low temperatures and for long-period waves at all temperatures. Paschen (l. c.) has noted the possibility of even such partial radiators as glass radiating as a black body at very low temperatures. This class of maxima is associated with the so-called continuous spectra.

II. A second class of maxima vary only slightly in position as the temperature is changed. These maxima are low and broad, but yet sharper and higher than the maxima in the spectra of perfect radiators at the same temperature. Such maxima we find commonly in the infra-red spectra of the crystalline salts, glass, water-vapour, and the like. The variation in the position of the emission maximum in the carbon-dioxide spectrum has been studied by Paschen *. He found the wave-length of the maximum varied about 3 per cent. as the temperature was raised from about 600° to 1700° absolute. We have not yet been able to study the emission maxima of the crystalline salts; but the work of Abramczyk † on the radiation from quartz indicates that we might expect such a variation in the maximum. The Kirchhoff law should hold in this region for this radiation, and if a plate of quartz does not absorb all the radiation emitted by a thinner plate at a higher temperature-as observed by Abramczyk-may we not most easily account for the fact by supposing that the positions of the emission and absorption maxima were different at the different temperatures? Whatever it is, the law of the variation is probably not a simple one; and this quite agrees with the mathematical expression developed later on.

III. A third class of emission maxima do not vary at all in position with the temperature, though in the regions of temperature and period in which they occur the isochromatic emission temperature function is probably the same as that which holds for perfect radiators. These maxima are, as a rule, narrow and sharp, and the emission in the region in which they occur is practically all confined to the maxima themselves. The emission at the maxima may be regarded as infinite in comparison with the emission at a short distance on either side of the maxima. This class of maxima (spectrum lines) are associated with high temperatures and short periods. They probably never occur in the same period-region with maxima of class I., at least when the radiator is an element or a chemically simple compound and when no chemical change occurs. We cannot draw such a distinction between maxima of types II. and III., nor is such a distinction drawn mathematically in the complete function.

*F. Paschen, Wied. Ann. 1. P. 440 (1893).

† Abramczyk, Wied. Ann. lxiv. p. 647 (1898).

Maxima of types II. and III. may be introduced into functions (8) and (9) by simply applying to them an inverse algebraic polynomial in 7 as a factor. The real roots of this factor would give infinite maxima, independent of the temperature of type III. And near each imaginary root would be a real maximum, finite and rather broad, of type II. But there are a number of conditions to be observed in the insertion of these maxima. In the first place, the emission-temperature function, which determines the isochromatic curves, must remain unaltered in form. Again, if all bodies emit like perfect radiators at low temperatures and long periods, and if these period-maxima are all of short period and occur at high temperatures, then the complete emission-function must return to the original form (8) or (9) for perfect radiators when we make these maxima zero in the function. And when we place these maxima equal to zero in the function, the form of the function must not depend upon the number of these maxima; so that the polar factor in the complete function must consist of a summation of simple factors of similar form, rather than a polynomial of high degree.

After unsuccessful trial of many possible polar factors, the form

was found to be satisfactory. This factor assumes that there are p maxima of type II. and q maxima of type III. We have then for our complete function, instead of (8) and (9),

in which we have written as one the two terms of the polar 1/2. The logarithmic derivatives of

factor above and

(10) and (11) give

considering but a single maximum of each type. In (13) we have written F(TT) for 7.4′(TT)/4(7T), remembering that

T.(TT) is again a function of the product TT. Maxima of type III. occur when these derivatives (12) and (13) are infinite, i. e. when TT and the negative sign occurs. For maxima of type II., these derivatives must equal zero: such a maximum is obtained by using the positive sign and finding the real root of an equation of the third degree in 7. For the value 7m=0, (12) and (13) give maxima of type I., TmarT=const., characteristic of perfect radiators. Also for T=0, (10) and (11) assume the forms (8) and (9), as they should. We may note in passing the coincidence of these maxima of the emission-period function with those of the reflexion-period and absorption-period functions, and that the maxima of the refraction-period and electromagnetic rotation-period functions are included among them.

Function (10) agrees well with the scant data at present available on the emission of partial radiators. Using the values of the constants n and B determined by Paschen and by Lummer and Pringsheim, giving 7m a value corresponding to the infra-red, (10) gives a curve closely resembling that obtained by Rubens and Aschkinass for carbon dioxide. Two maxima, 71 and 72, of different intensity near together, give curves similar to those obtained by Rosenthal† for quartz and mica. A number of maxima near together give the characteristics of an emission-band. For periods considerably greater than the greatest maxima of type II. or III. the effect of the presence of the several maxima is vanishingly small, and (10) gives the same emission-curve as (8). Nearer the greatest T, and within a few octaves of it, the emission by (10) is much less than that given by (8). The defect in the Wien-Planck-Paschen formula in this region was noted by Rosenthal. On the shorter period side of any maximum the emission by (10) falls off much more rapidly to a much smaller value than is given by (8). Thus, in the region of the lined spectra, the radiation is practically all confined to the lines themselves.

The polar maxima, T1, T2, ... Tm, may perhaps be identified with ionic, molecular, or atomic period. Mathematically considered, they may be functions of any argument except temperature. If we consider them hyperbolic functions of the time-that is, consider them damped wave-periods-the polar factor vanishes at intervals, and the integrated effect is that of a short faint continuous spectrum on either side of the spectrum-line. If we consider them functions of the ordinal numbers, we obtain a simply related series of lines. They * Rubens and Aschkinass, Wied. Ann. Ixiv. p. 595 (1898). † Rosenthal, Wied. Ann. lxviii. p. 796 (1899).

are known to be functions of the magnetic field, and the effect of the field upon their three space-components is well

known.

The constants C1, C2,...Cm determine the relative intensities of the maxima but not their form or position, since they do not occur in the derivatives of the function. Both the position and the form of the finite maxima depend on the values of B and n, and the latter at least depends upon the distribution of the ionic or molecular periods about the nean. Both these parameters appear to be a minimum for perfect radiators. B and perhaps n also are functions of the pressure when this is variable.

It is hoped that this rather desultory discussion of the critical properties of the complete emission function may prove useful and suggestive in the further investigation of the emission of partial radiators. It may be indicative of the possible great practical value of the discussion of physical functions from the standpoint and by the methods of modern mathematical function theory. The results obtained are of course only tentative at best, and liable to be greatly modified by new experimental evidence, yet the same may be said of elaborate analytical deductions.

Berkeley, California,
April 1901.

XXXIX. The Spectrum of Cyanogen. By E. C. C. BALY, F.I.C., and H. W. SYERS, M.A., M.D.* Na recent paper in the Philosophical Magazine on the Spectra of Carbon †, Professor Smithells pointed out the desirability of observing the vacuum-tube spectrum of cyanogen, with particular reference to the presence or absence of the spectra of carbon, inasmuch as their absence would materially support the explanation he puts forward as to their origin, namely, that the carbon-oxide spectrum is due to carbon dioxide, and the Swan spectrum to carbon monoxide, while the line spectrum is due to the element carbon.

One of us in a previous paper ‡, some years back, in dealing with the stratification of the electric discharge, described experiments which had been made on mixtures of hydrogen and carbon dioxide and referred to the spectra seen as those of hydrogen and carbon dioxide. This has since been criticised

Communicated by the Physical Society: read June 28, 1901. + Phil. Mag. April 1901.

Baly, Phil. Mag. xxxv. p. 200 (1893).

severely*, because in the general opinion of spectroscopists carbon dioxide has no spectrum at all.

A great number of experiments, however, had been previously carried out with a view of obtaining a mixture of hydrogen and carbon monoxide, or rather a mixture which showed only the Swan spectrum together with that of hydrogen. This was found at the time to be impossible, as the Swan spectrum was at once changed into the carbonoxide spectrum.

In order to obtain the Swan spectrum in a pure state, it was found necessary to fill a vacuum-tube with pure carbon monoxide, using the greatest precautions to eliminate all impurities, as the smallest trace of oxygen at once changed the spectrum to the carbon-oxide spectrum, which itself was always obtained when carbon dioxide was used.

These results were not published at the time as they did not actually bear on the work in hand; but in view of Professor Smithells's recent paper, it seems worth while to describe them and others more recently carried out, since they support very strongly the view he puts forward. There seems, indeed, no room for doubt that the true explanation of these spectra is that they are due to carbon monoxide and dioxide respectively, but at the same time it is easy to see how confusion could arise. In the first place, there is the extreme difficulty of obtaining a vacuum-tube containing pure carbon monoxide; and in the second place, a very small quantity of carbon dioxide in a mixture gives a very decided spectrum, and can easily mask that of carbon monoxide. These two facts can account for all the difficulty connected with the vacuum-tube spectra of the gases, because under ordinary circumstances carbon monoxide is changed so far into carbon dioxide that it shows none, or only very little, of the Swan spectrum, but practically entirely the carbon-oxide spectrum. Naturally, therefore, the carbonoxide spectrum was attributed to carbon monoxide.

As Professor Smithells has shown, if proper precautions are taken, then carbon monoxide gives the Swan spectrum, and when pure only the Swan spectrum. The chief difficulty lies in removing all the condensed air from the walls of the vacuum-tube, and all the occluded gases from the electrodes. This can be done quite easily by exhausting as far as possible, keeping the discharge passing and heating the vacuum-tube with a Bunsen-burner. The carbon monoxide must be made from formic acid and sulphuric acid, and should preferably

* Kayser, Handbuch der Spectroscopie, p. 198.