80 



C. V. L. Charlier 



As to o,', a^', Og', they are dependent on the moments of the second order, and 

 give the mean peculiar velocities o[ the stars parahel to the three axes of the sy- 

 stem Kj^. 



In the simplest case may be reduced to the first term in this serie. 

 Putting 



the surfaces of equal frequency are ellipsoids. Hence the name ^the ellipsoidal hy- 

 pothesis •>■> , which shall be the object of the discussion of this chapter. 



19. The mean value of Ü' is, according to (51) equal to — U^, h. e. the 

 mean value of a component of the relative velocity of a star is equal to the cor- 

 responding component of the stream-velocity diminished by the component of the 

 absolute velocity of the sun. For the sake of brevity I shall in the following, 

 generally, write 



Ü', r, w 



instead of, more completely, 



u' + u; - u,', r -f f; - f/, w -f w,' - w;, 



and similar abbreviations when the systems and are considered. 

 The expression for (pj^ may be written 



(52) ^^=--Re-''-\ 



where is a certain constant and 



(52*) /=^'f7'2-f B'F'2 + (7'TF'^ 



putting 



(52*^^) ^' = ;A-2' ^' = ;r^' ^' = ;r^- 



Oj Og 



The direction cosines of K.^ relative to are, according to § 9, given by 

 the scheme 





K,: 







X" Y" 



Z" 





Tji Tai 



Ï31 



K.: Y 



Ti 2 T22 



T32 



Z 



'!}3 T23 



T33 



SO that 



(53) F'==T2lC^+Y22^'+Ï23^> 



W" = ';,, C^+T32^+Ï33W^ 



and 



(53*) F=T,2C^" + Ï22^" + Ï,32>^"> 



W^^U, f7" + Ï23 ^" + T33 T^^"- 



