78 C. V. L. Charlier 



Aas = yui2 — ^oj.'' , 



(46) = zi'f' — .TCO, , 



äz = XiOij — yiüx- 



But 



X = COS s cos a 



(47) y = cos S sin a 



'Hence we have 



^ = sin S 



Aa; = — cos S sin aAa — sin S cos aAS, 

 Ay = cos S cos aAa — sin S sin aA§ , 

 A^' = cos S AS 



or 



Aa cos S = — sin aAx + cos aA«/, 



(48) AS = — cos aAx — sin aù^y , 

 . cos SAS = A0, 



to that 



Aa cos S = sin S cos a ■-]- sin S sin a w„ — cos S co^, 



(49) AS = — sin a w^,. -|- cos a co^ 



Comparing these expressions with (40), we get 



co_j- = o) cos Sq cos a^ = — A?; = — 0". 003386 , 

 0)^ = Ü) cos Sp sin = r sin s = — 0". 000398, 

 CO- = w sin Sq = — r cos s = + 0.000917. 



These equations give tlie same values for a^ and Sq as before. E'urther we get 



to = + 0". 003528. 



so that the node of the invariable plane on the plane of the Milky Way has a direct 

 motion amouting to 0". 003528 per year. 



This numerical value must naturally be considered as only provisory and ex- 

 presses principally the order of magnitude of the motion of the invariable plane. 



The total period for the revolution of the node amounts to some 370 millions 

 of years. 



