16 WICKSEIX, THE CORRELATION FUNCTION OP TYPE A, AND THE REGRESSION OF ITS CHARACTERISTICS. 



to be broken off in order that they may give the best possible fit to the ob- 

 served data. 



It was mentioned on p. 8 that the description given by a series of type A 

 of an arbitrary function is convergent if only two conditions are fulfilled. These 

 conditions are of such a character as to be very generally fulfilled by correlation 

 functions. 



The convergence of our series being secured, there arises the question of how 

 rapidly it converges. Naturally this rapidity may vary in a high degree. 



As has been pointed out by Pearson and others, there is not much use for a 

 theory requiring moments of higher than the 6:th order. Generally even the fourth 

 order is the limit. The reason for this lies in the great mean errors to which the 

 higher moments are exposed. But the parameters are functions of the moments. 

 And the characteristics of the i:th order do not contain moments of a higher order 

 than the i:th. Thus we should generally have to restrict the use of the equation 

 (B) to cases where terms multiplied by the coefficients of up to the fourth order 

 are enough to give a fairly good description of the correlation surface. Hence, we 

 should break off (B) with terms for which *-r-/ = 4. The resulting correlation func- 

 tion will be found to embrace a very large number of the cases occurring in 

 practice. 



However, one more step may be taken. As it is, the yl-functions are not me- 

 rely functions of interpolation. When applied to frequencies they have also a genetic 

 foundation which is afforded by the hypothesis of elementary errors. If this be taken 

 into account, it will be possible to learn something about the order of magnitude 

 of the different characteristics, as well as to include some more terms depending only 

 on moments of a lower order than the fourth. This was first shown by Edgeworth 1 

 in the case of the .4-function of one variate. Edgeworth found that a wider range 

 of applicability will be covered by the function 



(C) 



f' (£) = ?•(! 



L R ^V»^ 4- R &V»® 4- £ *>»& 



+ Pm d p 1-P* d ^ + 2 ' " d£« 



than if the function be broken off before the last term. The reason for this fact is 

 that if s be the number of elementary error-sources it will be found that 2 



1 



j] 3 is of the order of magnitude of 



Pi 



» » » » 



(j b >> » » » » 



jir, » » » » » 



Vs 



1 

 s 



l 



s 



1 



1 Edgeworth: The law of error. Cambr. Phil. Traus Vol. XX. 



2 As will be seen in the following pages this is really no necessari/ consequence of the hypothesis of 

 elementary errors. 



