•ÖO c. V. L Charlier 



in these coordinates are somewhat larger than - hut, indeed, of the same order 

 of magnitude as — the mean errors in the coordinates of the apex, given in § 13. 



26. Consider, for a certain square, the projections U and V of the hnear 

 velocities on a plane perpendicular to the line of vision. The curves of equal fre- 

 quency are, if the characteristics of the higher orders are neglected, ellipses, having 

 the equation 



(83) ^ + ÎT ^ constant. 



•"20 ■^'20'^'02 -^02 



If the system of coordinates is turned through an angle tp«, given hy 



-^'20 02 



the new coordinates (system are parallel to the axes of the ellipse. 



This angle cp^ determines the direction of the line of symmetry of the correlation 

 surface of (Î7, F). 



In like manner we find, that the direction of the line of symmetry of the 

 observed proper motions u and v is determined by an angle 9, given by 



(84) tg 2^ := '^"11 



^20 ^02 



If the velocities of the stars in the Milky Way are represented by the gene- 

 ralized law of Maxwell (eUipsoidal hypothesis of Schwaezschild) then the gr^at 

 circles on the sphere, defined by (84), all meet in a point, which properly may be 

 naujed the apparent vertex '). In like manner the great circles defined by (74) also 

 meet in a point, which may the named the true vertex ^). 



The named lines of symmetry will also meet in a point, if the stars move 

 into two streams, where — for each stream — the motions of the stars follow the 

 frequency law of Maxwell (twostream hypothesis of Kaptetn). 



It may be shown that apex, the apparent vertex and the true vertex lie on the 

 same great circle. 



.Substituting the values (20) of N^^, N^^ and we have indeed 



(85) tp- 2cD - 2^11 — %' - 1)^0^0 



^20 — ^02 — {Q — ^1 i^o — Po I 



whereas for the apparent vertex 



2v 



tg 2f = 



11 



^20 ^02 



These formulae are valid for any system of coordinates. Now suppose, for a 

 moment, that a system of coordinates be used, where apex is the pole. Then in all 

 squares we have x^^ — 0, and hence 



*) equal to the true vertex of Schwarzschild (and of Kapteyn). 



^) Owing to the reason named in § i6, true vertex does not coincide with the point, which 

 Kapteyn (and Schwakzschild) understand with tliat name. 



