November 29, 1894] 



NATURE 



ic; 



bution prevailing in the uarth. It can be demonstrated as a 

 mathematical fact that in the absence of terrestrial magnetic 

 observations wilhin and without the earth's surface, an infinite 

 number of different distributions is possible that will satisfy the 

 effects observed on the surface alone. 



In conclusion, it is my duty to make one more explanation. 

 Mr. Wilde understood from my first letter that I am .still in the 

 employ of the U.S. Coast and Geodetic .Survey. In view of 

 the fact stated by him, that he sent, at considerable trouble and 

 expense to himself, a duplicate of his magnetarium to the 

 " Survey," and, hence, it might appear that it was somewhat 

 discourteous in a member of the " Survey " to thus criticise him 

 publicly, I may say that I severed my connection with the 

 Survey two years ago, and that my criticisms have been made 

 without any knowledge whatsoever of what has been accom- 

 plished with it by the " .Survey." 



Friedenau bei Berlin, October 31. L. A. B.\uer. 



Boltzmann's Minimum Theorem. 



Mr. Culverwell's letter of October 25 ought lo have re- 

 ceived a much earlier answer. That it did not do so was owing 

 to purely accidental circumstances which I very much regret. 



In that letter Mr. Culverwell criticises my treatment of Boltz- 

 mann's familiar proposition concerning the properties of the H 

 function on the following grounds : — 



(i) The choice of the generalised coordinates. In investi- 

 gating the circumstances of a collision or an encounter between 

 two systems of molecules of m and it degrees of freedom re- 

 spectively, he sees a difficulty in my choice of the coordinates 

 as Qi, Q2 . . . Q,,„ y,, q-i . . ijn, where (y„ = a) deter- 

 mines a collision, or encounter. But supposing the requisite 

 number of degrees of freedom to be secured, the choice of the 

 independent variables is surely quite optional. I had myself 

 assumed this as self-evident, perhaps too hastily, but at any 

 rate Mr. G. H. Bryan has placed this proposition beyond doubt, 

 in the exhaustive report submitted by him to the British 

 Association last August. Take for example sets of plane 

 circular disks moving amongst each other in their own plane ; 

 here each pair of disks constitutes a material system, whose 

 position is completely determmed (assuming the orientation of 

 each separate disk to be indifferent) by the following four vari- 

 ables, viz. the two coordinates of the centre of one disk of the 

 pair, the distance p between their centres, and the inclination 

 of that distance to a line fixed in the plane ; this third variable 

 f is the ./„ of my proposition. 



(2) Mr. Culverwell objects that the general Boltzmann proposi- 

 tion j , always negative unless F/= Fy j, or H a minimum 



for one, and one distribution only, cannot be true, because if a 

 system were started from any initial configuration (P, Q), and 

 after the time /arrived at the configuration (/, (/)andthe definite 

 integral H were evaluated in these two configurations, the pro- 

 position asserts that the second H must be less than the first 

 H, or W, <H„, whence it would, follow by the same proposi- 

 tion that if at the end of the time / each velocity component 

 were reversed, the H^, must be less than 11/, and this, doubt- 

 less, I do assert. But Mr. Culverwell maintains that such an 

 .assertion is obviously untrue because at the end of the second 

 interval / the system has returned to its original condition, and 

 therefore W^i must be the same as H , and lo this proposition 

 I demur. 



Doubtless when a material system in a field of any conserva- 

 tive forces is started from any initial position and velocities, 

 arrives after a time / at another position and with other veloci- 

 ties, and here has each velocity reversed, it is true that at the 

 end of the next interval / it will be in its initial position, with 

 each velocity component reversed ; but it remains to be proved, 

 and cannot be asserted </ priori, that the Hj/ is equal to H,,, 

 and the only proposition available for the investigation is this 

 very proposition of Boltzmann's, which proves that H^ will be 

 less than II., and therefore less than H . 



Finally, Mr. Culverwell inquires, somewhat despondently, if 

 anyone will point out the use of the H function, and what i-^ 

 proved by it. 



I have already said in my second edition, that the proposi- 

 tion is not of my invention, and therefore that I have no 

 claim to answer this question with any authority, but to my 

 own mind the proposition appears certainly lo clear away one 



NO. 1309, VOL. 51] 



obvious difficulty. Without the aid of this proposition we are 

 enabled to assert that if F (p, q) were a function of the cg 

 ordinates and momenta of a molecule such that in the absence 

 of encounters -vith other moleciiks, F remains constant for all 

 time, then the form of F satisfying the condition F/= Y'f 

 must render F {p, q) ap dq a permanent law of distribution, 

 and therefore i( we can assert that F (/, (/) must of necesity be 

 of the form F (E) (E sum of potential and kinetic energy), 

 then the e'''^ law of distribution is certainly a permanent law 

 (neglecting, i.e., all but binary encounters) ; but in the absence 

 of this proposition, we cannot assert that the F/'= F'/' is 

 necessary as well as sufficient, because we cannot insist upon the 

 necessity of an exact compensation in the passage from the p ./ 

 to the /' (/' state, and conversely, t.aking place at eacii upurate 

 encounter. The H proposition, therefore, removes this element 

 of uncertainty, and reduces the question to that of the F (E) 

 restriction, because it proves that unless F/'= F'/' for eai/i 

 pair of encountering molecules, H and therefore F and / must 

 be a function of the time. As I have already asked for a dis- 

 proportionate share of your space, I will not enter upon the 

 question of the F (E) restriction now. 



H. W. Watson. 



I DID not exactly, as Mr. Burbury suggests, question Boltz- 

 mann's minimum theorem, but only the pages thereon in Dr. 

 Watson's " Kinetic Theory of Gases. " Indeed, I said that though 

 I had not seen Boltzmann's proof, I supposed it to be all 

 right. 



It appears from Mr. Burbury's letter that in order to prove 

 the theorem, even for the simple case of perfectly hard and 

 elastic spheres, some amount of assumption as to an average 

 state having been already attained must first be made, and of 

 course the .7 priori reasoning in my letter is not applicable to 

 such a theorem. Mr. Burbury's letter is exactly the kind of letter 

 I hoped to elicit, and if he can say what assumption in a general- 

 ised system will r-'place the assumption of equal distribution of 

 velocities in different direction in a system of hard spheres, he 

 will clear up the whole difficulty. Unfortunately, the case with 

 which he deals is one in which the error-law is known from other 

 considerations to be the only permanent state. 



I observe that Mr. Bryan, in his British Association Report, 

 quotes the oversight I pointed out in Dr. Watson's proof, with- 

 out making any criticism on it. 



Edwd. p. Culverwell. 



Trinity College, Dublin, November 24. 



The Alleged Absoluteness of Motions of Rotation. 



Prof. Gree.nhill, in his review of Mach's "Science of 

 Mechanics" (Nature, November 15), writes as if he dis- 

 approved of that author's not accepting "Newton's distinction 

 between the relativity of motion of translation and the 

 absoluteness of motion of rotation." He appears to think that 

 Mach would have obtained more insight into this distinction 

 from a study of Maxwell's "Matter and Motion." It might 

 more truly be said that Maxwell would have profited by a 

 perusal of Mach's book The latter finally refutes the paradox 

 contained in Newton's statement, and supported by .Maxwell, 

 and by so doing renders a great service to Mechanical Science. 

 He has disposed once and for all of "absolute rotation. " It is 

 high time that writers on Mechanics should revise the prelimin- 

 aries of their science so as to state their results in terms of 

 relative motion, whether of translation or rotation. This has 

 been partially done by Maxwell, and a further step has now 

 been taken by Mach. It is unfortunate that the reviewer in 

 drawing attention to this part of the book should have preferred 

 to stand by the prejudice he owes to Newton and Maxwell when 

 he might have done something to hasten its abandonment. 



\. E. H. Love. 



St. John's College, Cambridge, November 20. 



Mach s,ays truly (p. 237) "that precisely the apparently 

 simplest mechanical principles are of a very complicated 

 ch.aracter ; that these principles are founded on uncompleted 

 experiences, which can never be fully completed," &c. 



The modern student of theoretical mechinics is in a dilemma; 



