24-26] The Statistical Method 23 



The general solution of equation (22) is therefore seen to be 



log/= dj m (u 2 + v 2 + w*) + a 2 mu + s ww + 4 mw 4- a, ...... (24). 



The constants 2 , ot 3 , 4 , a., may be replaced by new ones and the solution 

 written in the form 



lo g/= a i m [( u ~ w o) 2 + ( v - v of + (W- W ) 2 ] + , 



or, if we still further change the constants, 



in which A, h, u , v , w are new arbitrary constants. 



26. By giving different values to these five constants we obtain all the 

 steady states which are possible for a gas. The different values of the con- 

 stants depend upon the different values of %i, S% 2 , S% 3 , S^ 4 , 2% 5 , i.e., upon 

 the total energy, momentum and mass of the gas. We proceed to deter- 

 mine the relations between these constants and the corresponding physical 

 quantities. 



The value per unit volume of any quantity % summed over all the 

 molecules is given by 



2% = v xAe-^^-^+^-^+^-^Wdudvdw ............ (26). 



If we write u u = u, 



V - V = V, 

 W W = W, 



this becomes transformed into 



' ..... (27), 



and if we further transform variables according to the scheme 



U = c sin 6 cos <f>\ 



v = csin sin .......................... (28), 



W = C COS 



the equation becomes 



.. ............... (29). 



If we take % = 1, % is the number of molecules per unit volume, and is 

 therefore equal to /'. Equation (29) accordingly becomes 







1 / 7T 



and since the value of the integral is known to be -j \/ ^ 3 this gives the 

 relation 



A=J k3 3 ... ...(30). 





