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



and should always be tested by constructing the curves for the marginal frequen- 

 cies by aid of (C). 



I. When the characteristics of the fourth order are of the same order of 

 magnitude as the characteristics of the third order we break off by the terms multi- 

 plied by (i^ for i + j = 4. 



II. When the characteristics of the fourtli order are of the same order of 

 magnitude] as the squares and products of the characteristics of the third order we 

 include also the terms multiplied by the coefficients (31). 



Now, naturally, with either sort of approximation the .4-function will give an 

 adequate description of the correlation function only within a limited domain of the 

 #.y-plane. Generally this domain will be found to extend över 3 times the dispersions 

 to either side of the mean. 



Probably both kinds of arrangement leading to approximations I and II may 

 occur in nature. From one point of view, however, I am inclined to regard I as the 

 general rule. Considerations which lead to case II, i. e. cases where the error-sources 

 have probability functions whose anormality does not depend on the number s of 

 sources, must lead to the conclusion that if s be infinitely great the correlation must 

 be normal. Consequently if such an arrangement occurred a case of skew correlation 

 could not be consistent with an assumption of s as infinite. But I think that the 

 fact that the observed correlation is skew forms no obstacle to this assumption. 

 Clearly there can be no limit to the number of ways in which any finite number of 

 sources may be subdivided; and as the theory requires the sources to be in- 

 dependent of each other, we must have recourse to these divisions. For instance, if 

 two def inite sources are dependent on each other, this dependence may be considered 

 as being due to the fact that a part of their error impulses have a common cause. 

 If then the part of the errors having this common cause be taken away from both 

 sources and attributed to this cause as a special source, we have three sources instead 

 of the two previously recognized. If these subdi visions are carried on far enough 

 the sources will be very great in number but will become to a greater extent independent 

 of each other. Clearly this method of reasoning depends on our regarding the errors 

 produced by any definite source of finite influence as being in their turn the effects 

 of a summation of a number of sub-sources. And this is the consequence of our 

 conception of the interdependence of two definite sources as caused by some sub- 

 sources being common to them. In my opinion the assumption of independent causes 

 must generally lead to our making divisions terminating in an infinite or very great 

 number of sub-sources. 



But if sources are assumed to be infinite in number their error distributions 

 must be infinitely skew in order that a skew correlation surface may result, because, 

 as the a vq are then given by observation and some of them are not verv small, 



the aj/^j must be of the order , and some of the coefficients of skewness and excess 



o 



C/" 1 ! ' + ' J — i 



\\ il hin the sources, v+r ! , are then of the order s - , as in the case of the Poisson limit. 



c) 'T' 

 a .. - 



