262 



MR. G. UDNY YULE ON THE ASSOCIATION 



number of positive groups of the wth and lower orders, including (U), the group of 



order zero ; that is, equal to 



n(n 1) n(n 1) ...(, m + 1) 



or 



" The number of independent frequencies of the mth order in n variables is 

 equal to the sum of the first (m +1) binomial coefficients." 



This gives the following expressions for the number of independent frequencies of 

 the second, third, fourth, and fifth orders- 



Order 2nd . 



3rd . 



4th . 

 5th 



2) 



6) 



lln 2 + 14n + 24) 

 25n 3 + 5n z + 94n + 120). 



The total number of frequencies of any order m is equal to the number of 

 positive frequencies of that order (see above) multiplied by 2"', since each letter, 

 A, B, &c., may be replaced by its negative, and this gives the following expressions 

 for the second to fifth orders : 



Order 2nd 



3rd 



4th 



5th 



2n(n 1) 

 fn(n- 



2)(n 3) 





It is evident from these expressions that, in the general case, the frequencies of any 

 order can never be expressed in terms of lower order frequencies. 



Table I. below gives the number of independent frequencies and the total number 

 from n = 2 to n = 6 and m = 2 to m = 6. 



TABLE I. 



