April i8, 1895] 



NA TURE 



;8i 



LETTERS TO THE EDITOR. 



Th: Editor does not hold himself responsible for opinions ex 

 pressed by his correspondents. Neither can he undertake 

 to return, or to correspond with the writers of, rejected 

 manuscripts intended for this or any other part of Nature. 

 No notice is taken of anonymous communications.] 



Professor Boltzmann's Letter on the Kinetic 

 Theory of Gases. 



In common, I am sure, with all the physical readers of 

 Nature, I have read Herr Bollzmann's letter with great 

 interest, .-^nd I am glad to observe that, though he appears to 

 think I differ from him, that part of his letter which chiefly 

 deals with my criticism on Dr. Watson's idea of what 

 " Boltzmann's Minimum Theorem " is, is simply putting 

 forward, with all his great authority, the view for which I 

 contended. Bat it is a liitle hard that Dr. Boltzmann should 

 represent me as endeavouring to disprove his theorem when I 

 expressly slated that while I did not know his proof, 1 supposed 

 that it was all right. True, I said that I found it hard to 

 conceive how any proof on the lines of Dr. Watson's could be 

 valid because that proof appeared to me to be a purely dynamical 

 proof, and I applied the reversibility argument to show that a 

 purely dynamical proof was impossible, io that the H-theorem 

 could not be a purely dynamical theorem ; and after indicating 

 the lines on which it appeared that there might be an average 

 dynamical theorem, I asked if some one would say what the 

 H-theorem really was. 



Thereupon Mr. Burbnry wrote a helpful letter, which he 

 followed up by a still more helpful correspondence, in which 

 verbal misunderstandings were gradually cleared away, which 

 showed that the proof of the Il-theorem considered as a 

 dynamical theorem, not as a theorem in probabilities, assumed 

 that in one respect the configuration was, before each set of 

 collisions, already perfectly average, and that this condition is 

 violated in the reversed motion ; so that the theorem, regarded 

 as a dynamical theorem, is not proved for configurations in 

 general, but for those possessing a certain amount of "average " 

 already — a restriction which comes to the same thing as the 

 limitation imposed by Prof. Boltzmann when he says the 

 theorem is not a dynamical theorem, but one in probabilities. 



Shortly after Mr. Burbury's letter appeared, Dr. Watson 

 wrote denying that the criticism from reversibility applied, 

 and claiming that the theorem was a general dynamical theorem, 

 in the sense that it applied to all configurations. Enlightened 

 by Mr. Burbury, I now see that Dr. Watson's reasoning is 

 not open to the objection that it proves a general dynamical 

 theorem ; but I cannot blame myself for thinking that it did, for 

 that was what Dr. Watson himself believed it to do, and what 

 his language naturally implies. Moreover, after perceiving the 

 oversight which vitiates the proof io its present form, I did not 

 examine it further. 



Prof. Boltzmann has misunderstood Mr. Burbury and me in 

 one or two particulars. He denies that there are as many con- 

 figurations lor which dHjdt is positive as there are for which it 

 is negative. He evidently thinks that we mean something 

 different from the bare meaning of the words, which are cer- 

 tainly true. It is easy to explain what we do not mean (I say 

 we, for I am sure Mr. Burbury will agree with me). Suppose 

 H=loto be the minimum value of H for a given system of 

 molecules, we do not mean that among all the configurations 

 for which H=:50, there areas many which will, if left to them- 

 selves, turn into configurations forwhich H =60, as will turn into 

 configurations for which H = 40 The illustration, which tomy 

 mind has most clearly removed the apparent contradiction 

 in the statement that there are as many configurations for which 

 H will increase as decrease, while yet the probability is that 

 H will on the whole decrease, is that of a j' turned upside down, 

 thus \. For every downward path there is an upward path.- 

 i.e. the reversed direction ; yet starting from the angle there 

 are two ways down for one way up, so that there is a greater 

 probability of going down than up. If in the reversibility argu- 

 ment one could assert, not merely that there are as many con- 

 figurations for which H tends to increase as to decrease, but that 

 for any given value of H there were as many configurations which 

 tend to increase as to decrease, then the conclusion that H was 

 ai likely to increase as to decrease could be deduced. But the 

 argument is quite invalid when we set off a configuration for 

 which H increases against one for which it decreases, although 



the values of H for each are different. As an illustration more 

 closely allied to the case of a gas, we might take a tree turned 

 upside down, with an infinite number of branches passing 

 through each point of its substance in all directions, there being 

 at every point more branches tending downward than upward 

 (because those whose tangents are horizontal may be said to 

 tend downward on each side), and every upward branch finally 

 turns downward and tends to become nearly horizontal at last, 

 when H is near its minimum value. 



To my mind this appears a far better way of meeting the 

 difficulty than Prof Boltzmann's illustration of the dice, for so 

 far as 1 can see, all that he has shown is that if you start from 

 an exceptionally high ordinate, i.e. one over the average, 

 you are likely, after a considerable time, to get to lower 

 ordinates in whichever direction you go, and an opponent 

 might answer that if you start from an exceptionally low 

 ordinate you are likely to get higher ones in whichever direc- 

 tion you go, and that there must of course be as many deviations 

 below the average as above it, so that if you start from an 

 arbitrary point in an arbitrary direction, you are just as likely to 

 get to higher as to lower ordinates. In point of fact this 

 appears to be the case for his curve, while it is not true for the 

 tree or for a gas. 



Prof Boltzmann must have put an entirely wrong construction 

 on something or other, which I suppose I have written, when 

 he says I object to the Maxwell Law of distribution because it 

 would ultimately lead to the total kinetic energy of the universe 

 being equally distributed among every degree of freedom of 

 every particle in the universe. Instead of considering that to 

 be a priori im-pxoh3h\e, I hold exactly the view put forward by 

 Prof. Boltzmann. 



With regard to the first portion of Prof. Boltzmann's letter, 

 there is so much that is speculative in it that any discussion 

 would occupy more space than I feel entitled to claim. 1 will 

 only say that the idea th at a gas takes years to con.c .0 thermal 

 equilibrium seems hardly consistent with vibrational portion 

 of the kinetic theory being of practical value, when applied to 

 gas which has only had a few hours to settle down. 



Edward P. Culverwell. 



Trinity College, Dublin, March 6. 



It seems to me that my meaning has not been expressed quite 

 clearly ; therefore, it may be worth while to add one remark. 

 Not for every curve, but only for the paiticular form of the H- 

 curve, disymmetrical in the upward and downward direction, 

 can it be proved that H has a tendency to decrease. This 

 particular form is very well illustrated by Mr. Culverwell's 

 suggestion of an inverted tree. The H-curve is composed of a 

 succession of such trees. Almost all these trees are extremely 

 low, and have branches very nearly horizontal. Here H has 

 nearly the minimum value. Only very few trees are higher, 

 and have branches inclined to the axis of abscisss, and the im- 

 probability ot such a tree increases enormously with its height. 

 The difficulty consists only in imagining all these branches 

 infinitely short. 



Finally there is the diflerence between the ordinary cases, 

 where H decreases or is near to its minimum value, and the very 

 rare cases, where H is far from the minimum value and still in- 

 creasing. In the last cases, H will reach, probably in a very 

 short time, a maximum value. Then it will decrease from that 

 value to the well-known minimum value. 



Paris, .'^pril 6. LUDWIG Boltz.maNN. 



The Recent Auroral Ph enomenon. 



On the evening of March 13, from 7.35 to 8.5, Greenwich 

 mean time, I was a spectator of the abnormal display of Aurora 

 Borealis which attracted so much attention at various places 

 throughout the country. It appeared here as a belt of light 

 spannmg nearly the whole sky in a great circle from east to 

 west. When first noticed by me at 7.35, the streak extended 

 from near the hind quarters of Leo to the head of Aries, or 

 from K..\. 169°, Decl. -I- 16" to K.A. 24', Decl. + 22°. 



At the time the streak was altogether cometary in appear- 

 ance, beginning in a fine point, but it gradually changed in 

 form, moving at the same time towards the south. Eventually 

 it also shortened so considerably that just before my last view 

 of it, it only extended from 7 Geminorum to 7 Ceti. Its greatest 

 breadth was about 12°. 



NO. 1329, VOL. 51] 



