CHAPTER IV. 



The determination of the frequency distribution of the 

 velocities in space from the proper motions. 



13. Dividing the sky with Charmer in 48 »Squares» of equal area symme- 

 trically distributed with regard to the equator, and letting the direction cosines 

 Ïj3> Ï23' Y33 be the direction cosines of the centre of gravity of the square, we 

 have as system of coordinates of each pair of diametrically situated squares the 

 system denoted by 1. Such a pair I will in the following refer to as simply a 

 »Square». 



Having computed the moments of the apparent proper motions of a square 

 we obtain by the equations (55) (55*) (55**) the moments of the linear cross motions 

 expressed in q and d- t . 



Now the general line of progress will be the following: 



By help of the equations (29) and the five first of the equations (17) we de- 

 termine the characteristics of the motion as projected on the square. Then by 

 equations (31) and (30*) we express those 12 characteristics in terms of the 31 

 characteristics of the motion in space referred to the system I of each square. 

 Hereafter by aid of equations (36) and (43) we get the characteristics of motion as 

 projected on the axes of system I expressed through the characteristics of the motion 

 referred to the system II. Accordingly we obtain from each square 12 equations 

 of condition between the 31 unknown characteristics of the motion in space referred 

 to system II. Evidently the problem of finding these 31 characteristics is mathe- 

 matically determinate if we have recourse to observations in three squares. For the 

 characteristics of the second order — that is the ellipsoid — even two squares are 

 enough. Now it will presently be shown that all the equations are linear, and na- 

 turally we then apply the method of least squares to the material from the whole 

 heavens. 



For the characteristics of the ellipsoid we thus compute normal equations for 

 six unknowns from 72 equations of condition. 



For the characteristics of the third order: 10 unknowns from 96 equations, 

 and for the characteristics of the fourth order: 15 unknowns from 120 equations. 



