The motion of the stars. 2 



3 



and 



(7) N\, = M{U) = X,, 



' Further we may write. 



(8) ^..^M[iu-x,y(v~y,yi 



From these formulée the fundamental relations between the moments are easily 

 derived. We thus get 



(^) ^ 20 ~ ^20 ~l~ ^0^^ ^20 ~ '■^ 20 



^02 ^02 "T ?/o ' ^02 ^02 ^1 



^ 11 ^ ^11 ^~ ^oVo' ^11 ~ ^ 11 ^oîfo' 



and similar formules for the moments of the linear velocities: 



(9*) 20 ~ -^20 -^0^' -^20 ~ 20 ^0^' 



0 1 

 ? 



0 ' 



N' — N -\- Y '-^ N ^ N' — Y 



02 -^'02 I -^0 ' -^'02 -^'02 ( 



2 



0 ' 



N' = N -\- X Y N = N' — \ Y 



11 -^'ll ^ ^0-^0' ^'ll 11 ''o-^o- 



Of the moments of the third and the fourth order it is sufficient here to con- 

 sider those with the indices 30, 03, 40 and 04. We have here to take the following 

 relations into consideration: 



(10) = + SVgo^:« + 



^ 03 ~ ^03 ~l~ ^^02!'o H~ ^0 ' 

 ^30 ~ ^ 30 20'^0 ~l~ ^'^0 ' 



^03 = %3 — 3v'o2«/o + 2^0^; 



and 



(10*) N\, = N,,-^3N,,X,+ X,\ 



N\, = N,,^3N,,Y,^ Y,\ 

 N,,= N\,-3N\,X, + 2X,^ 

 N,, = N',,-SN',,Y, + 2Y,\ 



Further is 



(11) v'^o = v^o + 4:v^^x^ -f ev^^a^o^ + 



^40 ~ ^ 40 SO'-^'O 20'''0 3aîQ* 



^4 = ^'04 - 4v'o3«/o + 6v',,y,2 — 3:(//. 



and 



(11*) N\, = N,, + ^N,,X, + 6N,,X,' + X,\ 



N'o, = + 4i^o3 + eN,, Y,^ + Y,\ 

 ^40 = ^'40 - ^N\,X, + (iN',,X,^ - 3X„*, 

 AT — N' 47V' Y 4- 6N' Y^ 3F* 



-'■'04 04 ^-^^ 03 0 I 02 0 ^ 0 • 



The relations between the moments of the observed proper motions and the 

 linear velocities is now found from (2). Supposing first that the velocities of the 

 stars are not dependent on the distance of the stars from the earth, this equatioi) 

 indeed immediately gives 



