m 



NA TURE 



[July 27, 1905 



matter how many degrees of freedom the resonator 

 possesses, or what the form of its potential energy. 

 Indeed, according to this argument, equation (2) is proved 

 for any dynamical system, e.g. the molecules of a gas. 



It is, however, known that equation (2), with Planclc's 

 meaning of fe, is true if, and only if, the energy of each 

 dynamical system is expressible as the sum of two squares. 

 It can, indeed, be shown directly that this latter condition 

 is exactly the condition that Prof. Planck's assumed basis 

 of probability calculations shall be a legitimate basis, i.e. 

 shall be independent of the time. Happily, this condition 

 of the energy being a sum of two squares may be sup- 

 posed to be satisfied by Planck's resonators, so that we 

 may regard equation (i) as true for such resonators. The 

 equation has, however, no physical meaning, owing to the 

 presence of the arbitrary small quantity c, and can acquire 

 a physical meaning only by putting 6 = 0. It then leads 

 merely to equation (2), which can be obtained much more 

 readily from the theorem of equipartition. 



Taking Mdv to be the law of radiation, where v is the 

 reciprocal of the period of vibration, Planck introduces 

 from his first paper the equation 



u = (8^^7f^)U (3) 



which in combination with equation (2) would lead to the 

 law of radiation, 



(87r/f/<-»)T^Vi- (4) 



and this, on replacing v by c/A., becomes 



S7r/t-T\-VA (5) 



which agrees with my own result. Planck arrives at 

 equation (3) by the help of his assumption of " naturliche 

 Strahlung," but I believe it will be found that this 

 " assumption " is capable of immediate proof by the 

 methods of statistical mechanics. Except for this, and the 

 other differences already stated, the way in which ex- 

 pression (5) has been reached in the present letter is 

 identical, as regards underlying physical conceptions, with 

 the way in which it has been obtained by Lord Rayleigh 

 and myself. 



Planck does not reach expression (5) at all, as he does not 

 pass from equation (i) to equation (2). Instead of putting 

 E = o, he puts t = hv, where h is a constant, and this leads 

 at once to his well known law of radiation. It will now 

 be clear why Planck's formula reduces to my own when 

 A = cx!. For taking X = 30 is the same thing as taking 

 v=o, or 6 = 0. 



The relation i = 'hv is assumed by Planck in order that 

 the law ultimately obtained may satisfy Wien's " displace- 

 ment law," I.e. may be of the form 



v'lc' f{■Ylv)d^, (6) 



This law is obtained by Wien from thermodynamical 

 considerations on the supposition that the energy of the 

 ether is in statistical equilibrium with that of matter at 

 a uniform temperature. The method of statistical 

 mechanics, however, enables us to go further and deter- 

 mine the form of the function /(T/>') ; it is found to be 

 Stt/I'iT/i'), so that Wien's law (6) reduces to the law given 

 by expression (4). In other words, Wien's law directs us 

 to take 6 = ;ii', but leaves h indeterminate, whereas 

 statistical mechanics gives us the further information that 

 the true value of h is /! = o. Indeed, this is sufficiently 

 obvious from general principles. The only way of elimin- 

 ating the arbitrary quantity t is by taking € = 0, and this 

 is the same as h — o. 



Thus it comes about that in Planck's final law 



^ — '— dK . (7) 



the value of h is left indeterminate; on putting 7i = o, the 

 value assigned to it by statistical mechanics, we arrive at 

 once at the law (5). 



The similarities and differences of Planck's method and 

 my own may perhaps be best summed up by saying that 

 the methods of both are in effect the methods of statistical 

 mechanics and of the theorem of equipartition of energy, 

 but that I carry the method further than Planck, since 

 Planck stops short of the step of putting /; = o. I venture 

 to express the opinion that it is not legitimate to stop 

 short at this point, as the hypotheses upon which Planck 

 has worked lead to the relation h = o as a necessary 

 consequence. 



NO. 1865, VOL. 72] 



Of course, I am aware that Planck's law is in good 

 agreement with experiment if h is given a value different 

 from zero, while my own law, obtained by putting h = o, 

 cannot possibly agree with experiment. This does not 

 alter my belief that the value /! = o is the only value vifhich 

 it is possible to take, my view being that the supposition 

 that the energy of the ether is in equilibrium with that 

 of matter is utterly erroneous in the tase of ether vibra- 

 tions of short wave-length under experimental conditions. 



J. H. Jeans. 



On the Spontaneous Action of Radium on Gelatin 

 Media. 



.SiNXE my communication to N'atlre on the subject of 

 the experiments in which I have been for some time past 

 engaged, my attention has been directed to the fact that 

 M. B. Dubois, in a speech at Lyons last November, stated 

 that he had obtained some microscopic bodies by the 

 action of radium salts on gelatin bouillon which had been 

 rendered " aseptic," but in what manner it is not stated. 



I write to direct attention to the fact, as also to add 

 that M. Dubois's experiments were quite unknown to me. 



Moreover, the theory that some elementary form of 

 life, far simpler than any hitherto observed, might exist 

 and perhaps be brought about artificially by " molecular 

 and atomic groupings and the groupings of electrons ". — 

 in virtue of some inherent property of the atoms of such 

 substances as radium — was pointed out in my article on the 

 " Radio-activity of Matter " in the Monthly Review, 

 November, 1903, whilst the experiments which I have been 

 carrying out to verify this view have been for a long time 

 known in Cambridge. 



Although I did not make a speech on the subject, I 

 demonstrated the growths to many people at the Cavendish 

 and Pathological laboratories early in the Michaelmas Term 

 last year. 



So momentous a result as it seemed required careful 

 confirmation, and much delay was also caused in taking 

 the opinions of various men of science before I ventured to 

 w'rite to you upon the subject. 



That M. Dubois's experiments have been made quite 

 independently I do not entertain the slightest doubt. 



Some critics have suggested that these forms I have 

 observed may be identified with the curious bodies obtained 

 by Quincke, Lehmann, Schenck, Leduc and others in 

 recent times, and by Rainey and Crosse more than half a 

 century ago ; but I do not think, at least so far as I 

 can at present judge, that there is sufficient reason for 

 so classifying them together. They seem to me to have 

 little in common except, perhaps, the scale of being to 

 which as microscopic forms they happen to belong. 



John Butler Burke. 



The Problem of the Random Walk. 



Can any of your readers refer me to a work wherein 

 I should find a solution of the following problem, or fail- 

 ing the knowledge of any existing solution provide me 

 with an original one? I should be extremely grateful for 

 aid in the matter. 



.\ man starts from a point O and walks I yards in a 

 straight line ; he then turns through any angle whatever 

 and walks another I yards in a second straight line. He 

 repeats this process n times. I require the probability that 

 after these n stretches he is at a distance between r and 

 r + Sr from his starting point, O. 



The problem is one of considerable interest, but I have 

 only succeeded in obtaining an integrated solution for (tco 

 stretches. I think, however, that a solution ought to be 

 found, if only in the form of a series in powers of i/n, 

 when n is large. Karl Pearson. 



The Gables, East Ilsley, Berks. 



British Archaeology and Philistinism. 



At the end of the second week in July two contracted 

 skeletons were found in a nurseryman's grounds near the 

 famous British camp at Leagrave, Luton. Both were 

 greatly contracted ; one, on its right side, had both arms 

 straight down, one under the body the other above ; the 

 other skeleton lay upon its left side, with the left hand 



