Mr. J. C. Maxwell on the Dynamical Theory of Gases. 141 



the axis of x. Then the volume included between the four planes 

 and the two cylindric surfaces will be Vbdbdcj>St. 



If this volume includes one of the molecules M 2 , then during 

 the time 8t there will be an encounter between Mj and M 2 , in 

 which b is between b and b-\-db,an& </> between (f> and $-\-d$>. 



Since there are ^Nj molecules similar to M 1 and dN 2 similar 

 to M 2 in unit of volume, the whole number of encounters of the 

 given kind between the two systems will be 



VM^S^N^N 2 . 



Now let Q be any property of the molecule M„ such as its 

 velocity in a given direction, the square or cube of that velocity 

 or any other property of the molecule which is altered in a 

 known manner by an encounter of the given kind, so that Q be- 

 comes Q' after the encounter, then during the time St a certain 

 number of the molecules of the first kind have Q changed to Q', 

 while the remainder retain the original value of Q, so that 



SQ^N, = (Q'- Q)V6«£67^N^N 2 , 

 or 



^^i = (Q'-Q)V^^N^N 2 . ... (3) 



Here - — ~— - 1 refers to the alteration in the sum of the values 

 ot 



of Q for the 6?N, molecules, due to their encounters of the given 



kind with the dN 2 molecules of the second sort. In order to 



determine the value of — k— -, the rate of alteration of Q among 



all the molecules of the first kind, we must perform the follow- 

 ing integrations : — 



1st, with respect to (j> from </> = to </> = 27T. 



2nd, with respect to b from Z> = to # = x . These operations 

 will give the results of the encounters of every kind between the 

 dNj and c?N 2 molecules. 



3rd, with respect to <?N 2 , ov f^^rj^d^drj^d^, 



4th, with respect to c?N t , or f^^^d^drj^d^. 



These operations require in general a knowledge of the forms 

 of ,£ and/ 2 . 



1st. Integration with respect to <£. 



Since the action between the molecules is the same in what- 

 ever plane it takes place, we shall first determine the value of 



1 {Q!—Q)d(f> in several cases, making Q some function of f, 77, 

 Jo 

 and t. 



