on the right hand of (82) never exceeds unity; but it becomes equal to unity, and the intensity of the scattered waves is therefore a maximum, when

When K is large, the lower roots of this, considered as an equation in ha, are easily seen to be real and to be very approximately equal to the real parts of the roots of (71). When the period of the incident waves is such that (83) is satisfied exactly, we have

If the incident waves be plane, the dissipation-ratio (68) takes the form

.(85).

(86).

If we compare this with (68), we find that in the case n = 1 the effect of synchronism. is to increase the dissipation in the ratio

put

9
(ka)-6.
4

The wave-length of maximum scattering is of course very sharply defined. If we

For example, in the case n = 1 the dissipation sinks to one-half of the maximum when the wave-length deviates from the critical value by the fraction (ka)3/K of itself.

The second type can be treated in a similar manner. Writing (32), with μ= 1, in the

form

U n

Un

g (ha)
G (ha) — ig (ha)

(90),

the equation G (ha) = 0 which determines the wave-lengths of maximum dissipation may be written

The lower roots (in ha) which satisfy this are very nearly the same as in the case of (78). When (91) is satisfied exactly we have

leading to the same formula (86), as before, for the dissipation-ratio when the incident waves are plane.

The definition is now less sharp than in the case of (89), in the ratio k2a2.

8. It remains to examine what sort of magnitudes must be attributed to the quantities a and K in order that our results may be comparable with ordinary optical relations.

Since ka (= 2πa/λ) must in any case be small, and since ha must in the case of synchronism satisfy (71) or (78) approximately, and must therefore be at least comparable with, it follows that if our molecules are to produce selective absorption within the range of the visible spectrum, the dielectric constant K (= h2/k2) must be a very large number.

Again, it appears from two distinct lines of argument* that in a gas composed of spherical dielectric molecules the index of refraction (1) for rays which are not specially absorbed is given by the formula

this

N denoting the number of molecules in unit volume. On our present hypothesis takes the simpler form

* Maxwell, Electricity, § 314; Lord Rayleigh, Phil. Mag., Dec. 1892, and April 1899. VOL. XVIII.

46

Hence if

1.0003, we have p = 2 × 10−4. This determines the product Na3, for a gas such as oxygen or nitrogen under ordinary atmospheric conditions, but not N or a separately. If in accordance with current mechanical estimates we take N = 2 x 1019, we find a = 1.3 × 10-8 cm. Hence if λ = 6 × 10−5 cm., we find

ka 1.4 × 10-3,

so that, if ha π, we must have

K = h2/k2 = 5 × 106.

In a dense medium composed of the same molecules the formula (98) is replaced by

where the accents refer to the altered circumstances. Comparing, we have

The fact that the refractive indices of various substances in the liquid and in the gaseous state have been found to accord fairly well with this formula shews that the observed moderate values of K' (= '2) for dense media, taken in the bulk, are incompatible with an enormous value of K for the individual molecules.

not

The formula (86) for the dissipation-ratio in the case of exact synchronism is independent of any special numerical estimates. It It can moreover be arrived at on widely different hypotheses as to the nature of a molecule and of the surrounding medium. Its unqualified application to an assemblage of molecules arranged at ordinary intervals may be doubtful, since with dissipation of such magnitude it may be necessary to take account of repeated reflections between the molecules. It is clear however that a gaseous medium of the constitution here imagined would be absolutely impenetrable to radiations of the critical wave-length.

As regards the falling off of the absorption in the neighbourhood of the maximum, the formula (95) in the case n=1 would (on the numerical data given above) make the absorption sink to one-half of the maximum when the wave-length varies only by '00,000,000,028 of its value. The formula (89) would give a still more rapid declension. The range of absorption in a gaseous assemblage must however be far wider than these results would indicate. So far as it is legitimate to assume that the molecules act independently, the law of enfeeblement of light traversing such a medium is ........... (101)†.

E = E1e ̄NIx

* This is Lorentz' result. Lord Rayleigh's investigations shew that it will hold approximately even if p' be not a very small fraction.

+ Lord Rayleigh, l. c.

We may inquire what value of the dissipation-ratio I would make the intensity diminish in the ratio 1/e in the distance of a wave-length. If we write

so that I'm denotes the maximum value of the dissipation-ratio for n = 1, the requisite value is given by

On our previous numerical assumptions this is about 4 × 10-7. The corresponding value of e in (95) is about 4 × 107. This is comparable with, although distinctly less than, the virtual variation of wave-length which takes place, on Doppler's principle, in a gas with moving molecules, and which is held to be sufficient to explain the actual breadths of the Fraunhofer lines. Having regard to the very much sharper definition which we meet with in the vibrations of the first type, and to the increase of sharpness (in each type) with the index n of the mode considered, it would appear that there is no prima facie difficulty in accounting, on our present hypothesis, for absorption-lines of such breadths as occur in the actual spectrum.

XIX. The Propagation of Waves of Elastic Displacement along a Helical Wire. By A. E. H. LovE, M.A., F.R.S., Sedleian Professor of Natural Philosophy in the University of Oxford.

[Received 4 December 1899.]

1. It is known that the modes of vibration of an elastic wire or rod which in the natural state is devoid of twist and has its elastic central line in the form of a plane curve fall into two classes: in the first class the displacement is in the plane of the wire and there is no twist; in the second class the displacement is at rightangles to the plane of the wire and is accompanied by twist. In particular for a naturally circular wire forming a complete circle when the section of the wire is circular and the material isotropic there are two modes of vibration with n wave-lengths to the circumference; these belong to the first and second of the above classes respectively, and their frequencies (p/2π) are given by the equations

where a is the radius of the circle formed by the wire, c the radius of the section, p, the mass per unit of length, E the Young's modulus and the Poisson's ratio of the material. These results may be interpreted as giving the velocities with which two types of waves travel round the circle.

So far little or nothing appears to be known about the modes of vibration of wires of which the central line in the natural state forms a curve of double curvature, except that the vibrations do not obviously fall into two classes related to the osculating plane in the same way as the two classes for a plane curve are related to the plane of the curve. The equation connecting the frequency with the wave-length when waves of elastic displacement are propagated along the wire has not been obtained; and although this equation would obviously be quadratic when rotatory inertia is neglected, and so would give two velocities of propagation for waves of a given length, it is by no means obvious what would be the distinguishing marks of the two kinds of waves with the same wavelength.