CHAPTER III 



The method of Charlier to express the moments of the linear 

 cross-motions through the moments of the 

 proper-motions. 



10. Taking out one region of the sky small enough to be regarded as plane 

 the stars in the region constitute a statistical population comparable as to their 

 projected motions. Taking as system of coordinates a system having its x-axis 

 directed in the plane of the equator towards growing right ascension and its «/-axis 

 towards growing declination, the components of the proper motions we denote by 

 u 4- oc 0 and v -f- y^. The components of the linear motion being U + V ~\~ Y 0 

 we have the equations 



if r is the distance of the star. 



If ,r 0 and ?/ 0 are the mean proper motions and A r 0 , Y 0 the mean linear 

 motions u, v and U, V are respectively the apparent and the linear peculiar 

 motions. 



By forming the moments we proceed in the following way : 

 Putting as moments about the origin 



(44) 



r . [u + x 0 ) = U+ X 0 

 r . (v 4- y 0 ) = V+Y 0 



v'v=.M{(u + x 0 )*{v + i/ 0 )J) 

 N' li = M((U+X 0 Y(V+ Y o n 



and as moments about the mean 



V y = M{u* vi) 

 Nr, - M( U* Vi), 



we derive the equations 



(45) 



v i i = v n ~i~ x o Vo 

 V 02 = V 02 + y\ 



