142 Mr. J. C. Maxwell on the Dynamical Theory of Gases. 

 (a) Let Q=£ and Q' = £' l5 then 



(£Wi)# = 



Jo 



Mj.+ M 

 M, 



r 



By transformation of coordinates we may derive from this 



+ i(M-M 2 )(? A + |^))87r S in 2 (9-8M,(f ,-«fo,-ifc)}, (6) 

 with similar expressions for the other quadratic functions of 



5) Let Q=? 1 (? 1 2 +'/ 1 2 + ? t 2 ), and Q'= fjff •+*/?+&; 



then putting 



!«+,»+-&«=y 1 «, &&+'w,+«i&=u, y+v+tf-W 



and (? ,-?,)« + ('?,-'?,)*+ (Si-a^V, we find 

 ' (ri V^_f l V 1 ^)^= s ^ M2 4^in^ { (f-? 1 )V^ + 2? 1 (U-V^)" 



+ (m^m J (8,r ■fr , '- a * si " 2 *^»« -fjv*- 



These are the principal functions of f, 77, f whose changes we 

 shall have to consider ; we shall indicate them by the symbols 

 a. i ft, or y, according as the function of the velocity is of one, 

 two, or three dimensions. 



2nd. Integration with respect to b. 



We have next to multiply these expressions by bdb, and to 

 integrate with respect to b from b = to Z> = co . We must bear 

 in mind that 6 is a function of b and V, and can only be deter- 



yen 



