UNIVERSITY OF CALIFORNIA 
 AT LOS ANGELES
 
 Cfarcnbon 
 
 KINETIC THEORY OF GASES 
 
 WATSON
 
 HENRY FROWDE 
 
 OXFORD UNIVERSITY PRESS WAREHOUSE 
 
 AMEN CORNER, E.G. 
 
 MACMILLAN & CO., 66 FIFTH AVENUE
 
 A TREATISE 
 
 KINETIC THEORY OF GASES 
 
 HENRY WILLIAM WATSON, D.Sc., F.R.S. 
 
 FORMERLY FELLOW OF TRINITY COLLEGE, CAMBRIDGE 
 
 SECOND EDITION 
 
 Ojfor* 
 AT THE CLARENDON PRESS 
 
 1893
 
 Ojcfori 
 
 PRINTED AT THE CLARENDON PRESS 
 
 BY HORACE HART, PRINTER TO THE UNIVERSITY 
 
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 I7S" 
 
 PREFACE TO FIRST EDITION 
 
 K THE idea of a Kinetic Theory of Gases originated with 
 " J. Bernouilli about the middle of the last century, but the 
 o first establishment of the theory on a scientific basis is due 
 to Professor Clausius. 
 
 During the last few years the theory has been greatly 
 
 developed by many physicists, especially by Professor 
 
 Clerk Maxwell in England and Professor Clausius and 
 
 Dr. Ludwig Boltzmann on the Continent ; and although 
 
 still beset by formidable difficulties, it has succeeded in 
 
 explaining most of the established laws of gases in so 
 
 ';- remarkable a manner as to render it well worthy of the 
 
 c attentive consideration of scientific men. 
 
 My object in the following short treatise is to make 
 the existing state of the theory more widely known by 
 presenting some of the scattered memoirs of the writers 
 I have mentioned in a systematic and continuous form, 
 in the hope that mathematicians may be induced to turn 
 their attention to the theory, and thus assist in removing, 
 if possible, the obstacles which yet remain in the way of 
 its complete establishment. 
 
 For the most part I have followed the method of 
 treatment adopted by Dr. Ludwig Boltzmann in some 
 very interesting memoirs contributed by him to the 
 
 t 
 
 303160
 
 vi Preface to First Edition. 
 
 Transactions of the Imperial Academy of Vienna,* but 
 in some cases I have varied this treatment for the sake 
 of greater conciseness or greater generality. 
 
 Thus, in place of Dr. L. Boltzmann's conception of a 
 molecule as a collection of mutually attracting particles, 
 I have substituted the more general conception of a 
 material system possessing a given number of degrees of 
 freedom, that is to say, a given number of generalised 
 coordinates. 
 
 Again, in the deduction of the second law of Thermo- 
 dynamics from the results of the Kinetic Theory, I felt 
 some difficulty in following Dr. Boltzmann's reasoning, 
 and I originally proposed to substitute for it a demon- 
 stration of my own, free from what appeared to me to 
 be the obscurities of Dr. Boltzmann's reasoning, but 
 applicable only to the case in which there were no 
 intermolecular actions. My friend Mr. S. H. Burbury, 
 formerly Fellow of St. John's College, Cambridge, to 
 whom I communicated my difficulties, has invented an 
 unexceptionable proof applicable to all cases, which he 
 published last January in the London, Edinburgh, and 
 Dublin Philosophical Magazine, and with his permission 
 I have adopted this proof in the following treatise. 
 
 To Professor Clerk Maxwell I am indebted for much 
 kind assistance, and especially for access to some of his 
 manuscript notes on this subject, from which I have taken 
 many valuable suggestions. 
 
 H. W. WATSON. 
 BEKKSWELL RECTOBT, COVENTRY: 
 Sept. 17, 1876. 
 
 * Sitzungsberichte der Kaiserlichen Akademie der Wissenscbaften Wien, 
 Band 63, 1871, und Band 66, 1872.
 
 PREFACE TO SECOND EDITION 
 
 MUCH interest has been manifested by physicists during 
 the last thirty years in the Kinetic Theory of Gases. 
 
 This interest has been greatly stimulated by the 
 researches of Maxwell in England and Boltzmann in 
 Germany ; but along with a growing conviction of the truth 
 of a Kinetic Theory in its general aspect, the particular 
 conclusions arrived at by these investigators concerning 
 the laws of permanent distribution have been received with 
 great distrust ; they have been subjected to hostile criti- 
 cism from mathematicians of such eminence that the mere 
 weight of their authority must almost be accepted as 
 conclusive, were there not room for the contention that 
 these criticisms are not really directed against the laws 
 themselves, as stated and applied in the Kinetic Theory, 
 but against certain aspects of those laws with which the 
 theory is not concerned. 
 
 The object of the original edition of this book was, as 
 stated in the preface thereto, to set forth in a more 
 systematic, and in some cases a more simple form, the
 
 viii Preface to Second Edition. 
 
 demonstrations of these laws contained chiefly in the 
 Transactions of the Royal Society of London, the Imperial 
 Academy of Vienna, and sundry papers in the Philosophical 
 Magazine and other scientific journals. Boltzmann's 
 methods, especially, were closely followed, with occasional 
 modifications, the most important of these being the 
 substitution, in the conception of a molecule, of a material 
 system with a determinate number of degrees of freedom, 
 for a determinate number of discrete atoms under the 
 action of mutual forces. In the present edition the 
 ground covered is substantially the same as in the original 
 edition, and is limited, as it was in that case, to the 
 investigation of the laws of permanent distribution, but 
 in much greater detail ; so much so, indeed, as to be liable 
 to the charge of undue prolixity, inasmuch as there is 
 spread over several propositions of increasing generality 
 that which might really have been condensed into the 
 single proposition of Art. 14. This more detailed treat- 
 ment, however, has been deliberately adopted, partly 
 because there is an historical interest in retaining the 
 steps by which the theory has actually been developed, but 
 mainly to avoid the extreme difficulty on the part of the 
 student in following out investigations of the great gener- 
 ality required in the single proposition treatment. Even 
 to the minds of advanced mathematicians the vagueness 
 imparted to the reasoning by the more condensed treat- 
 ment has been suggestive of inconclusiveness, while to the 
 learner it presents almost insuperable obstacles. The more 
 detailed treatment, with its continual repetitions, has the 
 advantage of investing each step of the process with a
 
 Preface to Second Edition. ix 
 
 distinct mechanical meaning, rendering the demonstration 
 easier to follow and more convincing when mastered. The 
 one essential condition is that the statistical treatment 
 should be always kept in view, the slightest attempt to 
 impart direction or control being fatal. 
 
 The investigations of the following pages appear to 
 point strongly to the conclusion that in a medium con- 
 sisting of sets of a very large number of similar consti- 
 tuents, called molecules, each molecule being a material 
 system with a definite number of degrees of freedom, and 
 all being in irregular and undirected motion, and in a field 
 of fixed central forces only, that is, free from finite distance 
 intermolecular forces, the physical properties of the medium 
 will be in entire accordance with the accepted laws of ideal 
 perfect gases, provided only that either the molecular density 
 of the medium or the minimum irreducible volume of each 
 molecule be so small that the total molecular volume per 
 unit of volume is a small fraction. 
 
 Should there be appreciable finite distance intermolecular 
 forces, it is obvious that in one respect the agreement in 
 physical properties of the medium and the ideal perfect gas 
 cannot be maintained, because there must be work done in 
 the former by compression or expansion, but even in such 
 a case (see Art. 21) the accordance may remain in the five 
 particulars enumerated in Art. 23. And under any circum- 
 stances it is true that the rarer the medium the more 
 nearly do its properties correspond with those of the ideal 
 gas, exactly as we find to be the case in the gases of 
 ordinary experience. 
 
 In conclusion, I desire to acknowledge the valuable help
 
 x Preface to Second Edition, 
 
 most kindly afforded me by friends in the preparation of 
 this Edition, among whom I would especially mention 
 Mr. S. H. Burbury, M.A., F.R.S., formerly Fellow of 
 St. John's College, Cambridge, and Mr. G. H. Bryan, 
 M.A., Fellow of St. Peter's College, Cambridge. 
 
 H. W. WATSON. 
 
 BERK SWELL RECTORY, COVENTRY, 
 June, 1893.
 
 INTRODUCTION 
 
 THE Kinetic Theory of Gases is based upon the concep- 
 tion of an infinitely large number of molecules in motion 
 in a given space with velocities of all degrees of intensity 
 and in all conceivable directions. These molecules, as will 
 be explained in the course of the following treatise, may 
 sometimes be regarded as smooth spheres, in which case we 
 shall only have to consider the motion of translation of the 
 centre of mass of each of them, or they may be regarded as 
 bodies of any form capable of any number of internal 
 vibrations. It is clear that the individual molecules in 
 such a system must be continually acting upon each other, 
 either in the way of collision, like the mutual impacts of 
 elastic spheres, or else in the more gradual way of mutual 
 attraction and repulsion ; such actions are called encounters. 
 
 It is easy to see that if encounters take place among a 
 great number of molecules, their velocities, even if origin- 
 ally equal, will become unequal, for, except under conditions 
 which can be only rarely satisfied, two molecules having 
 equal velocities before their encounter will have unequal 
 velocities after such encounter. Now, as long as we 
 have to deal with only two molecules, and have all the
 
 xii Introduction. 
 
 data of an encounter given us, we can calculate the result 
 of their mutual action ; but when we have to deal with 
 millions of molecules, each of which has millions of 
 encounters in a second, the complexity of the problem 
 seems to shut out all hope of a legitimate solution. 
 
 We are obliged therefore to abandon the strictly kinetic 
 method and to adopt the statistical method. 
 
 According to the strict kinetic or historical method as 
 applied to the case before us, we follow the whole course of 
 every individual molecule. We arrange our symbols so as 
 to be able to identify every molecule throughout its motion, 
 and the complete solution of the problem would enable us 
 to determine at any given instant the position and motion 
 of any given molecule from a knowledge of the positions 
 and motions of all the molecules in their initial state. 
 
 According to the statistical method, the state of the 
 system at any instant is ascertained by distributing the 
 molecules into groups, the definition of each group being 
 founded on some variable property of the molecules. Each 
 individual molecule is sometimes in one of these groups and 
 sometimes in another, but we make no attempt to follow 
 it ; we simply take account of the number of molecules 
 which at a given instant belong to each group. 
 
 Thus we may consider as a group those molecules which 
 at a given instant lie within a given region of space. 
 Molecules may pass into or out of this region, but we 
 confine our attention to the increase or diminution of the 
 number of molecules within it. Just as the population of 
 a watering-place, considered as a mere number, varies in 
 the same way whether its visitors return to it season after 
 season, or whether the annual gathering consists each year 
 of fresh individuals. Or we may form our group out of
 
 Introduction. xiii 
 
 those molecules which at a given instant have velocities 
 lying within given limits. When a molecule has an 
 encounter and changes its velocity, it passes out of one of 
 these groups and enters another ; but as other molecules 
 are also changing their velocities, the number of molecules 
 in each group varies little from a certain average value. 
 
 We thus meet with a new kind of regularity, the 
 regularity of averages, a regularity which, when we are 
 dealing with millions of millions of individuals, is so 
 unvarying that we are almost in danger of confounding it 
 with absolute uniformity. 
 
 Laplace, in his Theory of Probability, has given many 
 examples of this kind of statistical regularity, and has 
 shown how this regularity is consistent with the utmost 
 irregularity among the individual instances which are 
 enumerated in making up the results.* 
 
 These observations must be borne in mind in interpreting 
 the definitions laid down and the results arrived at in the 
 Kinetic Theory of Gases. 
 
 Thus, to refer to the illustrations already given, we shall 
 prove that the number of molecules lying within a certain 
 region of space, or the number of molecules having their 
 velocities within certain limits differing by some finite 
 quantity, is in each case a number bearing some finite ratio 
 to the total number of molecules in the mass under con- 
 sideration, and therefore infinitely large. But these results 
 are to be interpreted as average results. We do not assert 
 by them, nor are we capable of proving, that at any given 
 instant there is one single molecule satisfying either of 
 the required conditions, that is, comprised within either 
 of the contemplated groups. 
 
 * MS. notes by Professor Clerk Maxwell.
 
 xiv Introduction. 
 
 So, again, the density of the region in the neighbour- 
 hood of any point is defined as the limit of the quotient 
 of the number representing the aggregate masses of the 
 molecules within any volume containing the point, to the 
 number representing that volume, when the volume is 
 indefinitely diminished. In interpreting this definition 
 two things must be remembered. In the first place, 
 according to what has been said just now, we do not assert 
 and cannot prove that there is, as a matter of fact, any 
 particular number of molecules within the volume contain- 
 ing the given point, at any given instant ; and in the 
 second place, supposing we could prove that the number of 
 molecules within the volume was thus accurately deter- 
 mined, yet even so there could be no point within the 
 region at which the actual density of the matter had the 
 value determined by our definition ; for if the point were 
 within a molecule the actual density would be much 
 greater, and if it were not within a molecule the density 
 would be zero.
 
 KINETIC THEORY OF GASES 
 
 ART. 1 .] A very great number of smooth elastic spheres, 
 equal in every respect, are in motion within a region of space 
 of given volume, and therefore occasionally impinge upon 
 each other with various degrees of relative velocity and 
 in various directions. The space is so large in proportion 
 to the sum of the volumes of the spheres that the average 
 time during which any one sphere is moving free from 
 contact with any other is infinitely greater than the 
 average time during which it is in collision with some 
 other sphere*. Required to find the law according to 
 which the velocities must be distributed in order that such 
 distribution may be permanent. 
 
 Let N be the total number of spheres, and let 
 
 X (u, v, w] du dv dw 
 
 be the number of spheres whose component velocities 
 parallel to the axes are intermediate between 
 
 u and u + du, v and v + dv, w and w + dw 
 respectively. 
 
 * In the mathematical conception of a collision as an absolutely instan- 
 taneous phenomenon the proviso in the text is of course superfluous. An 
 actual physical collision must take some time, however short, and the 
 object of the proviso is to exclude the possibility, or rather to diminish 
 indefinitely the probability, of the occurrence of cases in which the 
 collision of one sphere with a second is not concluded before that with 
 a third commences. 
 
 B
 
 2 Law of permanent distribution in 
 
 If we change the variables and make c the resultant 
 velocity, the inclination of c to the axis of z t and < 
 that of the plane of cz to the plane of xz, the expression 
 given above will become 
 
 X (, v, iv) c 2 sin 6 d 6 d$ dc. 
 
 Let a spherical surface of radius unity be described round 
 the origin as centre, and let us write d<r for the element 
 sin 6d6d(f) on this spherical surface, then the last written 
 expression becomes 
 
 Since for the same magnitude of the resultant velocity 
 all directions must be equally probable, it follows that the 
 coefficient of dcda in this expression must be a function of 
 the resultant velocity c only, and therefore the number 
 of spheres having component velocities between u and 
 n + du, v and v + dv, w and w + dw must be 
 \j/ (c) du dv dw. 
 
 It is required, now, to find the form of \ft in order 
 that the value of this expression may be unaffected by 
 collisions. 
 
 We assume that in the permanent state the distribution 
 of the spheres throughout the space occupied by them 
 is homogeneous in all respects ; that is to say, on an 
 average of any long time there are the same number 
 of spheres in a given volume wherever that volume may be 
 situated, and the law of distribution of velocities is the 
 same throughout that volume as in the whole region under 
 consideration. 
 
 Hence the number of pairs of spheres having component 
 velocities between 
 
 ! and ?/ x + dit l , i\ and v-^ + dv L , w^ and w^ + dw l 
 for the one, and
 
 medium of equal spheres. No forces. 3 
 
 2 and w 2 + ^2 ' V 2 an d V 2 + dv. 2 , iff 2 and w 2 + ^? 2 
 
 for the other, and such that the lengths of the projections 
 of their line of centres upon the axes are between 
 
 x and x + d%, y and y + dy, z and z + dz t 
 must be proportional to 
 
 ^ f ( c i)^ f ( c z) d u \ di\ & w \ du 2 dv 2 dw 2 dx dy dz, . . (A) 
 
 where c l and c 2 are the resultant velocities. 
 
 Since the coefficient of dxdydz is independent of x, i/, z, 
 it follows that the number of pairs having- their velocities 
 between the above-mentioned limits and the projections 
 of their line of centres between x and x + dx, and dy, 
 and dz must be also represented by (A). 
 
 If dy and dz be infinitesimal in comparison with the 
 diameter of any sphere, the lines of centres are parallel 
 to the axis of x and the velocity of approach of these 
 centres is u-^u^. 
 
 If d be the sum of the radii of any two of the spheres, 
 or, what is the same thing in this case, the diameter 
 of any one of them, then it follows that any pair of spheres 
 having the x component of their line of centres less than 
 d + (u l u^) dt will encounter each other in the time dt\ 
 making therefore 
 
 SB = d and dx = (u^ u^dt, 
 
 it will follow that the number of collisions in time dt 
 of pairs of spheres with their lines of centres parallel to x 
 and velocities between the above limits is proportional to 
 
 u l -w 2 )dt. . . (B) 
 
 After collision let the different quantities involved in (B) 
 become /, c 2 ', /, &c., while x, y, and z, dy and dz remain 
 of course unchanged. 
 
 B 2
 
 4 Law of permanent distribution in 
 
 It follows therefore that if the values of these velocities 
 and component velocities had originally been within the 
 limits indicated by the accented letters with reversed signs, 
 the spheres would, after collision, have passed into the 
 original state as indicated by the unaccented letters. Call 
 these states E and F respectively. For permanence of 
 distribution therefore it is sufficient that the number 
 of collisions of pairs of spheres in state E during the time 
 dt should be equal to the number of collisions of pairs in 
 the state F during the same time ; and therefore that 
 
 Now /= u 2 , .'. du.i = du 2 , 
 
 <= i, dn'= fav 
 
 dv l = dv^, 
 dv. 2 = dv. 2 ', 
 
 Hence 
 
 du lt .. dw z dy dz (u^ u 2 ) du^.. . dw.^ dy dz (u.^u^) ; 
 
 *(<i)*W = *feOV'ta'). 
 
 and c* + c* = c l '* + c t '*. 
 
 This functional equation may be integrated as follows. 
 Let it be expressed in the form 
 
 and for c* and c.f write x and x\ and for <?/ 2 and c. z ' 2 write 
 y and^; then we have 
 
 where
 
 medium of equal spheres. No forces. 
 ' x (*) x (O = x (y) x 0* + '-.?) ; 
 
 therefore differentiating with respect to # and #', 
 
 and x (*) / (0 = X W x' 
 
 that is, , \ = constant ; 
 
 x(*) 
 
 , 
 
 Now the axis of a? may be taken in any direction. 
 Whence it follows that this form of \}/ will ensure the 
 permanence of distribution for all possible collisions. 
 
 Therefore the number of spheres with component veloci- 
 ties between u and u + dn, v and v + dv, w and w + cl it- 
 
 is Ae~ h<fl du dv dw, 
 
 employing 1 the notation used above. 
 
 Integrating with respect to da from to 477, we find for 
 the number of spheres with velocities between c and c + clc 
 the expression 
 
 c* dc. 
 
 Again, since the number with component velocities 
 between 
 
 it and n + du, v and v -f dv, w and w + dw 
 
 is Ae~ h (*+ *+ W 8 ) du dv dw, 
 
 or Z/Ae- hu *
 
 6 Mean velocity and mean kinetic energy. 
 
 it follows that the number of spheres having velocities 
 intermediate between u and u + du parallel to any fixed 
 line is 
 
 u f <?-** dv f " e~ dw ; 
 
 J- 00 J - *> 
 
 that is ^d-* Ma ^J; 
 
 where ^ may be determined by the equation 
 
 A 
 
 N$ 
 
 and therefore A = -, 
 
 
 
 that is to say, the number of spheres having velocities 
 between c and c + do is 
 
 If we multiply this expression by c, integrate the 
 product with regard to c from o to oo and divide by N, we 
 find the mean velocity of all the spheres to be 
 
 And, similarly, multiplying by c 2 instead of c, we find 
 the mean square of the velocities 'to be 
 
 and this is greater than the square of the mean velocity, as 
 it ought to be.
 
 Diagram of Velocities. 7 
 
 2.] Before we proceed further it will be useful to make 
 a formal statement of the method employed in the last 
 Article for denoting- the velocities of particles or material 
 points. 
 
 Instead of drawing straight lines from each particle 
 indicating- the magnitude and direction of the velocity of 
 that particle, we draw all such straight lines for all 
 particles from any assumed point taken as the origin. 
 
 This method is very useful when, as in investigations 
 like the present, we wish to compare the simultaneous 
 velocities of different particles as well as the successive 
 velocities of each particle separately. 
 
 We thus obtain a figure every point of which corre- 
 sponds to one of our particles, the velocity of that particle 
 is represented in magnitude and direction by a line drawn 
 from the origin to the corresponding point, and the relative 
 velocity of any two particles is also represented by the 
 line joining the points corresponding to these two particles. 
 If the system have a common velocity, then we must 
 suppose the position of the new origin of the diagram 
 of velocities with respect to the old origin to be so chosen 
 as to represent this velocity. 
 
 In studying the motion of the system it is found conve- 
 nient to divide the particles into groups according to 
 their velocities, those particles whose velocities lie within 
 certain limits with respect to magnitude and direction 
 being placed in the same group. 
 
 In the velocity diagram these particles are indicated at 
 once by the points which correspond to them being in- 
 cluded within a certain small region or elementary volume 
 of the diagram, the boundary of this region corresponding 
 to the given limits of velocity. 
 
 Thus, in the proposition just now considered, the state E 
 might be described as that in which the velocity points
 
 8 Law of permanent distribution in 
 
 are situated within the elementary volume 
 dudvdw or c 2 sin0 d9 b<f>dc 
 
 of the diagram of velocities. 
 
 We may also conveniently make use of the term 
 velocity-density to indicate the result of dividing the 
 number of particles whose velocities lie within the given 
 limits by the volume of the corresponding region in the 
 diagram of velocities. 
 
 3.] Let the .A 7 elastic spheres of the last Article be 
 replaced by two sets, one of ^spheres each having mass m. 
 and the other of N' spheres each having mass m' , and let 
 us find the law of distribution of the velocities of the N 
 and N' spheres respectively. 
 
 Exactly as in the first Article, it may be proved that 
 the number of the N spheres whose component velocities 
 parallel to the axes are intermediate between 
 
 u and u + du, v and v + di; w and w+dw 
 
 respectively must be 
 
 \l/ (c) du dv dw, 
 
 where <? 2 = u z + v z + ;2 ; 
 
 and that the number of the .A 7 ' spheres whose component 
 velocities are intermediate between 
 
 u' and u r + du', v and v' + dv r , w r and w' + dw' 
 must be ^i (e') du' dv' dw' ; 
 
 and that the number of collisions in time dt between 
 pairs of spheres, one from each set, with lines of centres 
 parallel to the axis of x and component velocities between 
 the above limits respectively, must be proportional to 
 
 . (A)
 
 medium of sets of spheres. No forces. 9 
 
 If the component velocities of these spheres after 
 collision be between 
 
 U and U+ JU, V and 7 + d V, W and 7F+ dW, 
 V and U' + dU', V and V' + dV, W and W' + dW 
 
 respectively, then, by the reasoning- employed in the last 
 case, we must have 
 
 \// (c) ^, (c') rfw . . . rfw' (w - ') 
 
 = *(C)* l (C')(IU...dW(U f -V\ . (B) 
 where F = r, JT= ;, F'= >', 7P= ;', 
 
 m + m 
 2m 
 
 rr . . 
 U'=<u'+ 
 
 whence we g-et m C 2 + TO'C 1 ' 2 = me 2 + m'c' 2 , 
 
 dU(W_ dU dlT_ _ (m - m'} 2 4mm' 
 flu' du du ' du' (m + m') 2 '- 
 
 and also 
 
 V'-U = -', 
 
 ,,dUdW dUclU\ 
 
 ;md dUdU = du du (, -= --- ; -- r-r ) =du dn . 
 \du du du du ' 
 
 So that equation (B) reduces to 
 
 where me 2 + m'c' 2 = mC* -\- m'C' 2 . 
 
 A functional equation of which, as before, the solution 
 may be shewn to be 
 
 _A?^ ^h*"'*" 
 
 t/r (c) = ^^ 2 and ^ (c') = ^e 2 .
 
 io Mean velocities and kinetic energies. 
 
 It must be remembered that Ji is here a constant of 
 different dimensions from the h of the former case. 
 
 Since the axis of x may be taken in any direction, 
 it follows that these forms of ^ and \j/ l will ensure the 
 permanence of distribution as far as collisions between 
 spheres of two different sets are concerned, and we have 
 already proved that they will ensure that permanence 
 so far as collisions between spheres of the same set are 
 concerned ; therefore it follows that the number of the 
 N spheres whose component velocities are intermediate 
 between 
 
 u and u -f du, v and v + dv, w and w + dw 
 
 _ A 2 
 must be Ae 2 du dv dw, 
 
 where c 2 = u 2 + v* + w 2 ; 
 
 and the number of the N' spheres whose component velo- 
 cities are intermediate between 
 
 u' and u' + du', v' and v' + dv', w' and w' + dw' 
 
 -* 
 must be Be 2 du' dv' dw', 
 
 where c 2 = u'' 2 + v' 2 + w' 2 ; 
 
 and it is clear that the same reasoning would hold what- 
 ever number of sets of such spheres the space might 
 contain. 
 
 Applying the results of the former case, and writing 
 
 instead of h, we get 
 
 N' ,m'h^ 
 3 jfj= ( ) : 
 
 rf ^ 2 > ' 
 
 and so on for any other sets which the region may contain.
 
 Distribution of relative velocities. 1 1 
 
 We also find that the mean velocity of each of the 
 N spheres is 
 
 JL /^ 
 
 V*' V ml 
 
 mh' 
 
 and similarly for the remaining sets of spheres. 
 
 Also the mean square of the velocity of each of the 
 N spheres is 
 
 and the mean kinetic energy 
 3_ 
 
 3 J 
 
 the last result being the same for each set of spheres. 
 
 4.] On the hypothesis of the last Article to find the 
 number of pairs of spheres, one being taken from the 
 N set and the other from the N f set, whose relative 
 velocities lie between given limits, and the number of 
 collisions in unit of time and unit of volume between 
 these sets of spheres. 
 
 Using the results of the last Article, writing for 
 and j for --- , and supposing the volume of the region 
 
 considered to be the unit of volume, we find that the number 
 of spheres of the N set which have their component velo- 
 cities parallel to x between the limits u and u +du is 
 
 N 
 ^e~*du; ...... (A) 
 
 a v TT 
 
 and that the number of spheres of the N' set which have 
 their component velocities parallel to x between the limits 
 + U and u + U+ dU is 
 
 M (u+uy 
 
 ^=>e~ P dU-, . . . . (B)
 
 12 Distribution of relative velocities. 
 
 and therefore the number of pairs of spheres fulfilling the 
 above-mentioned conditions is 
 
 Integrating with respect to it from oc to + oc , we 
 find for the total number of pairs of spheres whose relative 
 velocity parallel to x lies between U and U+dU, the 
 expression 
 
 x -^-^_ ^ 
 
 Comparing (C) with (A) we see that the distribution of 
 relative velocities follows a law of the same form as that of 
 absolute velocities, and therefore that the mean relative 
 velocity is the square root of the sum of the squares of the 
 mean absolute velocities in the two systems, and the mean 
 square of the relative velocity is the sum of the mean 
 squares of the absolute velocities. 
 
 It follows also of course that the number of pairs of 
 spheres, one from each set, whose relative velocities lie 
 between r and r + dr is 
 
 JW' * . e ~p r 2 dr ; 
 
 or, restoring to a and ft their values, 
 ^,-A^JYl _^ 
 
 5.] We proceed now to find the number of collisions in 
 unit of time and volume between the spheres of the N set 
 and those of the N' set. 
 
 Suppose a number N of equal spheres at rest to be 
 distributed in any manner throughout a unit of volume,
 
 Frequency of collisions, 13 
 
 and suppose that another sphere moves among them with 
 the velocity r. If a tubular surface be described having 
 for axis the path of the centre of this moving sphere, and 
 for its radius *, or the sum of the radii of the moving 
 sphere and one of the stationary spheres, the volume of the 
 surface thus generated in a unit of time by the moving 
 sphere will be nrs 2 . Hence the chance of the moving 
 sphere colliding with any one of the fixed spheres in a unit 
 of time must be irr# z , and the number of collisions in unit 
 of time between the moving sphere and stationary spheres 
 must be NirrfP. 
 
 The same results would hold good if we replaced the 
 stationary spheres by spheres moving with a common 
 velocity and the moving sphere had a velocity r relative to 
 each of them, that is to say, the chance of collision in unit 
 of time between the last-mentioned sphere and any one of 
 the former-mentioned spheres would be -nrtP. 
 
 Suppose now that there are two sets of N and N' spheres 
 in the unit of volume, and that the number of pairs of spheres 
 (one being taken from each set) whose relative velocities 
 are between r and r + dr is nn', then, if s be the sum of the 
 radii of each pair, it will follow that the chance of a 
 collision between any pair in unit of time is -nrs 2 , and 
 therefore that the total number of collisions in unit of 
 volume and unit of time is nn'nrs 2 . 
 
 But we have already seen that nn' is equal to 
 
 r 2 
 
 NN' _ 4 . e~ *+& r 2 dr, 
 
 VV(a 8 + j8)* 
 
 N and N' being the total number of spheres of each set in 
 unit- volume, and a 2 and /3 2 being written for . and ^ > 
 
 Wrl M n 
 
 as before. 
 
 Hence the number of collisions in unit-time between
 
 14 Frequency of collisions. 
 
 pairs of spheres whose relative velocity lies between ? and 
 r + dr is 
 
 Integrating from r = o to r = <w , we get 
 2NN' V*. -v/o^HSV 
 
 or 
 
 for the number of collisions in unit of time which take 
 place in unit of volume between spheres of the N and 
 If sets. 
 
 The number of collisions in the same time and volume 
 between spheres of the N set is therefore 
 
 
 and between spheres of the N r set the number is 
 
 Also, since the mean velocities in the two systems are 
 
 20^ 2f3 
 
 A/IT Vir 
 
 it follows that if ^ be the mean distance travelled by 
 a sphere of the N t set between each collision with a sphere
 
 Distribution in medium of circular disks. 1 5 
 
 of its own set and 1. 2 the corresponding mean distance for 
 the second set, then 
 
 Y = 77 ^ \/2 Sj* and j- -nN z A/ 2 * 2 2 , 
 *! ^ 
 
 s l and s 2 being diameters of spheres of the two sets. 
 
 Also, if Aj be the mean distance travelled by a sphere of 
 the JVj set without any collision whatever, 
 
 where j* r is sum of radii of spheres of the N^ and JV r sets, 
 with a similar expression for the mean distance travelled by 
 a sphere of any other set. 
 
 6.] Let the sets of spheres of the previous cases be 
 replaced by sets of perfectly smooth and elastic circular 
 disks with their centres of inertia not coincident with their 
 centres of figure, in motion within a plane area inclosed by 
 a perfectly elastic boundary, the number of disks in each 
 set being very great. The disks in each set are equal to 
 each other in every respect, while those in different sets 
 may differ in mass, area, and distance between the centre 
 of inertia and the centre of figure. 
 
 Each disk has in this case three degrees of freedom, 
 which may be expressed by the coordinates of the centre of 
 figure and the angle between the line joining the centres 
 of figure and inertia, and some fixed line, as the axis of x ; 
 or perhaps more conveniently by the perpendicular distance 
 of the centre of inertia from a line through that of figure 
 parallel to the x axis. There are three corresponding 
 momenta components measured, for each disk, by the 
 translational x and y velocities u and v of the centre of 
 inertia, and the angular velocity o> round an axis perpen-
 
 i6 Law of permanent distribution in 
 
 dicular to the disk. In this case it is clear that any 
 assumed distribution of momenta for the disks of each set 
 will, in the absence of all collisions, remain undisturbed, it 
 is required to investigate the conditions of such permanence 
 when there are collisions between two disks of the same or 
 different sets. 
 
 Let the expression 
 
 J\ (u, v, &>) du dv d <a 
 
 represent the number of the disks of any set with velocity- 
 components intermediate between u, u -f du ; v, v + dv ; 
 a), (D + dd) ; and let 
 
 be a corresponding- expression for any other set ; and let 
 us consider the effect of collisions, one at a time, between 
 disks of these two sets. 
 
 The number of pairs of such disks with velocity-compo- 
 nents, at any instant, between the aforesaid limits is 
 
 As in the case of the spheres we may, without loss 
 of generality, confine our attention t pairs whose line 
 of centres is parallel to any fixed line, say the axis of 
 x, then if p and P be the perpendiculars from the centres of 
 inertia upon that line of centres, the velocity of approach 
 of the points of contact of any pair, about to collide at the 
 instant considered, will be 
 
 And therefore the number of collisions in the short time dt 
 after the instant considered will be proportional to 
 
 { _ U(p o> - PQ,)} &t x/! (, v, co)/ 2 (U, 7, li) du. . .dl. 
 After collision let the several velocity-components be
 
 medium of circular disks. No forces. 17 
 
 represented by the same symbols as before collision with 
 dashes affixed, the coordinates remaining 1 unchanged, so 
 that the number of pairs of disks which pass from the state 
 u, u + du, Q,, ii + r/il to the state ', u' + dn'...Q,' t Qf + dQ,' 
 in the time dt will be proportional to 
 
 {u - U- (p* - Pfl)} / t (u, v, <o)/, ( U, F, li) du. . .da . dt. 
 
 But the number of disks, with any given velocity-com- 
 ponents at any instant, is assumed to be the same as the 
 number with those velocity-components reversed. Whence 
 it follows that the number of pairs of disks which pass from 
 the state vfii+d^...QfUf+dQ f to the state , u + dv...Q,, 
 in the time dt must be proportional to 
 
 Hence the condition for permanence, so far as collisions 
 between pairs of these two sets, one at a time are con- 
 cerned, will be satisfied by the condition that 
 
 (u-U-(pa>-PL))fi(u,v t w)f tji (U t F, 
 is equal to 
 
 Also the relative radial velocity of the points of collision 
 before impact must be equal, and of opposite sign, to the 
 same relative velocity after impact. 
 
 Therefore u - Up a> + P 12 = U'-u'-P&'+p a/, 
 i.e. u' + u-(U' +U)-j) (a/ + a>) + P(&' + il) = o. 
 Also we may prove that the differential products 
 
 du...da and du'...dQ,' 
 are numerically equal. 
 
 For if R be the measure of the impulse at the colliding 
 points, we have the equations 
 c
 
 i8 Law of permanent distribution in 
 
 ' ' ' 
 
 pR PR 
 
 a> - a) = ^F' *- Q = 
 
 whence by substitution in the equation 
 
 we get 
 
 2(<u-U-pa>-PQ.) u-Upu + PQ, 
 R= i_ ^_ -- ^I = " ~V -- .suppose; 
 ~m + M + mtf + MK 2 
 
 And the functional determinant giving the ratio between 
 the differential products, du'...dQ! and du...dL, becomes 
 
 I L_ 5 _^, JL., L* 
 
 i i p P 
 
 JTD' MD' MD' MI) 
 
 p p p 2 pP 
 
 pP ^ P* 
 
 MKW MK Z D' MK' 2 D' MK 2 D
 
 medium of circular disks. No forces. 19 
 
 that is 
 
 T _J_ _L P* p2 
 
 mD Ml) mk*D MK Z D 
 or i 2 or i . 
 
 So that the condition sought for is reduced to 
 
 /i(,",)/ a (0; r,ty =f l v',<')f2(U', F',a'), 
 
 leading, as in the two previous cases, to the result 
 
 TI being the kinetic energy of the disk of the one set and 
 T 2 that of the disk of the second set, h a constant for both 
 sets of disks, and A l , A 2 special constants for each of the 
 sets, determined in terms of h by the condition 
 
 err .-* 
 
 l e 
 
 where the integrations are in all cases between the limits 
 oo and + oc , and N^ and N 2 represent the number of 
 disks in each of the sets. 
 
 "With this law of distribution it follows that the state 
 remains permanent so far as collisions between disks of the 
 i and 2 sets alone are concerned. 
 
 Similarly it may be proved that the permanence of 
 the distribution, with corresponding laws for each separate 
 set of disks, would not be affected by collisions between 
 pairs of disks, one pair at a time, both selected from any 
 one set. Whence a law of distribution ensuring permanence 
 of distribution and unaffected by any collisions whatever, 
 with only two disks in each collision, is, that the number of 
 c 2
 
 20 Law of permanent distribution in 
 
 disks in any set with velocities between u, u + du, v,v + dv, 
 o>j CD, co + do* should be 
 
 Ae 2 dudvda, 
 
 where 
 
 A I I I e 2 dudv dw 
 
 is equal to the number of disks in that set. 
 
 7.] In Articles 1 and 2 we considered the cases in which 
 the constituents of the medium consisted either of a very 
 great number of smooth elastic spheres, equal to each other 
 in every respect, or of sets of such spheres, the mass 
 and volume of those of one set not being necessarily the 
 same as those of any other set, but the number of spheres 
 in each separate set being very great ; and we proved that 
 there would be a permanent state, unaffected by collisions, 
 provided the number of such constituents whose velocity- 
 components it, v, w were intermediate between u and u + du, 
 v and v + dv, w and w + dw at any instant were given by 
 the expression 
 
 Ae~ hT dudvdw, 
 
 T being the Kinetic Energy of the constituent, k a con- 
 stant for all the sets of constituents, and A a constant for 
 each particular set determined by the equation 
 
 A f f f e- hT dudvdw=N, 
 
 J 01 ' -01 '-oo 
 
 N being the total number of constituents in that set. 
 
 In Art. 6 a slightly varied hypothesis as to the con- 
 stituents led to a similar result, and it was proved that for 
 a medium in which the constituents were sets of smooth 
 perfectly elastic circular disks limited to motion in their own 
 plane, with the centre of figure of each disk not coincident
 
 medium of any rigid bodies. No forces. 21 
 
 with the centre of inertia, there would be permanence, 
 provided the number of constituents of each set, the velocity- 
 components of whose centre of inertia (u, v) lay between 
 u, u + du\ v, v + dv, while the angular velocity lay between 
 co and o) + d(ti, were given by the expression 
 
 Ae- hT du dv d a, 
 
 T being the Kinetic Energy of each disk, h a general 
 constant common to all the sets, and A a special constant 
 for each set determined by the equation 
 
 
 
 J -00 ^-00 J-< 
 
 where .2V is the number of disks of that particular set. 
 
 Before we proceed to the establishment of a general 
 proposition, including the particular cases just mentioned, 
 it will be necessary to prove the following theorem. 
 
 8.] Let there be any material system whose state is 
 defined by the n generalised coordinates q l . . . q n and the 
 corresponding components of momenta p l ... p n , and let the 
 system move in a field of conservative forces from the state 
 p l . . .p n , q l ...q n to the state p\ . . ,p' n , q\ . . . q' n in the time t, 
 so that each of the 2, n variables p\... q' n is a determinate 
 function of the 2n + i variables p l ...p n , ^...^n an d ^ 
 then if t be constant, the functional determinant 
 
 d pl'" dq n 
 
 is numerically equal to unity, or the multiple differentials 
 dp l ...dq n and dp^ ...dq' n are numerically equal to each 
 other.
 
 22 
 
 Law of permanent distribution in 
 
 For if S be the Hamiltonian Principal function of q l . . . 
 \. q' n and t, we have 2 n equations 
 
 dS dS\ 
 
 dS 
 
 = aR'-/-- 
 
 dS 
 
 the differential coefficients being partial. 
 
 Let us in any expression <j>(p'i ... q f ^ dp\. . ,dq r n change 
 the variables from p\ ...p' n q\ ... / to q l ... q n q\ ... /, 
 and we get 
 
 ...dq' n 
 
 where A = 
 
 '."*?<*/.' 
 
 by the use of the second set of equations (A) (above). 
 Similarly we get 
 
 d'P\ - d<} n = A'. dq l . . . dq' n 
 
 J*R 
 where A'= 
 
 dq' n dq n 
 
 by the use of the first set of equation (A)*". 
 
 Whence we get dp l ...dq n numerically equal to dp\...dq' n . 
 
 * The original Edition of this book contained a supposed demonstration 
 of a proposition analogous to that in the text but the reasoning was 
 fallacious. 
 
 In the proposition referred to the initial and final coordinates and 
 momenta were supposed to be connected by the condition E (the total 
 energy) constant and not t (the interval of time between the two states)
 
 medium of any rigid bodies. No forces. 23 
 
 9.] Let there be any number of sets of perfectly smooth 
 elastic rig-id bodies of any form, henceforth called molecules, 
 
 constant, and the demonstration was in all respects the same as that in the 
 text, with the substitution of the Hamiltonian Characteristic function A 
 for the Principal function 5, so that the result took the form 
 
 = 
 dpi...dq u (A')' 
 
 where (A) and (A') were the A and A' in the text with A substituted for 
 8, and therefore numerically equal to each other ; and so far the demon- 
 stration was correct, but in fact, in this case, A and A' are each separately 
 zero. 
 
 For the momenta components^ ... p n , with which the system must be 
 started, in order to pass from the initial position ^...Jn to the final 
 position q\...q'n with a given energy E must be the same for all values of 
 q'i...q'n, provided they be consistent with any configuration whatever 
 through which the system passes either before or after reaching the actual 
 final configuration, or 
 
 where q\, g' 2 , &c. are the time variations of q\, g-' 2 , &c. in the actual 
 motions, with similar equations whenp 2 , p 3 , &c. are substituted for p lf 
 Whence we have 
 
 dp L dp l 
 dq\ - dq' n 
 
 : : => 
 
 dp n dp n 
 
 or (A') = o, and similarly (A) = o. 
 
 Whence it follows that if the initial and final st-ites be connected by the 
 condition E constant, we can get no determinate relation between the 
 differential products dp\ ... dq' n and dp t ... dq n , and this conclusion might 
 have been anticipated, because from the equations of motion we get 2 n 
 equations among the 2+i variables p\...p' n q f i ...q' n > Pi---Pn> ?i -.-On 
 and t (the time of transit), and we need an additional relation between 
 them to determine each of the 2 n variables (p'j ... q' n } in terms of 
 
 But the condition E constant does not supply any such additional relation, 
 so that the problem remains indeterminate. The fallacy just noticed in the 
 original edition was first of all pointed out by Boltzmann in a communi- 
 cation to the Philosophical Magazine in 1882, when I privately suggested 
 to him the substitution of the proposition in the text. In a communication 
 to the same magazine in 1891, Lord Rayleigh independently advanced a 
 similar objection to that of Boltzmann against the original treatment and 
 proposed the same amendment.
 
 34 Law of permanent distribution in 
 
 in motion in a region inclosed within a perfectly elastic 
 boundary, let, also, the molecules in each set be exactly 
 similar to each other in all respects, while those of different 
 sets may differ in any respect, viz. as to shape, volume, 
 mass, &c., subject, at least, to the condition that the volume 
 of each molecule is infinitely small compared with that of 
 the inclosed region, and that the number of molecules 
 in each set is very large, then the state of the medium 
 so constituted will be permanent and unaffected by collisions 
 between two molecules, either of the same or different sets, 
 provided the law of distribution be such that the number 
 of molecules of any set whose momenta and coordinates 
 are intermediate between p L and A+^pj...^ and q m + dq m 
 be of the form 
 
 Ae- hT dp l ...dq m) 
 
 where m is the number of degrees of freedom of the 
 molecule and therefore lying between 3 and 6 inclusive, 
 T is the kinetic energy of the molecule, k a constant 
 the same for all sets, and A a special constant for each 
 set determined by the condition that 
 
 shall be equal to the number of molecules in the set con- 
 sidered, the integrations having taken over all possible 
 values of the variables. 
 
 In the first place it is evident that this law of distribu- 
 tion will be permanent, in the absence of collisions, because 
 in such a case T remains constant for all time for each 
 molecule, and so also does the multiple differential 
 
 dp r ..dq m 
 by the last Article. 
 
 Neither will the distribution be affected by collisions 
 between any pair of molecules either of the same or
 
 medium of any rigid bodies. No forces. 25 
 
 different sets. For, according to this law, the number 
 of molecules of any one set with coordinates intermediate 
 between q l and q^ + dq^ ... q m and q m + dq m} and momenta 
 between p 1 and p + dp^ ...p m and p m + dp m , is 
 
 Ae~ hT dq l ...dq m dp ... flp m . 
 
 Let us suppose m to have its greatest possible value 6, 
 and let q l , q. 2 , q z be the rectangular coordinates a, y, z of 
 the centre of inertia, and ^ 4 , q 5 , q 6 be the ordinary Eulerian 
 angular coordinates 6, $, \l/, determining the orientation of 
 the molecule about the centre of inertia, then if u, v, w be 
 the translational velocity-components of the centre of inertia 
 and Wj, <o 2 , w 3 the angular velocities about the principal 
 axes through that point and M the mass of each molecule, 
 Pi> Pz> Pa are equal to 3fw, JSfv, J/0, respectively, and 
 PiiPsiPs are eacn linear functions of o> 1? co 2 , o> 3 , so that 
 by elimination of p 1 ...p 6 the number of molecules of this 
 set with coordinates between #, x + dx...\f/, \{f + d\l/, and 
 translational and angular velocities between u, u + du . . . co 3 , 
 w 3 + do) 3 , is by the assumed law 
 
 where T = { u 2 + v 2 + w 2 + k* u* + 2 o> 2 2 + k* u, 3 2 } , 
 
 ^, k.^ k% being radii of gyration about the principal 
 centre of inertia axes. 
 
 Integrating for all values of x ... ty we get 
 A^ e~ HT du ... do> 3 
 
 for the number of molecules of the set in question with 
 velocities intermediate between u, u + du, ..., o> 3 , o> 3 + da> s , 
 T having the same value as before, and A l being determined 
 by the condition 
 
 I rrr. 
 
 where N is the number of molecules in the set in question.
 
 26 Law of permanent distribution in 
 
 This expression is perfectly general and applies to all 
 forms of the molecules and all numbers of degrees of 
 freedom ; for example, when they are smooth spheres the k 's 
 are all equal and w, 2 + co 2 2 + 3 2 for each sphere is constant, 
 so that the law reduces to the statement that 
 
 Ae~ hT du dv dw 
 
 is the number with translational velocities between u, u + du, 
 v, v + dv, w, w + dw, the number of degrees of freedom being 
 3 as in Arts. 1 and 3. 
 
 When the molecules are straight bars, one of the 's 
 becomes zero, and the corresponding u> does not affect 
 the state of the molecule, and the number of degrees of 
 freedom is 5, and so on. 
 
 By our assumed law of distribution for all the sets, 
 the number of pairs of molecules f any two sets, say the A 
 and B sets, which at any instant have their velocities 
 between the limits u, + ^,...ca 3 , o> a +do> 3 for the one, and 
 U, U+dU,... & 3 and % + ^Ii, for the other, is 
 
 and the rate of collision, per unit time, of these two 
 molecules is v, where v is the relative velocity of approach 
 of the colliding points, in the direction of the common 
 normal at these points, at the instant of collision, so that 
 the number of this particular class of collisions, between the 
 pairs, in the time dt, will be 
 
 ABe' h (T * +T * } du... dQ, s v dt. 
 
 After collision let the velocities be denoted by affixing 
 dashes to the symbols which denote the respective velocities 
 before collision, whence it follows that the number of pairs 
 of molecules, given by the last written expression, passes
 
 medium of any rigid bodies. No forces. 27 
 from the state u, tt + du.,.Q 3 , ii 3 + ^i! 3 to the state 
 
 in time dt, by this particular kind of collision. 
 
 It follows by reasoning in all respects similar to that 
 employed in the cases already treated that the number 
 of molecule pairs passing from the state 
 
 u', u' + du', . . . fl'g, il' s + </il' 3 
 to the state u, w + d...il 3 , il 3 + </i! 3 in the same time dt is 
 
 where v is the same function of u'... 1' 3 that v is of 
 n ...Q s since the coordinates are unchanged by collision, 
 i. e. where v is the reversed relative velocity of the colliding 
 points in the direction of the common normal. 
 But since the elasticity is perfect we have 
 
 also J 
 
 Therefore the condition sufficient for permanence, so far 
 as this particular kind of collision between the molecule 
 pair is concerned, is reduced to 
 
 du...dQ, 3 = du'...dL' y 
 
 But by D'Alembert's principle applied to impulsive 
 forces we have the equations 
 
 u'u _v' v _ Q' 3 - Q 3 p 
 
 A x A 2 A 12 
 
 where A t ... A 12 are known functions of the ^'s and Q's* 
 * It is easily seen that the condition of collision is 
 
 where ^ ... q n are all the coordinates of the two colliding molecules and 
 is a function of these coordinates which cannot become negative, whence it 
 
 will follow that \ lt A 2 &c. are proportional to , , &c.
 
 28 Law of permanent distribution in 
 
 Now v is a linear function of u . . . i! 3 of the form 
 
 and v' is the same function of '..., 1' 3 and is equal to 
 v, whence to determine R we have the equation 
 
 Whence we get the 6 equations 
 
 Therefore (writing 2 D for \ 1 n l + ... +A 12 /u 12 ) the ratio 
 of dv! ' ... dQ! to du...dL. A is equal to the determinant 
 A lMi A i^ 2 _ A i/*] 
 
 D ' ^ " i> 
 
 i 
 
 2) 
 
 Or the multiple differentials du...dQ 3 and du'...dL' 3 are 
 numerically equal to each other, and the assumed law 
 of distribution, if once established, is unaffected by collisions 
 between any pair of molecules whatever ; we have seen
 
 medium of any rigid bodies. No forces. 29 
 
 also that it is maintained in the absence of collisions, and 
 it is therefore permanent. 
 
 The number of what is here called kinds of collision 
 between any molecule pair is infinitely great ; in the case 
 of spheres these kinds of collisions differ from each other 
 only in the direction of the common normal to the surfaces 
 at the colliding- points, in a sphere and an ellipsoid they 
 will differ, not only in this circumstance, but also in the 
 orientation of the ellipsoid at the instant of collision, 
 and in the case of two ellipsoids they may differ in the 
 orientation of both ellipsoids, but this variety in the kinds 
 of collision does not affect the reasoning in the text. 
 
 10.] In the last Article the problem of collision between 
 two molecules has been treated by the assumptions ordinarily 
 employed in rigid dynamical problems of collision between 
 two perfectly elastic solids of any form, and all that is 
 necessary to the validity of the result arrived at, is the fact 
 of the independence of the quantities A 15 X 2 , &c. of the 
 momenta at the instant of collision, without reference 
 to the actual value of these A's. By a slight change 
 however in the language and conceptions employed, the 
 A's may be evaluated and the results arrived at may be 
 extended from rigid molecules with a maximum of 6 
 degrees of freedom each, to molecules of any form and 
 any number of degrees of freedom, as for example a chain 
 of r links, in which the number of degrees of freedom 
 is 2 r + 3, and other systems. 
 
 For, by Lagrange's equations, if T be the kinetic energy 
 of any system expressed as a quadratic function of the <fs 
 with coefficients functions of the ^'s, and U be the force 
 function, we have 
 
 d_ dT _dT_ dU 
 
 At dq dq ~~ dq
 
 30 Law of permanent distribution in 
 
 If in such a system there be finite change of velocities, 
 or momenta in a very short time t, without finite change 
 of coordinates, in other words a collision, we get by 
 integration 
 
 ,dT^_ ( dT^_ C^dU 
 
 ty) " yjJ ~J tl ~df 
 
 rti {U> 
 where t 2 t, = t, since a \-j-\ zero in the limit when 
 
 /ft v ?' 
 t 2 t^ is infinitely small, because in the general case 
 
 dT 
 
 -j- must throughout the integration be intermediate between 
 
 two finite quantities (see Routh's Rigid Dynamics (Art. 372, 
 Third Edition, Watson and Burbury's Generalised Coordinates 
 (Art. 12), &c.). 
 That is to say, 
 
 dU *' dU 
 
 Now collisions, or finite changes of momenta with un- 
 changed coordinates, can only be regarded as occurring 
 when the system arrives at a configuration determined 
 by such a condition as 
 
 where $ is a function of the q's which cannot change sign, 
 or in other words, the system and forces are such that there 
 is an infinitely great force resisting such change of sign. 
 
 Now U, the assumed force function of the forces, 
 is a determinate function of the n coordinates <?!,...,) 
 and therefore may be expressed as a determinate function 
 of </> and n I quantities chosen arbitrarily as < 15 < 2 ,.. </>_!, 
 and from the last paragraph it follows that if U be so 
 
 expressed -3 must be infinitely great when < = o, and
 
 medium of any constituents. No forces. 
 
 fhorp-frrr 
 
 therefore, when $ = o, > -=- &c. must be proportional to 
 * 
 
 and therefore we get n equations of the general form 
 
 . . _C**dU d<j>_d<fi C f >dU 
 P ~ P ~\ 1$ ' dq ~ Tq ' J tl Jtf 
 
 since is a function of the q's only and independent of 
 the q's. Whence it follows that A a , A 2 are proportional to 
 
 , -^-, &c., and therefore independent of the pg. 
 
 By supposing a system made <np of two molecules 
 of this more general constitution, with m and n degrees of 
 freedom, respectively, in the place of the 6 degrees of freedom 
 each of the last Article, we get, as in that Article, the 
 m + n I equations 
 
 P'lPl P\-P2 
 
 the Aj, A 2 , &c. being replaced by -j , --, &c, 
 
 Also since by our hypothesis there is no change of Kinetic 
 Energy, we have the additional equation 
 
 or since by the reasoning above given -3- is 
 
 . dU d<t> 
 equivalent to 3 -- , we get
 
 32 Law of permanent distribution in 
 
 Now ^ *+ q = A) and 2 ^ f= (*+) . 
 dq* ^dt } dq* \dt' 
 
 So that we get the equation 
 
 '= o, 
 
 the equivalent to v + v'= o of the last Article. 
 
 Whence it follows, by reasoning- in all respects the same 
 as that of the last Article, with the substitution of m and" n 
 for 6 and 6 respectively, that the functional determinant 
 expressing the ratio of the multiple differentials 
 
 dpi dp m+ and dp\ ... dp' m+n - 
 
 is numerically equal to unity, and since <j> measures the 
 rapidity of entry into the colliding state it will further 
 follow that a law of distribution for each set of molecules 
 thus generalised of the form 
 
 for the number of that set with momenta and coordinates 
 between p l and jt? 1 + dp l . . ,q m and q m + dq m will be permanent 
 in the absence of collisions, or in spite of collisions 
 between molecules (two at a time), either of the same or 
 different sets. 
 
 The expression of the law may also be conveniently 
 modified as in the last Article, for since T is an essentially 
 positive quadratic function of the p's with coefficients 
 functions of the #'s, it may, by suitable linear transforma- 
 tions of the p's, be expressed in the form 
 
 so that the law of distribution assumes the form 
 
 r, ... dr m dq,...
 
 medium of any constituents. Fixed centre forces. 33 
 
 And then integrating for all values of the ^'s, the law in 
 question becomes that the number of molecules of this set 
 for which the rs lie between r and r l + dr r ..r m , and r m + dr m 
 
 s 
 where h is constant for all sets and 
 
 is equal to the total number of molecules of this particular 
 set* 
 
 11.] Let there be any number of sets of perfectly smooth 
 and perfectly elastic molecules, of any form and any number 
 of degrees of freedom, in motion in a region inclosed within 
 a perfectly elastic boundary, as in the last Article, let also 
 the inclosed region be a field of forces towards fixed centres 
 acting on the molecules, then the state of the medium will 
 be permanent in the absence of collisions, and will be 
 unaffected by collisions between any two molecules, whether 
 of the same or different sets, provided the law of distribution 
 be such that the number of molecules in any set with 
 momenta components intermediate between p^ and p l + dp^ 
 Pm an( i Pm+dpm> an( l coordinates intermediate between 
 jj and q l + dfa ... q m and g M + dy m be 
 Ae~ hE d Pl ...dq m , 
 
 * In the treatment in the text the impulsive collision forces have been 
 regarded as ordinary force function forces which become infinite in the 
 colliding configuration. We may also regard the collision under the aspect 
 of an impact whose generalised components are J x , / 2 , &c., where Ii , I t , 
 &c. are subject to the condition 
 
 dd> , d<t> , d<t> , 
 
 whenever -7 dq\ + . do, + - dq n = o. 
 
 dq L H dq t dq n 
 
 leading to the same result, or
 
 34 Law of permanent distribution in 
 
 m being the number of degrees of freedom of the molecule, 
 E its total energy potential and kinetic, h a constant the 
 same for all sets of molecules, and A a constant for each 
 particular set determined by the condition that 
 
 is equal to the number of the molecules in the set con- 
 sidered, the integrations being taken over all possible values 
 of the p's and q's. 
 
 For, in the first place, it is clear that the law is permanent 
 in the absence of collisions, inasmuch as the values of E and 
 the multiple differential dp l . . . dq m are separately constant 
 in this case by the law of conservation of mechanical 
 energy, and by Art. 8. 
 
 In the next place, such permanence will be unaffected by 
 collisions between any two of the molecules, for the reasoning 
 of the last Article is in no respect affected by the substitu- 
 tion of E for T ; whence the proposition is proved. 
 
 In the previous Articles the position of a molecule did not 
 enter into consideration but only the values of the momenta 
 components, so that the condition of the molecule was fully 
 determined by the limits j 1} Pi + d/^, &c. of these com- 
 ponents, and these limits for each molecule are clearly inde- 
 pendent of those of all the rest. 
 
 It is otherwise however with the coordinates of position 
 of the centre of inertia of a molecule, for if the molecular 
 volumes be appreciable, these coordinates for any molecule 
 will not be independent of those of the rest, unless the sum 
 of the molecular volumes in a unit of volume be a very 
 small fraction. Since however the disregard of all but 
 binary collisions involves the same condition the reasoning 
 remains the same as before. 
 
 Again, the condition of interchange of the molecules
 
 medium of any constituents. Fixed centre forces. 35 
 
 after collision cannot of course be exactly satisfied in this 
 case, and the reasoning- would fail unless the size of the 
 molecules were so small that the potential was sensibly 
 unaltered for a distance comparable with the linear dimen- 
 sions of each molecule, or, at least, unless the variation of 
 the quantity h \ was inappreciable for such distances. 
 
 In all the cases, however, contemplated in this treatise 
 the Kinetic Energy largely preponderates over the poten- 
 tial, or ^Sx will be absolutely insensible for such variation 
 of x in comparison with h (x + T). 
 
 In dealing with the so-called rigid elastic molecule with 
 6 degrees of freedom as a maximum we regard it under the 
 ordinary conventional aspect of a body whose parts are 
 incapable of relative displacement, so that there is no gain 
 or loss of potential energy of mutual forces between these 
 parts, or, as they may be called, interatomic forces ; but in 
 passing to the more general conception of a molecule such 
 interatomic forces, with the corresponding variation of 
 interatomic potential, must be taken into account.* Their 
 presence will not affect the reasoning in the text because 
 for any single molecule Ae~ hE will be constant in the inter- 
 vals between collisions where E is the total Energy of the 
 molecule, and the differential product 
 
 <?Pl...<fy 
 
 will also be constant by Art. 8, while during collision all 
 the coordinates, and therefore also the interatomic potential 
 energy, are unchanged. It is to be remembered also that 
 no account is taken of other than binary collisions, in other 
 words, it is assumed that the chance of three or more mole- 
 cules being in collision simultaneously is infinitely less than 
 
 * For instance, our molecules might be conceived as made up of discrete 
 material particles acted on by mutual forces, or as bodies with a certain 
 limited number of degrees of freedom between the parts of which there are 
 still mutual forces but with a limited number of generalised components of 
 force. 
 
 D 2
 
 36 Law of permanent distribution 
 
 that of binary encounters, necessitating the postulate tacitly 
 made in all cases that the sum total of the volume of the 
 molecules in any region is very much less than the volume 
 of that region. 
 
 12.] Hitherto we have viewed the collisions between our 
 elastic molecules under the ordinary conventional aspect, in 
 which there is an infinitely large force acting for a time 
 absolutely insensible, and therefore accompanied by no 
 change of the coordinates. The truer view, as already pointed 
 out, would be to consider the force called into play by a 
 collision as very large but not absolutely infinite, and the 
 time of such collision as very small although not absolutely 
 evanescent, and such that changes of value of the coor- 
 dinates during collision are not absolutely excluded. The 
 collision viewed under such an aspect we call an encounter ; 
 if the number and size of the molecules be sufficiently 
 restricted it may still be true that the chance of a binary 
 encounter is infinitely larger than that of a multiple 
 encounter, and under such conditions all but binary 
 encounters may be neglected. We proceed now to show that 
 the e~ fc <x+ r ) law holds for encounters as well as for collisions. 
 
 If a material system in a field of conservative forces 
 having the n generalised coordinates q l ...q n and the cor- 
 responding momentum components p l ...f n move from the 
 state q l} Qi + dq^ .../ Pn+dp to the state 
 
 where the states are connected by the relation ^ n c[ n = C 
 (any constant), then the functional determinant . 
 
 -! <,-!
 
 when collisions replaced by encounters. 37 
 
 is numerically equal to * , where the dot indicates time 
 
 
 variation, or which is the same thing, 
 
 dp'i . . . d^H-i == ^r &P\ - fyn-i numerically. 
 
 For we know by Art. 8 that if the two states dashed 
 and undashed be connected by the relation t is constant, 
 where t is the interval of time taken to pass from one to 
 the other, 
 
 dp l ...dq n dp\ ... dq' n numerically. 
 
 From the equations of motion we can obtain two equa- 
 tions of the form 
 
 9* =/(/i Y> *). f. = *(?i ' 2> <). 
 where f and F are determinate functions. 
 
 Interchange the variables q n and t in the multiple 
 differential d Pl ...dg n 
 
 and it becomes 
 
 where jo\...q' n are constant in obtaining -^ 
 But -j- so obtained is equal to q n . 
 
 Cvv 
 
 And therefore 
 
 dpi . . . dq n = dpi . . . ^.j q n At. 
 
 Similarly, expressing q' n in terms of t, we get 
 
 HTf 
 
 dp\ ... ^' n = dp\ ... <//_! ^- , 
 
 dF 
 
 where jOj ... q n are constant in obtaining -=7, and therefore 
 
 <*c 
 
 303160
 
 38 Law of permanent distribution 
 
 therefore dp\ ... dq' n = dp\ ... dq' n _^ q' n dt numerically, 
 therefore 
 
 dpi . . . dq n _i q n dt = dj)\ . . .dq'^ q" n dt numerically, 
 
 because the time from jt ltt .q n to p\...q' n is the same 
 as that from jp\... q' n to J5 X . . . q n with reversed velocities. 
 Therefore 
 
 4ft... %-i 4* = d /i - <-i 2 numerically, 
 
 or 
 
 , a n . .. . 
 
 ^- numerically.* 
 _i q n 
 
 * This proposition has been proved by Boltzmann by the evaluation of 
 the functional determinant 
 
 '" dq n 
 
 dp,. '" dq n 
 
 and in a similar way by Burbury. 
 
 An interesting illustration is afforded by a trajectory traversing a closed 
 curve or surface. 
 
 If one of the two coordinates q^ , q t in the case of a plane trajectory, as 
 q t , be the distance of the point from the curve, then it will follow that 
 
 dpi dp 3 dq l _ fa ^ 
 
 For example, if the curve be a circle and the trajectory be referred to 
 polar coordinates r, 0, the centre of the circle being the pole, 
 
 d0 _dr 
 
 Then the proposition states that when r = r f , the determinant 
 
 dp, 
 
 dp^ 
 dp t 
 
 = -~ numerically. 
 
 dpi dp, d 
 
 A result which may be easily verified for a projectile.
 
 when collisions replaced by encounters. 39 
 
 In obtaining the ratio of the multiple differentials we 
 suppose p\... / to be found in terms of p . . . q n and t 
 from the 2n equations of motion 
 
 P\ = f\ (Pi ". ?> ." ?'n = fzn(P\ > 
 
 and then introduce the equation / = q n +C, regarding 
 both q n and C as constants, thereby lowering the number 
 of independent variables from a to aw I, and really 
 evaluating the functional determinant 
 
 dp\ 
 
 dp\ 
 
 13.] So far we have proved that when sets of molecules 
 are in motion in a region, the molecules in each set having 
 any number of degrees of freedom, and all being acted 
 on by forces, fixed-central or interatomic, then the law of 
 distribution 
 
 for any set of molecules, will be unaffected by collisions 
 regarded as of no sensible duration. 
 
 We may now prove that this permanence will be un- 
 affected by encounters between molecules, two at a time, 
 whether of the same or of different sets. 
 
 For if the coordinates of one of two molecules with m 
 degrees of freedom be denoted by q l . . . q m , and those of the 
 other with n degrees of freedom be denoted by 
 
 then the condition of an encounter between the two mole-
 
 40 Law of permanent distribution 
 
 cules may be regarded as defined by an equation among the 
 coordinates of the form 
 
 Let $ be taken for one of the coordinates, say $' JR+H of the 
 two molecules, then by reasoning exactly as in the previous 
 propositions, it will follow that if, in any encounter between 
 such a pair of molecules, jo, q pass to jy', q', the number of 
 pairs of molecules passing from the limits p and 
 q and q + dq to p' and p' + dp', q f and q' + dq' will be 
 
 in any short time dt, and also that the number passing 
 from the limits p' and (p' + djf), q' and (q' + dq') to 
 p and p + dp, q and q + dq in the same time, will be 
 
 *-*<*+*> dj i ... <*/+_! q' m+n dt. 
 
 And these numbers are equal by the last Article, whence the 
 proposition is proved. 
 
 Hence we conclude from this and the previous Articles 
 that with the most general conceivable construction of our 
 sets of molecules, subject at least to restrictions already 
 noticed as to number and size, the law of distribution, 
 
 for each set of molecules, will, when once attained, be a 
 permanent law. * 
 
 The total energy E is of the form x + T> where x is the 
 potential energy of the molecule in the force field, inclusive 
 of interatomic forces, at any instant, with the values which 
 the coordinates of that molecule have at that instant, and 
 is therefore a known function of the q's of that molecule, 
 T is the kinetic energy of the molecule, and therefore is 
 expressible in the form
 
 necessary as well as sufficient. 41 
 
 'M being- the mass, , v, w the component velocities of the 
 centre of inertia of the molecule, and r 4 . , . r m linear 
 functions of the jo's with coefficients functions of the 
 y's. Hence by simple transformations the number of mole- 
 cules of the set under consideration, with variables lying 
 between the limits 
 
 may be written in the form 
 
 and integrating for all values of the #'s, the total number 
 of molecules, of the set in question, with the u...r m 
 variables lying between the limits u, and u + du, ..., /,, 
 and r m + dr m becomes 
 
 -*<?+...+,,.) 
 Ae * du...dr m) 
 
 where k is a constant, the same for all the sets of molecules, 
 and A is a constant, differing for different sets and satisfy- 
 ing, for each set of N molecules, the equation 
 
 From this it follows that the average value of each of 
 
 x . Mu 2 Mv 2 Hr* m .1 
 
 the magnitudes > > ..., - -= is -7-, and that 
 
 22, 2, 2k 
 
 the average value of the total kinetic energy of each 
 
 molecule is -= 
 ih 
 
 14.] In the preceding Articles we have proved that the e~ hE 
 law of distribution is, subject to certain stated limitations, 
 sufficient for the permanence of distribution, we now pro- 
 ceed to prove, by means .of a proposition due to Boltzmann, 
 that this law is not only sufficient but also necessary.
 
 42 Law of permanent distribution 
 
 Let there be any number of sets of molecules circum- 
 stanced as in the preceding Articles, and let the distribution 
 of the coordinates and momenta of any one of the sets, with 
 m degrees of freedom, be such that the number of the 
 molecules of that set, with momenta and coordinates inter- 
 mediate between PJ and P l + dP 1 ... Q m and Q m + dQ m) is 
 F(P 1 ...Q m )dP 1 ...dQ m . 
 
 Also let the corresponding distribution for any other set 
 with (n] degrees of freedom be expressed by 
 
 F and f being any determinate functions. Let also any P' 
 and Q', p' and q' be connected with the corresponding 
 P and Q, p and q by the relation that P, Q, p, q are 
 changed into P', Q', p', q' respectively by an encounter 
 between the two molecules, and for brevity let F(P l . . . Q m ) 
 and F(P f 1 ... Q' m ) be denoted by F, F', with corresponding 
 meanings for / and /'. Then there exists a function H such 
 
 that the time variation -r- is always negative, unless the 
 condition 
 
 Ff= F'f 
 
 be satisfied for every combination, two and two, of the sets 
 of molecules, including pairs of molecules both of the same 
 set, that is, including the case 
 F(P 1 ...Q m ) F( Pl ... q m ) = F(P' 1 ... Q' m ) F(p\ ... /.). 
 
 In the first place, let us regard the medium as consisting 
 of only two sets of molecules with m and n degrees of free- 
 dom respectively, and with the distribution laws F and f, 
 then the number of encounters per unit of time between 
 pairs of molecules, one from each set, with momenta and 
 coordinates between P and P + dP t Q and Q + dQ, p and 
 p + dj), q and q + dq will be
 
 necessary as well as sufficient. 43 
 
 provided the coordinate q n be chosen so that q n = o is a 
 condition of the beginning or end of an encounter, and q n 
 be made equal to o wherever it occurs in f(j) 1 ... #). 
 And therefore the expression 
 
 is the number of pairs of molecules, one from each of these 
 sets passing from the state P, P + dP ... q, q + dq to the 
 state P f t I* + dP / . . . q\ q' + dq f per unit of time, when q n 
 is put equal to o in f. 
 
 Similarly the number of pairs passing from the state 
 P / ) P' + dP f ...q f , q' + dq* to the state P,P + dP...q, 
 per unit of time, will be 
 
 when q' n = o. 
 
 But in such a case, as proved in Art. 12, we have 
 
 So that the total diminution of the number of the m set 
 of molecules in the P, P + dP... Q, Q + dQ state, per unit 
 time, arising from collisions with the n set, will be 
 
 dP, ... dQ m fff...(Ff-F'f) d fl ... dq n _, q n , 
 where q n = q' n = o, that is to say, 
 
 = dP, ... dQ m fr... (F'f'-Ff) d Pl ... dq n _, q n , 
 
 "If"' ( F '<f- F f) 10 F - dp *'~ ^"-1 **> 
 
 q n and q' n being each equal to zero.
 
 Law of permanent distribution 
 By symmetry 
 
 ...f log/%... <^ 
 
 =jfjf . (F'f'-Ff) logfdP 1 ... %_, ,, 
 
 f 
 
 where ?tt = / tt = o. 
 Similarly 
 
 is equal to 
 
 jfjf ... (J 
 
 But since the integral ff^j log F dP 1 ... dQ 
 
 extends over all values of each P and Q, including each P 
 and Q', ^ follows that 
 
 and similarly
 
 necessary as well as sufficient. 45 
 
 = //... (F'f'-Ff] log ^dP,... d qn _, 2> 
 
 Ff 
 
 where the argument (F'fFf) log -~p is necessarily 
 
 negative if not zero. '* 
 
 Now let H be taken to represent 
 
 f 
 
 eJ FT 
 
 and is therefore essentially negative so far as -^r depends 
 
 upon encounters between molecules of the z and # sets, 
 unless the quantities Ff and -Z' 7 ',/' are equal to each other, 
 
 k-J. ^ 
 
 in which case -77- is zero, 
 ol 
 
 If we had taken If equal to 
 ff...F(P,Q){logF(P i Q)-i}dP l ...dQ m 
 
 , q) {logF( P)S )-i} d Pl ... dq m) 
 
 it would have followed by the same reasoning that -77 
 was negative unless 
 
 F(P, <2) F(p, q) =. F(P f ) Q') F(p', #'), 
 
 JJT 
 
 and therefore that -77- must have been negative unless 
 
 had been equal to
 
 46 Law of permanent distribution 
 
 so far as encounters between pairs of molecules each of the 
 m set were concerned, and similarly for the pairs each of 
 the n set. 
 
 And therefore generally, when there are any number of 
 sets of molecules, if the function H be written 
 
 2\ff...F(\ogF-i)dP l ...dQ 
 
 for all the forms of F and f in the different sets, the time 
 variation -^ will be necessarily negative unless the con- 
 
 09 
 
 dition 
 
 ). - (A) 
 
 for every binary encounter combination be satisfied ; that is to 
 
 7 TT 
 
 say, the quantity -j- tends to a minimum unless every one 
 
 of these conditions is separately satisfied whenever en- 
 counters take place in the medium, or equation (A) repre- 
 sents a necessary condition of permanence when encounters 
 take place. 
 
 In the absence of encounters each of the T^s must 
 
 /JJf 
 separately satisfy the condition -5- = o, because when there 
 
 are no encounters 
 
 must be independent of the time for each molecule 
 separately. 
 
 But the differential product 
 
 is known to be independent of the time, and therefore "F
 
 necessary as well as sufficient. 47 
 
 must be so. Hence, for permanence we must have for the 
 determination of each form of F the two conditions 
 
 ..) ...... (I) 
 
 independent of the time when no encounters ; 
 
 F(P l ... Q m )f( Pl ... ?) = F(P\ ... Q' m }f( P \ ... ? ' n )(2) 
 
 when there are encounters. 
 
 Now the only form of F which can be conceived as 
 satisfying the first condition is F(E), where E is the total 
 energy of the molecule, so that the second condition is 
 reduced to 
 
 where E PtQ + E p>q = 
 or F=Ae- hE p.Q, 
 
 Whence it follows that the e~ hE law is both sufficient 
 and necessary for permanence, provided the system be such 
 that every conceivable arrangement is also an attainable 
 arrangement. 
 
 If any restraint be introduced, whereby the attainment 
 of any of the conceivable arrangements is rendered impos- 
 sible, the reasoning of course fails, if the e~ hE arrangement 
 should be included in those thus rendered impossible of 
 attainment, otherwise the e~ HE law must still prevail 
 
 15.] From the preceding Article we learn that in the case 
 of a medium composed of sets of molecules such as we have 
 been considering, whatever be the assumed laws of distri- 
 bution, among the different sets, at any instant, such laws 
 cannot be permanent, unless certain conditions hold amongst 
 them, and that, unless these conditions be satisfied, a certain 
 function H, dependent upon these assumed laws of distri- 
 bution may be found, which is always, under the action
 
 48 Rate of return to permanent distribution 
 
 of encounters, tending to a minimum, and this minimum 
 being attained there is no further tendency to change, 
 or the state is permanent. 
 
 The time variation of ff, for any assumed laws of distri- 
 bution, consists, as we have seen, in an integral or sum of 
 a number of terms of the forms 
 
 where F and f depend upon the assumed laws of distri- 
 bution for any given pair of sets with, say, m and n degrees 
 of freedom each, q n is what may be called an encounter 
 function for two molecules of that pair, q n is the time varia- 
 tion of , and F' and f are the values of F and f after 
 an encounter. 
 
 j -fT 
 
 In thus determining the time variation -^- we have 
 
 limited the investigation to the changes in H from en- 
 counters only, because we are considering the question of 
 permanence with distribution laws F, f, &c., and it is clear, 
 a priori, that only such distribution laws as make each .Fand 
 f separately independent of the time need be taken into 
 account, for a distribution law which did not give 
 
 constant would be clearly inadmissible, and we know that 
 dP^ ...dQ m is constant, therefore Fmust be so. 
 
 Since the ZT-function, as thus found for any assumed 
 distribution laws, tends under the action of encounters to 
 diminish with the time at a rate which may be determined 
 as above shown, until the distribution laws have arrived 
 at the permanent state, we may find the rate at which the 
 medium starting from any assumed distribution laws, differ- 
 ing from those of the permanent state, tends to approach 
 that state.
 
 after disturbance. 49 
 
 As a simple illustration, let us consider a medium like 
 that of Art. 6, consisting 1 of a number of circular disks 
 equal to each other in all respects, with the centre of 
 inertia of each at the distance (c] from the centre of figure, 
 c being small in comparison with the radius of a disk ; the 
 notation employed will be the same as that of Art. 6, 
 except that the masses and radii of gyration of all the disks 
 will now be denoted by the same letters. 
 
 In this case we know that the law of distribution in the 
 permanent state is 
 
 , , 
 
 dudvdoo, 
 
 and we will denote this by F (u,v,<a) dudvdv or F dudvdio, 
 and suppose that at any instant the distribution law is 
 Fdudvdo), where 
 
 -^(M'+f'+M* 2 ." 3 ) 
 
 1 = Ae * 
 
 Let H and H be the JET-functions calculated for the 
 respective laws, then we know that 
 
 where K is some positive quantity which may be called the 
 
 J T7~ 
 
 disturbance, and our object is to find K and -5- 
 
 Since the total number N of the disks and the total 
 kinetic energy of the medium must be constant, we have 
 the following conditions : 
 
 rrr -^V-,.^*^) 
 A II IB * dudvdu 
 
 W V* tfttft
 
 50 Rate of return to permanent distribution 
 
 = NT, (a) 
 
 where T = (u z + v 2 + F a, 2 ), 
 
 and the bar denotes mean value. 
 From these equations we get 
 
 w '- 
 
 Also 
 H = /7YV(log F- 1) dudvda =fffflog Fdudvd^ - N 
 
 H Q 
 
 Therefore 
 
 K =ffff log Fdudvdo -[[[?<> log F dudvdu. 
 
 But F=Ae 2 ' = Ae~ M \ suppose. 
 
 .-. F log F=
 
 after disturbance. 51 
 
 / / / F log F du dv du = N log A Nh Q T. 
 
 Therefore 
 
 N log , because from integration it follows that 
 
 = N<- -~- + higher powers of (/* i) > 
 
 7 Jf" 
 
 In finding -7- let us, in the first place, confine our atten- 
 dt 
 
 7 TT 
 
 tion to the part of -jj arising from collisions of disks 
 
 whose lines of centres at the instant of collision are parallel 
 to the axis of x. 
 
 If s be the diameter of any disk, U and u, V and v, 
 1 and o>, the respective component translational and rota- 
 tional velocities at the instant of collision, the number 
 of the required kinds of collisions per unit of time is 
 
 JT7 
 
 (U ->* &) 2* (see Art. 5), and therefore the part of -5- 
 arising from such collisions is equal to 
 
 I (f...(Ff-F'f) 
 
 where the integrations with regard to U and u are to be 
 taken from u = o to u U, and from U = o to Z7 = oo.. 
 
 E 2,
 
 52 Rate of return to permanent distribution 
 
 TIT- 
 
 Also, since in the general expression for -=- we are con- 
 
 d/t 
 
 sidering all the pairs of disks of which the coordinates of 
 one relative to the other are s and any value of 6 from 
 
 - to + - , and in this restricted case only those pairs 
 
 for which these relative coordinates are s and any value of 
 from o to dd, it is clear that we must replace N z in this 
 
 case by N 2 , i.e. we must replace A 2 by 
 
 Also 
 
 writing T and t for the total kinetic energies of the disks 
 before collision, and T' and t' for the similar quantities after 
 
 collision ; 
 
 p 
 where ii r = li --(^7- -/) + Pli), 
 
 and P and p are perpendiculars from the Centres of Inertia 
 of the respective disks upon their line of centres, and there- 
 fore are, at least, of the same order of smallness as (<?). 
 If then c be small we have to a first approximation 
 
 a '= - W (U ~ U} > "' = " + -& (U -*l 
 and therefore, remembering that T + t = T' + if\ and
 
 neglecting c 2 , FfF'f is, to the same approximation, 
 equal to 
 
 after disturbance. 53 
 
 F'f is, to the same approximati 
 
 ^ 2 ^ -h T+t+i^l (a*+j) } r _ ^- ^* Ox-i)(pn- P )(tf-Oi 
 
 -A { r+f+,^7 (H2+" 2 ) } M , N / -n^ 
 ^ 2 e ' (fM-i}(Pa-p^(U- U ). 
 
 Also log (J7 / -^/) = -^( fl -i)(P Q-p) (U-u). 
 
 /7 3f 
 
 Therefore to the same approximation -?- is equal to 
 
 i.e. to 
 
 where P 2 is the mean value of P 2 or j) 2 , and therefore is of 
 the same order of smallness as c 2 . 
 
 The result of the integration is clearly of the form 
 L c2 
 
 where L is a numerical quantity whose exact value is of no 
 importance to the present investigation. 
 
 7 T7" 
 
 Therefore the value of this portion of ~ , remembering 
 that A 2 is here equal to 
 
 (10
 
 54 Rate of return to permanent distribution 
 becomes 
 
 7 irr 
 
 and the total value of -TT becomes 
 at, 
 
 C being a known numerical quantity. 
 
 Since K = N j ~ 6 + higher powers of (/* - 1)| , 
 
 we have, if /u I be small, 
 
 _ 6CT* c 2 
 
 W "^"F*- 
 
 or K=K Q e~ a ^'\ 
 
 6CNs Ns 
 
 where a = -= , i. e. a ex = 
 A ' V/& 
 
 That is to say, an initial distribution of this nature, in 
 
 MX? 
 which the mean value of -- &> 2 differs from the mean 
 
 ,Mu 2 Mv 2 ... 2 . A ,. 
 
 values or - or - , will reduce to the permanent dis- 
 
 tribution at a proportional rate -= 
 
 If the original distribution had been represented by 
 either of the laws 
 
 . _(r + j^*?) -,(r +M ^) 
 
 Ae ^ * ' or Ae ^ 2 /, 
 
 that is to say, if it had been a distribution in which the 
 
 mean values of u 2 and v z had been different from 
 
 a 2
 
 after disturbance. 55 
 
 each other, the value of K would have been the same as 
 
 j -rr 
 
 before, and the value of -=- would also have been the same 
 
 c z 
 
 with the omission of the fraction -.-^ 
 
 k* 
 
 Such a distribution then would, if \i I were small, be 
 represented by a disturbance function which, as in the last 
 case, would have diminished proportionally at the rate a 
 
 per unit of time, and therefore with the diminution of -r 
 
 the rate of approach to the state of permanence might con- 
 ceivably be infinitely greater with an initial small inequality 
 between the mean values of the translation component 
 portions of the kinetic energy than it is with an inequality 
 between either of these and the rotation component. 
 And whether the initial disturbance function be great or 
 small we always have its rate of change for a difference in 
 
 the mean values of u 2 and v 2 greater than that for 
 
 2 2, 
 
 a difference in the mean values of M , and either of the 
 
 2 
 
 c 2 
 former in a ratio comparable with the ratio -^ 
 
 Mr. Burbury and Professor Tait have, in the Transactions 
 of the Royal Societies of London and Edinburgh, given 
 similar investigations of the rate of subsidence of disturb- 
 ance in the case of a medium of two sets of elastic spheres, 
 the masses and number of spheres per unit volume of the 
 two sets being M, m and N } n respectively, and the dis- 
 turbance consisting in an initial small difference between 
 the h constants in the two sets. They have arrived by 
 independent treatment at the result 
 
 VA
 
 56 Pressure at any point in 
 
 for the proportional rate of subsidence, where j is the 
 
 Arithmetic mean of the initial values of the -,- constants. 
 
 h 
 
 16.] A number of sets of molecules are moving in a 
 bounded region, taken as unit of volume, as in Art. 13, the 
 masses and number of degrees of freedom of each molecule 
 of any set being denoted by M and m and the number of 
 molecules in the set by N, with suffixes attached to these 
 letters to distinguish different sets ; it is required to deter- 
 mine the action of one portion of the medium upon another, 
 or the pressure per unit of surface at any point. 
 
 Since the distributions of the molecules of different sets 
 are independent of each other, we will for the present confine 
 our attention to the set denoted by M, m, N t without suffixes. 
 
 Suppose that there are dN molecules of the set con- 
 sidered, whose component velocities of translation parallel 
 to the axis of x are between u and u + du. 
 
 The number of these molecules which cross the elementary 
 area dydz in time dt will be the same as the number of 
 the dN molecules whose centres of inertia are situated 
 within the elementary parallelepiped dxdydz in which dx 
 is equal to udt, and this number is 
 
 dN.u.dydzdt. 
 
 Each of these molecules carries across with it a mo- 
 mentum parallel to the axis of x equal to Mu ; the total 
 momentum parallel to the axis of x transferred across dydz 
 in the time dt is therefore 
 
 MdN**dydzdt. 
 
 If u be positive this is positive momentum transferred 
 from the negative to the positive side of the plane yz, and if u 
 be negative this is negative momentum similarly transferred 
 from the positive to the negative side of the same plane.
 
 medium of moving molecules. 57 
 
 In either case the result is that by the mere motion of 
 these molecules across the area dydz the positive momentum 
 parallel to the axis of x is diminished by the quantity 
 MdNu*dydz(tt on the negative side of the plane yz, and 
 increased by the same quantity on the positive side of that 
 plane in the time dt. 
 
 Hence the result is the same as a transference of positive 
 momentum parallel to the axis of x in time dt across the 
 
 00 
 
 area dydz equal to MdydzdfS u 2 dN, that is equal to 
 
 GO 
 
 dydzdtMNu 2 or to dydzdtpv?; ... (a) 
 
 where p is the density of the N set matter at the point 
 x, y, z, and 2 is the mean square of the velocities parallel 
 to the axis of x. 
 
 But u 2 is equal to , where v 2 is the mean square of the 
 velocities of the N set of molecules, and is equal, as we have 
 seen, to A. 
 
 Hence there is a transference of positive momentum from 
 the negative to the positive side of the plane yz across the 
 area dydz in the time dt equal to 
 
 7 7 7 ,i; 2 pdydzdt 
 
 pdydzdt-, or to * y m 
 
 Each separate molecule (whose component velocities of 
 translation are u, v, and w] carries across the same plane 
 momenta parallel to the axis of y and z equal to Mv and 
 Mw respectively, so that in the time dt there are carried 
 across the elementary area dydz momenta parallel to the 
 axes of y and z equal to 'SMuvdydzdt and ^Muwdydzdt 
 respectively.
 
 58 Pressure at any point in 
 
 It is clear from the symmetrical distribution of the 
 velocities that 2 Muv and 2 Muw are each equal to zero. 
 
 Therefore the resultant mutual action of the two portions 
 of the medium across the elementary area dydz in the time 
 
 _~2 
 
 dt is the transference of the momentum pdydzdt parallel 
 
 to the axis of x across this element from the negative to 
 the positive side. 
 
 If this mutual action, or as it is generally called this 
 pressure, when referred to unit of surface be denoted by the 
 symbol jo, we get the equation 
 
 pdydzdt = pdydzdt ; 
 
 3 
 
 or 
 
 Since the momenta parallel to y and z remain unaltered, 
 it follows that the mutual action or pressure between por- 
 tions of the medium separated by any plane is entirely 
 normal to that plane. 
 
 v 2 
 Since also the expression for p, or p , is independent 
 
 of the direction of the axis of x, it follows that the pressure 
 at any point of the medium is the same in all directions. 
 
 If we suppose the contiguous portions of the medium to 
 be separated by a material instead of an ideal plane, it will 
 clearly be necessary for the maintenance of the equilibrium 
 that there should be an action between this plane and the 
 adjacent portion of the medium exactly equivalent to the 
 transference of momenta estimated above. Hence the 
 pressure or force between the plane and medium is normal 
 to the plane, and its value per unit of time and surface is 
 
 r at any point. And this value remains unaffected by
 
 medium of moving molecules. 59 
 
 turning the plane of separation in any direction about the 
 point.* 
 
 When several different sets of molecules are present 
 tog-ether in the region under consideration, if p lt p 2 , &c. be 
 the densities of the matter of the different sets in the 
 neighbourhood of the point x, y, z, and if j^, j 2 , &c. be 
 the pressures at that point, defined as above, of the media 
 composed of these different sets, and if M lt M 2 , &c. be the 
 masses of the individual molecules of each of the sets, and 
 p the total pressure, we shall have 
 
 17.] We may prove that the value of p as determined 
 in the last Article satisfies the necessary condition of 
 equilibrium of the medium. 
 
 For consider the elementary parallelepiped datdydz situated 
 in the neighbourhood of the point x, y, z. 
 
 The positive momentum of the N set of molecule matter 
 within this element parallel to x is, as we have seen, 
 increased by the quantity pdydzdt in the time dt by trans- 
 ference of matter across the face dydz nearer to the plane 
 of yz> or, as we may call it, the left-hand face, and the x 
 momentum is in the same time diminished by the quantity 
 
 (p + j- dx\ dydzdt by transference across the right-hand 
 
 * It is important to distinguish between the velocity of agitation of the 
 molecules treated of in the preceding reasoning and that which we are 
 accustomed to consider as the velocity of the medium itself. This latter 
 velocity has been denned by Professor Clerk Maxwell as follows. If we 
 determine the motion of the centre of gravity of all the molecules within a 
 very small region surrounding a point in a medium, then the velocity of the 
 medium within that region is defined as the velocity of the centre of gravity 
 of all the molecules within that region. Should such a velocity exist in 
 the medium under consideration, we must suppose that our ideal plane of 
 separation moves with the same velocity, and therefore that the number of 
 molecules crossing it in any direction is on the average equal to the number 
 crossing it in the exactly opposite direction.
 
 60 Pressure at any point in 
 
 face. On the whole, therefore, the positive x momentum 
 within the element is diminished by the quantity 
 
 ^r dydzdxdt in the time dt. 
 
 It is necessary therefore for the permanent state of the 
 medium that the impressed force on the N molecule matter 
 within the element should produce a resultant momentum, 
 equal to the last found quantity, in the time dt, that is to 
 say, we should have 
 
 dp . 
 
 where X is the mean value of the x force on all the mole- 
 cules whose centres of inertia lie within dxdydz. 
 
 Now the number of the N set molecules whose m mo- 
 menta and m coordinates lie between 
 
 fi and p l + dp 1 ... p m and 
 q l and q^ + dq l ... q m and 
 is of the form 
 
 Let the coordinates be the x, y, z of the centre of inertia 
 and q ... q m , then the whole number whose centre of inertia 
 lie within the parallelepiped dxdydz is 
 
 A dxdydz fj. . . e~ h <*+ T > dp l . . . dp m d^... dq m , 
 
 and therefore the density p at the point x, y, z is of the 
 form 
 
 and 
 
 Tx
 
 medium of moving molecules. 61 
 
 Since T is equal to 
 
 and is therefore independent of x, y and z t 
 
 where X is the mean value of the x force on the molecules 
 within dxdydz for all values of the momenta and relative 
 coordinates of these molecules. 
 
 Therefore 
 
 = = i 
 
 18.] In the determination of ^ in Article 17 we have 
 only considered the transference of momentum which would 
 have taken place across the plane of separation owing to 
 the motion of separate molecules ; that is to say, supposing 
 one portion of the medium to be separated from the other 
 by a material boundary, the value of p already found is that 
 arising from the bombardment of the boundary by the 
 separately impinging molecules and assumes that the time 
 any one molecule is in collision with any other during 
 any measurable interval is so small as to be negligible; 
 when this condition is not satisfied, owing either to mole- 
 cular density or the minimum irreducible volume of 
 separate molecules, a correction is required ; to find this 
 we must prove the following propositions due to Clausius. 
 
 When any number of material particles are in motion 
 within a limited space so that the particles do not move 
 continually further and further from their original positions, 
 and provided also that the velocities of each particle do not 
 continually increase or decrease, such a system of material 
 particles is said to be in stationary motion*
 
 62 Properties of 
 
 If X, Y, Z be the component forces upon any one of a 
 system of particles in stationary motion, and x, y, and z be 
 the coordinates of such particle referred to any origin and 
 axes, and the quantity Xx+Yy + Zz be found for such 
 particle, then the mean value of the expression % 
 
 during any period of stationary motion of the system, where 
 2 represents summation for all the particles, is called the 
 Virial of the system. 
 
 When a system of material particles is in stationary 
 motion the mean kinetic energy of the system is equal to 
 its virial. 
 
 If x be any function of t, 
 
 d 2 , . d , dx\ fdx^- cPx 
 
 Let x be the coordinate parallel to the axis of x at the 
 time t of that one of the material particles whose mass is 
 M, and let X be the component force on that particle 
 parallel to the same axis, then 
 
 Hence equation (a) gives us 
 
 M ( dx_^_ Xx M d* 
 
 " ~ + ' 2( '' 
 
 M 
 
 where I ' I denotes the initial value of 7 , 
 L dt J u dt
 
 the Virial 63 
 
 Now j i (-17) dt and - \ Xxdt are the mean values 
 
 of (-T-) and Xx during the time t. 
 
 If the motion be strictly periodic and t be taken equal to 
 the length of a period or any multiple whatever of that 
 length, then the last term on the right-hand side becomes 
 zero ; and if the motion be not strictly periodic, still 
 
 from its stationary character as above defined, although 
 
 -*T 
 the coefficient of in the last term does not necessarily 
 
 become zero for all values of t however large, yet its value 
 .cannot increase indefinitely, but can only fluctuate within 
 certain limits, and therefore by sufficiently increasing the 
 value of t the last term on the right-hand side becomes 
 inappreciable and may be neglected, so that we obtain, in 
 either case, the equation 
 
 M ,TxJ 
 
 j:Gp" 
 
 and therefore by similar reasoning with respect to y and 
 
 and therefore for any system of particles, 
 
 *.* i 
 
 2 
 
 19.] Any number of particles being in motion in a given 
 region, to find the pressure referred to unit of surface at any
 
 64 Pressure at any point in 
 
 point within the medium, supposing the particles to be 
 acted on by fixed-central forces, and mutual forces between 
 the particles, any functions of the distance between them, 
 and which become infinitely large with the infinite diminu- 
 tion of this distance. 
 
 Let be the point in question, and about describe a 
 closed surface S of simple continuity, and take for the 
 origin of coordinates. 
 
 If the mutual action between the adjacent particles within 
 and without S be replaced by a force pdS acting normally 
 at every element dS, the distribution as to momenta and 
 coordinates of the molecules now confined within S will, on 
 the average of any finite time, be the same as in the actual 
 
 Now by the virial proposition, if M be the mass and v 2 
 the mean square velocity of any particle within S ; X, Y, Z 
 the components of the moving forces on that particle, and 
 x, y, z its coordinates, 
 
 the bar denoting mean values. 
 
 The part of the second term arising from pressure forces 
 on the surface element dS is 
 
 ^ pdS (x cos a +y cos /3 + z cos y), 
 
 where a, , y are the angles between the axes and the 
 normal at dS drawn inwards. 
 
 Therefore if r be the perpendicular from upon the 
 element dS, the portion of the pressure virial contributed 
 by the element dS becomes 
 
 and the virial equation becomes, therefore, 
 
 = 2 Mv 2 + 2 (Za+Yy + Zz),
 
 medium of moving molecules. 65 
 
 or if Fbe the volume included 'within S, and <Fthe ele- 
 mentary pyramidal volume with base dS and vertex 0, 
 
 3 2 (pd7) = 2 Mv 2 + S (Xx+Yy + Zz). 
 
 If S, and therefore V, be indefinitely diminished, we get 
 for the value of p at the point 0, being the value sought, 
 
 _ pv 2 T_ ^(Xx+Yy + Zz) 
 
 P ~ 3 3 y 
 
 The forces X, Y, Z include all the forces in the field 
 except the pressure forces, but since x, y, z become infinitely 
 small with the indefinite diminution of V it follows that 
 all the terms in Xx+Yy + Zz except those arising from 
 infinitely large values of X, Y, Z may be neglected, and 
 therefore that the only terms in Xtti+Yy + Zz to be re- 
 tained are those arising from the infinitely large mutual 
 forces of the particles within 8. 
 
 Now if #', y', z' and &", y", z" be the coordinates of two 
 particles M' and 31", and if R be the mutual moving force, 
 and r the distance between M f and 31", we have 
 
 r r 
 
 R being positive when repulsive, with corresponding values 
 for the other components ; so that 
 
 XV + Y'y' + Z'z' + X"x" + Y"y" + Z"z" 
 
 becomes ^ {(*"-*J + (y" -yj + (z"-z) 2 } > 
 or Rr, 
 
 with similar results for all other pairs, and the value of p 
 at is therefore given by the equation
 
 66 Pressure at any point in 
 
 the 22 denoting summation for every pair of particles 
 within V. 
 
 If the particles be replaced by molecules with irreducible 
 minimum volume, the equation 
 
 still holds as before, where X, Y, Z are the components of 
 the resultant force on each molecule, and the molecule is 
 replaced by a particle, of equal mass, at its centre of 
 inertia, but the direction of the mutual force between two 
 molecules does not generally lie accurately in the line joining 
 their centres of inertia. Since the dimensions of the mole- 
 cules are very small, this deviation is unimportant for a 
 pair at any measurable distance from each other, but it is 
 otherwise for contiguous or nearly contiguous molecules. 
 Hence the substitution of 22 (Rr) for 2 (Xx + Yy + Zz) for 
 the molecules within the small volume V is inadmissible, 
 except indeed for spherical molecules. 
 
 We must remember also that in passing from material 
 particles, or Boskovichian atoms, to molecules such as they 
 are generally conceived, namely with an irreducible mini- 
 mum volume however small, the indefinite diminution of 
 the volume of 8 is unattainable without destroying the 
 hypothesis of many mutually acting molecules being com- 
 prised within S. 
 
 In the case of such molecules, therefore, we may continue 
 to use the equation 
 
 for the determination of p, although we cannot give an 
 accurate mathematical meaning to the second term in the 
 right-hand side ; 2 (Rr) standing for 2 (Xx + Yy + Zz)* 
 
 * This is the same difficulty as that alluded to in the introduction as 
 occurring in the definition of density at any point in a medium of molecules 
 with irreducible minimum volume.
 
 medium of moving molecules. 67 
 
 If, for example, as will always be the case, in any 
 medium to which our reasoning is applicable, the irreducible 
 minimum volume of a molecule be quite inappreciable in 
 comparison with any measurable volume however small, 
 
 then the expression ~ - may be regarded as the ratio 
 
 of the Rr products for all pairs of molecules, within a sphere 
 about 0, of extremely small measurable radius, to the 
 volume of that sphere. 
 
 If our molecules, for instance, were hard elastic spheres, 
 each of mass M and radius *, and if N were the number of 
 these spheres contained within the small sphere about 0, 
 then, remembering that R and r at each encounter are 
 Mu and 2* respectively, where u is the relative velocity 
 in the line of centres, it follows from the reasoning of 
 Art. 5 that, omitting certain finite numerical factors, 
 22 (Kr) would be 
 
 > 
 
 and therefore lt v=Q " would in this case be, with 
 
 similar omission, 
 
 7J NM Ns* 
 #K=O ~j --- y **i 
 
 - total volume of inclosed spheres 
 or p0 z .# Fs=0 y 
 
 In such a case, therefore, the second term in the value of 
 p vanishes in comparison with the first term, whenever, 
 owing either to the smallness of the individual molecules 
 or the smallness of the number of such molecules per unit 
 of volume in the neighbourhood of the point 0, the total 
 minimum irreducible volume of the molecules per unit 
 volume near that point is very small, and therefore in either
 
 68 Pressure at any point of medium. 
 
 of these cases the value of p is sensibly reduced to the first 
 term, or 
 
 But, on either of the last-mentioned assumptions, the 
 e -ti(x+T) j aw o f distribution has been established for a 
 field of fixed central forces, and therefore by Art. 17 the 
 necessary condition of equilibrium 
 
 dp 
 
 -y- = pX 
 
 dx 
 also holds. 
 
 When neither one nor the other of the aforesaid assump- 
 tions is satisfied, we get 
 
 - -It 
 P ~ 3 3 
 
 which would not satisfy the condition 
 dp - n j 
 
 ~T~ == P -*- 
 
 dx 
 if the e- h <x+T) law held good. 
 
 But, unless either of these assumptions hold, the e~ h (x+ T ) 
 law has not been established, we conclude therefore that this 
 law of distribution is a necessary and sufficient condition of 
 permanence for a medium of spherical molecules in a field 
 of fixed central forces on either of these assumptions, 
 (i) that either the number of molecules in unit volume, 
 i.e. the molecule density, is very small ; (2) that the mole- 
 cules themselves are so small as to be practically material 
 points or Boskovichian atoms. 
 
 So far as these conditions are not fulfilled the e~ h( x +7 "> 
 law must be regarded as approximate, the more correct 
 statement of it being possibly of the form 
 ,-{+,*(}
 
 Law of distribution with intermolecular forces. 69 
 
 where E is x + T, $ some function, and /u a small function 
 of x, y> z, whose difference from zero depends upon the 
 extent of the departure from the aforesaid conditions. 
 
 The evaluation of 2S (Rr] for spherical molecules in the 
 above investigation proceeds on the assumption that the 
 distribution is uniform throughout the volume V when V 
 is very much diminished. Should the law of distribution 
 lead to a very rapid space variation of the density at any 
 point in the medium the reasoning is, so far, precarious. 
 
 It may be assumed that the results arrived at for hard 
 elastic spherical molecules hold good substantially for mole- 
 cules of any shape and constitution whatever. 
 
 20.] Hitherto we have confined our attention to elastic 
 molecules moving in a field of central forces, in which, there- 
 fore, the potential at every point is a determinate function 
 of the coordinates of that point the same for all times. 
 
 In such a case, viewing the problem under the ordinary 
 conventional aspect, we say that there are no forces in the 
 field except those from fixed centres, and no potential except 
 that of the fixed central forces. 
 
 But the truer view would be, as already mentioned 
 in Art. 12 above, to regard the molecules as subject to 
 mutual forces, in addition to those from fixed centres, 
 infinitely great during their action, but acting in each case 
 during intervals of time so very small that in any measure- 
 able length of time their average potential is absolutely 
 inappreciable in comparison with that of the fixed centre 
 forces, so that the \ of the above-mentioned law at any point 
 is still the x at that point of the fixed central forces only. 
 
 The treatment of the problem under this aspect is given 
 above in Art. 12, where the collision is replaced by the 
 encounter, that is to say, an exceptional action of forces 
 between the molecules lasting for a short but not abso- 
 lutely evanescent interval of time, and not inconsistent
 
 70 Law of permanent distribution 
 
 with changes of the coordinates defining the position, but 
 still subject to the condition that the chance of any mole- 
 cule being in encounter simultaneously with more than one 
 other molecule is infinitely small, and therefore that the 
 average potential of such encounter forces during any finite 
 interval of time is evanescent. 
 
 Such encounter forces, on the supposition of spherical 
 
 molecules, must clearly be of some such form as , ^-, 
 
 where r is the distance between the centres, s the sum of 
 their radii, and v a very large number, and indeed it is 
 
 W 7? 
 
 under this aspect that the evaluation of the term ^ 
 
 in Article 19 has been performed. 
 
 Now suppose that (the molecules being still for the 
 moment regarded as hard elastic spheres) there were mutual 
 
 forces between them of the form ^ , ^ , &c., where 
 
 r 2 r i 
 
 r is the distance between the centres and the indices of 
 r in the denominators are not infinitely great, then the 
 potential at any point P of the field would not be a 
 determinate function of the position of P, owing to the 
 motion of the molecules, but, if a permanent distribution 
 were attained, the potential at P of molecules remote from P 
 would not be sensibly affected by their motion but would 
 be, like that of the fixed central forces, a function of the 
 distance only, while the potential of the particles near to P 
 would depend upon the shifting positions of these particles 
 and would vary from time to time. In fact, if the /* in the 
 case of each of these forces were very small, a distance b 
 might be conceived ; it may be much greater than the 
 radius a of each sphere, but yet much smaller than any 
 measurable quantity, such that for two spheres whose 
 central distance was greater than 6 the mutual action was
 
 with fixed central and intermolecular forces. 7 1 
 
 absolutely negligible, and the potential at P of the infinitely 
 great number of spheres lying outside the sphere of radius 
 b described about P would have a value dependent upon 
 the permanent distribution but independent of the time, 
 while that of the spheres lying within the b sphere would 
 vary with the motion of such included spheres ; in point of 
 fact all the spheres lying within the b sphere would be in 
 an encounter with the spheres with centre at P and with 
 one another, in a field of practically fixed centre forces 
 arising from the given fixed centre forces and the sensibly 
 fixed centre forces of the spheres external to the b spheres. 
 
 If the medium density were so small that the chance of 
 more than two spheres being simultaneously in such 
 encounter was infinitely less than the chance of a binary 
 encounter, it would follow from the reasoning of Art. 12 
 that the e- h & +T ) law was a necessary and sufficient con- 
 dition of permanent distribution, where x includes the 
 potential of all the given fixed central forces at any point 
 and also the given fixed central forces arising from mole- 
 cules external to the b sphere about that point. 
 
 If the volume of a sphere of radius - described round 
 
 2 
 
 each spherical molecule and concentric with it were called 
 the effective volume of that sphere, the condition as above 
 enunciated would obviously be, that the number of spheres 
 in unit volume or the effective volume of each sphere must 
 be so small that the total of the effective volumes per unit 
 of volume was a very small fraction. 
 
 When this condition is satisfied it may be shown, much 
 
 V*^ /"> 
 
 as in the case of the hard elastic spheres, that lty=o ^~ 
 is insensible compared with ^ , provided the additional 
 
 J 
 
 condition of the preponderance of kinetic over potential
 
 72 Work done on 
 
 energy be maintained. The restrictions as to the number 
 and masses of the molecules thus introduced would necessi- 
 tate the part of x, depending upon intermqlecular forces, 
 being very small and indeed absolutely evanescent for 
 values of A greater than 2 or 3. 
 
 The reasoning may obviously be extended to molecules 
 other than spherical and with any number of degrees of 
 freedom. In this more general case there may be inter- 
 atomic forces whose potential must also be included in the 
 application of the e~ fe (x+ T ) law, but such forces do not enter 
 into the term 22 (Rr), or more correctly '%(Xx+ Yy + Zz}. 
 
 21.] If any number of sets of molecules axe inclosed in 
 a region under the circumstances of the preceding Articles, 
 to find the work done by the expansion of the medium, 
 uniformly in all directions, i.e. the increase of every linear 
 dimension bearing the same ratio to the original length of 
 that dimension. 
 
 Suppose that there is in the first place only one set of 
 molecules N in number, the mass of each being M and its 
 number of degrees of freedom being m, and suppose that 
 the force field is that of given fixed central and inter- 
 molecular forces. 
 
 If there are no fixed central forces the pressure p, at every 
 point of the bounding surface, is the same, and the required 
 work (dW] is equal therefore to pdV. 
 
 But in this case, as just now proved, we have 
 
 supposing each intermolecular force R to be positive when 
 it acts repulsively. 
 Therefore
 
 expansion or compression of medium. 73 
 
 but since linear expansion is rd of cubical expansion, we 
 have Y = j where r is any linear dimension. 
 Therefore, in this case, 
 
 dV N 
 
 where -~ is the mean value of -r^, and to be carefully 
 
 distinguished from -r^- \.* 
 
 If, however, there be external forces, and p therefore 
 be not constant over the surface, then we must use the 
 equation 
 
 2_pvdS=2 Hv 2 + 2 (Xx+Yy + Zz). 
 
 If p. be the ratio of the increase of each linear dimension 
 to the original length of that dimension, which is constant 
 
 dV 
 throughout and equal to ~, our equation becomes 
 
 = jit 2 Mv 2 + 2 (X 
 
 .e. ~. 
 
 as before. 
 
 * See also' a paper by Mr. S. H. Burbury, F.R.S., published in the Philo- 
 sophical Magazine, January, 1876.
 
 '74 Second law 
 
 The change of potential of interatomic forces is not 
 
 d\ 
 considered in this investigation, so that -r= refers only 
 
 to the intermolecular forces. 
 
 For there is no necessary connection between molecular 
 expansion and contraction and the expansion and contrac- 
 tion of the whole volume, and therefore on the average the 
 loss and gain of interatomic potential may be regarded as 
 balancing each other. 
 
 22.] A number of sets of molecules are in a given region 
 of space under the circumstances of previous Articles, the 
 field of force being perfectly unrestricted and including 
 interatomic, fixed-central, and intermolecular forces ; then, 
 if an increase of Energy equal to 8 Q be imparted to the 
 medium, there exists a function $ such that 
 
 or k 5 Q is a perfect differential provided the e~ hE law hold 
 good. 
 
 Suppose, in the first place, that the molecules are all of 
 one set, each with m degrees of freedom, and that there are 
 N of them. 
 
 When any molecule has its momenta and coordinates 
 between /? x and p l + dp l ...q m and q m + dq m , let that mole- 
 cule be said to be in the -A^ state, and for different cor- 
 responding limits of the variables let it be said to be in 
 the A. 2 , A 3 , &c. states respectively. Let the product of the 
 differentials of the m momenta in the A ly A. 2 , &c. states be 
 denoted by ds l , ds 2 , &c., and let the corresponding products 
 for the coordinates be denoted by d<r l} d<r 2 , &c. And let 
 also the potential and kinetic energies for these states be 
 Xl , x 2 , &c., T lt T 2) &c. 
 
 Then, since the number of molecules of this set in the A v 
 state is proportional to e~ h ^i +T ^ds l da- l with similar ex-
 
 of Thermodynamics. 75 
 
 pressions for other molecules, it will follow that the chance 
 of the first of the N molecules in the medium considered 
 being in the A l state in any length of time is also propor- 
 tional to e~ h bi +T Jds L d<r l with similar expressions for the 
 2 nd , 3 ri j & c - molecules being in the states A z , A 3 , &c., 
 provided always the total volumes of the N molecules be 
 inconsiderable in comparison with the whole volume of the 
 containing region, because if this proviso be not satisfied we 
 cannot regard the aforesaid chances as independent. Subject 
 therefore to this proviso, it will follow that the chance of the 
 ./V molecules being in the A l , A 2 . . . A N states respectively 
 at any instant will be 
 
 where \ an ^ ^ are the total Energies, Potential and 
 Kinetic, of all the molecules, ds being the product of all the 
 momenta differentials dp l . . . dp mlf contained in ds l ... ds K 
 above, and da- the product of all the coordinate differentials 
 in r/o-j ... da-ff above, except the 3 N differentials doa l ... dz$; 
 integrating therefore for all the momenta, we have the 
 chance that the Nm coordinates are between 
 
 equal to Pe~ hx d(rdx l ... dz N ^ 
 
 where P is a function of all the mN coordinates except the 
 $N coordinates x l ...z N , and da- is the product of these 
 remaining (m 3) N coordinates. 
 
 Hence it follows that the mean value, f, of any function 
 f of the position of the molecules, will be determined by 
 the equation 
 
 rrr 
 
 jjj...ff^i,^...i., 
 
 f f p / 
 
 L . . . dz N
 
 76 Second law 
 
 Suppose now that a small increase of energy 8 Q is 
 itaparted to this medium of molecules, then h and T will, in 
 general, vary. Let their increments be denoted by k and 
 8 V respectively. 
 
 The imparted energy 8 Q will be equal to 
 
 where T, X and W are the mean values of the kinetic and 
 potential energies of the medium and the work done by 
 expansion. 
 
 Now J?= M) = _ 
 
 and SF = j >~ - ^ 5 7, by last article. 
 
 Now let u = log ( iff. 
 then 
 
 
 Also 
 
 err 
 
 JJJ '" e Xi '" ZN 
 
 fff. . . Pe~ h ^d(rb(dx l ... dz N ] 
 
 l ... dz N
 
 of Thermodynamics. 77 
 
 where 8 (dx^ . . . dz N ) is the increase of the differential pro- 
 duct arising from the increase 8 V of V. 
 
 So that 8 . dx l = =- clx^ , 
 
 and 
 
 ?! . . . dz N ) = dx.i . . . dz N 8 dx^ + dx l . . . dz N 8 dx 2 + &c. 
 
 to 3 N terms 
 
 = JV =-.<?#! ... d' z N' 
 
 Whence .dF= - k 
 
 Substituting in (A) we have, therefore, 
 
 If there be any number of sets of molecules we have, 
 since the distribution of each set is independent of the 
 distributions of all the rest, 
 
 2 denoting summation for all values of m and N. 
 
 23.] We now proceed to summarise the results at which we 
 have arrived, concerning the physical properties of a medium 
 of sets of molecules moving about in a region of space very 
 much larger than the sum of the irreducible minimum 
 volumes of the molecules, and acted on by fixed central 
 and interatomic forces only. 
 
 On such a supposition the term U V=Q y disappears 
 
 from the value p, and there is no work done by or upon any 
 of the forces, except fixed central forces, on expansion or 
 compression of the medium, and the following physical
 
 78 Comparison of physical properties 
 
 properties of the medium have been proved to hold in a 
 state of permanent or stationary motion. 
 
 In the first place, we have proved that there is one 
 quantity which is the same for every set, namely, the mean 
 kinetic energy of translation of each of the molecules, or 
 * 
 ~ . Let this quantity be called r. 
 
 In the second place, we have found that the distributions 
 of the molecules of each set, both as to momenta and 
 position, is independent of the co-existence of the molecules 
 of the remaining- sets, and is in all the same as if that 
 particular set alone existed in the region considered. 
 
 We have also found that if p be the pressure referred 
 to unit of surface at any point in the medium on any 
 imaginary material surface drawn through the point, p the 
 
 density at that point, and therefore ^- the molecular 
 
 density, (M being the mass of each molecule of the set 
 considered), then 
 
 an equation which expressesithese third and fourth properties 
 of the medium ; viz., 
 
 Third, that when the mean kinetic energy of translation 
 remains constant the pressures varies directly as the density ; 
 and Fourth, when the density remains constant the pressure 
 varies as the mean kinetic energy of translation. 
 
 Fifth, we see from the value of p that whenever two 
 media composed of sets of different molecules are so related 
 that the pressure referred to unit of surface and the mean 
 kinetic energy of translation are the same in both media, 
 then the molecular densities are also the same in both 
 media. 
 
 If now we compare these results with the accepted
 
 of medium and perfect gases. 79 
 
 properties of an ideal perfect gas at rest, or of a mixture of 
 such gases, we know 
 
 First, that there is one physical property, viz. the tem- 
 perature, which must be the same for each gas at each point. 
 
 Second, from the law of Dalton, that the pressure and 
 density of each gas at every point must be the same as if 
 the other gases did not exist, or that each gas acts as a 
 vacuum to all the rest. 
 
 Third, from Boyle's law, that so long as the temperature 
 remains the same the pressure is proportional to the density. 
 
 Fourth, from Charles's and Gay Lussac's law, that when 
 the temperature and density both vary, the pressure varies 
 as the absolute temperature and the density conjointly. 
 
 Fifth, from Gay Lussac's law, that when two gases are 
 at the same pressure and temperature their atomic densities 
 are the same. 
 
 In these five particulars therefore, generally recognised 
 as summarising the physical properties of the ideal perfect 
 gas, if we replace the terms mean kinetic energy of trans- 
 lation and molecule, by absolute temperature and chemical 
 atom respectively, the physical properties of the moving 
 molecular medium become absolutely identical with those 
 of the ideal perfect gas. 
 
 In both eases, also, there is no change of potential energy 
 by expansion or contraction, and in both cases the known 
 thermodynamical law ir . 
 
 -^ = d$ 
 
 is established, the term equilibrium in the case of the 
 medium being interpreted as stationary motion or per- 
 manence of distribution.* 
 
 * The student will find little difficulty in proving the equation 
 
 8Q =N5r + S r(r^- 
 & well-known equation in thermodj'namics.
 
 8o Comparison of physical properties 
 
 The physical laws as above stated are generally regarded 
 as those of ideal perfect gases, although they are not quite 
 accurately satisfied by any gases actually existing. They 
 represent only limiting states to which actual gases approach 
 more closely as they more nearly correspond to perfect 
 
 For example, there is, in all actual gases, a slight depar- 
 ture from the laws of Boyle and Charles or Gay Lussac, 
 and there is also always a slight indication of change of 
 potential energy on compression or expansion ; and just so 
 in the molecular medium, unless either the molecular 
 density N, or else the irreducible molecular volume were 
 
 22 Tlr 
 
 so extremely small as to ensure the vanishing of the ^ 
 
 term, the e~ h & +T ) distribution is not rigorously proved, nor 
 the p = kpr equation established. It would appear there- 
 fore that up to this point we are in a position to infer with 
 very great confidence that a perfect gas, if such exists, is 
 accurately represented by a moving molecular medium in 
 the limiting state above indicated, and that all gases are 
 represented by a molecular medium nearly approaching to 
 that limiting state.* 
 
 24.] There is one physical property, however, in which 
 this agreement is not so complete, and this we proceed to 
 consider. 
 
 A homogeneous perfect gas being supposed to be con- 
 stituted of moving molecules with any given number of 
 degrees of freedom, required to find the ratio of the specific 
 heat at constant pressure to that at constant volume. 
 
 First suppose that when a small quantity bQ of heat is 
 
 * It appears from Art. 20 that the e~hE l aw o f distribution might 
 prevail when there are sensible intermolecular forces in the medium, and 
 therefore sensible gain or loss of potential Energy on compression or 
 expansion. Gases represented by such media would obey the five gaseous 
 laws above mentioned, even though their Energy were not entirely Kinetic.
 
 of medium and perfect gases. 81 
 
 imparted to a unit of mass of the gas the temperature is 
 raised from r to r + 8r, the volume remaining- constant. 
 
 Since the volume remains unchanged there is no external 
 work performed, and therefore the whole of this energy bQ 
 must be accounted for by increase of the total kinetic energy 
 and of the potential energy, therefore 
 
 where Sr and 8x are the increases of mean total kinetic and 
 potential energy respectively of any molecule, and N is the 
 number of molecules in the unit of mass. 
 
 We have proved that T, the mean total kinetic energy 
 
 of a molecule, is equal to > where m is the number of 
 
 3 
 
 degrees of freedom of the molecule ; 
 
 Next let the volume be allowed to increase so that the 
 pressure remains constant, and let b'Q, be the heat required 
 to raise the unit of mass from r to r + 8r in this case. 
 
 Since external work is performed equal to pbv we must 
 have in this case 
 
 *.-*tf +.)*-* 
 
 But we know that jov = r, and since p remains 
 constant it follows that 
 
 pftv = , 8r ; 
 3
 
 82 Comparison of physical properties 
 
 All that we know of -^ is that it must necessarily be 
 positive, and of m that it must be integral and not less 
 
 than 3 ; writing therefore e for 3 ~ in the calculated 
 
 ctt 
 
 value of the ratio of the specific heats, this becomes 
 m + e + 2 
 
 2 
 i + 
 
 There is one gas* for which the ratio is i| very approxi- 
 mately, giving us therefore m + e = 3, and thus agreeing 
 with the case of elastic spheres, where there are three degrees 
 of freedom, determined by the coordinates of the centre, so 
 that ^ = 3 and e = o. 
 
 For the majority of observed gases, however, the ratio is 
 1-408, giving us 
 
 = -408, 
 
 and m + e 4-9. 
 
 For a few gases the ratio is 1-3, whence we get 
 
 m + e 
 and m + e 6-6. 
 
 And for a few other gases the ratio falls as low as 1-26, 
 giving us 
 
 m + e 8, nearly. 
 
 * Mercury gas, for which the ratio in question has been determined 
 at 1.67.
 
 of medium and perfect gases. 83 
 
 One great difficulty in the establishment of the kinetic 
 theory of gases is to conceive a molecule so constituted as 
 to give a suitable positive and integral value for m in the 
 cases we have mentioned, especially for that most general 
 case of all in which 
 
 m + e = 4-9. 
 
 There is also a further difficulty arising from the follow- 
 ing considerations. 
 
 It is known that the light emitted by heated gas, so 
 long as the gas is of no great density, consists of rays of 
 one or more definite kinds of refrangibility, so that when 
 this light is examined by the spectroscope, the spectrum 
 produced consists of one or more bright lines, narrow and 
 distinct, the intervening spaces being dark. As the density 
 of the gas increases, these bright lines become broader and 
 the intervening spaces more luminous, until, as the gas 
 becomes very much condensed, a continuous spectrum is 
 produced. 
 
 If we conceive our gas to be represented by a number of 
 moving molecules, as in the hypothesis now under con- 
 sideration, the motion of translation or agitation of these 
 molecules is exceedingly irregular, the intervals between 
 successive encounters and the velocities of a molecule during 
 successive free paths not being subject to any law. It will 
 be different however with the internal motions or vibra- 
 tions of each molecule. When there is a long free path 
 very many such vibrations may take place in the interval 
 between successive encounters. At each encounter the 
 whole molecule is roughly shaken. During the free path 
 it vibrates according to its own laws, and these vibrations, 
 as is the case in every connected system, may be resolved 
 into a number of simple vibrations, the law of each of 
 which is that of the simple pendulum. At any instant the 
 G 2
 
 84 Comparison of physical properties 
 
 number of molecules in collision is negligible in comparison 
 with the whole number of molecules in the region under 
 consideration. And therefore at any instant we have a 
 collection of a very great number of bodies, all of which may 
 be regarded as performing precisely the same set of vibra- 
 tions. If these molecules be capable of communicating 
 their vibrations to the luminiferous ether, the result will be 
 light of one or more definite kinds of refrangibility, pro- 
 vided there be any light at all that is, provided that the 
 vibrations be of such a period as to belong to the luminous 
 part of the spectrum. As the density of this medium is 
 increased, the length of the free path of each molecule is 
 diminished, and since each fresh encounter disturbs the 
 regularity of the series of vibrations, we must no longer 
 regard the whole of the bodies or molecules, but only a 
 large majority of them at any instant, as performing the 
 same sets of vibrations ; the result therefore will be light 
 of one or more definite kinds of refrangibility, with a 
 mixture of fainter light of no definite refrangibility, or, 
 viewed under the spectroscope, bright lines of light, 
 along with a ground of diffused light forming a continuous 
 spectrum.* 
 
 In estimating the difficulty presented by these consider- 
 ations in the way of the establishment of the Kinetic 
 Theory of Gases, it must be borne in mind that they assume 
 a sensibly instantaneous return to the permanent state after 
 any disturbance of the medium. In that permanent state, 
 as we have shewn, there is an equal partition of the average 
 kinetic energy among the different degrees of freedom, 
 
 that is to say, the total kinetic energy is -- times the 
 kinetic energy of agitation, whence in the case of a perfect 
 
 * See Maxwell, ' Theory of Heat,' p. 306.
 
 of medium and perfect gases. 85 
 
 gas, pbv or dW ' dT, where dT is the total increase 
 m 
 
 of kinetic energy; and therefore the ratio of the specific 
 
 , 
 
 heats, or 7 - , becomes, as we have seen. 
 ell 
 
 dT dT(i + 
 ^ 
 
 dT+d x 
 
 dv 
 where e is written for the small positive quantity y^ or 
 
 d\ . dv 3 dv 
 
 3y% because - = -^- . - 
 
 dr ' dT m dr 
 
 But the rate at which the medium tends to the ultimate 
 state varies, or may vary, greatly with the nature of the 
 disturbance. For example, it has been shewn above, 
 Art. 15, that in a medium of plane circular disks, in each 
 of which the centre of inertia is at the distance c from the 
 centre of figure, an inequality between the mean kinetic 
 energy corresponding to the rotation and that correspond- 
 ing to either of the translational velocity components 
 may, if c be small, tend to disappear at an infinitely 
 slower rate than when the disturbance arose from an 
 inequality between the mean kinetic energies correspond- 
 ing to the translational velocity components themselves. 
 Now it is reasonable to believe that the first effect of 
 such a disturbance of the medium as an increase of heat, or 
 energy, would be shewn in the alteration of the agitation 
 energy of the molecules upon which the temperature 
 depends, and these molecules may be so constituted that 
 the time required before the equal partition of the perma-
 
 86 Comparison of physical properties 
 
 nent state is reached may be finite or even large ; and 
 during 1 such time the external work bW is bearing- to 8 T 
 the whole increase of kinetic energy a larger ratio than 
 
 ; or in other words, the apparent value of y, the ratio of 
 vn, 
 
 the specific heat as observed during this interval, would be 
 greater than the value 
 
 dT+d x 
 
 which it would have if the permanent state were reached 
 instantaneously. 
 
 In the case of the problem referred to in Art. 15 Pro- 
 fessor Tait finds that the difference of the average energies 
 of the two systems of spheres will on certain numerical 
 assumptions fall to -01 of its original value in x icr 9 of a 
 second, these assumptions being reasonable on the hypo- 
 thesis of certain gases being represented by a medium com- 
 posed of such spheres. 
 
 The principles involved are such as would lead to a 
 similar result in the investigation of the rate at which an 
 equality would be arrived at in the average kinetic energies 
 corresponding to the three components of agitation of either 
 set of spheres. 
 
 Replacing spheres by molecules we may, therefore, con- 
 clude that the equalisation of the components of agitation 
 energy in a medium of moving molecules representing a 
 perfect gas takes place, sensibly at least, instantaneously, 
 and it is on the assumption of such instantaneous equalisa- 
 tion among these components that the velocity of trans- 
 mission of sound waves and other physical properties of the 
 medium is found to agree with experimental results, but it 
 need not necessarily follow that there is this instantaneous
 
 of medium and perfect gases. 87 
 
 equalisation among the energies corresponding to all the 
 degrees of freedom, so that it is not necessarily a fatal 
 objection to the theory that a difficulty should exist which 
 is based solely upon the assumption of such universal 
 instantaneous equalisation. 
 
 THE END.
 
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