Mathematical Contributions to the Theory of Evolution. 243 



in terms of the ^n{n -\) coefficients of correlation, and a series of 

 new functions which we term the i"-functions. These satisfy the 

 difference equation : 



^nd the differential equation 



ax 



In fact 



n(n-l) ^ nin-\)(n-2){n-^) 

 In - X 21 1 ^' + 2-|2 



, , n{ii-\) (n-2r+\) 



The calculation of these functions is shown to be easy, and their 

 properties are investigated. In this manner the volume of a frequency 

 surface of the ??th order cut off by n planes parallel to the n co-ordi- 

 nate planes is shown to be capable of calculation, and its value is 

 determined in the numerical illustrations given for example of 1, 2, 

 3 up to 6-fold correlation. It may be noted that by putting = ^2 = 



^ = we have really obtained a result which enables us to 



find the " area "of a " spherical triangle " in ?i-f old hyperspace in terms 

 of a series ascending by powers and products of the cosines of the 

 angles between its faces. 



The application of these results to the correlation of characters not 

 quantitatively measurable, arises from the fact that the 71-fold integral 

 above given, and which we have shown how to evalute, measures the 

 total frequency beyond certain boundaries. We can observe, for 

 example, whether horses' coats are bay or darker (or chestnut or lighter), 

 whether eyes are grey or lighter (or, dark grey or darker). Thus by 

 forming mass frequencies instead of frequency distributions for small 

 changes of character, we can find equations to determine the correla- 

 tion. The probable error of such correlation, the convergency of the 

 series, and other points are investigated. 



3. Some discusssion is given to the problem of association, and 

 coefficients allied to Mr. Yule's coefficient of association but somewhat 

 closer m value to the coefficient of correlation are considered, and their 

 relative closeness measured. 



4. A number of illustrations of the new method are given from 

 heredity in horses, dogs, and man, and it is shown how normality of 

 frequency must even for such a character as stature"* only be looked 

 upon as a first approximation. 



An investigation is also made into the influence of superior stock 



* Cited by so many as an example of "normality." 



