The motion of the stars 73 



The mean errors in and (for a square) hence are nearly twice as large 

 for stars of the magnitude as for stars of the 5**^ magnitude. 



Taking the means of the mean errors in a and S for each magnitude we get: 



4'" : £4 = ± 0.3052, 

 5» : S5 = + 0.1664. 



The mean errors in t)-, Î7", F", ^■^W" are now found from (40). The 

 coefficient of these quantities being approximately of the same magnitude (= 32), 

 we conclude that the mean error is the same in each of the three components and 

 we have: 



£ (d^ IJ") --= £ [^^ V") = £ W") = 



S 



]/ 32.00 ' 



where for s we have to put or s^. 

 We thus get: 



4™ : £ U") = £ (^1 F") = £ W") = ± 0.0540 



5™: B{d-^U") = B{d-^r") = B{^,W") = ±0.02M. 



13. For deducing the mean errors in \>j,9, D and A we may proceed in the 

 following manner. Let x, y, z denote any rectangular coordinates (or velocities) 

 and let x^, y^, be their mean values and 0 the mean error in x, supposed to be 

 equal to the mean error in y or in ^. Then 



dxdydz 



gives the probability that x, y, z are included between the limits x±\dx, yjr\dy, 

 2 ±^dz respectively. 



If now polar-coordinates are introduced through the foriuulœ 



y = {J cos 0 cos 4" 

 x — p cos 6 sin 4» 

 = p sin 6 , 



which give 



!/o = Po cos 60 cos 4o 

 . iCo = Pq cos 6q sin 



^0 = Po sin 00 



then we find by the known transformation of this multiple integral that 

 P« . dy dQ d^ 



V2 



gives the probability that p, 0 and 4 are included between the limits p ± ^ dp , Q ± \ dQ 

 + ^d'^ respectively. 



From this expression we immediately find the values of the mean errors in 

 Po, 00 and ^p, namely 



Lunds Universitets Årsskrift. N. P. Afd. 2. Bd 8. 10 



