The motion of the stars 



71 



The equation for determiniug the position of the Apex, as well as the values 

 of I' and ài, now take the form : 



x,, = U" + Vai ^1 ^" + (^os £ cos 8 + sin e sin 8 sin a) — Ai sin S cos a, 



«/y = Tis'fl'i U" +722*^] ^" + T32'^i + 1' cos a sin s -f- sin «. 



It is, a priori, probable that the invariable plan of the planetary system has 

 a slow motion due to the attraction of the stars in the Milky Way. This motion, 

 if it exists, should probably consist in a conic motion of the axis of the invariable 

 plan about the axis of the ecliptic. If the observed values of and give such 

 values of V and A^ as are consistent with such an hypothesis, the rate of this mo- 

 tion of the invariable plan can be found from the values of T and ù^i. 



The main interest of such an investigation is that, from the motion of the 

 invariable plan, it should be possible to conclude the total mass of the stars in the 

 Milky Way, a coustant of the utmost importance for the discussion of the consti- 

 tution of our stellar system. 



10. Each square gives a relation of the form (41). Treating these 96 equa- 

 tions according to the method of least squares we obtain for each coordinate normal 

 equations where only one of the unknown quantities occurs. This simplification is 

 due to the symmetrical distribution of our 48 squares on the sphere. The motion 

 of the sun in space can hence be treated independently of the motion of the in- 

 variable plan. The value obtained for the position of the Apex has no influence 

 on tne conclusions regarding the motion of the invariable plan. 



11. We begin with the determination of the Apex. For obtaining a value of 

 the quantity Xj I treat separately the stars of the 4*^^ and of the ö*'^ magnitude. 



The proper motions in right ascension {x^^ give the following normal equations 



4'" 5™ 

 26.0000 !)■, L^" = - 0 1748 + 1.0022 

 22.0008 ^1 F" = — 18.1789 — 18.7731 



The declinations (y^) give: 



Ô.7680 x)-! f7" = — 1.1716 -f- 0.4310 

 9.6460 ô'j F" = — 9.0711 — 7.0178 

 32.5888 >r' = + 19.3751 -f 16.0758 



Taking together the proper motions in o. Mud §, we get 



31.7680 0- J f/" = — 1.3464 -f- 1.4332 



(40) 31.6468 0, F" = — 27.2500 — 25.7909 



32.5888 W" = + 19.3751 + 16.0758 



which give 



U" = — 0.04238 + 0.04511 



{>. F" = ~ 0.86107 — 0.81496 



■ö-, W" = -f- 0.59453 -|- 0.49329 



