The general Characteristics of the frequencyfunction of stellar movements 73 



To find the mean errors in the positions of the vertices we take recourse to 

 the equations (61) and (63). Suppose we have computed the second moments re- 

 ferred to a system of coordinates whose axes deviate by infinitesimal angles from 

 the axes of the ellipsoid. Then the moments 2V 110 , N U)1 and 2V„ n would be small 

 quantities of the first order. The roots s v s 2 , s 3 of the cubic (61) would then be 

 found to deviate from the moments N 200 , N 020 and N 002 by small quantities of the 

 second order. In equations (63) we then may write 



xv 200 1 — U > ""020 b l IY 02Q -'■"200' X, 0Ü2 *1 iy 002 1Y 200 



^ 2 o« - », = N 2 - 00 - N 020 ; N 020 - s 2 = 0; N 002 - s 2 = A T 002 - N 020 



^00 - », - ^200 - ^ ^020 - *, - ^020 - #00« ^00* -»3 = 0. 



Denoting by X and ß the longitude and latitude of the vertices in the system 

 of coordinates considered, we find, neglecting quantities of the second order, 



For vertex I For vertex II 

 tg \ = K = iîH^Hr- tg (90 -X 2 ) = J - x 2 ^ - 



iV 0 , n ftV 2 2 X 9nn — N n9(] 



v...„ - K 



For vertex III 



tg (90 — ß 3 ) cos X 3 = 

 tg(90 — ß 3 ) sinX 3 = - 



X. 



200 



2V n 



where <p 3 and tj> 3 are the longitudes of vertex III counted from the W-axis along 

 the UW- and FTT-planes. 



Neglecting mean errors multiplied with the small quantities iV 110 , N l0i and 

 N on we finally have 



s(X 1 ) = S (X 2 ) = — £ -i^— ; .(ßj-ifo) 



» (ß 2 ) = « (-y 



^200 —^020' N 200~ N i 



We now insert the values given in the next paragraph for the second moments 

 and their mean errors referred to system III and thus obtain 



a (XJ = s (X,) = P.?; E (ßj = , (<p 3 ) = 10.4; s (ß 2 ) = s (-y = 6°.8. 



Obviously we may now put it in the following way: The vertices lie as far 

 as the mean errors are concerned within small ellipses of the following descriptions. 



Vertex I. Within an ellips having its major axis parallell to the plane of the 

 Milky Way. The axes are a = 1°.7 b = 1°.4. 



Vertex II. Within an ellips having its major axis perpendicular to the plane 

 of the Milky Way. Here the axes are a = 6°.8 b = 1°.7. 



Lunds Universitets Årsskrift. N. F. Afd. 2. Bd 11. 10 



