Probabilities to the Movement of Gas-Molecules. 24:7 



distribution; (b) between molecules engaged in an encounter 

 (referred to below, III. 3). 



5. Controverted points. — Our first argument has a bearing 

 on some vexed questions. Thus with respect to "time- 

 averages" it would hardly occur to one impressed with 

 Laplace's theory of error to question Lord Rayleigh's state- 

 ment that " for a single particle the time-averages of (u 2 

 and IP) are equal, provided the averages be taken over a 

 sufficient length of time" (Phil. Mag. ser. 5, vol. xlix. (1900) 

 p. 108). Again as to the reversibility of the motion, consider 

 Galton's mechanical illustration of the law of error. Shot is 

 poured in through an aperture at the top of the apparatus and 

 comes out at the bottom after repeated collisions with inter- 

 posed obstacles in the form of the normal curve ('Heredity 

 and Genius/ p. 63; Yule, 'Theory of Statistics,' p. 298). 

 Xeed we trouble ourselves with the thought that if the shots 

 were dropped into the apparatus in numbers corresponding 

 at each point to the normal distribution (say by turning the 

 mechanism upside down) they might come out, after re- 

 passing the obstacles, in exactly the same arrangement as 

 that in which they first entered. Recognition of the leading- 

 law of Probability disposes the disciple of Laplace to accept 

 the warning of De Morgan, repeated by Tait with reference 

 to the Kinetic Theory of Gases (Transactions Roy. Soc. 

 Edinburgh, vol. xxxiii. p. 256): "No primary considera- 

 tions connected with the subject of probability can or ought to 

 be received if they depend upon the results of a complicated 

 mathematical analysis." 



II. The "H" Theorem. 



The second argument is based on the use of " H," defined 

 as the integral between extreme limits of flogf) where/ is 

 the sought frequency-function of the velocities (or the same 

 multiplied by, or with the addition of, a constant, I. c. p. 253). 

 The argument is to be distinguished from those in which 

 " H " is employed together with the premiss that the 

 frequency-distributions of the colliding molecules are inde- 

 pendent, e. g. by Boltzmann (' Gastheorie,' I. 5) and by 

 Jeans in \\is first appeal to the " H " theorem ('Dynamical 

 Theory of Gases,' 3rd ed., Art. 15 et seq.). Without that 

 assumption it is argued that the sought function / is that 

 which makes " H " a minimum subject to the constancy of 

 the energy (and momenta) of the system, 



1. Character of the argument. — The method is akin to the 

 use of Probabilities for the determination of constants per- 

 taining to functions of a given form when the data consist 



