18 WICKSELL, THE CORRELATION FUNCTION OF TYPE A, AND THE REGRESSION OF ITS CHARACTERISTICS. 



The usual form of the function is obtained by expanding the exponential and 

 putting 



Dt 7 i (xi) 



d l </i(Xi) 



If a more general definition of the characteristics af is desired it may be shown 

 that they are identical with the half-invariants of Thiele and that, whatever be the 

 form of fi(Xi), they are given by the identity 



„(i) „(«') „(») 



X. T a 4 — 2_ r n -) — *. r \ + . . . 



(26) e'-2 L 3 V = fi{xi)e*i*dxi 



Now instead of the characteristics A pq , a l} a 2 , r of the correlation function Char- 

 lter makes use of certain parameters a pq , which he shows are connected with the 

 former through the relations 



(27) a 20 = a\ ; a, l = r . a x a 2 ; a„, = er* 

 and the identity 



(28) ? e L4 



= 1 —(A 30 col + A 2[ o)]o>, ~\A n oj,al + A n ia\) + {A M io\ +^ 31 t(i*w 2 + 4 22 w"w| + i4 I3 o» 1 (tf" -f 4 or wJ)H 



For our purpose the most interesting feature of the theory lies in the way in 

 which the a pq and thus also the A pq are expressed through the a {i) of the error-sour- 

 ces. It may be shown that 



s s i 



j = 1 / - 1 i - 1 



and in general that 



n — Va ( l F/ ? 



i = l 



Denoting by t/,, the mean value of a p % q kflf for all the error-sources we have 



a, = sä, ; a u = sö JJ ; a„, = £«„, 



(29) a 30 = .9ö., ; rc 21 = «ä 2l ; a 12 == sä n ; a 03 = så as 



d 40 = SCL W ', Ct 3l = Sö 3 | *, fl 2 2 = SCI*,? ', #13 = Stt 13 J öf'o 4 == "S^u 1 • 



etc. 

 By equation (28) we find that 



