Stellar velocity distribution 51 



The variations caused by this rotation in the angular coordinates of a star now get 

 the form : 



(43) cos S A a = sin o cos a co. c 4- sin S sin a <% — cos o w ; , 



A 3 = — sin a i» x + cos a (% . 



Considering however the expression for the direction cosines 



Y 12 = — sin S cos a, 

 Y 22 = — sin S sin a, 

 Y 32 = cos 8 , 

 Yu = — sin a , 

 Ï21 = cos a ' 

 the formula? (44) may be written 



(44) cos 8 A a = — y !2 w * — Y22 co ;/ — Ï32 w * > 



A § = Yu W.r + Y21 <V 



The quantities w x , co (/ and u> 2 being known, the formula (42) gives the position 

 of the axis of rotation as well as the annual amount of this rotation. 



Assume further the observed means of the radial velocities to be affected 

 by a displacement — K — of objective existence or due to an error in the mea- 

 surements, the final equations get the following forms: 



Tu *i U o" + Ï21 »1 K" + Tsi »1 W o" — Ï12 °>* — T22 w i/ — Y32 w * — x o = °> 



(45) Ti, *, U 0 " + Y 22 »1 V 0 " + y,, », ^0" + Tu w * + Y21 <•>, - y 0 = 0, 

 T13 ^o" + T 23 V + Y33 ^o" + # -^o = 0. 



47. Forming the normal equations, we get from each of the equations (45) 

 three different systems: 



a. Normal equations from the motion in right ascension: 



[Tn Yu] »1 U 0 — [Tii* 0 ] = °> 

 [7mTm]», n " [T« x 0 ] = 0, 



(46) [Yi 2 Ti 2 ]^ +[Ti2^o] = 0. 

 [Xn Ï22] w <> + [T22 «0] = 0. 

 [Y32iT 32 ] w * + [T32 *oî = 0 



b. Normal equations from the motion in declination: 



[T 18 ï 12 R 





— [Yi2«/ 0 ] 



= 0, 



[T22 T22] »i 



V 



- [t 2 2 y 0 ] 



- 0, 



[Ï32 Y 32 ] »1 





— [T32 !'o] 



= 0, 



[Yi, TiJ ««« 





— ['In I/o) 



= 0, 



[T21 Y21] °>S 





— [T21 «/öl 



= 0, 



