628 



NATURE 



[October 29, 1891 



treatment of Laplace's pressure. Otberwise I should not, 

 in my answer, have referred to his b or to the unfortunate 

 results of comparing his formula with experiment. I 

 should, in fact, have contented myself with the acknow- 

 ledgment that you had given an accurate account of the 

 contents of a portion of Van der Waals's earlier chapters, 

 which I had carelessly missed on the first hasty perusal ; 

 and that these contents justified the expression 3K2'/2 as 

 the virial of Laplace's pressure. But to this I should 

 certainly have added that, even had I been fully cognizant 

 of that portion of the pamphlet when I wrote my paper, 

 I should probably not have modified (at least to any 

 serious extent) the passage you quoted. 



For (i) that passage contains the distinct statement that, 

 from the statical point of view, reasons " satisfactory on 

 the whole " were given by Van der Waals for regarding 

 Laplace's pressure as proportional to the square of the 

 density. And it would have been illogical on my part to 

 object, except on the ground of insufficient generality, to 

 the equation 



(/ + f^y^ = •2(;«0, 



though I might have regarded the mode of its establish- 

 ment as obscure or even doubtful. 



In fact, the equation which is one of the main features 

 of my own paper, viz. : — 



Pv -f -^ = i2(w«0 . (l + -^), 

 7/ + a V V ■\- aj 



includes it as the particular case when 



e = o, a = o. 



What I objected to was a totally different thing : — viz. 

 the above equation manipulated by the introduction of 

 the factor {v - b)\v in the left-hand member. 



Again (2) the equation 



Pk" - ^) = i2(w«^) 



is obtained in my paper (§ 64), and is there spoken of as 

 " perfectly legitimate" but only on the distinct condition 

 that 



where /3 is four times the sum of the volumes of the 

 particles (§ 30), ''^ be small in compariso7t with the other 

 terms in the [virial] equation^ As one of these terms is 

 -the quantity 2(/««'0/3 itself, this implies that for the truth 

 of the equation ^Iv must be a small fraction ; and it is 

 most certainly not so at the critical point of carbonic 

 acid, which furnished the first and one of the most 

 important cases for the application of the virial method. 

 In fact the equation above, when correctly obtained, 

 comes originally in the form (in which it ought to be 

 preserved) 



Pv = \%{mu^) . (i + 1^) ; 



again a particular case of my own equation, viz. when 



A = o, a = o, e ~ ^. 

 Here the factor i/v is (roughly) proportional to the 

 number of coUisions per particle per second, and it is in 

 -that capacity that it appears in the equation. As I said 

 in my former letter, it is impossible (at least with Van der 

 Waals's mode of interpreting 'S.{mu-)) to derive from this 

 a cubic in v; even when the term ajv^ is introduced as a 

 simple addition to p : — unless, for the express purpose of 

 obtaining the indispensable cubic, we write vl{v - /3) 

 in place of {v + ^)lv, on the right-hand side ; which is, 

 practically, what Van der Waals does. The true mode of 

 getting a cubic here, if we keep to Van der Waals's inter- 

 pretation of l,{mu^), is to write ^/(v - y) instead of 0/v. 

 This can, to a certain extent at least, be justified ; the 

 other method can not. 



On the question of the introduction by Van der Waals 

 of the factor (v- b)/v, whether or not it is applied alike to 



NO. 1 I 48, VOL. 44] 



p and to K, I regret to find that our views must continue 

 to differ. For it appears to me that when once the various 

 terms of the virial equation have been correctly extracted 

 from the expression 2(Rr), we have no right to modify any 

 of them. There seems therefore to be no doubt what- 

 ever that the procedure in Van der Waflls's sixth chapter 

 is entirely wrong in principle :— except in so far as (in the 

 German version) he borrows some correct expressions 

 from Lorentz. The meanings of v and of p, in the term 

 pv of the virial equation, are (from the very beginning of 

 the inquiry) definitely assigned as total volume and ex- 

 ternal pressure : — so that this term cannot in any way be 

 altered. No more can the term 2(;«m-)/3, or the ratio of 

 these two terms. Van der Waals's argument seems (for 

 his pamphlet is everywhere somewhat obscure) to be that 

 (when there is no molecular force) in consequence of the 

 finite diameters of the particles the pressure, for a g^iveti 

 amount of kinetic energy, will be greater than if these 

 were mere points. Perfectly true :— but we must seek the 

 expression for this increase of pressure in the remaining 

 parts of the term 2(Rr), and 7tot artificially introduce it 

 by diminishing the multiplier of / in a term already 

 definitely extracted. And further, if this procedure of 

 Van der Waals were allowed to pass without protest in so 

 far as the term pv is concerned, I think that we should 

 logically be forced to treat the term Y^v (not to the same 

 but) to a very different factor : — for hereiht. consideration 

 of the finite volumes of the particles would appear to call 

 for a reduced rather than an increased value of K ; and 

 therefore analogy would require a multiplication of the 

 term K'z/by some such expression as {v + ■y)/V, where y is 

 essentially positive. — ^ Yours very truly, 



P. G. Tait. 

 Edinburgh, 17/10/91. 



NOTES. 

 To-day the Senate of Cambridge University will decide 

 whether official inquiry shall be made as to the expediency of 

 allowing alternatives for one of the two classical languages in the 

 Previous Examination, either to all students or to any classes of 

 students other than those already exempted. Everyone who 

 devotes attention to questions connected with the higher educa- 

 tion recognizes the importance of the i-;sue, and the discussion 

 of the subject has been followed with wide-spread interest. 



The ordinary general meeting of the Institution of Mechanical 

 Engineers began yesterday evening, and will be continued this 

 evening, at 25 Great George Street, Westminster. The papers 

 to be read and discussed, as we have already stated, are by Mr. 

 Samuel Bjswell and Prof. W, C. Roberts- Austen, F.R.S. 



The Geologists' Association will hold a conversazione at 

 University College, Gower Street, on Friday evening, Novem- 

 ber 6. Members are invited to send exhibits, and to let the 

 secretary know the nature of the object or objects they propose 

 to show. 



At the meeting of the Royal Horticultural Society in the 

 Drill Hall, Westminster, on Tuesday, there was an interesting 

 display of autumn foliage arranged for jKSihetic effect. A lec- 

 ture was delivered by Mr. H. J. Veitch, who urged that trees 

 and shrubs in gardens and plantations should be selected, not 

 only with a view to their summer beauty, but also with regard to 

 their autumn hues ; and he had many suggestions to offer as to 

 the various ways in which these hues may be most effectively 

 contrasted. 



Prof. Boys has arranged his apparatus for the repetition of 

 the Cavendish experiment in the basement of the Clarendon 

 Laboratory, Oxford. The experiment will be proceeded with 

 immediately. 



We regret to have to record the death of Dr. Philip Herbert 

 Carpenter, F.R.S., the fourth son of the late Dr. W, B. Car- 



