Probabilities to the Movement of Gas-Molecules. 251 



first try to find linear functions for U and u in terms of 

 U' and u\ say U = AU'-fIW and u=aW + bu'. The 

 Liouville condition gives one equation for the coefficients, 

 viz. Ab — dB = l. Two more equations are given by the 

 identity MU + mw = MU' -\-mu f : and five more by the identity 

 MU 4 -t-wnfc 4 =MU' 4 + raw' 4 . Though these eight equations 

 are not all independent, it will not be possible to find values 

 of the variables which satisfy them all; even admitting 

 imaginary roots. If the value of U, and likewise v., in terms 

 of U' and u is other than linear, it appears necessary to 

 increase the number of equations, with the result that a 

 solution is unattainable. 



It is therefore significant that when the conditions im- 

 posed are the conservation of energy and of momentum, the 

 condition (13) is fulfilled by the expressions for U and u in 

 terms of U' and u' in the simple case of one dimension (1) ; 

 and likewise in more complicated collisions and encounters. 



1. Dense gases. — The analogy of general statistics suggests 

 a warning against the danger of supposing that the premiss 

 of independence (above II. 1) ignored by the second argument 

 is otiose. Consider a normal surface formed by the super- 

 position of n two-dimensioned elements with laws of frequency 



& = &(fi,V), &=*.(&%) (14) 



Let (if,., 117,.), ( 2 f r , 2 Vr) • • • be successive concurrent values 

 of the variables pertaining to <j> r ; and let 



«#= s £i+s?2+ •••+*?», s y— s 'n\-\- s r n 2 + ...+ s r} n ; 



the %'s and the ?; 5 s and accordingly the aggregate #'s and y's 

 being measured from their respective average values (Camb. 

 Phil. Trans, p. 116, 1904). Given that the mean value of 

 Maj' + my 2 is constant, we might conclude by the second 

 argument that the frequency-distribution of x and y was of 

 the form Const, exp — \(M.x 2 + my 2 ). But if the values 

 of £ and y in each element are not independent, the distri- 

 bution of the a?'s and y's will not be of that form, but of one 

 like (2) above, containing the product of the variables in 

 the index. Yet the argument is not fallacious ; it gives the 

 right answer relating to the data, which is all that can be 

 expected from Probabilities (Cp. Keynes, 'Probability/ ch. 1, 

 et passim) . The answer would be true on average in the 

 long run of different instances if positive and negative 

 correlation were equally probable (a supposition perhaps not 

 relevant here). It is true of any particular case on the 

 supposition that we no longer tabulate each y against the x 

 with which that y is formed concurrently, but having dispersed 



