216 RADIATION 18. Kirchhoff's addition to Stewart's result may be given as follows. Let radiation r, of the same particular wave- length as that spoken of in 14, fall on the substance ; er of it will be absorbed, and (1 - e)r transmitted. This will be recruited by the radiation of the substance itself, so that the whole amount for that particular wave-length becomes (1 - e)r + eft, or r - e(r - K). Thus the radiation is weakened only when H<r, a condition which requires that the source (even if it be a black body) should be at a higher temperature than the absorbing substance ( 4, above). But the converse is, of course, not necessarily true. This part of the subject, as well as the special work of Kirchhoff and of Bunsen, belongs properly to spectrum analysis (see SPECTROSCOPY). 19. From the extension of Prevost's theory, obtained in either of the ways just explained, we see at once how the constancy of the radiation in an enclosure is maintained. In the neighbourhood of and perpendicular to the surfaces of a black body it is wholly due to radiation, near a transparent body wholly to transmission. A body which reflects must to the same extent be deficient in its radia- tion and transmission ; thus a perfect reflector can neither radiate nor transmit. And a body which polarizes by reflexion must supply by radiation what is requisite to render the whole radiation unpolarized. A body, such as a plate of tourmaline, which polarizes transmitted light, must radiate light polarized in the same plane as that which it absorbs. Kirchhoff and Stewart independently gave this beautiful application. 20. Empirical formulae representing more or less closely the law of cooling of bodies, whether by radiation alone or by simultaneous radiation and convection, have at least an Mstoric interest. What is called Newton's Law of Cool- ing was employed by Fourier in his Theorie Analytigue de la Chcdeur. Here the rate of surface-loss was taken as proportional to the excess of temperature over surrounding bodies. For small differences of temperature it is accurate enough in its applications, such as to the corrections for loss of heat in experimental determinations of specific heat, <fec., but it was soon found to give results much below the truth, even when the excess of temperature was only 10 C. 21. Dulong and Petit, by carefully noting the rate of cooling of the bulb of a large thermometer enclosed in a metallic vessel with blackened walls, from which the air had been as far as possible extracted and which was main- tained at a constant temperature, were led to propound the exponential formula Aat + B to represent the radia- tion from a black surface at temperature t. As this is an exponential formula, we may take t as representing absolute temperature, for the only result will be a definite change of value of the constant A. Hence if t Q be the temper- ature of the enclosure, the rate of loss of heat should be A(a. 1 - afo), or Aa^a'-to _ i). The quantity A was found by them to depend on the nature of the radiating surface, but a was found to have the constant value 1*0077. As the approximate accuracy of this expression was verified by the experiments of De la Provostaye and Desains for temperature differences up to 200 C., it may be well to point out two of its consequences. (1) For a given differ- ence of temperatures the radiation is an exponential func- tion of the lower (or of the higher) temperature. (2) For a given temperature of the enclosure the radiation is as (1-0077)*- 1, or 0(1+0-00380+ . . . ), where is the temperature excess of the cooling body. Thus the New- tonian law gives 4 per cent, too little at 10 C. of difference. 22. Dulong and Petit have also given an empirical formula for the rate of loss by simultaneous radiation and convection. This is of a highly artificial character, the part due to radiation being as in the last section, while that due to convection is independent of it, and also of the nature of the surface of the cooling body. It is found to be proportional to a power of the pressure of the surrounding gas (the power depending on the nature of the gas), and also to a definite power of the temperature excess. The reader must be referred to French treatises, especially that of Desains, for further information. 23. Our knowledge of the numerical rate of surface- emission is as yet scanty, but the following data, due to Nicol, 1 may be useful in approximate calculations. Loss in heat units (1 ft) water raised 1 C. in temperature) per square foot per minute, from Bright copper 1'09 0'51 0'42 Blackened copper 2 "03 1 '46 1 '35. The temperatures of body and enclosure were 58 C. and 8 C., and the pressure of contained air in the three columns was about 30, 4, and 0'4 inches of mercury respectively. The enclosure was blackened. 24. Scanty as is our knowledge of radiation, it is not at all surprising that that of convection should be almost nit, except as regards some of its practical applications. Here we have to deal with a problem of hydrokinetics of a character, even in common cases, of far higher difficulty than many hydrokinetic problems of which not even ap- proximate solutions have been obtained. 25. What is called Doppler's Principle (LIGHT, vol. xiv. p. 614) has more recently 2 led Stewart to some curious speculations, which a simple example will easily explain. Suppose two parallel plates of the same substance, per- fectly transparent except to one definite wave-length, to be moving towards or from one another. Each, we pre- sume, will radiate as before, and on that account cool ; but the radiation which reaches either is no longer of the kind which alone it can absorb, whether it come directly from the other, or is part of its own or of the other's radiation reflected from the enclosure. Hence it would appear that relative motion is incompatible with temper- ature equilibrium in an enclosure, and thus that there must be some effect analogous to resistance to the motion. We may get over this difficulty if we adopt the former speculation of Stewart, referred to in brackets in 13 above. For this would lead to the result that, as soon as either of the bodies has cooled, ever so slightly, the radia- tion in the enclosure should become that belonging to a black body of a slightly higher temperature than before, and thus the plates would be furnished with radiation which they could at once absorb, and be gradually heated to their former temperature. 26. A very recent speculation, founded by Boltzmann 3 upon some ideas due to Bartoli, is closely connected in principle with that just mentioned. This speculation is highly interesting, because it leads to an expression for the amount of the whole radiation from a black body in terms of its absolute temperature. Boltzmann's investigation may be put, as follows, in an exceedingly simple form. It was pointed out by Clerk Maxwell, as a result of his electro-magnetic theory of light, that radiation falling on the surface of a body must produce a certain pressure. It is easy to see (most simply by the analogy of the virial equation, MECHANICS, vol. xv. p. 719) that the measure of the pressure per square unit on the surface of an impervi- ous enclosure, in which there is thermal equilibrium, must be one-third of the whole energy of radiation per cubic unit of the enclosed space. We may now consider a re- versible engine conveying heat from one black body to another at a different temperature, by operations alternately of the isothermal and the adiabatic character (THERMO- DYNAMICS), which consist in altering the volume of the en- 1 Proc. JR. S. E., vii. 1870, p. 206. 2 Brit. Assoc. Report, 1871. 3 Wiedemann's Ann. , 1884, xxii.
