30 REPORT— 1873. 



bution of velocity .amon^ the molecules of any given portion of the gas. He 

 assumes, as Olausius (at least in his earlier investig-ations) did, that the velocities 

 of all the molecules are equal, vrhereas I hold, as I tirst stated in the Philosophical 

 Magazine for .January 1860, that they are distributed according to the same law as 

 errors of observation are distributed according to the received theory of such en-ors. 

 It is easy to show that if the velocities are all equal at any instant they will 

 become unequal as socn as encounters of any kind, whether collisions or " perihelion 

 passages," take place. The demonstration of the actual law of distribution was 

 given by me in an improved form in my paper on the " Dynamical Theory of Gases," 

 Phil. Trans. 18(36, and Phil. Mag. 18iJ7 ; and the far more elaborate investigation 

 of Boltzmann has led him to the same result. I am greatly indebted to Boltzmann 

 for the method used in the latter part of the following sketch of the general 

 investigation. 



Let perfectly elastic molecules of different kinds be in motion within a vessel 

 with perfectly elastic sides, and let each kind of molecules be acted on by forces 

 which have a potential the form of which may be different for different kinds of 

 molecules. 



Let X, y, s, be the coordinates of a molecule, M, and ^, rj, ( the components of 

 its velocity, and let it be required to determine the luuuber of molecules of a given 

 kind which, on an average, have their coordinates between .r and x-\-dx, y and 

 y-^dy, z and s+ffc, and also their component velocities between ^ and |+(/^, v and 

 r}+drj, and f and C+d(. This number must depend on the coordinates and the 

 components of velocities and on the differences of the limits of these quantities. 

 We may therefore write it 



<?N=/(jr, y, s, ^, v, f) dx dy dz d$ dr, d(. (1) 



We _shall begin by investigating the manner in which this quantity depends 

 on the components of velocity, before we proceed to determine in what way it 

 depends on tlie coordinates. 



If we distinguish by suffixes the quantities corresponding to different kinds of 

 molecules, the whole number of molecules of the first and second kind within a 

 given space, which have velocities within given limits, may be written 



fi{ki,nuQdix,(kx,<Ki=ni, (2) 



and 



fAi.,r,,,Qdl,,dr,,,dC, = », (.3) 



The number of pairs which can be formed by taking one molecule of each kind 



is M, Wj- 



Let a pair of molecides encounter each other, and after the encounter let their 

 component velocities be ^,', i;,', f/ and l^', »;,_,', f^'- The nature of the encounter is 

 completely defined when we know I, — ^u Vi^^v d—Ci the velocity of the second 

 molecule relative to the first before the encounter, and .r,^— a;,, y., — ?/i, s^ — c, the 

 position of the centre of the second molecule relative to the first at the instant of 

 the encounter. When these quantities are given, ^/— |^,', r^.^'—q^', and f ,'— C/i the 

 components of the relative velocity after the encounter, are determinable.' 



Hence, putting a, (3, y for these relative velocities, and «, b, c for the relative 

 positions, we find for the number of molecules of the first kind having velocities 

 between the limits |i and ^,-f f/^, &c., which encounter molecules of the second 

 kind having velocities between the limits ^., and t^ + d^, &c., in such a way that 

 the relative velocities lie between a and a+da, &c., and the relative positions be- 

 tween a and a-\-da, &c. 



/i (^1 Vu CO d^ <^1 dC-A (L, V2^ Q d^ dq d(. (j> [abc «,3y) da dh dc da dfidy; . (4) 



and after the encounter the velocity of Mi will be between the limits |i ' and 

 ^\+d^, &c., and that of JM.^ between the limits ^.^ and ^^ + d^, &c. 



The differences of the limits of velocity are equal for both kinds of molecules, 

 and that both before and after the encouuter. 



When the state of motion of the sj'steni is in its permanent condition, as manv 

 pairs of molecules must change their velocities from V,, V. to V,', V ' as froiii 



[ 



