Probabilities to the Movement of Gas-Molecules. 251 



normal** Accordingly ^„, when n is indefinitely great, 

 becomes normal. 



There will, indeed, be some interdependence between the 

 elements which constitute the set of velocities after a great 

 number of collisions. But presumably this correlation will 

 be of that slight degree which is not fatal to the approximate 

 fulfilment of the normal law f. If N, the total number 

 4)f molecules clashing with each other, is sufficiently large, 

 any degree of approximation to the normal law can be 

 secured. For consider any particular degree of approxi- 

 mation — say, that which is effected by a linear combination 

 of a large number, v (e. g. a thousand), of elements taken at 

 random from an immense number N. And suppose a group 

 numbering /jl (e. g. a thousand) of compounds thus con- 

 stituted, each out of a different set of elements. Such a 

 group will be presented by the values of U (and likewise 

 of u) after a certain number (logy/log 2) of collisions, if N, 

 the number initially distributed according to any law of 

 frequency, is sufficiently large. It suffices that fiv elements 

 taken at random from among N should all be different — 



N — 1 IS — 2 N — LLV 



that is, that -^f— x XT — . . . x — ^— should be very approxi- 



JN jN IN 



mately =1. Since 



^-sH^D—fHsf)* W^ + iygN nearly, 



the condition will be satisfied if /n 2 v 2 is small with respect 

 to N (e. g. jj?v 2 a million million, and N a trillion). The 

 law then will be approximately fulfilled by a group of 

 velocities after a certain, not very great, number of 

 collisions. With these let the remaining N — \xv velocities, 

 after about the same number of collisions, now be grouped. 

 So far as these superadded specimens are formed from 

 entirely different sets (of the initial N) they render the 

 grouping more approximately normal in the degree to be 

 expected from the addition of independent observations. 

 So far as the elements which go to the N— /zv velocities 

 are not quite distinct, they will enhance the fulfilment 



* At any time, say vAt, after an epoch taken as initial, the velocities 

 ti x u-t . . . successively encountered by an average piston of mass M occur 

 with a frequency that is more nearly normal than were the corre- 

 sponding velocities at the initial epoch ; and accordingly the collisions 

 at the later period are more conducive to the normal distribution 

 of the U's. 



t Cp. 'Journal of the Koyal Statistical Societv,' vol. lxix. (1900) 

 p. 500, and vol. lxxxi. (1918) p. 624. 



S2 



