684 



are (he consequence of star-streams, or may be ascribed to an equi- 

 valent non-splierical distribution of tlie individual motions, which 

 we might call sytsti'inatic jyroper uiotluiis of tlie second land '), are 

 excluded from our investigation. 



In the fust place, then, the dependence of the pai'allax upon the 

 galactic latitude must be expressed in a simple formula; for the 

 derivation of this we have used the table given by Kaptkyn and 

 Wkeks-Ma ill their paper Fuhl. Groningen 24, 15 In that table 

 values for the mean parallax are given for the magnitudes 3.0 to 

 11.0, and for galactic latitudes: between — 20^ and -|- 20°, between 

 ± 20' and ± 40° and between ± 40° and ± 90°. For all mag- 

 nitudes the same ratio is assumed: between -^5 and .t„ and with 

 sufficient arcurac}' for our purpose — the table is given as "quite 

 provisional" — we could put: :rii-=^ Ji^il -\- c shf if). 



The three columns of Kapteyn and Weersma's table were assumed 



to apply to gal. latitudes of ± 10°, ± 30° and ± 60°, and it appeared 



that the coefficient c must be given a value between 0.60 and 0.70 



We assumed therefore 



,T,3 ^ -T-j (1 -j- 0.65 sin" jj) 



or 



K 



Ra = 



^ 1 + 0.65 siti' [i 



The relation assumed by EddiiNgton is equivalent to a formula of 

 the same form with .c = 0.60. 



Our value for R must now be substituted in the equations for 

 the systematic proper motion, whereby, for the present, we confined 

 ourselves to the terms de[)eiideiit upon a precessioii-cori-ection and 

 upon the parallactic motion. 



The usual equations are 



A' . Y 



H-j cos d =z Li III CON Ö {- L. n sin (f sin a -| sm a cos u 



li R 



Z X , F . . 



UC =z — — cos d -h ^ n cos a ~\ sin d cos a -\ sin d sin a 



' R ^ ^ R R 



Sui)stituting in these the value of R, expressed in R^, and after- 

 wards, according to the formula 



sin 3 z:= sin d cos i — cos d sin (<t ~&)s!n i 



V The frequency-surface may be more general than the eUipsoid, bul must, 

 according to our definition, have a centre, as the part of the movement that 

 depends upon the spherical place (Systematic Prop. mot. 1st liindj is subtracted 

 from the total movement. 



