150 



NA TURE 



[December 14, 1893 



a body acted upon by two forces whose direction and intensity 

 may be conveniently represented by the adjacent sides of a 

 parallelogram. The body really moves along the diagonal ; 

 virtually it has a double track, one along each of the adjacent 

 sides of the parallelogram. In like manner, in our hydro- 

 carbon-oxygen system, we may picture two compelling 

 forces, viz. the tendency of oxygen to combine with carbon, and 

 of oxygen to unite with hydrogen. You may, if you please (and 

 this seems a fascinating exercise to Dr. Armsrong), shut your 

 ey^s alternately to each force and say, " first the hydrogen gets 

 ail the oxygen, and then the carbon snatches some from it," or you 

 may just as well put it the other way about. I simply recorded 

 the fact that the carbon got the most of the oxygen, with an 

 explicit reference to the fact that I was dealing with the cooled 

 products. Before my experiments were published Br. Arm- 

 strong thought that hydrogen got most of the oxygen. He had 

 actually tried to persuade Sir G. G. Stokes and many others to 

 this effect. When my paper appeared he sought to discount the 

 facts it contained by flights of polysyllables worthy of a great 

 statesman. 1 Is it not strange that he should now turn to rend 

 the man who relieved him from what he so abhors — a dogma? 



The assumption that the products which are collected Irom a 

 flime may have totally altered in kind during a minute fraction 

 of a second is perfectly gratuitous, and a similar assumption 

 might be made about nearly every reaction in chemistry. Dr. 

 Armstrong might just as well forbid me to say that zinc and hot 

 sulphuric acid give sulphur dioxide, as to say that when a 

 hydrocarbon burns with limited oxygen, the carbon has the pre- 

 ference. 



I will not trespass on your space with a discussion of the 

 liberation of carbon in luminous flames. Dr. Armstrong's con- 

 tentions on that matter are of preci-ely the same character as 

 those I have dealt with above. Dr. Frankland has promised 

 us some new evidence in favour of his theory, I trust I have 

 always treated this theory with respect. I am not bigoted on 

 the subject, though I await Dr. Frankland's promised publication 

 with the same sort of feelings as those with which a Neapolitan 

 might look forward to the reawakening of Vesuvius. 



1 have now, I hope, given an adequate reply to the question 

 of scientific fact raised by Dr. Armstrong. I will not say much 

 about the imputations he casts upon my scientific honesty. It 

 ought not to be a light thing for a man in Dr. Armstrong's posi- 

 tion to accuse a scientific worker of deluding an audience into 

 unsound opinions by means of dazzling experiments, of playing 

 to the gallery ; of doing, in short, just those things which are 

 most repugnant to the conscience of an earnest investigator. 

 After yeirs of personal friendship. I know Dr. Armstrong's 

 idiosyncrasies very well, and they are, I imagine, pretty well 

 Icnown to the scientific world in general. I feel compelled, none 

 the less, to ask him either to justify or withdraw his aspersions. 

 I make no appeal ad miscricordiam, and seek no comforting 

 eulogy. It is a duty Dr. Armstrong owes no less to the scientific 

 world than to myself to state clearly and precisely how I have 

 departed from the standards of diffidence, deliberation, and exac- 

 titude that are becoming to a man who is honestly seeking to 

 expound the truth. This, at any rate, is a matter the settle- 

 ment of which is not contingent upon the arrival of that chemical 

 millennium when we shall recognise "chemical interchange 

 and electrolysis as interchangeable equivalent terms " ; and I 

 have the right to ask for an immediate and definite reply to my 

 demand. Arthur Smithells. 



December 2. 



The Second Law of Thermodynamics. 



Clausius' supposed deduction of the second law from the 

 ordinary equations of dynamics in the form 



30 



T 



= 23 log {lY) 



has been discussed at length by Messrs. Larmor and Bryan in 



1 Here is a quotat'on fro n one of tiis letters to Sir G. G. Stakes : " Re- 

 garding the inieractions in flames as consisting in a series of simultan^aui 

 and consecutive expbsions, of whicti \va can only examine the fi ill s:eady 

 state, it seems to me that the phenoaiena are necessarily of an excessively 

 cjmplex character, and that their appreciati)n ani successful interpretation 

 must tax our powers of mental analysis in a high degree. It will cartainly 

 he unwise at present to infer ihu the oxidation of the hydroca-bo.is, or the 

 separation of carbon anl also of hydrogen from them, takes place entirely 

 in any one way." This seems to me like saying of a fall downstiirs, thit 

 it is -'a series of simultaneous and C)nie:utive" bumps, &c. so diffi :ult ta 

 trace out and presenting so many possible varieties of motion that it is hirdly 

 -afe to call it a f-ill down stairs at all. 



NO, 1259, VOL. 49J 



their Report on Thermodynamics for the British Association. 

 They accept the deduction on condition that the system be 

 conservative, that is, that the exterral as well as the internal 

 forces a.cting on it are to be derived from a potential. 

 Now it is admitted that the equation 



^ = 29 log (eT) 



can be proved for a conservative system with the meaning 

 of i given in the report. But in order that this equation, how- 

 ever true, may express the second law, T and « must be inde- 

 pendent variables, or (which is the same thing when there is only 

 one controllable coordinate v) T and v must be independent 

 variables. 



Now the second law implies comparison of two states, in 

 either of which a substance can exist permanently. So if we 

 seek to prove the law, or an analogous law, for a dynamical 

 system, it is essential that we should compare two states of the 

 system in either of which it is in stationary fnotion. One state 

 may have the variables T and v, and the other may have 

 T -f 3 T and v + d v, but there must be stationary motion with 

 either pair of values. If then K be the virial of all the forces, 

 external as well as internal, the Clausian equation, K = 2 T 

 must hold. But if the system be conservative, as Larmor and 

 Bryan assume, K is a determinate function of v, and the virial 

 equation constitutes a relation between T and v, so that only 

 one of them is independent. For example, a fixed quantity of 

 gas in equilibrium in a vertical cylinder under a piston of mass 

 "w" acted on by gravity. Clearly if T be given, v, the 

 volume, is determinate, or we have only one independent vari- 

 able. If VI be disposable, you may make T and v vary in- 

 dependently, but then the system is not conservative. It 

 seems to me that Larmor and Bryan's equation does not express 

 the second Law. 



Prof. J. J. Thomson, in his "Application of Dynamics to 

 Physics and Chemistry," pages 95-103, proves the second law on 

 a certain assumption. And Boltzmann, " Uber die Mechan- 

 ischen Analogien des zweiten Ilauptsatzes," has proved it on, I 

 think, the same assuinption. In order to show clearly the 

 nature of the assumption I will begin a proof as follows, treat- 

 ing only the case of T temperature, and v volume. If x denote 

 the potential of all the conservative forces, p the external force 

 necessary to maintain v constant, we have T being the mean 

 kinetic energy of one of N molecules 



3Q =N3T -f 3x + P^v. 

 But by the virial equation 



^Qy dx 



pdv = SNT^ 



'fdv, 



iv 



dx d — 



in which -1^ is to be distinguished from -;rx ^s explained in 



Watson's " Kinetic Theory of Gases. 

 Therefore 



dv' 



3Q = N3T 



|NT3 log e/ -f 3x - '^Sz', 



dv 



3Q 

 T 



N31ogT-F|\31og 



now make 



('X - ^f^4 



(a definite time), then 



3 1ogT 

 and therefore 



5Q 



T^ 



13 log V = 23 log (jT), 



?g = 2N9 1og(^T)-^,L(ax"- J>) 



Now J. J. Thomson assumes (p. 97) that x, in his notation 

 V, is to be a function of v only, whence it follows that 



9x - fj^ 



o, 



as he says, and so 



?^ =2N3Iog(jT), 



which I submit as a fo-m o' his result (118). I understand 

 Boltzmann in the treat se above cited to make the same 



