198 Prof. R. Clausius on the Theorem of the Mean Ergal, 



Integrating this from 5" = 0to^ = oo, we get, on the left-hand 

 side, according to (91), logtt; and on the right-hand side, ac- 

 cording to (84) we have to suppose 



py(*)&=i, 



Jo 

 so that the equation changes into 



log tt= log'(16mc«) -2 — - - W~n ft z*~^~f{z)dz. (104) 



n c Jo 



In order to calculate the integral which occurs in equations 



(103) and (104), a special assumption would have to be made 



concerning the function f 3 which determines the ratio of the 



velocities of the different points. Taking Maxwell's law as 



basis, according to which the expression a /_e - ^ 2 , given in 

 (85), would have to be substituted for f{z), we should obtain 



\ z* ~f{z)dz = xf - \ z^e-i^dz, 

 by which we should arrive at a gamma function, namely 



/*°° n-l 1 n-l Q™ O 



J^/^-^r-^. . (io5) 



For the following developments, however, it is not necessary 

 to know the value of that integral ; we may be content to use 

 an abbreviated symbol for it ; and for this purpose we will 

 put 



Z= (z^fWdz • (106) 



The two equations then become : — 



9 R n ~ 1 

 h=-Z"xV~^; -} 



n c 



_1 R l >' • ( 107 ) 

 logu = log(16mc 2 )-2^— -Z"w~n. I 

 n c J 



We will now apply these equations to the three directions 

 of the coordinates singly; and form the sum of the three 

 equations which thereby arise out of each of them, thus : — 



---2 /l 1 IX"- 1 



*,+VfV-!v(i + i + J)»-| 



log (u,lMl 3 )=log[(16m) 3 c^] -2— Z/Sf 1 +i + -Wi 



n \c x c 2 c 3 / 



We multiply the first of these equations by iNm. The left 



