594 Mr. J. H. Jeans on the Conditions 



Then a solution of our equations (20) and (21) will be seen 

 to be 



logft= X (*=l,2...co). . . . (22) 



Further, the difference between this value for log p k and 

 the most general solution for log p k which is such as to satisfy 

 equations (20) and (21) must be a quantity satisfying the 

 conditions satisfied by ^. In other words, the most general 

 solution of our equations consists of the superposition of 

 solutions of the type of (22). Let ^fr, -% 2 , Xv • • • X* De ^de- 

 pendent quantities, each satisfying the conditions already 

 postulated for ^, and let it be supposed that there are no 

 other such quantities, then the most general solution of the 

 equations of steady motion will be 



}og Pk =A 1 Xi + A 2X 2+ • • • +A 5 %„ 

 in which A x A 2 . . . A., are independent and, so far, arbitrary 

 constants. 



§ 15. The quantities A x A 2 . . . A s can be uniquely deter- 

 mined from a knowledge of the values of S% 1; £% 2 , • • • the 

 summation extending throughout the gas, and the various 

 sums accordingly each remaining constant throughout the 

 motion of the gas. Hence for a given mass of gas (the 

 values of ^ 1? X% 2 • ■ • being given) there is a unique solu- 

 tion for a steady state, provided that a steady state is 

 possible. 



§ 1(>. Let us next examine what quantities satisfy the con- 

 ditions assumed for ^. Firstly, if we take %i = l for a single 

 molecule, %i = 2 for a double molecule, &c, we see that ^ 

 satisfies the requisite conditions, and 2% is proportional to 

 the total mass of gas. Again, if we take ^2 == 2.E, where E 

 is the total energy of the molecule or system of molecules 

 (including, if necessary, the potential energy in an external 

 field of force), we see that % 3 satisfies these conditions. As 

 other obvious instances we may suppose ^ to represent the 

 amounts of translational or rotational momentum. As a 

 final instance we may consider imaginary molecules which 

 are capable of carrying a charge of electricity, and we notice 

 that the amount of this charge would be a possible value 

 for % . 



For instance, if we have a number of electrically charged 

 spheres each of mass m and capacity C, inclosed in a vessel 

 of which the velocity is (w , v , w ), the solution will be 

 found to be 



_ h \~ m ((u-ii )Z+(v-v n+{w-u- y-!)+ I (Q-Qo) 2 ] 



p = Ae L v ' *J J , 



in which Q is the mean value of Q, the electric charge. 





