28 WICKSELL, THE CORRELATION FUNCTION OF TY1'E A, AND THE REGRESSION OF ITS CHARACTERIST1CS 



If v 'tlS)n De computed by the aid of the tables of /?,-(*,) and |* be subtracted we 

 may construct the curves of scedasticity. The curve will be applicable within the 

 whole range where the ^4-function is applicable. 



Within the ranges discussed in the preceding paragraph the form (33) for the 

 denominators may be used. Hence, neglecting quantities of an order already dis- 

 regarded, and arranging according to powers of rj we find, writing the additions to 

 be made in case II within square brackets, 



(38) I and [II] v' 2 ^) n = 1 — r 2 — 2p„ + 6r£ 3 — 12r 2 p' 0i + [3ft, + Qp„p tl - 3Qrp 03 p l2 i 45r 2 #.] ~ 

 — ri{2p n — 6rtt u +12r*p ,) + 



+ r, ä (r 8 + 2p„ — \2rp lt + 3Qr*p 0i -[6p{ 2 + 6p' 031 >' 21 - 72r ( >' 03 + p u l44rVi,]) - 

 - v *(2rp i2 -6r*p 03 ) + 

 + ^(2^ 13 -8r 2 / S 04 + [/Ji s -18r/? 03 ^ 2 + 45r»/»;,]) 



But from (34*) we conclude that 



^=f^ 2 -6r^^ 03 + 9r 2 ^ 3 ] + 

 + ij(2rp ta — 9i*p o ,) + 

 + ^(,.2 _ Gr/: » i:i + 24r2/A)i _ [ 2/ j» f _ 36r/* 03 ft 2 + 90r 2 ft ,]) 



-f(2r/* 12 -6r s /* 03 ) 



+ rf[2 r p i3 - 8r% 4 + [fl,- 18rft,/*„ + 45r»/*:,]). 



Hence, it follows that 



(40) I „,(£)„ = ffj(|) - 1 - r 2 - 2p„ + Brp ia - 12r-p'„ , 



-iy(2/? ai — 4r/? 18 + 6r"/Jot) 

 + rf(2p„ — 6rp ia + 12t*p M ). 



(40) II r,(i-), / = fff / (i-)= 1 -r 2 — 2p 22 + 6rp 13 -l2r*p 0t + 2pl 2 + 6p (l3 p 2l -24rp 03 p i2 + 36r*til a - 



-r ] (2p n — årp i2 + &r 2 p 03 ) + 

 + ,f (2 p„ - 6 r ft, + 12 r 9 - (i , - 4 p] a - 6 /S 03 p ai + 36 r M p l3 - 54 r 2 ftj . 



In both cases the higher powers of jj have coefficients that are of an order 

 already neglected. Thus the curves of scedasticity in an approximation to the fourth 

 order will be parabolic. 



Formula (38) may be checked in the same manner as in the preceding para- 

 graph if we remember that 





(4i: 



00 CO 



— 00 — 00 — 00 



00 00 Cl) 



[ri*>' 2 (£) v F 3 (r})dr)=f fayF^dSdrj- - 2p, 



