_M-,r, MK. (*. UDNY YULE OX THK ASSOCIATION 



Similarly all the other frequencies may be calculated, and we get set (2). 



If we take (ABCD) = 1 1 we get set (3). All three sets are consistent with the set 

 of third order, and possess equality of contraries. The state of affairs is precisely the 

 opposite of that illustrated by Table II., where only one set of third order frequencies 

 could be obtained, consistent with the given set of the second order, and possessing 

 equality of contraries.* 



It should be noted, however, that the possible number of fourth order sets in such 

 a case as the present is not infinite, for certain limits are imposed by the fact that 

 negative frequencies are impossible. Thus, if we take 



(ABCD) = 60 

 we have 



(ABC5) = 58 - 60 = - 2 



or if (ABCD) = 3 



(ABC8) = 58 - 3 = 55 

 (AByS) = 53 55 = 2, 



so (ABCD) must lie at all events between the limits 58 and 5. 



13. It follows from what we have proved that a state of complete equality of 

 contraries, in which this state subsists for groups of all orders, is not and cannot be 

 an artificial state created by choice of the points of division between A and a, B and /8, 

 and so on, but must arise from some real and natural symmetry in the distribution of 

 frequency. In dealing then with the next problem, to find the number of independent 

 frequencies of any order in the case of complete equality of contraries, we must not 

 rashly apply the formulae obtained (by extrapolation, as it were) to an empirical case 

 in which we only know that the condition subsists for a few low orders. 



The general result we arrived at was that the number of independent frequencies of 

 the with order in n variables was given by the sum of the first (m + 1) terms of the 

 series 



1.2 1.2.3 



In this expression we may now strike out alternate terms commencing with n, for 

 these represent frequencies of odd order which can be expressed in terms of the next 

 lower order of frequencies, and so do not give any independent data. This leaves the 

 series 



j + (* - 1) , *_- DO* ~ )( ~ 3) 

 1.2 1.2.3.4 



Abfc 4/4/00. I only noticed in reading Tables II. and III. in proof that the theorem holds " If 

 two set* of ultimate mth order frequencies are both consistent with a given set of wTTth order, the 

 differences between corresponding pairs of with order frequencies are numerically constant." It is 

 noted below (pp. 272, 273) that this holds for second order frequencies. The theorem is proved at once 

 by expanding the (m - l)th order frequencies in terms of the two sets of the mth order. 



