Fep.kuarv 28, 1895] 



NATURE 



415 



case is, that Hj lies in a part of the curve ascending to 

 or iiesccrding from a higher summit. Then Ihe ordinates on 

 <he one side of H, will be greater, and on the other less than 

 H]. But because higher summits are so extremely improbable, 

 the first case will be the most probable, and if we choise an 

 ordinate of given magnitude Hj guided by haphazard in the 

 curve, it will be not certain, but very probable, that the ordinate 

 decreases if we go in either direction. 



We will now assume, with Mr. Culverwell, a gas in a given 

 state. If in this stale II is greater than H (min.) it will be not 

 certain, but very probable, that H decreases and finally reaches 

 not exactly but veiy nearly the value H (min.). and the same 

 is true at all subsequent instants of time. If in an inter- 

 mediate state we reverse all velocities, we get an exceptional 

 case, where H increases for a certain time and then decreases 

 again. But the existence of such cases does not disprove our 

 theorem. On the contrary, the theory of probability itself 

 shows that the probability of such cases is not mathematically 

 zero, only extremely small. 



Hence .Mr. Burbury is wrong, if he concede.s that II increases 

 in as many cases as it decreases, and Mr. Culverwell is also 

 wrong, it he says that all that any proof can show is, that 

 taking all values of dW dl got from taking all the configura- 

 tions which approach towards a permanent state, and all the 

 configura'ions which recede from it, and then striking some 

 average, dWidt would be negative. On the contrary, we have 

 shown the possibility that II may have a tendency to decrease, 

 whether we pass to the former or to the latter configurations. 

 What I proved in my papers is as follows : It is extremely 

 probable that II is very near to its minimum value ; if it is 

 greater, it may increase or decrease, but the probability that 

 it decreases is always greater. Thus, if I obtain a certain 

 value for dHjdl, this result does not hold for every time- 

 element dl, but is only an aveiaije value. But the greater 

 the number of molecules, the smaller is the lime-inteival dl 

 for which the result holds good. 



I will not here repeat the proofs given in my papers; I will 

 only show that just the same takes place in the much simpler 

 case of dice. We will make an indefinitely long series of throws 

 with a die. Let A, be the number of times of throwing the 

 number I, among the first 6n throws. A, the number of times of 

 throwing I, among all the throws between the second and the 

 (6« + i)th inclusive, and so on. Let us construct a series of points 

 n a plane, the successive abscissae of which are 



I 2 3 



the ordinates of which are 



'•'0-'l-"-i:^-'1 



let us call this series of points the " P-curve." If « is a large 

 number, the greater proportion of the ordinates of this new 

 curve will be very small. Bat the P-curve (like the afore- 

 mentioned Hcurve) has summits which are higher than the 

 ordinary course of the curve. Let us now consider all the 

 points of the P-curve, whose ordinates are exactly = i. We 

 will call these points "the points B." Since for each point 

 y = (A/« - i)-, therefore for the points B we have .\ = 2h ; these 

 points mark, therefore, the case where, by chance, we have thrown 

 the number 1 in 211 out of 6« throws. If n is at all large, 

 that is extremely improbable, but never absolutely impossible. 

 Let 2/ be a number much smaller than h, and let us go forward 

 from the abscissa of each point 1! thr ugh a distance = 6:-jn in 

 the direction of .<■ positive. We shall probably meet a point, 

 the ordinate of which <i. The probability that »e meet an 

 ordinate >I is extremely small, but not zero. By reasoning in 

 the same manner as Mr. Culverwell, we might Iielieve that if 

 we go backward {i.e. in the direction of x negative) from the 

 abscissa of each point B through a distance = ^vjii, it would 

 be probable that we should meet ordinates >i. But this in- 

 ference is not correct. Whether we go in the positive or in the 

 negative direction the ordinates will probably decrease. 



We can even calculate the probable diminution of y. We 

 have seen that for every point B we have A = 2h (i.e. 2« throws 

 out of 6h turning up i). If we move in the positive or negative 

 direction along the axis of jr through ihe distance i/«, we exclude 

 one of the 6// throws, and we include a new one. When we 

 move forward through the distance 6j','«, we have excluded 6t' 



NO. 1322. VOL. 51] 



of the original throws, and included 6f others. Among Ihe 

 excluded throws we have probally 2v, among the included ones 

 V throws of the number I. Therefore the probable oiminution 

 of A is V, the probable diminution r,\ y is zvjn approximately. 

 Because the variation of .v was 6t'/«, we may write 

 dy I 



~Jx ~ "3" 

 But this is not an ordinary difTerential coefficient. Il is only the 

 average ratio of the increase ai y to the corresponding increase 

 of X for all points, whose ordinates are = I. The P-curve 

 belongs to the large class of curves which have nowhere a 

 uniquely defined tangent. Even at the top of each summit Ihe 

 tangent is not parallel to ihe jr-axis, but is undefined. In 

 other words, the chord joining two points on the curve does not 

 tend towards a definite limiting position when one of the two 

 points approaches and ultimately coincides with the other.' The 

 same applies to the H-curve in the Theory of Gases. If I 

 find a certain negative value for d\\ 'dl, that does not define the 

 tangent of the curve in the ordinary sense, but il is only an 

 average value. 



§ 3. Mr. Culverwell says that my theorem cannot be true 

 because if it were lt"ue every atom of the universe would have 

 the same average :■/,.- viva, and all energy would be dissipated. 

 I find, on the contrary, that this argument only lends to con- 

 firm my theorem, which requires only that in the course of 

 time the universe must tend to a state where the average vis 

 viva of every atom is the same and all energy is dissipated, 

 and that is indeed the case. But if we ask why this state is 

 not yet reached, we again come to a "Salisburian mystery." 

 I will conclude this paper with an idea of my old assistant, 

 Dr. Schuetz. 



We assume that the whole universe is, and rests for ever, in 

 thermal equilibrium. The probability that one (only one) part 

 of the universe is in a certain slate, is the smaller the further this 

 state is from thermal equilibrium ; but this probability is 

 greater, the greater is the universe itself. If we assume 

 the universe great enough, we can make the probability of 

 one relatively smrdl part being in any given state (however 

 far from the state ol thermal equilibrium), as great as we 

 please. We can also make the probability great that, though 

 the whole universe is in thermal equilibrium, our world is in 

 its present slate. It may be said that the world is so far from 

 thermal equililrium that we cannot imagine the improbability 

 of such a slate. But can we imagine, on the other side, how 

 small a jart of the whole universe this world is ? Assuming the 

 universe great enough, the probability that such a small part of 

 it as our world should be in its present slate, is no longer 

 small. 



If this assumption were correct, our world would return 

 more and more to thermal equilibrium ; but because the whole 

 universe is so great, il might be probable that at some 

 future lime some oiher world might deviale as far from 

 ihetmal equilibr'.um as our world does at present. Then the 

 afore-meniioned Hcurve would form a representation of what 

 lakes place in ihe universe. The summits of the curve would 

 represent the worlds where visible motion and life exist. 



LUDWIG BOLTZMANN. 

 Imperial University of Vienna. 



Oysters and Typhoid. 



Willi reference 10 the article " Oysters and Typhoid," which 

 appeared in your last issue, it may interest your readers to know 

 that De Giaxa investigated some years ago the behaviour ol the 

 typhoid bacillus in sea-water, both in its natural and sterilised 

 condition. He found that in ordinary sea-water the typhoid 

 bacillus suffered very considerably in the competilion with 

 ihe numerous other water bacteria present, but it was still 

 identified on the ninth day after it was first introduced. In 

 sea-waier in which all other bacteria had been d<-stroyed, the 

 typhoid bacillus was delected in very appreciable numbers on 

 the twenly-fifth day. More recently, however, the existence of 

 typhoid bacilli in sierilissd sea-water has been examined by 

 Cassedebal, and his resulis are not in accord wiih those obtained 

 by Giaxa. Cassedebat found that whilst many pathogenic 



I Sec Ulisse Diiii. " GrundKlfen fiir cine T hcorie der FtuictioDcn einer 

 rccllcn Vcriinderlichcn" ('I'cubncr,, 1S92, ? 136), or Weierstiass, Jcurnai 

 fur die Malhemattk^ Band 79, p. 29. 



