308 Mr. S. H. Burbury on some Problems 



the integral being over the whole spherical surfaces, that is 

 for all directions of P'Cp'. 

 Also 



log (F/) = log {F(OP). f(Op} 



--A(1TF=V. + ,gL ,) + X, + * + log(N,(f J(^f} 



And therefore 



jjrfSlog(F/)(F/'-F/) 



:N«(^ ) S (^)V* (^tvh - kp!)47r|rT - |2 _ (XT - yiK 



the last factor being the square of the mean minus the mean 

 of the square of X P + x p> for all directions of PCp. 



Hence we can find -=- in the form 

 at 



dt 



= 4tt — !— Nn( — ( — I dV.Y* \ oIUtts^Wx 

 m \ w J \ ir J J J 



x e-*( M+ » V2 +MTS p 2 ) tf^ sin . }XT^_ (X+^|h 



in which R is written for p. 



m 



Examples. 



14. Diffusion. — Two reservoirs, A and B, are connected by 

 a uniform horizontal tube. In A is a mixture of two gases, 

 gas M and gas m, in certain proportions. In B is a mixture 

 of the same two gases in different proportions. The tempera- 

 ture and pressure of the mixture are the same in either reser- 

 voir and at all points in the tube between them. Gas M is 

 of greater density in A than in B. 



Then a stream flows through the tube, of M from A to B, 

 of m from B to A. If the proportions in which the gases are 

 mixed in the reservoirs be maintained constant, as if for 

 instance the reservoirs were of infinite extent, the stream 

 becomes steady. 



