30 VVICKSELL, THE CORRELATION FUNCTION OF TYPE A, AND THE REGRESSION OF ITS CHARACTERISTICS. 



To check \ve have as before to multiply by F 2 (i.)d> l and rjF.ir^d^ and integrate. 

 If vve remember that 



p' 3 ^)r,F 2 (v)dri=- 1 fpFd^did^ -ii-;, 



(45) 



— 00 — QC 



CO 00 00 



— 00 — ao 



the result will be found satisfactory. 



Now we have from the preceding formulae 



o\ (I) ■ i n = (t ia — Sr /)'„ 3 — r- p u + 3 r 3 £ 0i + 



+ r](r-r a -3(i ta -2r(} i2 +l2r(} 0i + 9rzii i3 -2*r*lij 

 - , 2 ( P ' 12 + 2r& t -3rft 3 - 5r*/?, 2 + 9r 3 /? 03 ) + 

 + rf (/?„ + 2 r (3 22 - 4 r p 0i - 7 r s />' 13 + 16 r 2 /?„„) , 



and 



(46) 



|^ = f (3r 8 /?n-9r3 /?„) + 



-i?*( + 3r"/5f I3 -9f»/» ,) + 



| ,/(3rVn-12rVo,). 

 Thus, according to (42) 



v, (£)„ - — 6/y S0 + 6rp 2i — 6r 2 /?i2 + 6 r»/S„ + 



+ , ; (6 / > > 3 1 -12r/>' 22 +I8»- 2 /i 13 -24r' ( y 04 ). 



This fonnula gives the regression of the moments of the third order in the 

 arrays, when the correlation is only moderately skew. When the correlation is con- 

 siderably anormal formula (43) should be used, and the quantities 3o?,(§)§ n + |,% which 

 are found from I (32*) and I (37), subtracted. The range of applicability is the 

 same as for formula I (34*) and I (40). When the indices are written backwards 

 and £ is put for Tl the formula gives the value of v 8 {r})^. 



(10) The regression of the characterisiics of the fourth order. 



We have here 



00 00 



v'M) n - fa F (I, r))d§: [f (|, ,;) d B . 



Further the moment about the mean £",, of the array is given by 

 (47) v t (|)„ = v\ (|)„ - 4 v, (I), . f„ - 6 o, (|) |jj - |« . 



