56 



C. V. L. Charlier 



5. In the following table II I put together the values of the characteristics for 

 the different squares. The means and y^, as well as the dispersions and Oy, 

 axe here (as in Tab. I) expressed in class-breadths. For obtaining their values in 

 seconds of are we have to multipl}^ by 0".05. 



The values of and «/^ — giving the mean displacement of the stars for 

 each square — determine the motion of the solar system in space. These values 

 shall be discussed beneath. 



The mean values of the dispersions, a, for the different groups are the follow- 



ing: 



Mean o^. Mean g„ 



.0 1/ 



Stars of the 4*1^ magnitude. . . . 1.426 + 0.068 1.249 + 0.076 



» » » 5*^^ » .... 1,284 + 0.057 1.209 + 0.050 



All-stars < 6™ 1.414 + 0.068 1.319 + 0.044 



The stars of the 4*^ magnitude give a greater value for o than those of the 

 5*^ magnitude, as ought to be the case owing to their different mean distances. The 

 mean errors of the dispersions are however so large that it is not convenient to 

 compute the proportion between the distances from these values. 



A glance at the correlation tables suffices for convincing us that the proper 

 motions are by no means symmetrically distributed about the mean values of x 

 and y. Generally the correlation table is considerably prolonged in a certain di- 

 rection, which shall be called here the line of symmetry of the correlation table. 

 Its position can be found analytically from the moments by the formula 



(28) tg2<p=-^^, 



where © signifies the angle between the axis of symmetry and the axis of x. The 

 formule (28) determines two directions in right angle to another. The choice be- 

 tween these directions is given by the sign of Vj^ (or r, which has the same sign 

 as VjJ. It Vjj is positive we have to take a direction in the quadrant, if v^^ is 

 negative the line of symmetry lies in the second quadrant. 



Tracing the lines of symmetry for the different squares on a globe, we find 

 that they, prolonged, nearly go through the same point of the sphere, the apparent 

 vertex ^). This point does not coincide with the apex of the solar motions. This is 

 what can, indeed, be easily inferred from the correlation tables. The direction, for 

 each square, to the apex is, narhely, found through drawing a line from the origin 

 — indicated by the point of intersection of the horizontal and the vertical lines in 

 each diagram — to the centre of gravity of the correlation table (a centre nume- 

 rically given by the values of x^ and A glance at the tables shows that this 

 direction generally does not coincide with the line of symmetry. This important 



') We shall find beneath that this point does not coincide with what Kapteyn defines by 

 that name. 



