32 WICKSELL, THE CORRELATION FUNCTION OF TYrE A, AND THE REGKESSION OF ITS CHARACTERISTICS. 



ofå)& = -r i {- 2r(i l2 + 6r*i1 t3 + 2r 3 /S 13 -6r*/S 03 ) f 



+ J? 2( r 2_ r 4_ 6r/ s i3 _ 2r 2 / ? 22 + 24r 2 /* 04 + 12r 8 /? 13 — 36r»fri)- 

 -»/ 3 (2r/i' 12 — 6r 3 /y , + 2r ä ^ 2I - 6r 3 ^ 12 + 12r*/? 03 ) + 

 + ,/(2r^ 13 -8rVo, + 2r*p„ — ar*p ia + 20r*/? ?4 ); 

 ^ = -i? 3 (-4r s + I2r* /*„,) + 



+ r / *(r 1 -12»- 8 /? 13 + 48rVoJ- 

 ^4 f »/J| i — l 2^0,) + 



On account of (47) \ve now have for the moments of the fourth order in the 

 /,-arrays, the formula 



r*(f)„ = 3 + 24/9, - 12& 2 - 6r s + 3r* + 36r/>\ 3 - 24r/J„ + 36r 8 ^.,,— 72*-»/8 ftl - 60H/8,, + 96*%, - 



(51) - / (12/? 21 — 24r/? ls - 12r«/?„ + 36rVo3 + 24r» i * ia -36r*j* M ) + 

 + 9 «(12/?„-36r/!? 1 ,- 12ry 22 + 72r»fti • 36r 3 /? 13 -72r*Ä,4). 



This is the formula of regression of the fourth order moments. Its range of 

 applicability is seen from the discussions of paragraph 7. 



However, the characteristics of the fourth order in the arrays are not directly 

 proportional to y« (£),,. Instead it is the quantities 



>'«(£)* — 3*50?) = *,(!)„ , 



that are proportional to those characteristics. 

 As from (40 I) it follows that 



ff*(|) = l — 2r*+'t* — 4/? 22 + 10rp i3 -24r*p 0i + 4r*/J 22 — 12r 3 /? 13 + 24rVoi — 



(52) -/ (4/? 21 - 8r/? I1 + 12rVo3-** , Vn+ 8rV, 3 - 12»*/W + 

 + , ; *(4^ 22 - 12r/» ]a + 24r»/?oi — *»*i*« + 12»- 3 /*,* - 24r*/?»<), 



we find, with the same approximation as in the case of formulae (34* I), (40 I), (46) 

 and (51), 



^(£)„=24& — 24r/? 31 + 24rV 22 — 24rV M + 24r*/? 04 . 



This formula vvill be applicable wherever formulae (34* I), (40* I), (46) and 

 (51) are applicable, that is, on the whole when the skewness of the correlation is 

 moderate. If formula (48) be used instead, and the quantity 4 /' 3 (i%i\, + 6^^(1)1^ + 

 + '£* + 3o^(£) subtracted, the value of l t (|) (/ obtained will be found valid vvithin the 

 whole of the domain where the ^4-function is valid. 



If the indices be written backwards and r and 5 are interchanged we obtain 

 the formula for /.,(>;)■ 



