CHAPTER IV. 



The üf-theorem. 



26. AVhen Maxwell (1859) first took account, in the kinetic tlieory of the 

 gases, of the different velocities of the molecules in a gas, he was led to the law, 

 which bears his name, that, for a gas in equilibrium, the frequenc}^ function (/) 

 of the components of velocity must necessary take the form 



u'^ -\- -\- 



f= Ce 



supposing all particles to have the same mass. 



The demonstrations he has given at different times of this fundamental theorem 

 are, however, by no means successful. Even later attempts to give a direct proof 

 of the theorem have scarcely been satisfactory. 



It was hence an essential improvement when Boltzmann (1872) with the help 

 of his so called i?-theorem attacked the problem from an essentially new point of 

 . view, thus throwing light upon the matter mathematically as well as physically, and 

 simultaneously giving a new and rigorous proof of the theorem of Maxwell. 



In the researches of the physicists on this theorem there is generally to be 

 taken into account the special circumstances under which the gases are met with 

 in their laboratory experiments. The gases are usually confined in vessels, and not 

 exposed to outer forceg (except gravity) etc. 



In the astronomical applications we must, however, think of a free assembly 

 of particles (stars), where the density distribution at the outset is arbitrary, and 

 where no other limits are given than those determined b}' the motion and the 

 mutual attraction of the particles. 



27. Let / be our frequency function so that 



de^ d^ 



denotes the number of stars which at the time t lie between the limits 



^ ^ ^tX) 



z±\ds 



