, p. 260.

§ Compare W. Thomson's calculation (Liebig's Ann. vol. clvii. p. 54), from which the number of physical molecules in the same volume would be, at most, 1023.

Phil. Mag. S. 4. Vol. 47. No. 311. March 1874.

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examining the question for myself, I came to the conclusion that the theory of gratings, as usually presented, is encumbered with a good deal that may properly be regarded as extraneous.

One of the first things to be noticed is the extraordinary precision required in the ruling. The difference of wave-length of the two sodium-lines is about a thousandth part. If, therefore, we suppose that one grating has 1000 lines in the space where another has 1001, it is evident that the first grating would produce the same deviation for the less-refrangible D line that the second would produce for the more-refrangible D line. We have only to suppose the two combined into one in order to see that, in a grating required to resolve the D line, there must be no systematic irregularity to the extent of a thousandth part of the small interval. Single lines may, of course, be out of position to a much larger amount. It is easy to see, too, that the same accuracy is required, whatever be the order of the spectrum examined.

The precision of ruling actually attained in gratings is very great. In the 3000 Nobert it is certain that the average interval between the lines does not vary by a six-thousandth part in passing from one half of the grating to the other; for the D lines, when well defined, do not appear so broad as a sixth part of the space separating them.

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In considering the influence of the number of lines (n) and the order of the spectrum (m), we will suppose that the ruling is accurate, and that plane waves are incident perpendicularly upon the face of the grating whose width is represented in the figure by A B. But inasmuch as a large part of the phenomenon covered by the usual mathematical investigation depends upon the limitation of the grating at A and B, we shall find it convenient to take first the simple case of an aperture represented by A B, and afterwards to consider the influence of the ruling.

D

In the perpendicular direction BC all the secondary waves emanating from AB are in complete agreement of phase, and their resultant accordingly attains its highest possible value. In a direction B P, making with BC a very small angle, the agreement of phase will be distur If BP be so drawn that the projection of A B upon it is equal to X, the phases of the secondary waves will be distributed uniformly over a complete period,

and the resultant will therefore be nil. The same result must ensue whenever BD is an exact multiple of X.

For the intermediate directions we require a little more calculation.

The phase of the resultant will always correspond with that of the secondary wave which issues from the middle of the aperture. If a denotes the retardation of any element with respect to this one, the amplitude of the resultant is given by

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where R is the relative retardation of the extreme parts A and B, or, on integration,

Ꭱ R

sin ÷
2 2

This expression gives the magnitude of the resultant amplitude compared with that in the principal direction B C, where all the components agree in phase.

X

P

B

The composition of elementary vibrations whose phases vary uniformly within certain limits may be illustrated by a mechanical analogy. Each elementary vibration is represented by a force proportional to the element of circular arc P Q and acting at O along a direction O P, making with a fixed line of reference OX an angle corresponding to the phase of the vibration. The force may be supposed to be due to the attraction of the arc on a particle placed at O. The group of vibrations is thus represented by the group of forces whose directions are distributed uniformly through the angle A OB; and the resultant of the forces, found by resolving in the ordinary way, represents on the same system the resultant of the

vibrations. In the present case AOB corresponds to R, and

the integrated expression sin

Ꭱ R
2 2

denotes the ratio of the re

sultant force to the aggregate of its components calculated without allowance for the difference of direction-that is, as if the whole attracting mass were concentrated at X.

According to what has been already explained, (sin÷)

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angles between the vanishing directions; but it will be sufficient for our present purpose to note that the principal maximum (R=0) is unity, and that the others do not differ greatly from 212 212 212 &c. It is evident that on either side of

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the principal direction the illumination falls off with great rapidity. If A B is 1 (inch) and λ= 1 the angle C B P corresponding to the first minimum is only about 5".

40,000

The image of an infinitely narrow line of light (whose length is perpendicular to the plane of the diagram), as formed by an object-glass with aperture A B, is thus a series of parallel stripes, composed of a central narrow band whose illumination varies from a maximum in the middle to zero at the edges, enclosed by parallel bands of rapidly decreasing illumination. We have now to examine the effect of the ruling.

For the sake of simplicity, we will take first the case of a grating composed of transparent bars of width a, alternating with opaque bars of width d, and consider the central image or spectrum of zero order. In the principal direction, BC, the secondary waves are, as before, in complete agreement, but the amplitude is diminished by the ruling in the ratio a: a+d. In another direction, making a small angle with BC such that the relative retardation of A and B amounts to a few wave-lengths, it is easy to see that the mode of interference is the same as if there were no ruling. For example, when the direction is such that the projection of A B upon it amounts to one wave-length, the elementary components neutralize one another, because their phases are on the whole distributed symmetrically, though discontinuously, round the entire circumference. The only effect of the ruling is to diminish the amplitude in the ratio a: a+d; and except for the difference in illumination, the appearance of a line of light is exactly the same as if the aperture were perfectly free.

The lateral images occur in such directions that the projection of the element a+d of the grating upon them is an exact multiple of λ. The effect of each element of the grating is then the same; and unless this vanishes on account of a particular adjustment of the ratio a: d, the resultant amplitude becomes comparatively very great. These directions, in which the retardation between A and B is exactly mnλ, may be called the principal directions. On either side of any one of them the illumi

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