The motion of the stars. 2 



5 



or 



è N = iy N' 9- X ^ 



d' ^ N = — V 1 — 



"1 -^'20 „2 20 \ ^ „2] '^fi 



= v\,-\X,\ from (18) 



= ^0 + ^0' — '9'2^o''> from (9 

 But from (17) and (7) we deduce 



"1 10 1 -'^O '10 "''O' 



hence we have 



"2-^^20 '20 1^2 ^1 "^0 



From (16*) we have, however, 

 SO that, finally, 



I 



Q. 



In this manner we get the following formulae: 



Moments of the first order: 



(19) \X, = x,, 



^1 1^0 = ^0- 



Moments of the second order: 



(20) ^^^iSr^^_v,,g'-(l-g')V, 



*i'^n = ^1 (1 — 2') ^0 No- 

 where 



(21) q = l:q\ 



Moments of the third order: 



(22) = - 3v,, q' (I - q'^) + (2 - 'àq' + g'^*), 

 ^r' ^03 = - 3v„, ^0 5' (1 - g'^) + (2 - 3g' + g'=*). 



Moments 0/ /Ae fourth order: 



(23) »,W,,=v,„g'«--4v3,^o9'='(l-g'3) + 6v,oV2Hl-VM-g''^)-.Xo*(3-6g' + 4g'«-.7'^ 

 *iX4=V?' -4vo3M"(l-?") + 6W?'(l-2g'^+3'-Y-J/o^(3 -6g' + 4(/='-<^'«). 



For obtaining the value of the characteristic named the excess of the correlation 

 surfaces, we need to have the expression for N^^ — '^N^^^ and iV^^ — 32Vyg^, which are: 



(24) ^^^[N,,-^N,,^)=v,,q^-^v,,x,q'\l~q')^Q^^^^ 



-V(6 - 12g' + 3./H4g'-W), 



^AN,,---^N,,')--^^,,q'---^^>,M%\-q'^^^^^^ 



-^/(6-~12g' + 3^'^+4î'--g'«) 



From there formulée we derive the expressions for the sJcewness — Sx and Sy — 

 and for the excess — Ex and Ey — which are 



