April Cj, 1874] 



NATURE 



443 



them in the gross ; and after a sufficiently long time 

 from the supposed initial arrangement the difference of 

 energy in any two equal volumes, each containing a very 

 great number of molecules, must bear a very small pro- 

 portion to the whole amount in either ; or, more strictly 

 speaking, the probability of the difference of energy ex- 

 ceeding any stated finite proportion of the whole energy 

 in either is very small. Suppose now the temperature to 

 have become thus very approximately equalised at a certain 

 time from the beginning, and let the motion of every 

 particle become instantaneously reversed. Each molecule 

 will retrace its former path, and at the end of a second 

 interval of time, equal to the former, every molecule 

 will be in the same position, and moving with the same 

 velocity, as at the beginning ; so that the given initial 

 unequal distribution of temperature will again be found, 

 with only the difference that each particle is moving in 

 the direction reverse to that of its initial motion. This 

 difference will not prevent an instantaneous subsequent 

 commencement of equalisation, which, with entirely 

 different paths for the individual molecules, will go on in 

 the average according to the same law as that which took 

 place immediately after the system was first left to itself. 

 By merely looking on crowds of molecules, and reckon- 

 ing their energy in the gross, we could not discover that 

 in the very special case we have just considered the 

 progress was towards a succession of states in which the 

 distribution of energy deviates more and more from uni- 

 fonnity up to a certain time. The number of molecules 

 being finite, it is clear that small finite deviations from 

 absolute precision in the reversal we have supposed 

 would not obviate the resulting disequalisation of the 

 distribution of energy. But the greater the number of 

 molecules, the shorter will be the time during which the 

 disequalising will continue ; and it is only when we regard 

 the number of molecules as practically infinite that we 

 can regard spontaneous disequalisation as practically im- 

 possible. And, in point of tact, if any finite number of 

 perfectly elastic molecules, however great, be given 

 in motion in the interior of a perfectly rigid vessel, and be 

 left for a sufficiently long tmie undisturbed except by 

 mutual impacts and collisions against the sides of the 

 containing vessel, it must happen over and over again that 

 (for example) something more than nine-tenths of the 

 whole energy shall be in one half of the vessel, and less 

 than one-tenth of the whole energy in the other half 

 But if the number of molecules be very great, this will 

 happen enormously less frequently than that something 

 more than 6-ioths shall be in one half, and somethmg 

 less than 4-ioths in the other. Taking as unit of time 

 the average interval of free motion between consecutive 

 collisions, it is easily seen that the probability of there 

 being something more than any stated percentage of 

 excess above the half of the energy in one halt of the 

 vessel during the unit of time, from a stated instant, is 

 smaller the greater the diinensions of the vessel and the 

 greater the stated percentage. It is a strange but never- 

 theless a true conception of the old well-known law of the 

 conduction of heat to say that it is very improbable that 

 in the course of 1,000 years one half the bar of iron shall 

 of itself become warmer by a degree than the other half; 

 and that the probability of this happening before 1,000,000 

 years pass is 1,000 tines as great as that it will happen in 

 the course of 1,000 years, and that it certainly will happen 

 in the course of some very long time. But let it be re- 

 membered that we have supposed the bar to be covered 

 with an impermeable varnish. Do away with this im- 

 possible ideal, and believe the number of molecules in 

 the universe to be infinite ; then we may say one half of 

 the bar wilt never become warmer than the other, except 

 by the agency of external sources of heat or cold. This 

 one instance suffices to explain the philosophy of the 

 foundation on which the theory of the dissipation of 

 energy rests. 



Take however another case in which the probability may 

 be readily calculated. Let a hermetically-sealed glass jar 

 of air contain 2,000,000,000,000 molecules of oxygen, and 

 8,000,000,000,000 molecules of nitrogen. If examined any 

 time in the infinitely distant future, what is the number of 

 chances against one that all the molecules of oxygen and 

 none of nitrogen shall be found in one stited part of the 

 vessel equal in volume to i-5th of the whole? The 

 number expressing the answer in the Arabic notation has 

 about 2,i73,220,ooo,oooofplacesof wholenumbers. On the 

 other hand the chance against there being exactly 2-ioths 

 of the whole number of particles of nitrogen, and at the 

 same time exactly 2-ioths of the whole number of particles 

 of oxygen in the first specified part of the vessel is only 

 4021 X 10' to I. 



[Appendix.— Calculalion of Probability 7-especting Diffu- 

 sion of Gases^ 

 For simplicity I suppose the sphere of action of each molecule 

 to be infinitely small in comparison with its average distance from 

 its nearest neighbour : thus, the sum of the volumes of the spheres 

 of action of all the molecules will be infinitely small in proportion 

 to the whole volume of tiie containing vessel. For brevity, space 

 external to the sphere of action of every molecule will be called 

 free space : and a mulecule will be said to be in free space at any 

 time when its sphere of action is wholly in free space ; that is to 

 say, when its sphere of action does not overlap the sphere of 

 action of any other molecule. Let A, 5 denote any two particu- 

 lar portions of the Avhole containing vessel, and let a, b be the 

 volumes of those portions. The chance that at any mstant one 



individual molecule of whichever gas shall be in^ is ;,how- 



ever many or few other molecules there may be in A at the same 

 time ; because its chances of being in any specified portions of 

 free space are proportional to their volumes ; and, according to 

 our supposition, even if all the other molecules were in A, the 

 volume of free space in it would not be sensibly diminished by 

 their presence. The chance that of 11 molecules in the whole 

 space there shall be z stated individuals in A, and that the other 

 II -- i molecules shall be at the same time in B, is 



Hence the probability of the number of molecules in ,•/ being 

 exactly /, and m B exactly n - i, irrespectively of individuals, is 

 a fraction havhig for denominator (a + b)", and for numerator 

 the term involving rt^^" " ' in the expansion of this binomial; 

 that is to say it is — 



n(n- I) . . . . {n-i + i) / a \i / b \n ~ i 

 1.2 .... i \a + b) \a + b) 



If we call this 7^ we have 



^7^ 11 — i a rr. 



^-■+- =7n/. ^'■+' 



Hence 7} is the greatest term if / is the smallest integer which 

 makes 



n — i b 

 < - 



2 -H I 



a 



this is to say, if ; is the smallest integer which exceeds 



"0 + b a + b 



Hence if a and b are commensurable the greatest term is that for 

 which 



a 



I = H 



a + b 



To apply these results to the cases considered in the preceding 

 article, put in the first place 



« = 2 X lo'2 

 this being the number of p.irticles of oxygen ; and let i = h. 

 Thus, for the probability that all the particles ol oxygen shall be 

 in A, we find 



/ a \8 X loi= 

 \a + b) 



