

20 WICKSELL, THE CORRELATION FUNCTION OF TYPE A, AND THE REGRESSION OF 1TS CHARACTERISTICS. 



a (>) 

 which. measure the skewness and excess of the error distribution within the source Q iy 



p + g_j 



would have to be of the order s - 



Now this is not impossible. We shall here give an example. 



If we assume that the a - ; are discontinuous and may be writteii x i = e i v i wheré 

 ti is a constant and v t any integer, and if we further assume that f L (a-j) is given by 

 the Poisson limit to the point binomial 



«i /*(«*£*) = c -: > 



it may be shown that 



a <i) = Ui) e l- a (*) = ^) e ? ; «(/) = ;,(*) £ * ; etc. 



and thus 



l£±£>ft.-(Fis) 



1 \P + ?-2 



As (vl () =Ä (,) é? can at most be of the order - we should find, assuming e t to be not 



s 



p+q t 

 very small, y p l) +q to be of the order s 2 



In such a case it is easily seen that the (J pq may all be oj the same order oj 

 magnitude, or at least that their magnitudes do not decrease wilh increasing number 

 of error sources. 



Indeed, if y p % q for any p + q are in all the sources of the same order of 



magnitude, it may be seen that y pq will be of the same order. Thus y pq should 



p + q j 

 be of the order of s 2 ' , and consequently the characteristics should be of the or- 

 der of unity. Or at least they do not disappear if s be taken infinitely great. 

 In such a case, hovvever, the arrangement may well be such that for other reasons 

 than the greatness of 5 the (1 pq fall off in magnitude when p + q is taken greater and 

 greater, but then there is not the least necessity to suppose that /? p2 for p + q = 4 and 

 6 are of the same order of magnitude as p 2 pq for p + q = 3. On the contrary, when 

 yl-correlation does arise from such an arrangement of the elementary errors, we 

 should be justified in breaking off the series with the terms for which p-\-q = 4:. 

 Really, in practice, the excess has often been found to be of the same order as the 

 skewness, not as the square of the skewness. 



The other case to be regarded is the one where the skewness and excess of the 

 error distributions are thought to be of an order of magnitude that is the same whether 



