May 1 8, 1905J 



NATURE 



55 



coordinates of a point, the wliole system of points con- 

 stitutes a cubic array of volume-density unity. If R be 

 the distance of any point from the origin, 



R2^f + „2+f'; 

 and the number of points between R and R + dR. equal, 

 to the included volume, is 



47rR=dR. 

 Hence tlie number of modes corresponding to d\ is 



4,r(2//a)»fd/, 

 or in terms of \ 



47r.S/\\-'rfA (7) 



If c be the Ivinetic energy in each mode, then the kinetic 

 energy corresponding to d\ and to unit of volume is 



327rcA-*d\ (8) 



Since, as in the case of the string, we are dealing with 

 transverse vibrations, and since the whole energy is the 

 double of the kinetic energy, we have finally 



i28.7r.e.A-'d\ (9) 



as the total energy of radiation per unit of volume corre- 

 sponding to the interval from K to X + dX, and in (q) e is 

 proportional to the absolute temperature 9. 



Apart from the numerical coefficient, this is the formula 

 which I gave in the paper referred to as probably repre- 

 senting the truth when A is large, in place of the quite 

 different form then generally accepted. The suggestion was 

 soon confirmed by Rubens and Kurlbaum, and a little later 

 Planck (Dnidf Ann., vol. iv. p. 553. iqoi) put forward his 

 theoretical formula, which seems to agree very well with 

 the experimental facts. This contains two constants, h 

 and k, besides c, the velocity of light. In terms of A it is 



E,A = ?^'- '^^— . . (10) 



\» ,,f ////.AS _ , 



reducing when A is great to 



EdA=8vr/cflA-'dA (11) 



in agreement with {9). E dA here denotes the volume- 

 density of the energy of radiation corresponding to dA. 



.■^ very remarkable feature in Planck's work is the con- 

 nection which he finds between radiation and molecular 

 constants. If N be the number of gaseous molecules in 

 a cubic centimetre at 0° C. and under a pressure of one 

 atmosphere, 



/.= L^'321i°! ... . (,2) 



273N 

 Though I failed to notice it in the earlier paper, it is 

 evident that (9) leads to a similar connection. For e, 

 representing the kinetic energy of a single mode at 

 temperature 6, may be identified with one-third of the 

 average kinetic energy of a gaseous molecule at that 

 temperature. In the virial equation, if N be the total 

 number of molecules, 



;j/j' = 2i"A-= = ^Xc, 

 so that 



c = pv!2S (13) 



If we apply this to one cubic centimetre of a gas under 

 standard conditions, " N has the meaning above specified, 

 i'=i, and f= 1013x10" C.G.S. .Accordingly, at 0° C. 



t'= 1013X 10'' 2\, 

 and at f)" 



e = i:^3Xio^ .... (,4) 



Introducing this into (g), we get as the number of ergs 

 per cubic centimetre of radiation 



64.7r. 1-013. 10'. W.dA / -1 



27^n:^' ■ ■ ■ ■ ' 



8 being measured in centigrade degrees. This result is 

 eight times as large as that found by Planck. If we re- 

 tain the estimate of radiation used in his calculations, we 

 should deduce a value of N eight times as great as his, 

 and probably greater than can be accepted. 



A critical comparison of the two processes would be of 

 interest, but not having succeeded in following Planck's 

 reasoning I am unable to undertake it. .\s applying to 

 all wave-lengths, his formula would have the greater value 

 if satisfactorily established. On the other hand, the 

 reasoning which leads to (15) is very simple, and this 

 formula appears to me to be a necessary consequence of 



NO. 1855, VOL. 72] 



the law of equipartition as laid down by Boltzmann and 

 .Maxwell. My difficulty is to understand how another 

 process, also based upon Boltzmann 's ideas, can lead to a 

 different result. 



.According to (i.^), if it were applicable to all wave- 

 lengths, the total energy of radiation at a given tempera- 

 lure would be infinite, and this is an inevitable consequence 

 cf applying the law of equipartition to a uniform structure- 

 less medium. If we were dealing with elastic solid balls 

 cclliding with one another and with the walls of a contain- 

 ing vessel of similar constitution, energy, initially wholly 

 translational, would be slowly converted into vibrational 

 forms of continually higher and higher degrees of sub- 

 division. If the solid were structureless, this process would 

 have no limit ; but on an atomic theory a limit might be 

 reached when the subdivisions no longer included more 

 than a single molecule. The energy, originally mechanical, 

 would then have become entirely thermal. 



Can we escape from the difficulties, into which we have 

 been led, by appealing to the slowness with which equi- 

 partition may establish itself? .According to this view, the 

 energy of radiation within an enclosure at given tempera- 

 ture would, indeed, increase without limit, but the rate of 

 increase after a short time would be very slow. If a 

 small aperture is suddenly made, the escaping radiation 

 depends at first upon how long the enclosure has been 

 complete. In this case we lose the advantage formerly 

 available of dividing the modes into two sharply separated 

 groups. Here, on the contrary, we have always to con- 

 sider vibrations of such wave-lengths as to bear an inter- 

 mediate character. The kind of radiation escaping from a 

 small perforation must depend upon the size of the per- 

 foration. 



-Again, does the postulated slowness of transformation 

 really obtain? Red light falling upon the blackened face 

 of a thermopile is absorbed, and the instrument rapidly 

 indicates a rise of temperature. Vibrational energy is 

 readily converted into translational energy. Why, then, does 

 the thermopile not itself shine in the dark? 



It seems to me that we must admit the failure of the 

 law of equipartition in these extreme cases. If this is so, 

 it is obviously of great importance to ascertain the reason. 

 I have on a former occasion {Phil. Mag., vol. xlix. p. 118, 

 1900) expressed my dissatisfaction with the way in which 

 great potential energy is dealt with in the general theory 

 leading to the law of equipartition. Ravleigh. 



May 6. 



The Cleavage of Slates. 



In his critique of Dr. Becker's theory of slaty cleavage 

 in N.^TURE of May 4, " A. H." says that it is substantially 

 the same as mine, and rightly objects that, " if the 

 cleavage plane were a plane of shearing it would corre- 

 spond with a circular section of the ellipsoid " of distor- 

 tion. It is true that I made that suggestion in the body 

 of niv first paper on cleavage in the Geological Magazine, 

 1884, but in a postscript to that paper I stated that a 

 conversation with Mr. Harker had led me to the con- 

 clusion that the cleavage surfaces are determined by the 

 position of the principal axes of the ellipsoids of distortion 

 produced by a shearing movement, and to this view I have 

 ever since adhered. 



" .A. H." says that " there are many slates in which the 

 strain ellipsoid is actually presented in deformed spherical 

 concretions or colour-spots." Is this certain? Is it not 

 probable that these discolorations took place after the rock 

 became a slate? In that case the chemical influence 

 emanating from the foreign particle, usually obvious in the 

 centre of the spot, found the greatest conductivity in the 

 direction of the longest axis of the ellipsoid, the next 

 greatest along the mean axis, and very little along the 

 least. It is from this property of little conductivity across 

 the cleavage that slates are eminently suited for roofing. I 

 have a piece of a school slate with two sharply defined oval 

 patches, of each of which the two diameters are 25 mm. 

 and 16 mm. The thickness of the slate is less than 4 mm., 

 and yet the discoloration does not pass through to the 

 other side. If these spots are sections of ellipsoids formed 

 out of spheres bv compression, the resulting condensation 

 must have been incredibly enormous. The spots in Borrow- 

 dale slates are of a different character from spots of dis- 



