40 THE DYNAMICAL THEORY OF OASES. 



Theae are the principal functions of (, r), f whose changes we shall have to 

 consider; we shall indicate them by the symbols a, /8, or y, according as the 

 function of the velocity is of one, two, or three dimensions. 



2nd. Integration with respect to b. 



We have next to multiply these expressions by bdb, and to integrate with 

 respect to 6 from 6 = to b = <x> . We must bear in mind that 6 is a function 

 of 6 and V, and can only be determined when the law of force is known. 

 In the expressions which we have to deal with, 6 occurs under two forms 

 only, namely, sin' 6 and sin 1 20. If, therefore, we can find the two values of 



B,= \ tirbdbBm*0, and B t = f irbdb sin* 26 (8), 



Jo Jo 



we can integrate all the expressions with respect to 6. 



B l and B t will be functions of V only, the form of which we can determine 

 only in particular cases, after we have found 6 as a function of b and V. 



Determination of B for certain laws of Force. 



Let us assume that the force between the molecules M t and M t is repul- 

 sive and varies inversely as the nth power of the distance between them, the 

 value of the moving force at distance unity being K, then we find by the 

 equation of central orbits, 



n-l\aj 

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



time : x is therefore a numerical quantity ; a is also a numerical quantity and 

 is given by the equation 





The limits of integration are x = and x = x', where x' is the least positive 

 root of the equation 



