26 WICKSELL, THE CORRELATION FUNCTION OF TYPE A, AND THE REGRESSION OF ITS CHARACTER1STICS. 



fourth and higher powers of rj will have coefficients of an order already neglected. 

 The result will be the following: Introducing the notations 



(34) 

 we find 



(34*) I 

 (34*) II 



»30 — ' K21 "VPsO} r 4» PSI * r i'i<>\ 



r 03 = Äi — 3r ('o, ; r 0i = /? 13 — 4 *■/*„< , 



I» = »'os + ', (*" — 3r 04 ) — jfr 08 + ;/V 04 , 

 % = »'so + !(r — 3r 40 ) — | 8 r 30 + £ 8 r 40 ; 



^ = n, 3 + ', [r — 3 (r 0t — 4/? 03 r 03 )] — r 2 r 03 + r 3 (r 04 — 6£ 03 r 0S ) , 

 % = »'30 + II> - 3(r 4U — 4£,, r 80 )] — i'-V 30 + i" 3 (r 40 — 6/? 30 r 30 ). 



These are the regression formulae for moderately skew correlation. Their range 

 of applicability will as a rule amount to some 2 or 3 times the dispersions on either 

 side of the mean. When the correlation is considerably anormal we had better use 

 the original forms (32*) (I) and (II). Compare, however, the discussion of the 

 numerical examples. 



The coefficients r 30 , r 03 ; r^, r 04 , ivhich must disappear when the regression is linear, 

 we shall call the correlation coefficients of the third and fourth order. 



Expressed in the variates observed we have 



(35) I 



x v — m, = o t r 03 + (y — m 2 ) ~ (r — 3r 01 ) — {y — m,) 2 ° : l r 03 + {y — m 2 f '\ r 0i , 



O ., (7„ I) „ 



y. r — ra, = a 2 r 30 + (x — m 2 ) — (r — 3r 40 ) — {x — mj 2 -| r 30 + (x — w,) 3 -f r 4 „ ; 



o, o. a, 



(35) II 



x y — m l = a t r 0H I (y — m 2 ) ~ [r — S(r 0i — 4& 3 r os )] — (y — w 2 ) 2 ^ r 03 + (y — »»,)»-; (r 04 — 0^ 03 ?- n3 ) , 

 v/,. — m, = o\r, + (x — mj-r |> — 3 (t t0 — i^ 30 r 30 )] — (x— m,) ! -* r 30 + (.r— ra,) 3 -f (r 40 — 6^ so r, ) . 



To test the formulae we may make use of the following method. First we 

 remember that 



<x oo oo 



| F 2 (rj)dr t = 1 ; j ^(^ = 0; j /,^ 2 (/>// = 1; 



— oo — co — CO 



(36) 



00 00 



jr?F 2 (ri)dr t = - 6/? 03 : | »/^(i,)^ = 24/S 04 + 3, 



