176 



NA TURE 



[December 20, 1894 



Now let P' . . . ^' be made to pass through all values from 

 which, 9' and ^' being suitably chosen, they can assume after 

 encounter the given values V . . . V + a? . . . ii . ■ . q -h tfq. 

 The final values of 6 and ^ will varj', but all possible 

 values of 9 and <p must appear for some or other of the 

 values through which P' . . . if pass, and therefore we shall by 

 ihis process obtain the whole number of pairs which are in the 

 state V . . . q after encounter, without restriction of the state 

 which they had before encounter. It will be, namely : 



d^(Kidpdq 



//'S'/'d?-dQ;dp'dq' 



ff dVdCj'dp'dq' 



But the number which are in the state P . . . y now is 



dfdCldpdqYf. 



Therefore, as the result of encounter,, it is increased by an 

 amount proportional to 



d?dqdpdq f [{¥■/■ -F/)dP'dq'dp-dq: 



From this point, thanks to the labours of Boltzmann and 

 Watson, the proof is easy, and I need not repeat it, that 



JIT 



— is negative or zero. 

 dt ^ 



I have assumed condition A. I do not say that that is the 

 only assumption that will answer the purpose. But it is suffi- 

 cient. And it is, I think, the most useful assumption, because 

 the distribution of coordinates assumed to exist is that which 

 would tend to be produced by any disturbances acting on the 

 system (rom without. 



The proof in this form is not open to the obiection that by 

 reversing the velocities we can prove two mutually contra- 

 dictory propositions. 



Oh, that now my friend would write a book, and point out 

 with regard to these assumptions what more is necessary, or 

 what le.-s sufficient. S. H. Burbury. 



Lincoln's Inn, December 5. 



P. S. — Dr. I.armor describes the reverse motions as the 

 " exceptions which do not disprove the rule." I would apply 

 the maxim Exccptio probat re^itlam in a slightly different sense. 

 They are the exceptions whicfi put the rule to the proof. They 

 compel you to define accurately the limits wiihin which the 

 rule holds. When that has been done lor Boltzmann's law (if 

 it has not been di>ne already), it will be lime to consider how 

 far the cases which fall within the law are more important than 

 those which fall without it. S. H. B. 



December 15. 



The presence of any assumption in Dr. Watson's able proof 

 of Bilizniann's Minimum Theorem might easily be overlooked ; 

 but if Mr. Culverwell will apply his test of reversing the motions 

 in each sepaiate stage of the proof, he will unearth the assump- 

 tion at once. On the top of p. 43 Dr. Watson says : 



" And therefore the expression 



FA/P; . . . r/y„-, -;„ 



is the number of pairs of molecules, one from each of these sets, 

 passing from the slate P, P-fr/P... q, q + dq to the slate 

 F', V + dV' . . . </', q' + dq' per unit of lime, where g„ is put 

 equal to o my'." 



Now let the motion of every molecule be reversed as Mr. 

 Culverwell suggests. It will be convenient to speak of the two 

 stales as the iniartenl.J and accented stales, and we shall thus 

 have the assumption that the expression 



F/,/P, . . . dq.,^^ ,/„ 



(which is also equal lo 



F/rfP,' . . . <//„_, '/'„ bul ml to V'f'd?i' . . . dq'„-^ 1),,') 



.«hall reprc»enl ihe number of pairs of molecules passing back 

 from the acctnlcd to Ihe unaccented slate, and this number will 

 depend on F ami J, the frequencies of distribution v/iic/i the 

 molecules are about to have after ihc colli ions ha-.'i taken place. 



If ihi< a^sumplion be made we doubtless shall have a case 

 io which II lends to a maximum instead of a minimum, and if 

 Mr. Culverwell endows his molecules with the power of 

 forelhouglii and Ihe prediction regarding their future stale 

 necessary 10 enable them to regulate their movements according 

 to this suppositious law, then Dr. Watson's proof, and indeed 



No, 1312, VOL. 51] 



any proof, will necessarily fall to the ground. If however the 

 motions of ihe molecules are allowed to take their own natural 

 course, and nothing special is known about them, the only 

 reasonable assumption to make is that the number of pairs 

 passing from Ihe accented lo the unaccented state per unit 

 time is 



F7VP/...^/„_,/,., 



and this assumption is actually made by Dr. Watson in the 

 next few lines of his proof th.it H lends to a minimum. 



What Mr. Culvervvell's objection shows, then, is that it is 

 possible to conceive the molecules of a gas so projected that 

 they would not tend lo assume the Bollzmann-Maxwell 

 distribution. 



But practically it wouH be impossible to project the 

 molecules in their reversed motions with sutiicient accuracy to 

 enable ihem lo retrace their steps for more than a very few 

 collisions, just as, if we try placing a number of pool balls in a 

 straight line on a billiard table at distances of a foot or two 

 apart, we find it impossible to project the first ball with 

 sulTicient accuracy fur each liall to strike the next in front all 

 down the line if there are many balls. 



The question of the choice of coordinates has been so fully 

 dealt wilh by Dr. Watson that I need say nothing more. 

 However, if Mr. Culverwell prefers, he may tiansform from 

 Dr. Watson's Qj . . . q,, 10 any other vaiiables defining the 

 position of ihe/afrof molecules, prozided that he -works wilh 

 the corresponding generalised momenta instead ol Pj , . . /„, 

 and he will have no difiiculty in choosing one of his new 

 variables to be such that it vanishes at an encounter. 



I think Loreniz's paper (" Sitzungsherichte der Wiener 

 Akademie," 18S7, p. 115) affords the fullest account of the 

 assump'icins underlying the proof of the Minimum Theorem. 



Cambridge, December 5. G. 11. Brya.n. 



Science and History. 



I SEE by your review of the National in the last number of 

 NATt;RE, p. 162, that Pr.if. G. W. Prolhero, in his "Address 

 on History," takes occasion to notice Buckle's " History of 

 Civilisation." " Buckle," he says, " in illustrating his theory 

 that national character depends largely upon food, attributes 

 the weakness of the Hindoos to an alnmst exclusive diet of rice. 

 A striking bul misleading generalisation, for, as Sir H. Maine 

 has pointed out, the great majority of Hintoos never eat rice at 

 all" Buckle, ho*cvrr, never said anything of the kind; and 

 since no author wrote more clearly than he did, it is evident 

 that the Professor, like many before him, has not taken Ihis 

 extr.ict at first hand. 



What Buckle did say was : that rice, millet, or whatever the 

 Hindoos fed on, was grown with little trouble and in abund- 

 ance ; that the climate made clothes superfluous; that living 

 was consequently cheap, and that hence the population increased 

 beyond the demand for labour ; labour was ill-rewarded, and 

 the population became practically enslaved. I put the aigu- 

 ment very shortly and inadt:<|uately, for anyone may see it fully 

 sel forth in the "History of Civilisation,'" 185S, vol. i. p|i. 



63-74- 



Sir II. Maine utterly failed lo perceive that whatever might 

 have been the fool that the Hindoos lived upon, it m.ade no 

 difference to the argument provided that that food was cheap. 

 He was further wrong in his slalemcnl that the Hindoos did not 

 feed on rice, as it used to be a far more usual article of diet 

 than in later limes ; bul his worst mistake was to limit the 

 argument to the people of India, who were only one people, out 

 ol many, used lo illustrate ihe iioint. 



Alfred IL Hum. 



London, December 18. 



Geometry in Schools. 



Asa mathematical teacher of long experience, I wish lo stale 

 that I thoroughly agree with Prof Hcnrici that experimental 

 geometry should lie taught antecedently to and concurrently 

 with a rigorous deductive course. 



Teachers who have lo introduce young students to ihc siudy 

 of deductive geometry (lo begin Euclid, as it is called) are 

 conlronicd with iwo difficulties. Their pupils in many cases (l) 

 have never been seriously taught lo reason about anyll ing ; 

 (2) have no slock of geometrical ideas to rea-on aljout. The 

 attempts made in kindergartens to give sound notions of form 



