L2 



Sven Wieksell 



and 



11 ^400 



Ii b 310 



\1 12 i? 220 



Ii #o*o 



Ii #30, 



[2 12 B 202 



Ii 5 004 

 Ii 5 013 

 12 12 5 022 

 |8 5 031 

 [2 5 211 



ii B**, 

 Ii Bl„ 



'4 0 0 ,J/ i 

 = V 2 2 0 2V, 



= v, an — 3v, 



0 2 0 

 ^ V 0 2 0 

 "'2 0 0 



2v s 101 

 3 

 3 

 3v 

 2v 

 3v 

 2v 



0 0 2 

 0 0 2 



'200 

 'o 2 0 



by which the expressions of the characteristics to the fourth order in terms of the 

 moments are completed. 



5. We here introduce the following notations 



T* — i — = Riß '4%, y, z ), 



dx l dy 3 d2 k r ' 



where is seen to be a polynom of the degree i -j- j '4- k in x, y, 0, and now 

 may write 



(19) F(x, y, z) = ?{x, y,e){l + Yi B m in- 



putting v 200 = cs x s ; v 02 o = a // 2 ; V 002 = ^ 2 ' 



(20) 



= fa, 



a X * Vy' 1 o z k 



a x , a y and a z are the dispersions of x, y and 8; r xy , r xz , r yz are the correlation 

 coefficients; further, for i-\-j-\-k = 3, ß,p are the coefficients of skewness and for 

 i -j- j -j- 1c = 4, ßy/, are called the coefficients of excess. They all constitute the 

 characteristics to the fourth order of the frequency-function. To discriminate them 

 from the parameters A, B, C, D, E, F, Byk, which also are called characteristics, 

 we will always refer to them as the ß-characteristics. 



Regarding x, y, z and F(x, y, z) as coordinates of a 4-dimensioual system we 

 introduce the dispersions o. c a y v z as units of x, y, s and define the following 

 normal coordinates. 



