270 Prof. F. Y. Edgeworth on the Application of 



Of the terms on the right some disappear because affected 

 with first powers of v and tj, the means whereof are respec- 

 tively zero, e. g., Xv r v s ; others for that reason combined 

 with the presumption that the v's and 77's are independent *, 

 e. g., ^vt] ; others because the number of the combinations 

 is of an inferior order compared with that of some which 

 are retained, e. g., Xv^ as compared with %v r 2 v s 2 f. There 

 are retained when n is large, only combinations equatable 

 to powers of [2u 2 ] and [2^ 2 ], say of 7c 1 and k 2 . Thus 



[V 2 ] = Ai% + 2BA 2 +....+ A 2 2 * 2 + 2B 2 £ 2 2 . 

 Also \_u' 2 ~\ =k 1 (by the conservation of energy). Therefore 



^ 1 = A 1 2 /. 1 + 2B 1 2 ^ 2 + ...+A 2 2 ^ + 2B 2 2 ^ 2 + ..., 



with a like equation for k 2 . 



Differentiating with respect to /q and k 2 separately, we 

 find A 1 = A 2 = 1, and every other coefficient =0. Thus u\ 

 and likewise v', is shown to be in effect a sum of numerous 

 independently fluctuating, randomly selected constituents ; 

 and accordingly each of these velocities is distributed 

 according to the normal law. 



To generalize this reasoning : consider first a simplified 

 example of two correlated co-ordinates. Imagine corpuscles 

 each consisting of two links AB and BO, as in fig. 3 ; each 



Fisr. 3. 



rod is of negligible mass, but there is a nucleus of mass 

 m at the extremity of each, at B and at C. There are two 

 degrees of freedom, the link BC turning about a joint at B, 



* Op. T ait, cited above, p. 252. Clerk Maxwell's similar judgment 

 (Phil. Mag. 1860) will surely not be questioned with respect to the 

 free molecules here considered. 



t For the proof of these propositions, see the proof of the law of error 

 given by the present writer in Camb. Phil. Tians. loc. cit. Part I. § 1. 



