CHAPTER II. 



The different systems of coordinates to which the velocities 

 are referred. 



8. In the present investigation all velocities and results are referred to right- 

 angular Cartesian systems of coordinates. In this paragraph these will be defined. 

 The denominations of the systems as well as the direction cosines of their axes are 

 taken from the memoir of Charlier x . 



The following three systems of coordinates are used : 



The system K v having its three axes coincident with those of the velocity ellipsoid, 

 the positive X-axis directed towards the principal vertex 2 . This 

 system will be further defined in connection with the ellipsoidal 

 hypothesis. 



The system K 2 is the usual astronomical system of coordinates. The X Y plane 

 coincides with equator, the positive X-axis is directed toward the equi- 

 noctium and the positive Z-axis towards the north pole. 



The system K 3 is determined through the position of each square. It has its posi- 

 tive Z-axis directed towards the centre of gravity of the square. 

 The positive X and Y axes are directed towards increasing right 

 ascension and declination. 



This latter system of coordinates is not quite identical with the system K A 

 used by Charlier in the cited memoir. There he assumes the positive X and Y axes 

 to be directed towards decreasing right ascension and declination. 



1 Meddelande fr. Lunds Astr. Obs. Serie II. No. 9. 



2 I will in the following use the indication principal vertex for the point corresponding to 

 the antivertex of Kapteyn. The addendum principal is necessary as the velocity distribution is 

 expressed through a three axial ellipsoid. 



