264 



MR. G. UDNY YULE ON THE ASSOCIATION 



Equations (2) and (3) are evidently quite special relations. The set of arbitrary 

 frequencies in Table II. below is drawn up to illustrate the theorem for the case of 



TABLE II. 



second and third order frequencies. Column 4 gives a set of second order frequencies 

 for which equality of contraries subsists, the numbers for this having been in other 

 respects written down at random. These give the first order frequencies of Column 2. 

 If we now proceed to calculate the third order frequencies by equations of the above 

 form, 



2 (ABC) = (AB) -f (BC) - (aC), 







that is, using the figures of Column 4, 



2 (ABC) = 27 + 59-50 



= 36 

 (ABC) = 18, 



we get a set of frequencies, Column 6, for which equality of contraries subsists. 



If we take, however, an arbitrary value for (ABC), say 15, and calculate the 

 remaining frequencies of the same order from it, we get a set of third order fre- 

 quencies (Column 7 of Table II.) for which equality of contraries does not subsist, 

 but which is equally consistent with the second order frequencies. 



12. Now apply precisely the same method to a group of the fourth order. We 

 get finally 



(a/3yS) = (a/3y) - (a/3D) + (aCD) - (BCD) -f (ABCD). 



But if equality of contraries subsist for the third order groups, we have by the 

 theorems of 11, 10 



2 {(a/3y) + (aCD)} = (a/3) + (/3y) - (Ay) + (aC) + (CD) - (AD) 

 = (a/3) + (/3y) + (CD) - (AD) 





