68 



C. V. L. Charlier 



Retaining the same value of k as before {k = 3.o) we now get 



q = 1 .2674 



and 



q' = 0.6325. 



It was found that such a variation in the value of q only shghtly influenced 

 the resulting values of the moments N„ . It suffices to give the values of the 

 skewness and the excess and to compare them with the values given in tab. V. 



Taking that into account I have based the following discussion of the motion 

 of the stars on the values of the moments given in Tab. IV. 



9. Determination of the apex of the motion of the sun. We begin with the 

 moments of the first order, which give the motion of the sun. 



We have in the following to consider three different systems of rectangular 

 coordinates, which I shall denote as K^, and K^. 



The system refers to the three principal axes of the velocity ellipsoid de- 

 fined below. The axis of z is directed against the vertex. 



The system is identical to the usual astronomical equatorial system of 

 coordinates; the axis of z being directed against the mean pole 1900.0, the axis of 

 X against the vernal equinox. 



The system has the axis of ^ directed against the centre of gravity of any 

 of the 48 squares, in which we have divided the celestial sphere. The axis of «/, 

 in this system, lies in the plan through the pole and the axis of the system K^. 

 Hence, the axis of x is parallel to the equator. 



Further I shall denote all coordinates and velocities referred to K^^ by one 

 comma at the top (X', Y', Z', U\ F', W a. s. f.); coordinates and velocities refer- 

 red to K.2 by two commas (X", Y", Z", U", F", W" a. s. f.), and coordinates 

 and velocities referred to without commas (X, Y, Z, U, F, W a. s. f.). 



Let Z7", F", IF" denote the components, in the system K^, of the relative 

 velocity of a star with regard to the sun, and Ü, F, W the components of the 

 same velocity in the system K^, then 



C^" = T,if^ + T„F+T.3l^, 

 (39) V" =^'i,,U+'i,,V +'i,,W, 



W" = Tsi U + T3, V + T33 



or 



U = '!,,U" + F" + Ys> W", 

 (39*) V=^,,U" ^^i,,V" + -!,,W", 



where the direction cosines 7.. have the following values, expressed in the right 

 ascension (a) and the dechnation (S) of the centre of a square : 



