CHAPTER I. 



The fundamental differential equation in statistical 



mechanics. 



1. Let us consider a part of space limited by a parallelopipedon having the 

 sides Ax, Ay and As. The centre of the parallelopipedon is supposed to have the 

 coordinates x, y, s. 



The sides Ax, Ay, Az are supposed to be small, butjiot infinitely small The 

 parallelopipedon must, indeed, be so great that a relatively large number of stars 

 or molecules have place within it. Where stars are concerned, we know from 

 Medd. 70 that the number of stars upon an average for the whole Galaxy may be 

 estimated at 0.25 per cub-siriometer, or at 250 stars per 1000 cub-sir. It is hence 

 admissible to think of Ax, Ay, Az as being of the magnitude 10 sir. 



As to the gases we know that the number of molecules amounts to 30X10'^ 

 per cub-mm. We may then use values of Ax, Ay, As of the order 10"* mm = O.l jj,. 



Within the space 



A9. = Ax Ay Az 



we suppose that there are stars having velocities in all possible directions. Let 

 u, V, w be the components of velocity and consider the stars within the parallelo- 

 pipedon 



Js = Au Av Aw 



having the centre u, v, w. The stars within zJe have then velocities between the 

 limits u ± ^ Au, v ± ^ Av, w ± ^ Aw. 



Even the quantities Au, Av, Aw may not be taken as small as we please. 

 In discussions regarding the stars we may assume, for instance, that these quantities 

 are of the order 1 sir. per stellar year (1 sir./st.). 



Let us now evaluate the number of stars situated within AÜ as well as within 

 Js. This number is evidently, supposing the density approximately continuous, 

 proportional to Ail As. and may be expressed through 



(1) A=fAQAe, 



where / is a function of x, y, z and u, v, iv as well as of the time t: 



f=-- f(t; X, y, z; II, V, ir). 

 * Compare Bokel: Introduction géométrique à quelques théories physiques. 



