Probabilities to the Movement of Gas-Molecules. 260 



conceives *. We are now concerned only with the free 

 molecules. Their number per unit urea and their mean 

 energy are to be considered constant. 



I. The velocity (in either direction) of any one in 

 particular may be considered as depending on an immense 

 number of preceding velocities, as in the simpler examples. 

 But the dependence is not now known to be linear. Let 

 / be the function whereby ti/ s the present velocity in the 

 horizontal direction of any particular molecule, is connected 

 with the velocities of n molecules at a prior epoch. Say 



(( r'=f(\ u r, tf<v-n n r> l v r, 2 l 'r---n r r), 



where the variables involved in the functions are velocities 

 at the prior epoch. Expanding/in ascending powers of the 

 variable, write 



«; = a l Sur + b l Su/ + c ] Zi( > z +...+a£r r + b 2 Zv ) 2 +... 



+ terms involving combinations of the t^'s and of the t»'s 

 inter sp, and of the w's with the v's. Here the symbol X is 

 for the moment used to denote loeighted sums ; since with 

 respect to any particular value of u/ a particular coefficient 

 attaches to each it. Similarly interpreted, 



u r ' 2 =• a^Xur 2 + 2a i 2 '2 l u r u s + by%u* + ... + a 2 2 Xv r 2 + 2a 2 2 ^v r v s 



+ tVXr, 4 + • ■ • + 2a l a 2 %i< r r s 4- . . .. 



But in forming the mean value of u, or any of its powers, it 

 is proper to treat one value of u as on a par with another. 

 Accordingly we may write 



[M /2 ]=a 1 2 [X^l+2a 1 2 [2w r w s ] + ...+a 2 2 Ltt' 2 i + :.. 



+ 2a 1 a 2 [lu r v s ], 



where 2 is interpreted as usual, and square brackets denote 

 mean values. Put a=w/n, rj = v/n, A = na, 



[u' 2 -] = AftXf] + 2A X 2 [Sv rVs ] + . . .A,* [V] + . • ■ 



+ 2A 1 A 2 ['£vr'n 8 '] i 

 and we have (neglecting some quantities of the order 1//?) 



[ M ' 2 ] = A 1 2 [2v 2 ] + 2A 1 2 [2^J+...A 2 2 [S^] + ... 



* According- to his" Assumption B," ' Kinetic Theory of Gases,' p. 11 

 et passim. That within the spheres of influence the distribution of 

 velocities must be normal may be concluded by the use of an argument 

 like our second. Presumably the whole class of molecules within such 

 regions may be broken up into species characterized by the stage of 

 approach — some couples for instance at the initial stage, others as close 

 together as possible, at the very acme of the encounter. To each stage 

 there would pertain a different coefficient of correlation. 



