100 



c. V. L. Charlier 



8 Vdn/ 



or accordiug to I (52) 



(99) w,,. = 1. 3. 5 . . . 2^ ^2,.^-r.„ + irU'^_ 



The odd moments evidently vanish. 



The formula (99) allows to compute the moments of any order. It suffices 

 here to consider the moments of the second and tlie fourth order, from which the 

 values of the dispersion and of the. excess are obtained. We get 



The excess — E — is obtained from the formula 



E 



which gives 



(100) 



We draw from this formula at once an important conclusion: 

 If the law of Maxwell is strictly applicable to the motion of the stars and if 

 thus, the mean velocity of the stars is inversely proportional to the square-root of the 

 masses, then the frequency curve of the velocities must, necessarily, have a positive 

 excess. 



36. It is possible to proceed a httle further. It is a plausible assumption 

 that there exists a certain correlation between the masses of the stars and their 

 luminosity of such a form that a larger mass generally corresponds to a greater 

 luminosity. This is plausible from two reasons: first to a larger mass corresponds, 

 generally, a larger surface and, hence a greater amount of light, and secondly larger 

 masses are more slowly cooling and preserve longer a high temperature. Let H 

 be the absolute brightness of a star (= brightness at the distance of one Siriometer), 

 then we may put as a simple first approximation, 



(101) fl" = constant X 



where c is a parameter, the numerical value of which ought to be larger than 2/3. 

 (This value corresponds to a brightness proportional to the surface of the star). 



Introducing the absolute magnitude (M) of the star and the logaritmus {z) of 

 the mass, this relation may be written : 



(102) ^ = constans —%M, 

 where 



(102«) . ^^09^ 



From (102) it follows a simple relation between the dispersion in z [L) and 

 the dispersion in M (which has been denoted in these memoirs .by o^, namely 



(103) L = xo^, 



