Summary. 



If all the stars possessed the same luminosity, then all stars of the same 

 apparent magnitude would be situated at the same distance from the sun. In such 

 a case the distribution of the apparent proper motions would be proportional to the 

 distribution of the true linear velocities of the stars. The problem of finding the 

 frequency distribution of the stellar velocities would then be solved — or at least 

 mathematically determined — as soon as we know the apparent proper motions of 

 the stars in the different parts of the sky. 



We know that, in reality, the luminosity of the stars is far from being constant. 

 Removing" the stars at a distance of a Siriometer, the magnitude of the stars (in 

 this memoir called the absolute magnitude) has a statistical dispersion (a^) of some 

 two or three classes of magnitude, perhaps — though not probably — still more. 



This variation in the absolute magnitudes of the stars makes the distribution 

 of the apparent proper motions of the stars essentially different from the frequency 

 distribution of the (projected) linear velocities. The principal difference ought to 

 be that, in the frequency distribution of the apparent proper motions, the very 

 small proper motions occur, proportionally, much more frequently than in the 

 distribution of the linear velocities. Using the language of mathematical statistics 

 we may express the same thing so, that the frequency curve of the proper motions, 

 say, in right ascension ought to have, generally, a positive excess, even if the fre- 

 quency curve of the corresponding linear velocities had a normal form. 



The first problem met with in discussing the motion of the stars is, hence, to 

 deduce, from the distribution of the observed propei- motions in a and S, the fre- 

 quency surface of the corresponding projected linear velocities. In order to solve 

 this problem we ought to know the number of stars of different apparent magni- 

 tudes and the dispersion [o^ of the absolute magnitudes. A mathematical analysis 

 shows that the caracteri sties of the linear velocities may be derived from those of 

 the proper motions ivith the help of a single parameter [q). This parameter is by 

 the formula 



q = e 



related to the constants h and X^, defined in the first of these memoirs here cited 

 as I. (bis a numerical constant = 0.4605). As to h its value is, relatively, well known. 



