bands which Kayser & Runge* designate respectively as the second and third cyanogen bands in the arc-spectrum of carbon in air.

Besides these bands, a number of single lines appear which are common to all the gases. Among these, the most promi

nent are:

3652

4048

4080

4360

In the tube filled with cyanogen many of the same lines that appear in tubes with metallic terminals, filled with nitrogen, are present besides the above. This is true to a certain extent of the other gases; and it is not surprising, since it is to be expected that some atmospheric air is always present as an impurity,

The conclusions to be drawn from these experiments with steady currents are similar to those we have deduced from condenser-discharges. When elementary gases are introduced into tubes with carbon terminals, and exhausted to a pressure of 1-2 mm., and are submitted to continuous currents, we obtain the spectrum of carbon, or some compound of carbon. From the above results very little information can be obtained as to what this compound is. The same spectrum is obtained whatever gas is introduced into the tube; and, moreover, this is the same spectrum which is given by gaseous carbon compounds in tubes with metallic terminals. What, then, are the conclusions to be drawn from the present stage of my investigation with gases submitted to powerful electric discharges? It seems to me that they are as follows:1. Hydrogen is an insulator.

2. The passage of electricity through hydrogen, nitrogen, oxygen, and their gaseous compounds is conditioned by the water-vapour present.

3. The dissociation of this water-vapour in the ease of tubes filled apparently with pure hydrogen, under the effect of a strong steady current of electricity, shows an electrolytic action closely analogous to that of the voltaic cell. In the case of electrolytic copper terminals in an atmosphere of hydrogen, pure copper is deposited from the negative terminal, and a suboxide of copper at the positive terminal.

4. Under the effect of powerful condenser-discharges,

* Abhandlungen der Akademie der Wissen, zu Berlin, 1889.

oxygen is set free from commercial aluminium and magnesium. 5. Certain carbon bands are always present in glass tubes filled with hydrogen, nitrogen, oxygen, and ammonia gas, notwithstanding the greatest care which may have been taken in submitting them to a high temperature during the process of exhausting, when powerful discharges are employed.

6. The brilliancy of the light of tubes filled with hydrogen diminishes as the process of the dissociation of watervapour goes on and the resistance of the tube increases. It is possible to raise such a tube to the a-ray stage from a pressure of 1-2 mm. merely by the application of a strong steady current.

7. The x-rays excited by the application of a steady current are due to the radiations set up by the dissociation of highly rarefied water-vapour.

Jefferson Physical Laboratory,

Harvard University, Camb., U.S.A.

THE

XXXVIII. On the Complete Emission Function.
By P. G. NUTTING*.

HE relation between emission, temperature, and wavelength, developed theoretically by Wient and by Planck, and empirically by Paschen §, expresses the amount of the radiation from a black or perfect radiator, in regions of temperature and period in which the emission-period function is continuous. Von Köveslighety || has given a function by which the emission of a limited class of substances may be represented in the optical region. By means of the modern theory of functions, a function more general than either may be developed; expressing the emission of all waveperiods, of both complete and partial radiators, as well in the lined as in the banded and continuous spectra. The Wien formula will be developed by the same method, as a preliminary step in the development of the complete function.

The intensity of the emission from a body, being a function of the entirely independent arguments, temperature and wave-period, we may construct each function separately and * Communicated by the Author.

+ W. Wien, Wied. Ann. lviii. p. 662 (1896). †M. Planck, Berl. Berichte, May 1899.

§ F. Paschen, Wied. Ann. lviii. p. 491 (1896).

R. v. Köveslighety, Astr. Nachr. No. 2805, cxvii. p. 330 (1887); Math. u. Nat. Berichte Ungarn, xvi. p. 40 (1899); Beibl. 1900, p. 1280.

then combine them in any manner such that each argument shall enter the function of the other as a parameter, without affecting its form. Consider first the emission as a function of the temperature, the wave-period being a parameter. Experimental evidence indicates that this function is finite and continuous for all periods and for all substances in all conditions, for all values of the argument from zero to positive infinity. The only real root of the function is zero. The first derivative of the function is always positive and has no real roots except zero and infinity, nor has it real maxima nor minima, at least when there is no change of chemical phase. Neither the function nor its first or second derivatives has apparently finite, real roots at temperatures of fusion or vaporization. We reject all polygenic and automorphic functions, as well as elliptic and circular functions of real or complex period, for the function is finite and continuous with a derivative, in all finite regions.

The inverse exponential

is a simple function satisfying the above conditions. It is of course not the only function satisfying them, but it is probably the simplest in form, and further, it is unrestricted, elastic, and easily meets other conditions to be imposed later on. There are but few algebraic functions whose only real root is zero and whose derivative has no real roots except zero and infinity. A limited class of the form

E‚=T′′ƒ1(T)/ƒ2(T),

(2) however, do satisfy the conditions, provided f1 and f1⁄2 have no real roots, n is greater than unity, and the degree of the numerator is the same as the degree of the denominator. The parameters a and b in (1) may be functions of any argument whatever except temperature. We do not regard them as functions of the temperature, for if we replace either a, b, or T in (1) by f(T), we get an emission which is either not zero at zero of temperature, or else has a finite maximum or minimum.

The emission as a function of the period of the emitted radiation is a single-valued, continuous function. It has the value zero at zero and infinity, and only at these points. For a black or perfect radiator, it has a single maximum varying with the temperature. Its first derivative has real roots only at zero, infinity, and one finite point. The inverse exponential

is probably the simplest and most general function satisfying these conditions. In this function, (7) must have no real root unless it be zero, and its derivative none except perhaps zero and infinity. A class of algebraic functions of the form E, AT'1(T)/02(T)

(4) satisfy the above conditions, provided, and 2 have no real roots and the denominator is of higher degree than the

numerator.

For perfect radiators we have then for the complete function, combining (1) and (3),

The combination of (2) with (3) does not satisfy the conditions imposed upon (1) and (2).

Another property of the emission from a perfect radiator shown by a number of investigators* to hold well experimentally is that the product of the absolute temperature and the period of the emission maximum is a constant. Mathematically this means that the derivative of the emission function with respect to the period is expressible as a function. of the product period times temperature. This condition rules out the combination (1)-(4), determines (7) in (5) to be T, and simplifies (6) to

Paschen and Lummer and Pringsheim (l. c.) found also that the emission corresponding to the period of maximum emission was proportional to a power of the absolute temperature. The condition limits j(T) in (7) to a simple power of T which may be included in the factor T".

Provided then there are no maxima independent of the temperature in the emission spectrum of any body, any formula representing the emission is probably included in one of the two following forms :

*Cf. Paschen, Sitz. K.A. W. Berlin, April 27, 1899; Lummer & Pringsheim, Verh. d. D. Ph. Ges. i. p. 227 (1899); Paschen, Ann. Physik, iv. p. 294 (1901).

The parameters A and n in (8) and the parameter B have been limited only to positive real quantities, 4, and 2 must have no positive real roots, n in (9) is real and positive and equal to the degree of p, less the degree of $1, and v is less than n. Both functions show the logarithmic congruency observed by Paschen, and both give Stefan's law as particular cases. Function (8) may be easily identified with the formulas of Wien, Planck, and Paschen. Köveslighety's function

E=A72T4/(72T2 + c2)2

is a particular form of (9). It is, however, not in agreement with Stefan's law; for by it the total radiation-the integral of Edr from zero to infinity-is proportional to T instead of T. If we extend 1 and 2 to include exponential as well as algebraic functions, as we may without violating the conditions imposed, the Wien function (8) becomes a particular form of (9), in which n=0) and v is negative, since it is less than n. The emission formula of Weber *,

E-CA-2aT-1/6TA?

and of W. Michelson †,

E=CT3/2x-6e-a/λ=T,

are open to objection, for both give infinite emission at very high temperatures, even of waves of long period. Also the derivative of each is infinite for T=0, instead of zero of high order as it should be. Michelson's formula, further, does not give the AT const. law now so well established.

m

=

Extension to Partial Radiators.

In order that the complete emission function may express the emission of imperfect radiators, such as quartz, carbon dioxide, or sodium vapour, we must impose upon it at least two other kinds of emission maxima besides that maximum varying inversely as the temperature which it already has. The complete function must then contain at least three distinct types of emission maxima, namely:

I. Maxima which vary in position according to the Amax.T=const. law. This constant appears to be different for different sources, being about 8 x 10-12 deg, sec. for black or inciosed sources and larger for others . This class of maxima appears to be characteristic of all substances at

*Weber, Berl. Sitzb. ii. p. 938 (1888).

† W. Michelson, Journ. d. Phys. (2) vi. p. 467 (1887).

Cf. Lummer and Pringsheim, er. d. D. Ph. Ges. i. p. 222 (1899) ; H. Rubens, Wied. Ann. lxix. p. 580 (1899).