Preface. The mathem atical part of the following treatise was worked ont and 

 written down in the spring of 1916. The essential part of it was read before the 

 astronomical seminary at Lund May 2:d 1916. As, however, it was considered of impor- 

 tance that the work should also contain some numerical examples, it was not pos- 

 sible to publish at once. Other work that came in between has further delayed the 

 publication. 



In his well known work On the general theory of skeiv correlation and non-linear 

 regression Professor Karl Pearson deals with the problem of finding the regression 

 curves of two correlated variates. He also introduces some new and very important 

 conceptions which he names the clisy and the scedasticity . Admirable as his study 

 is, and of general application, it suffers, nevertheless, from some disadvantages. As 

 Pearson himself points out, there is a disadvantage in the very generality of his 

 starting points. As we see them the objections against the theory are chiefly the 

 following: — Theoretically there is no means of making sure of the convergency of 

 the developments; practically there is no means of determining the probability of 

 an individual deviation within assigned limits from the regression curves. Thus while 

 the theory gives the mean of the one variate for any fixed value of the other, it 

 does not give any adequate measure of the extent to which the first variate dusters 

 about this mean, that is of the strength of the correlation. The reason for this lies 

 in the fact that the method makes no assumptions as to the mathematical form of 

 the correlation function. Thus while the method may be extended so as to give the 

 standard deviation (scedasticity) and skewness (clisy) of the distribution of the one 

 variate for any fixed value of the other, it cannot give the form of this distribution 

 since it is not certain, in the general case, that these two characteristics are sufficient 

 to describe that distribution. 



The reason why the method is founded on a basis so broad as not to give 

 these details lies in the fact, several times pointed out by Pearson, that his 

 general theories of skew variation have so far defied every attempt at their exten- 

 sion to correlation surfaces. Hence, for Pearson no mathematical correlation func- 

 tion was available except the normal funtion of Bravais, for which he naturally had 

 no use here, as the theory was to give methods for skew correlation. 



As is well known, when Pearson invented his general theory of skew varia- 

 tion his main idea was that the variation should be given by the binomial and 



