62 



Sven Wicksell 



The mean errors of the characteristics of the fourth order it is somewhat more 

 complicate to derive as they are compound functions of the moments. The method 

 proposed by Charlier is the following: 



If by Vj^ajj, x 2 ) we denote the mean of the quantity x 1 x 2 we have for the 

 mean error of the function h(x 1 , x 2 , x s , ... x s ) 



(") m = \f i ( d ±y 5 (,f + 2 £ ft. !* Vli(a%i ^ 



* it, \dXiJ y ' ^ i, 3 tidXi dxj 111 31 



Now by the fourth characteristics the moments of the fourth and second orders 

 play the part of the quantities x 1 ,x 3 ,..x t in the above function h. Thus we get 

 the mean errors of the fourth characteristics expressed in terms of the second mo- 

 ments, the mean errors of the fourth and second moments and the v n of these 

 moments. These mean errors are taken from equations (71) and (74) and for the 

 quantity v n between two moments Charlier gives the general formula 



(78) v u (tf„, N p , q ,)=±(N p+p , q + q ,^N p . q N p , q ,). 



Now by the formula (17), (71), (74*), (77) and (78) we obtain 

 1 



241/ N 



<B*x) = -^7^ K 6^.o ^o 2 + 18xV| 0 N\ , 



(75*) ,(»„) = — i=- \'4N\ 0 Nl 2 + 16iV 20 N~ N\, + 4N\ 



s(B 13 ) = V6N t0 N> 0i + \SNl 2 N\] 

 61/ N 



b(B 04 ) = 1 -= \/2TN^ 2 



04 24V-N 02 



and for the ß-characteristics 



»(PJ = 77"= = J (ftJ e(ß 22 ) = -tA- 1/4 + lör" + 4r* 



As the excess in x is E x — 3ß 40 and similarly for y we again come back to 

 a well known formula viz: 



„ l/ü.375 0.612372 



