One unit means a distance of 24.6 mm. which gives for the average distance 25.2 mm. and for the ratio of the wave-lengths of the two lines 1.0000212.

Closely connected with the preceding investigations is the study of the effect of the temperature, thickness, and density of the source on the composition of the radiations, as shown by the symmetrical or unsymmetrical broadening of the spectral lines and the consequent shifting of their mean position. This question has quite recently been taken up by H. Ebert and the results he has already obtained are very promising. The principal effects noted are: first, the shortening of the difference of path at which interference can be observed; secondly, the shifting of the fringes as the mean wave length changes. Ebert has shown that the interference method is far more delicate than the spectroscopic; and by its means he has established two conclusions which, if verified, are of the greatest importance-namely; first, that the chief factor in the broadening of the spectral lines is the increase in density of the radiating body; secondly, that the broadening, in all the cases examined is unsymmetrical-causing a displacement of the line toward the red end of the spectrum. The importance of these conclusions, in their relation to the proper motions of the heavenly bodies and their physical condition, can hardly be overestimated. The value of results of this kind would, however, be much enhanced if it were possible to find a quantitative relation between the density of the radiating substance and the nature of its radiations. In the case of hydrogen enclosed in a vacuum tube this could readily be accomplished. It may, however, be objected that it would be difficult in this case to separate the effects of increased density from those due to the consequent increase in the temperature of the spark. The problem of the temperature of the electric discharge in rarefied gases is one which has not yet been solved. In fact it may seriously be questioned whether in this case temperature has anything to do with the accompanying phenomena of light; and it

appears to me much more reasonable to suppose that the vibratory motion of the molecules is not produced by collisions at all but rather by the sudden release of tension in the surrounding ether.

Whether true or not, the results obtained and interpreted by this hypothesis would be of great interest. The method could be applied directly to any substance, mercury for instance, for which the relation between the temperature and the pressure is known. For substances for which this relation has not been established, as sodium, thallium, etc., the density may be found by heating the substance in a tube closed with plane parallel glass ends and measuring its index of refraction. The density will be very approximately proportional to the excess of this index over unity.

Aside from its application to this problem, it seems highly probable that this method of measuring the density and pressure of vapors may be made to yield excellent results in cases where these are far too small to be measured directly.

It may not be entirely out of place in this connection to present a few observations concerning the causes of the broadening of the spectral lines. It seems to me that by a thorough and systematic study and discussion of this phenomenon we have a possible means of materially increasing our knowledge of a subject, of which we are at present in almost total ignorance: namely, the real action of the forces and motions of vibrating atoms and of the ether which transmits these vibrations in the form of light.

The possible causes of the broadening of spectral lines are as follows:

First, the addition of vibrations of periods differing from the normal period, due to the influence of neighboring molecules; secondly, the effect on the wave length due to the velocity of the molecules.

It is evident on considering the second cause, that it could not possibly account for more than a small fraction of the effects observed. For example, to effect a change in wave-length corresponding to the difference between the two sodium lines, would require velocities of the order of three hundred thousand meters per second, over a hundred times as great as the velocities given by the kinetic theory. But it is also clear that when a gas is so rarefied that the first cause cannot possibly produce any perceptible effect, the second cause would be quite sufficient to limit the fineness of the lines, and to impose a definite limit to the difference of path at which interference is visible; and it is worthy of note

that the actual limits observed are of the same order of magnitude as those given by the kinetic theory.

There is still a third cause which might limit this distance, but which would not have any effect in broadening the lines; namely, the diminution in the amplitude of the vibrations after collision. There must be such a diminution and it would evidently be the more marked the more rapidly the energy was transferred to the ether, that is, the brighter the light. If the effects due to this cause alone could be separated from the others it would be possible to measure the diminution in amplitude and therefore the rate of transference of the energy. Thus it may be shown that a vibrating sodium atom gives up to the surrounding ether less than six millionths of its energy at every oscillation.

Returning to the first and chief cause of broadening, it may be remarked that the universal opinion of scientific men seems to be that during collisions between the molecules the vibrations are entirely "irregular ;" and the longer the collisions last in proportion to the time between collisions, the more intense will be the light due to these "irregular" vibrations, and consequently the broader the lines and the more impure the light.

The following consideration would seem to show that this explanation will not hold.

If, in the refractometer, so frequently referred to, white light be used, all phenomena of interference are lost to sight when the difference of path exceeds a few wave lengths, for the well-known reason that the fringes due to the infinite number of different kinds of light are superposed, thus producing a uniform illumination. If now this light be analyzed by a spectroscope, the spectrum will be traversed by well-marked interference fringes which are the finer and closer, the greater the difference of path of the interfering pencils. Now, I have observed such interference fringes in the white light from the incandescent carbons of an arc light when this difference amounted to thirty thousand waves. And it may be added that this limit was reached by the closeness of the lines rather than by their indistinctness.

It seems to me that the only conclusion which can be drawn from this experiment is that in the light from an incandescent solid the vibrations must be isochronous for at least thirty thousand waves. The same observation applies also to the so-called "irregular" vibrations of the broadened sodium lines, for the same limit (about thirty thousand waves was also observed in this case). The

inference seems irresistible that the broadening is not caused by "irregular" vibrations, but by the addition of vibrations whose intensity is greater the nearer their period is to that of the normal vibrations and which may be almost if not quite as regular as the normal vibrations themselves.

If these conclusions be granted we must profoundly modify our conception of radiation in solids and liquids, or at least that part of it which supposes that such radiation produces a continuous spectrum because the molecules have no "free path," and, therefore, no proper periodic vibrations.

What, then, is the nature of the effect produced by the collision of molecules? If it be to produce or reinforce vibrations differing from the normal type, it must be granted that these new vibrations are isochronous. If so, they must be due either to a change in the form or in the mass of the molecule itself produced by collision, such changes tending to revert back to the type when the frequency of the collisions is not too great. The only alternative is to suppose that the molecules differ among themselves, either in form or weight. In this case, the molecules agreeing most nearly with the type and hence having a proper period differing but little from the normal would be more easily set in vibration than the others, or their vibrations once started would outlast the others. Accordingly, when a gas is very much rarefied, the collisions are few, hence only the typical vibrations persist; but when the collisions are frequent the other vibrations are also sustained.

I fear I have wandered so far from the subject of this address, if such a name be at all appropriate, ever to return; and, though many other interesting and important applications of light-waves clamor for recognition, I fear they would be wearisome even to

enumerate.

I fear also that it may with some justice be said that I have made a plea for my own instruments and theories, rather than "a plea for light waves;" and still more that I have presented many crude and half digested ideas, when it would have been more to the purpose to present facts and results of diligent study and careful experiment.

In extenuation and in conclusion I can only hope that if I have created the slightest interest in the matters here presented for your consideration, if there be any chance that even a few of the seeds may germinate, grow, blossom and bring forth fruit worthy of acceptance, my purpose will be accomplished.

PAPERS READ.

ON THE DIFFUSION OF HEAT IN HOMOGENEOUS RECTANGULAR MASSES, WITH SPECIAL REFERENCE TO BARS USED AS STANDARDS OF LENGTH. By R. S. WOODWARD, U. S. Geological Survey, Washington, D. C.

[ABSTRACT.1]

THIS paper discusses the laws of diffusion of heat in rectangular masses of any dimensions, and aims to give to the various problems that arise, solutions which may be readily used in computing numerical results.

The assumptions on which the work is based are the following: (1) that the mass has initially a uniform temperature; (2) that it cools or heats in a medium of sensibly constant temperature; (3) that the exterior and interior conductivity and thermal capacity of the mass remain constant.

Starting from Fourier's solution of the general problem defined above, the obstacles met in applying that solution are pointed out. Fourier's method requires a certain number of the roots of the three transcendental equations which express the boundary conditions of the mass. The new solution either avoids the difficulty of determining those roots altogether, or makes use of the first root only of each equation. Incidentally, however, methods of computing the roots are given.

The pervading idea of the investigation is this, viz. to separate the terms independent of from those dependent on the exterior conductivity, or emissivity. In accordance with this idea, the problem divides itself into two cases, in the first of which the ratio formed by the product of the emissivity and a linear dimension of the mass is less than unity, and in the second of which that ratio is greater than unity.

Formulas for certain average temperatures of special interest relative to standards of length are given, viz. : the average temperature of the whole mass, of any face and of the axis of a bar.

Special attention is given to the needs of the computer in the derivation and arrangement of formulas, and the application of nearly every formula is illustrated by a numerical example.

1 The paper will be published in full in Annals of Mathematics, Vol. 4, No. 4.