Probabilities to the Movement of Gas- Mole rules, 271 



and the link AB turning about a fixed vertical pin at A ; 

 the same pin for all the corpuscles, which move in horizontal 

 planes that are indefinitely close to each other. Each 

 nucleus repels any other within a minute circle of influence 

 with a force depending on the horizontal distance only, 

 according to some unknown law. Let the angles made by 

 AB and BC with the horizontal be $ and i/r respectively. 

 The present velocities of any molecule <fi' and ^' are con- 

 sidered as each an undetermined function of previous <£'s 

 and ty's ; the two velocities of any molecule among the 

 progenitors being not now independent of each other. 

 Identifying the mean energy of a molecule as deduced (by 

 expansion of the functions) from that consideration with 

 what it is as given by dynamical theory, we find by paritv 

 of reasoning that <£', the present velocity of a molecule, is in 

 effect the (weighted) sum of innumerable prior 0'sand yjr's, and 

 likewise -<jr', of yjr's and <£'s. Accordingly, the distribution of cf>' 

 and yjr' is normal ; but as among the constituents, each pair, 

 e. g., <f> r and yfr r are now correlated, the exponent of the 

 resulting error-function may be expected to involve the 

 product, as well as the squares of velocities *. 



As before, we may pass from the distribution of the 

 " universe " to that of the genus f. 



Parity of reasoning is applicable to molecular motion of 

 the most general character; admitting movements of trans- 

 lation and other degrees of freedom. 



II. The result is reached more readily by the second 

 argument with regard to the free molecules moving (with 

 any number of degrees of freedom) in the space outside the 

 spheres of influence. We have now to determine the law of 

 distribution f, so that Xflogf should be a minimum, subject 

 to the conditions that 2/=const.,2/T= [T], where X is used 

 to denote integration with respect to all the velocities (or 

 components of momentum), but not the co-ordinates. T is 

 the quadratic expression in terms of the velocities (or com- 

 ponents of momentum) for any assigned values of the 

 co-ordinates, say 



= A n Q 1 2 + 2A 12 Q 1 Q 2 + A 22 Q 2 2 -f ...+«n7i 2 + 2rt 12 7^ 2 ..., 



* As to the formation of correlated compounds from correlated 

 elements, and other propositions implied in this paragraph, see Camb. 

 Phil. Trans, he. cit. p. 116 ct seq. 

 Cp. above, p. i'63. 



