supposed Irreversible Processes. 479 



bution (1) component velocities between the limits 

 U... U-MU, 



y ... v +dY, 



There we will call F spheres, or spheres of the class F. The 

 relative velocity, R, of any / with any F has components 

 W _XJ, v -V, 10 — W. The number of pairs, each containing 

 an / and an F, is f¥ da dv dw dXJ dV dW, or shortly /F da. 



10. Now adjacent to every/ sphere construct an element 

 of volume thus: — About its centre describe a spherical surface 

 of radius a (the diameter of a sphere). And the direction of 

 R being polar axis, let 0, </> be the usual coordinates defining 

 a point on that surface. Then the element of volume shall be 



a? sin cos 6 dd d<j> Rdt = drj dt. 

 It may be on either side of the / sphere according as 



6 > or < - . And it shall be so taken that an F sphere whose 



z 



centre is within drj dt is approaching the / sphere. 



If we neglect the rare cases in which one such element 

 contains the centre of more than one F sphere, the number 

 of these elements which contain an F sphere is, on average, 



/F da dt] dt. 



Its actual value at the initial instant will generally differ from 

 this average, because it depends on the chance distribution (2) . 

 Let this actual value be 



kfFdadydt, 



where k is a positive quantity whose mean value is unity, and 

 for which a value nearer to unity is more probable than one 

 more remote. 



11. Every one of these F spheres whose centre is within 

 one of the elements dr) dt will within the time dt after the 

 initial instant collide with the / sphere to which the element 

 belongs. We here neglect, as we may, the chance that one 

 of the two spheres whose collision with each other is arranged 

 may disappoint its partner by previously colliding with some 

 third sphere. The actual number of such collisions in the 

 time dt is therefore k/F da drj dt, and its average value is 

 J'F da dv dt. 



Each of these collisions diminishes the number of the class 

 /by one. 



