Probabilities to the Movement of Gas-Molecules. 249 



Whence log n r = log ?z_ r == Const. —XMr 2 . 



Put U = rAU ; X = A(AU) 2 ; ft 2 (AU) 2 = N2E, 



and there results for the distribution of U, 



AU Const, exp —AMU 2 ; . . . . (12) 



where A is to be determined by treating AU ns a differential, 

 multiplying (12) by U 2 and integrating with respect to U 

 between +oo and -co, and equating the result to N2E 

 fthe outside constant securing that Xn=K). By parity i£ 

 U and u are selected by a random sortition from an indefinite 

 number of values demarcated respectively by the small finite 

 differences AU and An, subject to the condition 



2 2 (MU 2 + mu 2 ) AU Au = constant, 



the most probable distribution of! U and u— the one which 

 in a prolonged series of trials will be most frequently attained, 

 and around which the sets of N will hover — is 



A UAu Const, exp -X(MU 2 + «w 2 ). . . (11) 



When differentials are substituted for finite differences 

 this may be described as the function which minimizes "H/' 

 subject to the imposed condition (expressed as an integral). 



2. Legitimacy of the argument. — May we regard the 

 velocities in a molecular medley as analogously determined 

 by a sortition subject to the conditions imposed by Dynamics? 

 The high authority of Professor Jeans may be appealed to 

 in favour of this view (' Dynamical Theory of Gases/ 3rd ed. 

 § 51 et passim). He confirms too the connected proposition 

 that the distribution of velocities in the medley tends to and 

 hovers about some ultimate form. Tait has thus expressed 

 the presumption in favour of this proposition: "Everyone 

 who considers the subject .... must come to the conclusion 

 that continual collisions among our set of elastic spheres will 

 .... produce a state of things in which the percentage of 

 the whole which have at any moment any distinctive pro- 

 perty must (after many collisions) tend towards a definite 

 numerical value; from which it will never afterwards 

 markedly depart" — with certain reservations (Trans. Roy. 

 Soc. Edinb. vol. xxxiii. p. 67). Professor Jeans confirms 

 this presumption when he shows, with the aid of Liouville's 

 theorem, that certain incidents adverse to the presumption 

 are not to be apprehended (Encyclopaedia Britannica, ed. 11, 

 Art. Molecule, p. 658 ; Phil. Mag. ser. 6, vol. vi. (1903) 

 p. 722). 



