52 Walter Gyllenberg 



c. Normal equations from tbe motion in the line of sight: 



[Tis Tis] U 0 " — [ T]3 * 0 ] = 0, 



(48) [y 23 Y 23 ] V-[Y2 8 ^o]=0, 

 [y 33 Tss] W 0 " - [t 38 * 0 ] = 0, 



In the last equation the coefficient of K denotes the number of squares. The 

 equations (48) are indeed identical with (5) chapter II. 



Solving the three systems simultaneously, we may multiply the last system 

 with Then we get 



48 U 0 " - [ Tll x 0 ] - [ Tl2 y 0 ] - [ Yl , 0 O ] » ± = 0, 



(49) 48 ,\ V 0 " - [ Til x 0 ] - [y 22 y J - [t 28 * 0 ] » t = Ü, 



48^-TTo" -[f„yo]-tr..*J»i = o, 



and 



[Yi2 Tu + Tu Tu] w x + [y 12 x 0 ] — [y„ yj = 0, 

 (49*) [y 22 y 22 + Tgi Y 2 i] + [Tu »o] — [Y21 2/ 0 ] = °- 



[T32 T 82 ] w ^ + [Ys2 x o\ = °- 



48X-[* 0 ] = 0. 



48. As seen from what precedes the solution of the motion of the invariable 

 plane as well as of the constant K is quite independent of the solution of the apex 

 direction and the solar motion. This will however only occur when equal weights 

 are given to all squares. 



If the weights however vary in any way or if the stars are treated separately 

 the equations get the form: 



P < l> i U 0 " — [yh px 0 ] — [y 12 py 0 ] — [y 13 P^} *, = 0, 



(50) P * x V 0 " — [y 21 px 0 ] — [y 22 py 0 ] — [y 23 P*>o] *i = 0, 



P ^ Wo " _ [ Ya2 pyj _ [ Ï3s p g o ] ^ = 0, 



Here |> denotes the weight in a single square aud 



P = £p, (i= 1, 2 . . . 48) 



The weights being assumed equal to the numbers of stars we get 

 p = n, P=N, 



where n is the number of stars in a single square and N the total number. 



In the formula (50) the unknown quantity K is rejected in the present solution 

 because the velocities are already corrected for this constant. 



