256 Prof. F. Y. Edge worth on the Application of 



let x be sum of in elements, each a random specimen of 

 a group characterized by some definite law o£ frequency, 

 not all the laws identical, but each having the same mean 

 square of deviation, say W\m*. Then /, the sought 



frequency-function of x, is such that I flogfdx is a 



minimum, subject to the conditions \ fdx = const., and 



x 2 fdx = ^c 2 . Whence (if the average is zero) by 



parity of reasoning f(x) is proved to be of the form 

 Aexp.-.^/c 2 . Q.E.D. 



III. The proof by way of the " H " theorem is sometimes 

 introduced by a less complete proof which shows only that 

 the normal distribution of velocities is sufficient, not that it 

 is necessary, for stability f . This proof is readily adapted 

 to the case of motion in one dimension. Let NF(U) 

 represent the average number of pistons (of mass M) with 

 velocity between U and U + AU, say of the class U, which 

 occur at any time between any two points on the line 

 distant from each other by a unit of length ; and let f(u) 

 be similarly defined. Measure a short distance Ax from 

 the position of every molecule of class U throughout a 

 tract of unit length, to the right or left according as 

 U > or < it, any assigned velocit}^. Say J]>u: and put 

 Ax=rAt(U — u), where At is an interval of time so short 

 that the number of molecules making more than one 

 collision during that instant may be neglected. Then the 

 expected number of collisions produced by a specimen of 

 class U overtaking one of class u, during the instant At, is 



T3At(\J-u)ATJAiiF(U)f(u). . . . (12) 



That is the number of members lost to class IT by collisions 

 of that type during At. Let the respective velocities 

 resulting from such a collision be U' and u' (v! >U / ). 

 When a molecule of class v! overtakes one of class U', 

 the resulting velocities are the original it and U. Accord- 

 ingly J, the loss to class U through collisions of the first 



* The reasoning- is readily extended to the more general case in which 

 the mean square increases with the number of the elements ; the elements 

 have not all the same mean square, and are not measured from their mean 

 value. 



f Cp. Watson, ' Kinetic Theory of Gases,' § 14 et seq. 



% See below, p. 258 (16). 



