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3. Interim Report on Cosmic Dust.

4. Report on Underground Temperature.-See Reports, p. 75.

5. Report on the Sizes of the Pages of Periodicals.-See Reports, p. 77.

6. Report on the Comparison and Reduction of Magnetic Observation. See Reports, p. 209.

7. Interim Report on the Comparison of Magnetic Standards.
See Reports, p. 79.

FRIDAY, SEPTEMBER 13.

A joint Meeting with Section B.

The following Papers were read:

1. The Refraction and Viscosity of Argon and Helium.
By Lord RAYLEIGH, Sec. R.S.

As compared with dry air, the refraction (μ-1) of argon is 0·961, and that of helium (prepared by Professor Ramsay) is as low as 0.146.

Dry air being again taken as the standard, the viscosity of argon is 1.21, and that of helium is 0.96.

2. On Specific Refraction and the Periodic Law, with reference to Argon and other Elements. By Dr. J. H. GLADSTONE, F.R.S.

In 1869, 1877, and 1883 the author had shown that the specific refractive energies of the metallic elements are usually in the inverse order of their combining proportions, and that the specific refractive energies of the elements in general are to a certain extent a periodic function of their atomic weights.

The present communication refers to some developments of these old observations.

(1) Argon. The specific refractive energy of argon gas, as reckoned from Lord Rayleigh's data, is 0.159. Deeley had suggested that this property might throw light upon the question whether the atomic weight is about 20 or double that figure. The following are the specific refractive energies of the elements with atomic weights between 12 and 23, with the insertion of argon. Carbon, 0.417; nitrogen, 0-236; oxygen, 0·194; fluorine, 003 (?); argon, 0·159; sodium, 0.209. Argon appears to be here in place on the rise which follows the great descent from carbon to fluorine. It does not equally well fit the neighbourhood of calcium, 0.250. If the atomic weight be 19.94, the molecular refraction will be 3·15, which is almost the same number as that for oxygen gas, 3·10, or nitrogen gas, 3.30.

(2) The fact that the specific refractive energies of the univalent metals are generally inversely as the square roots of their atomic weights is confirmed by further research, the product of the two being about 13. The same observation is now extended to the earthy metals in the second column of Mendeléeff's table, the products in that case being fully 14. The rule does not apply to the halogens in column 7. As to column 8, iron, palladium, platinum, and gold all give products

1895.

R R

which are far higher. This confirms the belief that gold is not rightly placed in column 1.

(3) It is known that the refraction of a salt when dissolved in water is often slightly modified by the proportional amount of the solvent. The author and Mr. Hibbert have recently found that salts of the metallic elements in columns 1 and 2 of Mendeléeff's table show an increased refraction on dilution, those of metals in column 8 a diminished refraction.

3. A Discussion 'On the Evidence to be Gathered as to the Simple or Compound character of a Gas, from the Constitution of its Spectrum,' was opened by Professor A. Schuster and Lord Rayleigh, and the following Papers were read :

4. The Constituents of Cleveite Gas. By C. RUNGE and F. PASCHEN.

As the spectrum of the gas contains two sets of lines, each consisting of three 'series,' and no other lines, we may, according to the analogy of other spectra, draw the conclusion that it consists of two, and not more than two, elements. The yellow line D, belongs to the heavier of the two elements, which therefore should alone be called helium.

We have separated the two elements to a certain extent by a method of diffusion, the lighter constituent streaming more easily through a plug of asbestos. It was shown that the lines in the visible and in the ultra-red part of the spectrum ascribed to the heavier constituent are less intense relatively to the other lines the earlier the stream of the gas is cut off.

The same conclusion that the gas consists of two elements may also be drawn, first from the spectrum of the sun's limb, where the stronger lines of the heavier constituent are always.present, while the stronger lines of the lighter constituent are only seen once in every four times. On the other hand in the spectrum of Nova Auriga at its first appearance we have the opposite case, the lines of the lighter constituent being far more prominent.

On Motions competent to produce Groups of Lines which have been observed in Actual Spectra. By G. JOHNSTONE STONEY, M.A., D.Sc., F.R.S.

In most of the spectra that consist of lines very remarkable groups present themselves, in which the lines are seen to be associated into definite series. In such cases, except under special circumstances, we may safely presume that all the lines of a group arise from the motion of a single electron in every molecule of the gas.

Very striking examples of such groups are present in the absorption spectrum of oxygen and in the bright line spectrum of carbon. The oxygen of the earth's atmosphere produces the great A group of double lines in the solar spectrum, as well as the very similar great B group, and the a group. It also produces a group more refrangible than D, about which we know less. This group is much fainter than the others, and it is only under exceptional circumstances that it can be seen at all in the solar spectrum. Each of the other three groups can be distinguished into two sub-groups, which from their appearance have been called a head and a train. The general features of these three groups are the same, and Mr. Higgs has made a careful geometrical analysis of one of them, the great B group. From his analysis we may infer that the head and the train are due to motions in the molecules which are distinct, although related to one another. This conclusion receives further support from the circumstance that in the double lines of 'the head' it is the violet component of each pair which is the stronger, while in the train it is the red component of each pair which is the stronger.

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In a paper in the Scientific Transactions of the Royal Dublin Society' for 1891, p. 563, the present author pointed out that, if we proceed on the probable

Proc. Roy. Soc., 1893, p. 00.

supposition that the motion of each electron is an orbit of some kind going on within the molecules, it can be shown that the partials of the motion of the electron which causes the lines are elliptic partials, and that where an elliptic partial suffers an apsidal perturbation, it divides into two circular sub-partials, giving rise to the two constituents of a double line. We may infer from this that the sub-partials corresponding to the red constituents of the fourteen or more double lines of the train of B are circular motions revolving one way, and that all the violet constituents of these double lines result from circular motions revolving the other way.

In order to advance beyond this point it is necessary to make two further hypotheses which probably are both true. Two hypotheses must here be ventured upon because observations with the spectroscope give us no information as to the phases of the elliptic partials or the planes in which they lie. One hypothesis that recommends itself is that the circular sub-partials belonging to a connected series of double lines, e.g., to the train of the great B group, lie in one plane. Another hypothesis which we may venture to make, as a preliminary working hypothesis, is that the amplitude of the motion of the electron has its maximum value at starting, i.e., when that event has occurred at the close of a struggle between two molecules which has set up the motion of the electron, which continues during the comparative repose of the quiet, undisturbed journey in which the molecule is indulged after its encounter.

With these assumptions it is possible to synthesise all the motions causing the red constituents of the double lines into one motion, which is, however, not circular, but a slowly contracting spiral; and a similar resultant spiral motion turning the opposite way is furnished by the sub-partials forming the violet constituents. While these spirals are being traversed the radii or semi-amplitudes of the circular motions of which they are composed, and which correspond to the individual lines in the spectrum, may become shorter or longer owing to the escape of energy to the æther, or absorption of energy from it; so that the actual orbits are spirals which may be somewhat inside or somewhat outside those which result from the assump. tion that the radii retain their length. These two spiral motions combine at each instant into a single elliptic motion so elongated that it is nearly a linear vibration; and this elliptic motion continues to represent what occurs, if subjected to the five following perturbations :

1. A decrease of amplitude.

2. A diminution of periodic time.

3. A slow apsidal motion in a direction opposite to that in which the revolution of the electron in the orbit takes place.

4. A slight fluttering motion which may be represented by a very shallow wave running rapidly round the ellipse.

5. A further slight modification of the form of the ellipse which takes the form of a secular perturbation.

Accordingly we arrive at the conclusion that an elliptic motion undergoing these perturbations is such a motion of an electron as would produce the entire series of lines in the train of B. A similar motion would produce the train of A, of a, and of each of the other similar groups, if such exist in the spectrum of oxygen. These elliptic motions undergoing perturbations may be appropriately called mega-partials in their relation to the actual orbit described in oxygen by the electron that produces all these trains of lines, since that orbit is the resultant which we should get by superposing the motions in these few mega-partials.

A similar treatment applied to the head' of any of the oxygen groups shows that it, too, arises from an elliptic motion subject to perturbations, the chief differences being in the law connecting the falling-off of amplitude and the periodic time; that the quick, fluttering perturbation is absent; and that the apsida! motion takes place in the opposite direction. In oxygen the strength of the lines of each sub-group fades out towards the red. When the fading is in this direction, the periodic time decreases as the amplitude falls off. Whereas when, as in the carbon spectrum, the lines fade out towards the violet, the periodic time becomes longer as the amplitude decreases. And, finally, if the lines present themselves,

when plotted on a map of oscillation frequencies, as disposed symmetrically on either side of a common centre, this indicates that the periodic time continues unchanged during the shortening of the amplitude.

This suggests the cause of the width of spectral lines in general, so far as their width is not merely apparent, i.e., due to the Doppler effect of the translational motions of the molecules, or to the breadth of the slit of the spectroscope. The rest of the width of the line, as seen, is its true physical width, and seems to be due to the interchange of energy between the molecule and the æther. This leads to diminished amplitude; and this reduction of the amplitude may be accompanied by either a reduction, or an increase, or a persistence unaltered of the periodic time, according to the way in which the motion of the electron is dynamically associated with the rest of the events which go on within the molecule. If the periodic time decreases, this gives rise to a ruling fading out towards the red; if there be an increase of the periodic time, the shading is towards the violet; while if the line fades out both ways symmetrically, there is no change in the periodic time. The relative intensities and the spacings of the lines of the ruling depend on the law which connects the escape of energy and the shortening of the semi-amplitude, and in its turn this law depends on the dynamical relations in which the parts of the molecule stand to one another. The excessively fine rulings of which the widths of individual lines consist can probably not be seen otherwise than as a shading, unless perhaps in some very few exceptional instances, owing to their being blurred together by the Doppler effect.

We have attributed these very fine rulings to the interchange of energy with the æther. On the other hand, the more conspicuous rulings, such as those we have been studying in oxygen and carbon, seem to be associated with the transference of energy from one motion within the molecule to another. This may be briefly described by saying that the widths of the individual lines and their being in various ways shaded off are due to radiation, while that they are arranged in series is due to conduction.

SATURDAY, SEPTEMBER 14.

The Section was divided into two Departments.

The following Papers and Reports were read :-
:-

Department 1. MATHEMATICS.

1. On the Translational and Vibrational Energies of Vibrators after Impacts on fixed Walls. By Lord KELVIN, Pres. R.S.

2. On Bicyclic Vortex Aggregates. By Professor W. M. HICKS, F.R.S.

The author showed that in any case of a vortex aggregate in which the motion takes place in planes through an axis, and symmetrical about this axis, another state of motion was possible with the same current and vortex sheets, but in which the motion was in circles round the axis, and in which the fluid external to the aggregate remained at rest. These two motions can be superposed with a resulting steady motion, and the two cyclic constants independent of each other.

3. On Hill's Spherical Vortex. By Professor W. M. HICKS, F.R.S. The author showed that it is possible to build up a compound spherical vortex, consisting of shells in which the rotation is oppositely directed in successive shells.

The vorticity and size of each shell must satisfy definite relations. When the vorticity of the fluid is everywhere of the same magnitude the ratio of the (n + 1)th radius to the nth satisfies an algebraical relation of the form

4λn (xn2 + X'n + 1) = 3x3 (în + 1),

where λ = 1 − λn-1 (1 − 2′′n−1).

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The ratio of the volumes of the shells for the first three are 1, 1:3:13, 1·341.

4. On a Dynamical Top.1 By G. T. WALKER, M.A.

The author exhibited a top in the shape of a fattened ellipsoid with a central circular portion movable, and arranged so as to be clamped with the lines of curvature inclined to the axes of dynamical symmetry. In this condition a rotation communicated to the top when placed on a sheet of plate glass sets up oscillations which reverse the direction of motion: these reversals may, under favourable conditions, be as many in number as fifteen. A vertical tap administered at the end of an axis of symmetry gives rise to angular velocity, of which the sign depends on the difference between the periods of longitudinal and transverse vibrations, as well as on the angular deviation of the movable portion.

5. Suggestions as to Matter and Gravitation in Professor Hicks's
Cellular Vortex Theory. By C. V. BURTON, D.Sc.

6. On the Graphical Representation of the Partition of Numbers.
By Major P. A. MACMAHON, F.R.S.

7. On a New Canon Arithmeticus.

By Lt.-Col. ALLAN CUNNINGHAM, R.E., Hon. Fell. King's Coll. Lond. This is a series of tables, drawn up precisely like Jacobi's 'Canon Arithmeticus,' giving the solution of the congruence 2R (mod. p and mod. m) for all prime moduli (p) <1,000, and also for all moduli m <1,000, where m is a power of a prime. There are two tables to each modulus, p or m. The left table shows the remainders (R) to a given index (a); the right table shows the index (x) to a given remainder (R).

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Uses of Tables.

1. To find remainder R after dividing 2 by p or m (x, p, m being given). 2. To find index a such that 2÷p or m may leave remainder R (R,p, m being given).

3. To find whether a given number N is exactly divisible by a given prime p, or power of prime m; and, if not, what remainder (R) is left.

4. To find whether a given number N is exactly divisible, or leaves a given remainder (R) after division, by any prime or power of a prime <1,000. 5. To find all the primitive roots of a given prime (p), and all the roots which are residues of a given order e of a given prime p, when 2 is a primitive root of p; and to find all the roots which are residues of a given order e of a given prime p when 2 is a residue of order not >e.

Jacobi's Canon gives the solution of g=R (mod. p and mod m) as above, except that y is a certain primitive root of p. His table is better for case 5 above

whenever 2 is not a primitive root of p; but the new canon to base 2 is much more convenient for the more practical uses 2 and 3 above. His canon is well described in Cayley's Report on Mathematical Tables in the British Association Report of 1876: the description applies, with slight obvious modification, to the new canon.

1 The paper will be published in the Quarterly Journal of Mathematics.

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