II. 



Moments of the second order. The ellipsoidal hypothesis. 



18. General form of the frequency function of the linear velocities. Let, in the 

 system of coordiuates of § 9, 



U ' V ' W ' 



0 ' 0 ' ''0 



be the components of the absolute velocity of the sun. 

 Let furtlier 



u\ r, w 



denote the components of the velocity of a star relative to the sun, so that 



are the components of the absolute velocity of a star. 

 Further we denote by 



the components of the mean velocity of the stars. 



The mean velocity of the stars may suitably be called the ^^ream-velocity 

 of the stars. 



If now 



(50) N'^,(U\r',W')dU'dV'dW' 



denotes the number of stars having a relative velocity with the components U' ± \dTJ' , 

 V'±\dV\ W±\dW'. I have shown in my lectures on mathematical statistics, 

 that the function <ï>j, being supposed to belong to the A-type, can be expressed in 

 the form 



&> — ,r (TT' V WM I ßäoo ^^'f' i ^ßäi o ^''Ti i ^ßgo, 8^^, 



where 



_ {V + Up' - Ih'T- _ (!''+ Fq' -Fi')2 _ {W'+W»' - TFi')2 



Oj Og (2ît)/- 



and ßgoQ, ßgjQ, ß^oi etc. are certain functions of the moments of the frequency function. 



