It is this problem (p. 148) which gives rise to Mr. Burbury's final difficulty, and which leads him to the conclusion that the hypotheses made by Prof. Gibbs concerning the mechanical systems are not sufficient to serve as a basis of rational thermodynamics. Perhaps I may best show why Prof. Gibbs' conclusions seem to me legitimate by restating his demonstration, as I understand it, in slightly different form and with special reference to the objections which have been brought against it. Let us consider an ensemble of systems not in statistical equilibrium. An ensemble is in statistical equilibrium if, during any interval of time, as many systems enter any fixed element of extension-in-phase as leave it during the same interval; the density-in-phase in any fixed element (finite or infinitesimal) does not change with the time. But when the ensemble is not in statistical equilibrium the density in fixed elements of extension-inphase will vary with the time, and therefore the value of n associated with a fixed finite element (as explained above) will obviously vary. The question is whether the average value, 7, for the whole ensemble, as determined by the use of these elements, will increase or decrease with the time. At a certain initial instant t' let the density be distributed in any arbitrary manner throughout the extension-in-phase, that is, at this instant, we may consider D (or ŋ) to be given as "an arbitrary function of the phase." Later, of course, it will be a function of the phase and of the elapsed time. With this initial distribution given we may now choose a system of fixed finite elements of extension-in-phase, DV, small enough so that the density may be regarded as sensibly constant throughout any one of them. At a later time (unless the motion in phase is of a highly special and relatively improbable kind) the systems which were together in one element at t' will not all be in a single element. Thus some of the systems which were at t' in the element DV may now be in the element DV", but they will have mixed with them systems which, at t', occupied other elements, DV,', DV2', &c. If, now, we ascertain the average density in DV" and take its logarithm (7), thus assuming that ʼn has a constant value for all the systems in the element, we shall, by Theorem IX. of Chapter XI., get a less value than if we took the actual values of n which the separate systems have. But the actual values of ʼn for the separate systems are those which they have brought with them into DV", and are the same which they had in their scattered condition in DV1', DV,', &c. at the instant t'. Therefore the value of which we have obtained by averaging the density in the element

η

ๆ

DV" at t" is less than that based on the same systems as they were at t'; and, as this is true for every element, the average value of for the whole ensemble so determined is less at t than at t. This diminution in the average is wholly the result of mixing, in the elements, systems each of which preserves a constant value of n. It may be well illustrated by the hydrodynamical case which Prof. Gibbs uses earlier in the same chapter (p. 146), in which a cylindrical mass of liquid is imagined, one sector of 90° being black and the rest white. If it is given a motion of rotation about the axis of the cylinder, in which the angular velocity is any function of the distance from the axis (except in the highly special case when this function is a constant), "the black and white portions would become drawn out into thin ribbons which would be wound spirally about the axis." At any given instant we might choose a system of finite elements of volume so small that the density of the black colour in any one would be constant, either unity or zero; but at a later time the same elements would each contain a mixture of black and white, provided of course that the motion continues long enough.

But the chief difficulty is that the analytical demonstration will work either forward or backward in time, as Mr. Burbury points out. Assuming that the motion of the systems, or of the liquid in the illustration, extends backward in time uninterruptedly, we shall inevitably come to a prior time, say t1, at which the average ŋ in an element is less than at t', or at which the black and white portions of the liquid are better mixed than at the later epoch. Prof. Gibbs has not overlooked this fact, and has, I think, given the true solution of the difficulty, although so briefly that the true import of his remarks may easily be overlooked. He says (p. 150) “It is to be observed that if the average index of probability [7] in an ensemble may be said in some sense to have a less value at one time than at another, it is not necessarily priority in time which determines the greater average index. If a distribution, which is not one of statistical equilibrium, should be given for a time t', and the distribution at an earlier time t" should be defined as that given by the corresponding phases", if we increase the interval leaving t' fixed and taking t" at an earlier and earlier date, the distribution at t" will in general approach a limiting distribution which is in statistical equilibrium. The determining difference in such cases is that between a definite distribution at a definite time and the

*Italics are mine.

limit of a varying distribution when the moment considered is carried either forward or backward indefinitely.

"But while the distinction of prior and subsequent events may be immaterial with respect to mathematical fictions, it is quite otherwise with respect to the events of the real world. It should not be forgotten, when our ensembles are chosen to illustrate the probabilities of events in the real world, that while the probabilities of subsequent events may often be determined from the probabilities of prior events, it is rarely the case that probabilities of prior events can be determined from those of subsequent events, for we are rarely justified in excluding the consideration of the antecedent probability of the prior events."

Let us consider first the rotating cylinder of liquid. It is quite true, if we imagine the motion to be continued backward in time from the instant when we have the black sector according to the same distribution of angular velocity along the radius, that we shall have the black and white portions more and more drawn out into thin ribbons the further back we go. But that there should have been the nice adjustment of distribution of colouring matter and of angular velocity necessary in order that these ribbons should, at a given instant, resolve themselves into the black and white sectors is exceedingly improbable. In all reasonable probability such a distribution is essentially an initial one, that is one produced by outside causes and not by antecedent motion of the same type; as a matter of fact we do not separate liquids by stirring them.

In the same manner the thermodynamic states of natural bodies which correspond to ensembles not in statistical equilibrium are, except in very improbable cases, produced by external causes. In Chapter XIII., in which Prof. Gibbs considers the effects of external influences upon an ensemble of systems, he shows that an ensemble originally in statistical equilibrium will have that equilibrium disturbed by changes in the "external coordinates." This is truly an initial state, and thereafter the mean value of ʼn will decrease* as time goes on; that it would also decrease if we supposed the same motions carried back into the past is of no practical importance since that is not the way in which the existing state of things has arisen. An ensemble may unquestionably be arranged, as to distribution and motion, so that its shall increase, but only up to a certain time, after which it will decrease; that such a preliminary arrangement should

It is to be remembered that it is -n which is analogous to entropy.

fortuitously occur is exceedingly improbable. If there were something like an attraction in phase between systems with the same values of 7, some sorting effect might be expected to occur, but scarcely otherwise; and thus, as it appears to me, instead of requiring an assumption to make ʼn decrease, we should require an assumption to keep it from decreasing.

III. On Deflexions of the Plumb-line in India. By Rev. O. FISHER, M.A., F.G.S., Hon. Fellow of Jesus College. Cambridge, and of King's College, London

ARLY in 1902 I received, kindly sent to me by the author at the suggestion of my friend Mr. Oldham, Superintendent of the Geological Survey of India, a Report on the attraction of the Himalaya Mountains upon the plumb-line in India †. The observed phenomena led the Surveyors to suspect the existence of what they term a hidden chain of excessive density, traversing India from Balasore near the mouth of the Hooghly to Jodhpur in Rajputana, and underlying Manata and Bhopal. The position of this supposed chain is given in chart No. 6 of the Report. It appears however that, if such a chain exists as to cause deflexion of the plumb-line towards it, its presence ought likewise to be betrayed by its influence upon the pendulum; because gravity ought to be locally increased above it.

Now, in the 'Account of the Great Trigonometrical Survey of India', we learn that there is no escape from the conclusion that there is a more or less marked negative variation of gravity over the whole of the Indian continent. These variations are tabulated in my 'Physics of the Earth's Crust '§, and it appears that there is a slight increase of density at Kalianpur in latitude 24° 7', which is in the position of the supposed hidden chain, but at the same time there is another nearly of the same amount at Usira in latitude 26° 57', so that if the one affect the plumb-line the other ought to do so in the same manner. The deficiency in the vibration number calculated for the sea-level, which is due to diminished

*Communicated by the Author.

By Major S. G. Burrard, R.E., Superintendent of the Trigonometrical Surveys of India. Dehra Dun, 1901. Calcutta, 1879. By General Walker.

§ 2nd ed. p. 208.

density at the stations in that part of the meridian as com-pared with Punnæ, is :—

Prefixed to the Report is given a cross-section of outer Himalayan ranges on the meridian of 77° 25' to the scale of one inch to four miles. This was constructed by Col. St. G. C. Gore, R.E., Surveyor-General of India. It appears from this section that through a distance of 124 miles the summits rise fairly regularly from the plains to the height of 18,000 feet, so that as far as attraction is concerned the outer ranges may be taken to be approximately represented by an inclined plane, whose base is 124 miles, and angle of elevation 1° 31′ 28′′. Beyond these ranges lies the Tibetan plateau, estimated to be on an average three miles high and 400 miles across. To facilitate calculation I suppose the entire area to be rectangular, and to extend to an equal distance on each side of the meridian of the station. Pratt estimated the area to be equal to that of a circle of radius 335 miles *. This would make the length of the rectangular area about 880 miles. I suppose this mass to have been accumulated.

*Figure of the Earth,' 4th ed. art. 201.