60 



Sven Wicksell 



VN 



(71) 



]/ N 



/ö % 



moments Jfy m and iV 2 ;, 2m may be expressed in terms 

 we wish to fiud the mean errors of the moments and cha- 

 racteristics to the fourth order we must needs find the expressions for the above 

 moments to the eighth order. To the fourth order they are already given in for- 

 mulae (14) and (15). For the sixth and eighth orders I use the following method. * 



and, furthermore, 

 of N. n , JV„, N na . 



the 

 As 



Putting as in § 14 



we know that 



(72) 



I ! 



dx dy e 



By differentiating this equation three and four times with regard to the para 

 meters A ± , B t , F 1 , and multiplying with , all the moments of the sixth resp 

 eighth order come forth. We obtain, taking note of equations (28*) and (29): 



N 61 = 16N M *N U 

 N iO = 3N 2l *N 02 +\2N 20 N u * 

 (73) N 3a = 9N 20 N 02 N u .+ 6N n s 



N u = 'ÔN 02 *"N S0 +l2N 0S N n * 

 N u = 15^ 02 2 N u 



K (t =H)»N 20 < 



N 62 = 15N 20 *N 02 + 90N 20 *N n * 

 (73*) N lt = 92T M - N w * + 72tf 20 N 02 iV n 2 + UN tl * 



N 2e =lbN 2(l N 02 *+ 90 N 02 >N n > 

 2V 08 = 10ÖN 02 \ 



and the remaining moments of the eighth order, which are not needed here, are as 

 easily derived. 



Introducing these expressions into equation (69) and remembering the formulae 

 (14) and (15) we get 



* Compare § 4. 



