DISCUSSION OF THE CRITICAL CASES.

315

345. When is sufficiently small, all the u's are exceedingly large positive quantities; for since uj+1=∞, it follows from (7) and (14) that

so that u; can always be made as large a positive quantity as we please, by taking 7 small enough; whence it follows from (7) and (13), that all the u's can be made positive, provided is small enough. But if u is positive, we see from (11), that the signs of x; and x must be different; accordingly when 7 is very small, each shell is moving in the opposite direction to the two adjacent ones; also when u; is large, the numerical value of x- must be very much greater than that of xi.

These considerations show, that when the period is exceedingly small, the vibrations of each shell, and also those of the outer massless envelop, which is supposed to be rigidly connected with the ether, are executed in opposite directions; and that the amplitudes of the vibrations of successive shells diminish with great rapidity, as we proceed inwards into the molecule.

It follows from (7) and (13) that as 7 increases, a; diminishes, whilst C+1/+1 increases; accordingly when 7 has sufficiently increased, u; will be zero. Now when u is zero, ui-=∞; and will therefore have passed through zero, and have changed sign for some value of T, less than that for which u; became zero. therefore follows, that as 7 increases from a value for which all the u's are positive, u will be the first quantity which vanishes and changes sign, and that u, will be the next and so on.

It

346. In the problem we are considering, the motion is supposed to be produced by means of a forced vibration of amplitude , and therefore when a1 = ∞, u,= 0; but u, will also vanish when =0, hence the critical period for which u, vanishes is the least period of the free vibrations of the system, when the massless envelop is motionless. As soon as T exceeds the first critical period, u will become negative; and consequently the first shell and the massless envelop, will be moving in the same directions, whilst all the other shells will be moving in opposite directions. If now be supposed to still further increase, u, will diminish and finally vanish, in which case u1 = − ∞, x1 = 0. This is the second critical case; and the period of vibration is equal to the period of the free vibrations of the system when m, is fixed, and all the

other shells are vibrating in opposite directions; and this period is the least period of the possible free vibrations of the system under these conditions. The remaining critical cases.

can be

From this equation we see, that 1/u- is a fraction, whose numerator is a linear function, and whose denominator is a quadratic function of 72. It therefore follows that 1/u, or -x/C, is a fraction whose numerator is a (j-1)th, and whose denominator is a jth function of 72. Since u is zero, when 7 is equal to any one of the j periods of the free vibrations of the system, when the envelop is held fixed, it follows that the denominator of 1/u, is expressible as the product of factors of the form K2; -T2, where K1, K2 κ are the above mentioned free periods.

...

The value of 1/u, may therefore be resolved into partial fractions, and may accordingly be expressed in the form

where 91, 92

Writing for a moment D; for x2;/72 — 1, (18) becomes

-1

D1

D2

=

dr- (q1/D1 + q2/D2+..............)2

(19).

Now is the amplitude of the th shell; if therefore we denote the actual displacement by x', we shall have

x';= x; sin 2πt/T,
αι

provided t be measured from the epoch at which each shell passes through its mean position. The energy in this particular configuration will be wholly kinetic; whence remembering that the mass of each shell is equal to mi/47, it follows that if E be the total energy,

-1

so that R-1 denotes the ratio of the whole energy of the molecule to that of the first shell; then (19) becomes

Hence if R, denote the value of R when 7 = ki, we obtain

348. Let us now imagine a medium, whose structure is represented by a very large number of molecules of the kind we have been considering; and let us suppose, that the interstices between the molecules are filled with ether, which is assumed to be a medium, whose motion is governed by the same equations as those of an elastic solid. Then if we confine our attention to a small element of the medium, which contains molecules and ether surrounding them, and for simplicity consider the propagation of waves parallel to the axis of x, the equation of motion of a particle of ether will be

Integrating this equation throughout the volume of the element, we obtain

℗ [[fïó dwdyde = n [[fdw' dydz], – n [[ƒdu' dyd: ].........(22). SSS

P

The first surface integral on the right-hand side is to be taken over the outer boundary of the element, whilst the second is to be taken over the boundaries of each of the molecules. Let w be the mean value of w' within the element; then the values of w' at the points + dx and x-x will be

The second integral represents the resultant of the forces, which each molecule exerts upon the ether; and if we represent the force due to a single molecule by 47°C ( — w') dx'dy'dz', it follows that we may represent the resultant force due to all the molecules within the element by 47°C (-w) dx dy dz, where is the mean value of the displacements of all the molecules. The equation of motion therefore becomes

Now (n/p) is the velocity of light in vacuo, whence the lefthand side is equal to μ2, where μ is the index of refraction. Hence if we substitute the value of /§ from (21), and write q, for CRK/m, we shall finally obtain

350. This equation determines the index of refraction in terms of the period. To apply it to ordinary dispersion, we shall write it in the form

From the manner in which this result has been obtained, it follows that K1, K... are in ascending order of magnitude. Now in the case of ordinary dispersion, μ increases as the period diminishes, whence we must have T> and <; also the quantities 92, qs must be inappreciable, and q1 must be very slightly less than unity. Under these circumstances, we approximately obtain

-

If we omit the term (1-q1) 7, this expression is the same as Cauchy's dispersion formula, which agrees fairly well with experim etteler has however shown that for certain sub

DISPERSION AND ABSORPTION.

319

351. In order to apply (24) for the purpose of illustrating anomalous dispersion, it will be sufficient to confine our attention to the terms involving 1 and 2.

Differentiating (24) with respect to us, we obtain

When T=0, dμ3/dr2 is negative; but when = ∞, it will not be negative unless q+q<1; since however μ increases as diminishes, we may suppose that the various quantities are so related, that dμ2/dr2 is negative throughout that part of the spectrum, which is capable of being observed. And if we suppose q is very nearly equal to unity, and q2<1-q, this will be the case for a considerable range of values of 7.

Let us further suppose that >; then when T is excessively small, μ2 will be less than unity; as 7 increases, μ2 will diminish to zero, and will then become a negative quantity. When μ is negative, the velocity of the waves will be imaginary, and consequently waves whose periods produce this result, are incapable of being propagated in the medium, and absorption will take place. When т=1, μ2=-∞; and as soon as т> K1, μ becomes a very large positive quantity, and regular refraction begins to take place. As further increases, μ2 diminishes, until it vanishes and changes sign. A second absorption band accordingly commences, and continues until T>2, when regular refraction begins again, and so continues until T = ∞.

352. The following figure will serve to give the reader a general idea of the value of μ. The abscissæ represent the values of 72, and the ordinates the values of μ2. The dotted lines AC, BD, EF are the lines T = K1, T = K2, μ = 1, and the upper parts of the curves Pp, Qq represent the visible portion of the spectrum produced by a prism filled with a substance, which produces anomalous dispersion, and has an absorption band in the green.

The portion Ee may be supposed to consist of waves whose periods are too short to be observed, then comes an absorption band, and beyond A a region of highly refrangible ultra-violet light commences. The line H in the spectrum may be supposed to commence at P; and a band of light accordingly becomes visible, which continues through the indigo to the blue, and in