Let I be the moment of inertia of the disk about its axis of oscillation. Let K be the oscillatory couple, of period 27. applied to the disk.

Then the equation of motion of the disk is expressed by

(I + B1) + ΑΩ = K. .

გი
at

(20)

B1 therefore represents an addition to the inertia of the disk and A is a damping coefficient due to the expending of energy in the production of waves. Two cases are of interest, namely, when R is small compared to the wave-length and when R is correspondingly great.

The expansions of K, and K1 are

Considering first the addition to the inertia, we have

(21)

so that B1 is independent of the period and the wave-length. Imagine a hemisphere of air to become rigid on one side of the disk, and to oscillate with the disk.

The moment of inertia of this hemisphere about the diameter of oscillation is B1.

Hence the increase of inertia is represented by 1π times the inertia of such a hemisphere.

The damping term is observed to be extremely small, a result which is to be expected in the absence of viscosity, but its magnitude has not been hitherto determined.

In the case of the disk oscillating normally to itself the damping term is proportional to R'.

When is large we require the asymptotic expansions of J。 J1 K。 K1.

The above solution is interesting as compared with that for the disk oscillating normally to itself. The great difference between the two solutions is due to the fact that, in the problem just solved, the air on opposite sides of the diameter of oscillation is in opposite phases of its motion.

For small values of the series K, and K, are both rapidly convergent. Even when =2 only four terms of each series. are required to obtain values correct to three decimal places. The following short table may be found useful:

XLVI. The Curves of the Periodic Law.-II. By W. M. THORNTON, Professor of Electrical Engineering in Armstrong College, Newcastle-on-Tyne *.

1.

HE rise and fall of the densities of the elements shown in fig. 1 resembles the side elevation of a somewhat irregular spiral drawn upon a transparent cone. The end elevation of such a curve of uniform angular pitch is a logarithmic spiral. This possibly suggested the scheme of representation of the elements on such a spiral given by Dr. Johnstone Stoney, in which the densities of the argon group were forecast t. Sir William Crookes, in his Presidential Address to the Chemical Section of the British Association in 1886, arranged the elements symmetrically about a line through those of maximum density, and obtained a zigzag curve dividing the elements in a remarkable manner. The present note deals with the minor fluctuations of the density curve.

The periods of the oscillations in fig. 1 increase with distance from the origin, but not so uniformly as they do on the cone spiral, where they are strictly proportional to it. Any regular periodic curve of this kind can be represented by a combination of harmonic curves chosen by inspection or analysis. The larger dotted curve in fig. 1 is that which *Communicated by the Author.

Phil. Mag. [6] iv. pp. 411, 504 (1902).

appears to fit the observed curve of densities best as a fundamental, amplitude and period varying. It is the representation of that great periodic change in the configuration of the outer electrons in an atom under their own forces which a successful theory of atomic formation must explain. It is not the graph of the central force of the nucleus, for this, being proportional to the nuclear mass, changes by uniform steps, and is represented by the dotted straight centre line.

On the fundamental curve there is superposed one of double frequency, the amplitude and phase of which in fig. 1 are chosen by inspection. This curve can be regarded by analogy as representing some form of retardation in the change of atomic volume of the nature of hysteresis*-that is, a change of arrangement caused by the resultant force to which the fundamental is due, always occurring so as to produce the same deviation from the mean position in successive half periods.

In fig. 2, moving from right to left, the effect of the

double period at a is to make the resultant density less than normal, taking the fundamental curve to represent the normal type of change. At b it is greater, at e less, and at d greater.

2. The physical and chemical properties of the elements appear to depend as much upon the minor periodic change as upon the fundamental. In order to bring out the differences between them, curves are given below in which the ordinates are densities, and abscissæ the projections of the major harmonic curve on the vertical axis. Thus in fig. 5, to take a clear example, the horizontal range from 3 to 11 5 arbitrary units corresponds to m n on the vertical axis of fig. 2, where

* Phil. Mag. xxxiv. pp. 70-75 (July 1917).

this represents the oscillations of the fundamental in that part of the curve of densities. The abscissæ being drawn to the same scale as the densities, the centre line of the loops is inclined at 45 degrees. If the density rose and fell in every case proportionally to the fundamental alone the loops would close on to the centre line.

The first curve (fig. 3) contains uranium, thorium, and radium. Of these, the first two have densities greater than normal, as shown by the position of the minor periodic curve drawn above the mean line in fig. 1. Radium is somewhat below it. Densities greater than normal in reference to the major periodic curve may be called over-saturated; radium is, on this view, under-saturated.

The second group (fig. 4) begins with bismuth and ends with caesium, passing through the region of rare earths. This loop differs from the rest in having double flexure on the descending side. The elements there are alternately overand under-saturated twice in falling from the maximum to the minimum. This appears to have had a marked influence on the densities of the rare elements. Those whose densities are known--erbium 47, lanthanum 61, praseodymium 647, cerium 6 68, neodymium 6.95, samarium 7·75--all lie on a small part of the curve in the neighbourhood of cerium. It would seem that, after tantalum, the elements which should lie immediately below it in fig. 4 are unable to form in the usual manner, and precipitate into a group having normal atomic weights, so that they take their places in the descending scale of elements, but abnormally small densities.