July 26, 1877] 



NATURE 



243 



the limits which define the group, being, of course, greater 

 when these limits are wide than when they are narrow. But 

 it also depends on the character of the group, that is to 

 say, the particular set of mean values of the conditions 

 which entitle a body to be ranked in the group. 



It appears from the investigation that if </> be any 

 property of a body, such that if <^i and <^^ are its values 

 or two bodies before an encounter, and <I>i and 4>.^ its 

 values after the encounter, and if under all circumstances 

 ^^ -|- <^2 = *, 4- *2, and if the number ,of bodies in each 



— it^ 

 group varies as e , then the distribution of the bodies 

 in the groups will not be altered by the encounters 

 between the bodies. 



Now if we make </> equal to the sum of the kinetic and 

 the potential energy of each body, the quantity <^, + '\>i is 

 not altered, either by an encounter between the two 

 bodies or by external forces acting on them ; so that a 



distribution according to the values of the function e 

 will satisfy the condition of permanence. 



The most general case is that given in the seventh pro- 

 position. The bodies are no longer supposed to be 

 smooth rigid-elastic spheres, but molecules, that is to 

 say, material systems consisting of any number of parts 

 acting on each other with forces of any kind consistent 

 with the principle of the conservation of energy. The 

 molecules of any one kind are supposed j to have in 

 degrees of freedom, this number being, in general, dif- 

 ferent in different kinds. 



It is also assumed in the enunciation that all the forces 

 in the system are either forces tending to fixed centres 

 and functions of the distances from these centres, or else 

 forces acting between the parts of the same molecule, 

 thus excluding forces acting between one molecule and 

 another except during the encounter of two molecules. 

 This restriction, however, does not appear necessary, and 

 indeed it is easy to remove it. 



For the result of the proposition is to prove that if we 

 define the group {A) of molecules as consisting of those 

 whose generalised co-ordinates {q) are between certain 

 limits {q and q -f- dq), and whose generalised momenta 

 (/) are between certain other limits (/ and/ + dp), then 

 the average number of molecules in the group is — 



A c dp^ . . . dp^^^ dq-i . . . dq^^^, 



where A is a constant which is the same for all groups 

 of molecules of the same kind, but is different for dif- 

 erent kinds of molecules in the same mixture, but /i is 

 the same for all kinds of molecules, x 's the poten- 

 tial energy, and T the kinetic energy of a molecule when 

 in the state {A), and dp-^ . . . dp^^ are the differentials of 

 the components of momentum, and dq-^ . . . d q^^ the dif- 

 ferentials of the co-ordinates. The continued product 

 of these differentials specifies the extent of the group. 



By integrating this expression with respect to any one 

 of the variables, we may ascertain the average number 

 of molecules in a larger group, in which that variable 

 does not form a ground of subdivision. For instance, if 

 we integrate with respect to all the co-ordinates, we 

 arrive at a group consisting of all the molecules whose 

 momenta are between certain limits, or by integrating 

 with respect to the momenta we form a group of mole- 

 cules whose configuration lies within certain limits. 



In this way we obtain two very important results : — 



1. The average kinetic energy of a molecule is — y 



where m is the number of degrees of freedom of the 

 molecule. This is independent of the position of 

 the molecule. 



2. The average number of molecules whose configura- 

 tion lies between certain limits is — 



— Ax 



A e dqi . . . dq^^, 

 where x is the potential energy of the molecule, arising 

 from forces either internal to the molecule or tending to 

 fixed centres, but (according to Mr. Watson) excluding 

 intermolecular forces. 



But as our definition of a molecule is of the most 

 general kind, nothing is easier than to take into account 

 any intermolecular forces by simply including within 

 our " molecule " all those molecules between which 

 intermolecular forces are exerted. 



For instance, there is nothing to prevent us from defin- 

 ing as a molecule a material system consisting of one 

 atom in Sirius, .another in Arcturus, and a third in Alde- 

 baran. If the universe is supposed to have attained that 

 condition of thermal equilibrium to which alone these 

 propositions apply, the average kinetic energy of each of 



these atoms will be — , because each has three degrees 



2 //, 

 of freedom. 



That of the system of three atoms will [be the sum of 



the kinetic energies of the three atoms, namely, - 



We might obtain the same result from the consideration 

 that this system has nine degrees of freedom. 



The centre of mass of the three atoms is a mathe- 

 matical point at an immense dis'ance from any of 

 them. It has, of course, three degrees of freedom, 

 and the kinetic energy of a material particle whose mass 

 is the sum of the masses of the atoms and which moves 



as the centre of mass does is ^, 

 2/1. 



The value of the kinetic energy of the centre of mass 

 will be the same for any system of atoms provided that 

 every atom of the system is liable to encounters with 

 atoms not belonging to the system. Of course if we take 

 into our " molecule " all the atoms of a material system 

 unconnected with any other system, its centre of mass 

 will not be agitated at all by the mutual actions of the 

 atoms during their encounters. 



And here we must notice a point to which Mr. Watson 

 has adverted only in a note at the foot of p. 20 — the 

 definition of the motion of the medium as distinguished 

 from the motion of agitation of the molecules. For this 

 is connected with the weak point of the demonstration 

 and shows us the way to strengthen it. 



The weak point of the demonstration is the tacit assump- 

 tion that the sum of the potential and kinetic" energies of a 

 pair of molecules is the only function which does not 

 change during their encounter. For there are other 

 quantities which are not altered by the mutual action of 

 t o bodies, such as their masses themselves, the sum of 

 their momenta resolved in any given direction, and their 

 angular momenta about any fixed axis. 



Hence if instead of T = \ M (//^ -\-v^ + -w^) we write 

 in the expression for the distribution of velocities — 



