The general Characteristics of the frequencyfunction of stellar movements 



63 



25. Now, to derive the mean errors of the characteristics of the motion in 

 space computed for, say, the galactic system of coordinates we may proceed hy a 

 method similar to that generally in use for least squares solutions Let us by if 

 denote a characteristic of the i:th order referred to the galactic system. It is clear 

 that by a least squares solution the values found for the characteristics if are the 

 same whether we apply the solution to equations of condition connecting the squares 

 with the galactic system directly, or, as has actually been done,' first go by the system 

 of the equator. Furthermore we should have found the same values of if if the 

 squares had been chosen equally symmetrical to the galactic plane as the actual 

 squares to the equator. Then we should also have obtained normal equations for 

 the if having the same left membra as those of Chapter VI, as these membra de- 

 pend only on the direction cosines of the squares. 



We shall suppose that we have applied the least squares method to such a 

 galactic system of squares, which, to avoid confusion, we will call the (/-squares. 

 Then between each //-square and the galactic system of coordinates we have equa- 

 tions of condition of the form : 



+ <2 W + < 3 * 3 + + ••■ + = ^ 



*iK\ €l + «22 6« -h "23 6" + + ••• + a Br€?) = Ii 



(79) 



W, 6? + < ^ + < i% + + ... + a» if) = 

 Having adopted a value of q, the yfj' are the » observed » characteristics of the 

 linear * motion as projected on the ,9-square. Naturally by the multinomial theorem 

 we have 



_ 3 .4 . 5 . ... (3 4- i — 1) _ 2.3. 4 . ... (2 + i — 1 ) 



Ii ,9 ~" ~~T± 



On account of the choice of the (/-squares the coefficients a% are the same as the 

 coefficients tabulated in tables II, III and IV so that: 



< — «* ; < = h > «8 « ï a- . 



(80) <> = a k ; a; 3 ; = 6, ; < = c, ; a% d k , 



<*=«*; <= 6 *; <= c *; °6*=<- 



The normal equations are now the same as in Chapter VI. A glance at them 

 will show that, with an approximation sufficient for our purposes, we may write them 



( 81 ) *U%«ip]^ = KpVn- 



Here 



[a' lp ] = 2 {a 2 lp 4- a\ v + aj, + < p ) , [a lp ijÇ] = Ï (a lp r,« + a 2p ij« . . . a . kj« ), 

 the sum being taken over all the squares **. 



* By the linear motion of a square we then mean the angular motion of a star placed at a 

 distance corresponding to the mean parallax of the square. 



** The index (t) of the a jp is here omitted for the sake of convenience. 



