64 Sven Wick8ell 



According to (77) the sought mean errors of the characteristics will be 



[ L%%J e ^^- la lp a lp ] 



In the method of least squares the »observations» are generally supposed to be 

 independent of each other so that the last term in the right membrura disappears. 

 Furthermore, if weights of the form 



had been applied to the equations (79) the formula (82) would take the familiar form 

 (82*) ^%%]^(^^) 2 = ^- 2 . 



Now, the mathematical treatment of the questions connected with finding the 

 mean errors is brought to an end, a treatment, which for the sake of completeness 

 I have considered appropriate to carry out at some length. Evidently the numeri- 

 cal determinations may be taken more in a way to obtain only the order of 

 magnitude. The question is then only: which are the values to be assigned to the 

 quantities which are a sort of general mean errors of the characteristics of the 

 i;th order in a ^-square? 



Denoting by Yi? the direction cosines of a ^-square we have for the second 

 moments in the square the equations 



^2 0 = ''"il ^ 200 H - T I 1 ^0 20 

 (83) N 0 g = "I i2 V 0 0 Y 2 2 ^ T 'o 2 0 ~T" 7 32 "^0 0 2 



^11 = Y 1 1 'C 1 2 0^ 's 0 0 ^ 0 2 0 )' 



and denoting by ß and X galactic latitude and longitude we have 

 Yj, = — sin X, y 21 = cos X , y 12 = — sin ß cos X , y 22 = — sm ß sin ^ , Y 32 = cos ß • 

 Putting q = 0.75 we see from the following chapter that 



K N< 200 = 2 - 4243 *i iY 0 2 0 = l - mo *ï iV 00 2 = 0.7482. 



Now Yu Yi2 ^ s tne coefficient Yi of table II, which we see is generally small, never 

 exceeding the value 0.3368. 



Hence N u is always smaller than 0.45 and generally much smaller. Compared 

 to N 20 and 2V 02 we will neglect the effect of N n . For jV 20 and iV 02 we will take 

 their mean values which are obtained by multiplying the equations (83) with 

 cos ßdX dß and integrating over the whole sphere. 



Thus we find 



»« M(N 20 ) = Vi *i (N% 00 + iv;'. 20 ) = 1.75 



»\ M(N 02 ) = '/ 6 *; (JV» 00 + iV' 0 ' 20 + 4 JV' 00> ) = Los 

 and the mean number of stars in a square 



