1. The fundamental equation of Stellar Statistics. Consider some 

 character (as brightness, motion etc.) of a star and suppose this character to have 

 different degrees of intensity for different stars. Let E denote the degree of inten- 

 sity of this character and 



'ß[E) dE 



tiie relative frequency (= probability) of stars having this character in the degree 



E ± V, dE. 



Observed from the earth the character in question njay appear to possess 

 the intensity e. Then, generally, e is a function of the distance (r) of the star 

 from the eartli and furthermore a function of E. Hence 



e = e(E, r) or E = E{e, r). 



We shall deduce an expression for the frequency of stars with an ob- 

 served intensity of the character within the limits e ± Vs de. 



The expression sought I shall call the fundamental equation of stellar 

 statistics. 



Take two concentric spheres with the radii r and r -j- dr. The space 

 between these spheres has the volume 'iizr'^dr. If a portion of this space is cut 

 off by a cone having its apex in the centre of the sphere and having the sohd 

 angle œ of any form, then the space between the spheres intercepted by the cone 

 has the volume 



cor^ dr. 



Let D be the density of the stars, then the number of stars in the element 

 of volume considered is 



D . (ür^dr. 



[D is generally a function of r). Of these stars 



D . wr^ dr . '£{E) dE 

 have the absolute intensity of the character considered in the degree 



E{^ E ± Vî dE). 



