18 



Sven Wicksei] 



that the ellipsoid obtained by only employing the position angles, regardless of the 

 closeness of approximation, belongs as well to the apparent as to the linear motions, 

 while the correlation ellipsoid has analytically different form for the two kinds 

 of motion. 



8. In the following we will refer to four different kinds of systems of coor- 

 dinates. We denote them with the numbers I, II, III, IV, and they are defined 

 as follows: 



I. A system having its If -axis directed toward a point of the northern 

 hemisphere and its K-axis in direction of growing northern declination. 



Referring to this system we use the notations 



U,V,W.A, B, G, D, E, F . B m , N ijk . 



II. The usual astronomical system of coordinates having its f7-axis directed 

 toward the vernal equinox and its W-axis toward the noithpole of the equator. 



The notations are here 



U", V", W" . A", B'\ G\ D", E", F" . Bfo, 



III. A galactic system of coordinates having its TT-axis directed toward the 

 north galactic pole and its U-axis towards the point a = 270° 3 = — 15°. 



Here we use notations with the index (.?). 



IV. The system of vertices, or the system where the function / has the form 



f=A'U' 2 + B' V' 2 + G W'\ 

 using here notations with index '. 



It will be seen in the practical part that HI and IV only slightly differ. 

 The direction cosines of the systems referred to the system II are given ac- 

 cording to the following schemes 



I — II 



U" 



V" 



W" 



III- II 



U" 



V" 



W" 



IV— II 



U" 



V" 



W" 



Ü 



T11 



V21 



Tbi 





a n 



«21 



«31 



XT 



£ 11 



hi 



S 31 



V 



V12 



V22 



V32 









«32 



V 



£ 12 



£ 22 



£ 32 



w 



Vis 



V23 



V33 



WW 



«13 



«23 



«33 



W' 



£ 13 



£ 23 



£ 33 



9. It will now be our concern to express the characteristics of the motion 

 as referred to one system of coordinates in terms of the characteristics as referred 

 to another system of coordinates. This constitutes the problem of the rotation of 

 a frequency function and involves the chief algebraical labour of this memoir. 



We take the case of transforming the frequency function of the components 

 of motion in system II into the frequency function of system I. The relative 



