4 



c. V. L. Charlier 



(12) ^'.. = ^.^.N'.., 

 where we have used the abbreviation 



(13) -a- =ijf^^ 



r' 



The mean value of 1 : r' may be calculated from the investigations on the con- 

 stitution of the Milky Way. We have indeed, with the notations of I, 



+ CD 



(14) a{m) ilf,„ = j dy e''' \ (y) [m + yl 



— CO 



where 



+ 00 



(14*) a[m) ^fdy\ [y] % {m + y) 



— CD 



and it is supposed that all the stars considered have the magnitude m ± ^ dm. 

 Using the form of (y) and ^ (m -f y) given in I, we hence deduce 



(15) - ^^ = M^I^^^ = e^'^ + ^'\ 

 where 



(15*) ß (^^ (1 _ Xi) + m^Xj — wîXj), 



T = i^^PX,(l -XJ. 



Writing further 



we have 



(16) ^ = pq\ 

 which formula also may be written: 



(16*) d^ = V/'"- 



Introducing this value of in (12), we hence have 



(17) V^ = ^/+^q^+^^-^^^N'^. 



from which formula we conclude that the moments — and then also the characte- 

 ristics — of the linear velocities can be obtained from the observed values of the 

 proper motions in a and §, as soon as the values of the both parameters and q 

 can be determined. I shall show below how this determination can take place. 



3. Relations between the moments and the characteristics of the })roper motions and 

 the characteristics of the linear velocities. These relations are directly obtained from 

 the formulée of the preceding paragraph. As an instance of the procedure to be 

 followed I give here, in detail the calculations for deducing the value of N^^^. 



We have from (12) 



(18) ^2^;o = vV 

 According to (9*) we have 



-^'20 ■'-^ to -^0 



