(19) 



and 



(19*) 



= Mj + 



'1 '1 



2m,; C to 

 mj + w?2 



, 2m„ C CO 



^ + JWg 



?^ = 



î^o = — 



2w?, C w 



TOj + Wj, 



2w^ Ç CÜ 

 4- wig 



2mj C w 



+ ^2 



Statistical mechanics 

 — 6 sin a + cos ß cos a 



i cos a 4" ""l cos ß sin a 

 — Y] sin ß 



17 



C sin ß cos a) , 

 C sin ß sin a) , 

 ■ C cos ß) ; 



— ? sin a + Y) cos ß cos a — C sin ß cos a) , 



i cos a +• I'] cos ß sin a — C sin ß sin a) , 



■q sin ß — C cos ß) . 



11. The integral invariant in statistical mechanics. 



Let F be any function of b, ']j and of the velocities u^, v^, w^; u^, v^, tv^ 

 hefore the passage, and consider tlie integral 



(20) j F dh d'\ du^ dvj dw^ du.^ dv^ div^ 



taken over a certain dominion of the variables. 



If in this integral the variables h, 4", , ^2 ®*c- ^"^^ exchanged for the variables 

 h' , '!>', , etc. after the passage, we get 



(21) S = j I . F db' d<!^' du^' dv^' du\' du^ dv^ dw^ 



where the integration is to be performed over a dominion of the new variables 

 corresponding to the dominion of the old variables in the integral (20). Here I is 

 the so called lacobiana, or the determinant of substitution, which more fully may 

 be denoted: 



(b, '\) ; u^, v^, ; ti^, v^, 



(22) 



h' , 'Y; u^, Wj'; u./, w^' 



In the kinetic theory of the gases it is shown that this determinant has the 

 value 1. Using a terminology introduced by Poincaré, we may express the same 

 fact by the statement that the integral 



j db d<^ du^ dv^ dw^ du.^ dv^ dw^ 



is an integral invariant at the passage. 



The demonstration of this important theorem is however generally incomplete 

 and, especially in the fundamental treatise of Boltzmann (»Vorlesungen über Gas- 

 theorie»), erroneous. In the following I give a direct and elementary proof of the 

 theorem. 



Lunds Universitets Årsskrift. N. F. Avd. 2. Bd 28. 



