The motion of the stars 



105 



M{r) ' 



Simultaneously with the determination of the apex I try to derive the motion 

 of the invariable plane of the planetary system. The observed mean proper motions 

 of the stars indicate, in addition to the translatory movement of the system of 

 coordinates caused by the motion of the sun, a rotation of the system of coordinates 

 about an axis, having the coordinates 



= 186".7, 



s„ = + 150.1. 



This point nearly coincides with the pole of the Galaxy (a = 191", S = -f 27"). 

 We hence draw the conclusions: l:o that the correction to the constant of precession 

 used by Boss (that of Newcomb) is insignificant, 2:o that the invariable plane moves 

 with constant inclination to the Milky Way. The result of the numerical computa- 

 tion is that the node of the invariable plane on the plane of the Milky Way has a 

 direct motion amouting to 0". 003528 per year. 



In the second chapter I pass on to consider the frequency distribution of the 

 linear velocities. According to the theory of mathematical statistics the general 

 form — if it is of type A (compare Meddelanden N:o 4 Serie II) — of a frequency 

 distribution in three variables [x, y, z) is: 



i =3 



where (p^^, the so-called normal function, when the system of coordinates is suitably 

 chosen, has the form 



Here a-Q, y^, ^q, a^, «g, a^, as well as A..^, denote certain constants that may 

 be determined from the moments of the function . 



The constants , and depend on the moments of the first order; a^, 

 and on moments of the second order and, generally, A^.^^ on moments of the 

 ordar i -\- j -\- k and on moments of lower orders. 



The first term in the above expression, h. e. 'f^^, gives the approximate 

 value of $j . 



When this term alone is taken into consideration the surfaces of equal fre- 

 quency are^ ellipsoids. Considering the stellar velocities, the theory based on this 

 term alloue is called the ellipsoidal hypothesis. 



The problem is to determine the direction cosines and the length of these 

 axes. Through graphical considerations it was found that the velocity ellipsoid is 

 nearly an ellipsoid of revolution. The numerical computations were based on this 

 assumption. 



Lands Uiiiv:s Årsskrift. N. F. Afd. 2. Bd 8. 14 



