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

 which occur, 



Further, in 2 can be substituted for the product m v m (k \ and since 

 the number of the combinations of N masses in twos is equal to 

 ^N(N — 1), for which we can write -JN 2 on account of the high 

 value of N, if r 2 denotes the arithmetic mean of all occurring 

 values of r 2 , we can put 



2m„7w ft r s = ^ N 2 #z 2 r 2 . 

 Z 



Inserting these values in equation (78) and dividing both sides 

 by 2, we get 



1 n ^ = n™zT 2 . 

 2 2 2 



In this the equality of the vires viva is expressed ; for the pro- 

 duct N-^-v 2 is the vis viva of the N molecules, and - N ^-r 2 may 



z z z 



be regarded as the vis viva of the \ N material points which have 

 also the masses m } and with which the mean value of r 2 is the 

 same as with the combinations of two molecules each. 



We have consequently here also arrived at the same result as 

 with the motion considered in the preceding section, viz. that 

 the J N material points moving in the given manner in the vessel 

 have the same mean ergal and the same mean vis viva as the N mo- 

 lecules flying through one another and striking against one another 

 in the volume V, and that, so far as these quantities are concerned, 

 the one motion can be replaced by the other. 



For the vessel we again choose a rectangular parallelepipedal 

 form, by which the periodicity of the variations of the coordinates 

 is attained. As, however, the expression N47rp 2 , to which the 

 superficies of the vessel must be equal, represents a very large 

 surface for the space-content V— N|-7rp 3 , we must assume that 

 one side at least of the parallelepiped is very small. Call this 

 side a. The ratio of the other two sides to each other we can 

 select as we please ; hence we will assume them to be equal, and 

 denote each of them by b. Then the space-content of the pa- 

 rallelepiped will be ab^, and its superficies 2& 2 -f 4>ab ; and putting 

 these quantities equal to the two above mentioned, we obtain for 

 the determination of a and b the equations : — 



«*=y-- N J«* I "... . (79) 



2^ + 4^ = N47r /3 2 . J 



For abbreviation, we will in these equations employ the sym- 

 bol S (before used) for the total surface of all the spheres of 



