Mr. J. C. Maxwell on the Dynamical Theory of Gases. 143 



mined when the law of force is known. In the expressions which 

 we have to deal with, occurs under two forms only, namely 

 sin 2 and sin 2 20. If, therefore, we can find the values of 



f* oo r% oo 



B = I 4firbdb sin 2 0, and B 2 = \ irbdb sin 2 20, 



Jo Jo 



(8) 



we can integrate all the expressions with respect to b. 



B x and B 2 will be functions of V only, the form of which we 

 can determine only in particular cases, after we have found as 

 a function of b and V. 



Determination of for certain laws of Force. 



Let us assume that the force between the molecules M x and 

 M 2 is repulsive and varies inversely as the nth power of the dis- 

 tance between them, the value of the moving force at distance 

 unity being K, then we find by the equation of central orbits, 



■ * -=r ,. *. ,.,_ „ . • o) 



Jo 



2 



sA-^-A©" 



where x= -, or the ratio of b to the distance of the molecules 

 r 



at a given time : x is therefore a numerical quantity ; a is also a 



numerical quantity and is given by the equation 



a=b \K(K+Mj (10) 



The limits of integration are x = and x — x\ where x' is the 

 least positive root of the equation 



i-* 2 - .i-ffy-^o (ii) 



n — \\u/ 



It is evident that is a function of a and n, and when n is 

 known may be expressed as a function of a only. 

 Also 



Mb "= \Ym l u, ) " d " ; • • • ' (12) 



so that if we put 



A 1= f 4W«sin 2 0, A 8 =l ir*doi$m*20, . (13) 



• Jo Jo 



A x and A 2 will be definite numerical quantities which may be 



