50 



Walter Gyllenberg 



General formulae for determining the apex direction and related 

 constants. 



45. Let, as before, the relative velocity of the sun in the system K 2 be ex- 

 pressed through the three components 



U 0 '\ F 0 ", W 0 ". 



Assume further the displacement of the mean in each square to be caused by 

 the sun's motion, then these displacements 



may be expressed in linear measure through 



(39) ?/o = ^I r o. 



Here $ 1 denotes the mean of the reciprocal value of the distances, so that 



(40) »i=^(})- 



The third coordinate, the mean from the radial velocity observations may be 

 considered as indepedent of the distances. 



Knowing the three volocity components in the K s system, we may according 

 to the scheme page 13 express them in relation to the system K 2 so that 



Yn », U 0 " + T« ». V + t„ », W 0 " -x 0 = 0, 



(41) T» », ^o" + T 22 », F- a " + TS », WV' - y 0 = o, 

 Ti3 U Q " + t 23 K 0 " + t 38 ^o" - *o = 0- 



46. It is however possible to generalize these formulae. 



At first we may assume that the node of the invariable plane has a retro- 

 grade motion in relation to a fixed plane '. Charlier has deduced the necessary 

 expressions, but owing to the new definition of the positive axes I will here re- 

 print the complete formula. 



Let the velocity of rotation about the instantaneous axis be to, then we have 



tà x = eu cos § 0 cos a 0 , 



(42) w ;/ = to cos 8 0 sin a 0 , 

 wc = to sin 8 0 



1 Se Charlier loc. cit. pages 76—78. 



