The inotidii of the atars 



61 



denotes the luunber of stars at the distance r having an absolute magnitude equal 

 to ilf (± I (IM) and velocity components equal to Ü {±}dU) and V (± I dV). 



If here the apparent magnitude, m, and the apparent proper motions u and v 

 are introduced by the formulae 



M = m — 5 log r, 

 U — nr, V = vr, 



which give 



clM = dm ; dU — rdu , dV = rdv , 



then 



Do) ccg {m — 5 log r) cp^ {ru, rv) dr dm du dv 



denotes the number of stars at the distance r having an apparent magnitude equal 

 to m and proper motions (say in a and S) equal to u and v. 

 Let now 



$ (m, M, v) dm du dv 



denote the total number of stars, in the area w, of the apparent magnitude m, 

 having apparent proper motions equal to u and v, and let 



4>j(m, Ü, V)dmdUdV 



denote the total number of stars, in w, of the apparent magnitude m, having linear 

 velocities equal to U and F, then 



5 log r) 'f, (U,V) 

 5 log r) œ, [rii, rv). 



If U and V are indépendant of r — as must here be assumed as a first 

 approximation — then 4>i ()w, U, V) is obviously proportional to fi(U, V). On the 

 other hand we have not, generally, <I> (m, «, v) proportional to {u, v). 



Supposing the function f,{U, V) to be known, we are able to derive from 



(30) the form of ^ {m, u, v). We may, for instance, accept for '^^{U, V) the form 

 of Maxwell for the motion of the molecules of a gas. Even under this simple 

 supposition the evaluation of the integral (30) is rather difficult. Instead of directly 

 calculating 4>, we may, however, proceed in an indirect manner, and evaluate the 

 moments of <ï> instead of the function itself. From the moments we might then 

 determine the form of the frequency function in usual manner. The moments can, 

 however, always be found from (30). Let, inded, v^j be the moment, about the 

 origin, of the order i in ii and j in v, then 



(31) v'ij = jj dudv ^ (m, u, v) u''v^. 



— 00 



Substituting here for ^ {m, u, v) its value (30) and integrating first regarding u 

 and V and afterwards regarding r, then the value of v'y can be found, if for cp^ ( Ü, V) 



(29) 



whereas 

 (30) 



<ï>j {m, U, F) = üj I drr^I)(pQ{in — 



CK 



<[>(««, u, v) = CO I drr'^ B'fQ{m 



