Mr. J. C. Maxwell on the Dynamical Theory of Gases. 145 



The paths described by mo- 

 lecules about a centre of force 

 S, repelling inversely as the 

 fifth power of the distance, are 

 given in the figure. 



The molecules are supposed 

 to be originally moving with 

 equal velocities in parallel paths, 

 and the way in which their de- 

 flections depend on the distance 

 of the path from S is shown by 

 the different curves in the figure. 



3rd. Integration with respect 

 to d~N ± . 

 We have now to integrate 

 expressions involving various 

 functions of £ , rj, £ and V with 

 respect to all the molecules of 

 the second sort. We may write the expression to be integrated 



n-5 



QYn- ^M^VzQ^ 2 d%d^ 



iff' 



where Q is some function of £, rj, £, &c., already determined, 

 and / 2 is the function which indicates the distribution of velocity 

 among the molecules of the second kind. 



In the case in which n = 5, V disappears, and we may write 

 the result of integration 



where Q is the mean value of Q for all the molecules of the 

 second kind, and N 2 is the number of those molecules. 



If, however, n is not equal to 5, so tLat V does not disappear, 

 we should require to know the form of the function/^ before we 

 could proceed further with the integration. 



The only case in which I have determined the form of this 

 function is that of one or more kinds of molecules which have by 

 their continual encounters brought about a distribution of velo- 

 city such that the number of molecules whose velocity lies within 

 given limits remains constant. In the Philosophical Magazine 

 for January 1860, I have given an investigation of this case, 

 founded on the assumption that the probability of a molecule 

 having a velocity resolved parallel to a? lying between given limits 

 is not in any way affected by the knowledge that the molecule 

 has a given velocity resolved parallel to y. As this assumption 

 may appear precarious, I shall now determine the form of the 

 function in a different manner. 



[To be continued.] 



Phil Mag. S. 4. Vol. 35. No. 235. Feb. 1868. L 



