242 



NATURE 



\yuty 26, 1877 



THE KINETIC THEORY OF GASES 

 A Treatise on the Kinetic Theory of Gases. By Henry 

 William Watson, M.A., formerly Fellow of Trinity 

 College, Cambridge. (Oxford : Clarendon Press, 1876.) 



THIS book does not profess to treat of all that has 

 been written about the kinetic theory of gases. It 

 discusses the ultimate average condition of a material 

 system, consisting of a very great number of parts in 

 motion within a confined space, and it follows for the 

 most part the methods of investigation given by 

 Boltzmann. The discussion is arranged in the form of 

 thirteen propositions, in which the different cases are 

 considered in the order of their complexity. In the 

 earlier propositions the moving bodies are supposed to 

 be rigid-elastic spheres acting on each other only by 

 impact, afterwards external forces are introduced, and 

 finally the bodies are supposed to be material systems, the 

 parts of which are held together by any system of forces 

 consistent with the principle of the conservation of energy. 



The ultimate average condition of such a system is 

 investigated in a very satisfactory manner in this book. 

 No part of mathematical science requires more careful 

 handling than that which treats of probabilities and 

 averages. Mathematicians, whose competence to deal 

 with other questions is undoubted, have fallen into errors 

 in treating of probabilities, and even the validity of certain 

 methods of proof is still apparently an open question. 



Besides this, some of the consequences to which these 

 theorems lead us are so startling that we are not 

 prepared to admit them without an unanswerable proof, 

 and of the investigations already given, some are so short 

 and incomplete, and others so long and roundabout, that 

 it requires no ordinary exercise both of penetration and of 

 patience to find out whether they are proofs at all. Mr. 

 Watson has conferred a great benefit on the students of 

 the kinetic theory by placing before them in a series of 

 distinct propositions, none of them too long for the mind 

 to grasp, all the necessary steps leading to the result, and 

 none of the superfluous evolutions in which the mental 

 energy of the student is so often dissipated. The book, 

 as we have said before, is confined to the investigation of 

 the ultimate average condition of the system, and does 

 not discuss the processes of diffusion by which that 

 ultimate condition is attained, such as the inter-diffusion 

 of gases, the diffusion of momentum by viscosity, and the 

 diffusion of energy by thermal conduction. These have 

 been recently treated in a larger work,'^ to which we may 

 have occasion to refer. 



There are two very different methods of defining and 

 investigating the state of a complex material system. 

 According to the strict dynamical method the particles of 

 the system are defined in any sufficient manner, as, for 

 instance, by their co-ordinates at a given epoch, and the 

 position of any particle at any other time is then defined 

 by its co-ordinates, expressed as functions of the time, 

 the form of these functions being different from particle 

 to particle, and not necessarily -continuous in passing from 

 one particle to another which was contiguous to it in the 

 initial configuration. 



According to this method our analysis enables us to 

 trace every particle throughout its whole course, and there- 

 fore we can apply the laws of motion in all their strictness. 



■ "Die kinetische Theorie der Gase in elementarcr Darstelluiig, mit 

 mathematischen Zusiitzen. Voa Dr. Oskar Emil Meyer, Professor de 

 Physik an der Unlversitat Breslau. (Breslau, 1877.)^ 



The application of this method to systems consisting 

 of large numbers of bodies is out of the question. We 

 therefore make use of another method which we may call 

 the statistical method, on account of its analogy with the 

 methods employed in dealing with the fluctuations of a 

 large population. 



We divide the bodies of the system into groups accord- 

 ing to their position, their velocity, or any other property 

 belonging to them, and we fix our attention not on the 

 bodies themselves, but on the number belonging at any 

 instant to one particular group. This number is, of 

 course, subject to change on account of bodies entering 

 or leaving the group, and we have therefore to study the 

 conditions under which bodies enter or leave the group, 

 and in so doing we must follow the course of the bodies 

 according to the dynamical method. But as soon as the 

 process is over, when the body has fairly entered the 

 group or left it, we withdraw our attention from the body, 

 and if it should come before us again we treat it as a new 

 body, just as the turnstile at an exhibition counts the 

 visitors who enter without respect to what they have 

 done or are going to do, or whether they have passed 

 through the turnstile before. 



The first mode of grouping the bodies of the system is 

 to class those together which, at a given time, are in a 

 given region of space. This is called grouping according 

 to configuration, and what we learn from it is the distri- 

 bution of the positions or co-ordinates of the bodies in 

 space. 



The second mode of grouping is that according to 

 velocity. The best way to understand this is to suppose 

 a diagram of velocities constructed by drawing from a 

 given point as origin a system of vectors representing in 

 direction and magnitude the velocities of the different 

 bodies. The extremities of these vectors are called the 

 velocity-points of the bodies to which they correspond, 

 and by grouping the bodies according to the regions 

 of the diagram in which their velocity-points lie, we 

 learn from the numbers in the groups the distribution of 

 velocities among the bodies. 



In like manner we may form groups defined in any 

 other way, as, for instance, those pairs of bodies whose 

 distance from one another lies between given limits, and 

 by confining within sufficiently narrow limits the values 

 of all the properties of the bodies which form the group, 

 we may consider all the bodies belonging to the group as 

 practically in the same state. Whether at a given instant 

 any body actually belongs to the group is, of course, 

 another question. 



The object of study in the statistical method is the 

 probable number of bodies in each group. We may get 

 rid of the idea of probability by supposing the system to 

 continue under the same conditions for a very long time. 

 During this time many bodies will enter the group, stay 

 in it for a certain time, and then leave it. If we add 

 together the times of residence within the group af all 

 these bodies, and divide the sum by the whole time ol 

 observation, we obtain a numerical quantity which we 

 may call the average number of bodies in the group. The 

 longer the time of observation, the nearer does this 

 number approach to what we have called the probable 

 number of bodies in the group. 



The average number of bodies in a group depends on 



