304: Mr. S. H. Burbury on some Problems 



law of mutual action is known. Call this the second state. 

 The number of pairs, M and m, in the second state, at any- 

 instant is F/// R /2 dR' dS' dg' dff. 



Now by a known theorem (Boltzmann, Sitzungsberichte, 

 Vienna, 1871 ; Watson, ' Kinetic Theory of Gases,' Prop. III.), 



R 2 dR dS dq d6 = R' 2 dB! d& do* dff, 

 or, since 



dq= —pdt, and dq'= —p l dt, 



R 2 dR dS dd pdt = R /2 dB! dS' d& p'dt. 



The number of pairs which in unit of volume and time dt 

 pass by their mutual action from the first to the second state 

 is 



Fj,f p WdRdSdOpdt; 



and the number which so pass from the second state to the 



'Fj> , f p , md'R'd$'dd'p'dt. 

 If therefore 



the number of pairs which pass from the first to the second 

 state is equal to the number which pass from the second to the 

 first in the same time, and this being true for all positions of 

 C and of P, the distribution of velocities is unaffected by the 

 mutual action of M and m. 



If Fp' fp'zfiF-pfp, then we form — as before, and it will as 

 before consist of a series of terms, each of the form 



logJ4(Fp'/ P '-F P /„), 



and is therefore necessarily negative until for given C, F F >f pl 

 becomes equal to Fp/ p for all directions through C. 



8. The complete expression for -=- in case of elastic spheres 

 is 



JjV?<MS f ^RttsVR 2 UdSlog(F/){F7 / -F/}, 



where JJ^S denotes integration over a spherical shell described 

 with radii R . . . R + cZR about the point f tj f as centre, and 

 F/ 7 denotes the mean value of F/ for all positions of the 

 diameter of that shell. 



