440 Dr. Macfarlane's Analysis of Relationships. 



Suppose we have a relationship of the second order, as cc. 

 We may write : — 



2ccU=2(m+/>cU, (1) 



= 2(m+/) 2 ccU, (2) 



= X(m+ffccV (3) 



In the case of (1), the terms in m+/may be inserted before 

 the first c, or between the two c's, or after the second c. In 

 the case of (2), the terms of (m+f) 2 , which are mm, mffm, 

 ff, may be inserted either in the first and second places, or in 

 the first and third places, or in the second and third places. 

 In the case of (3) the terms, which are mmm, mmf, mfm, &c, 

 can be inserted in only one manner. Hence the number of 

 forms which cc can take (counting cc itself as one) is 27. 



To find the number of elementary relationships of the nth 

 Order. 



By an elementary relationship is meant a single relation- 

 ship, in contrast to a relationship expressing the coexistence 

 of several single relationships. The nth order has 2 rt general 

 relationships. Consider any one of these. A distinction of 

 sex can be introduced before each c or c~ l and after the last. 

 Hence the number of different ways in which a distinction of 

 sex can be r times introduced is equal to the number of com- 

 binations of n + 1 things r together. The number of different 

 relationships obtained by the expansion of a term in which a 

 distinction of sex has been r times introduced is 2''. Hence 

 the number of terms for one general notion is 



that is, 3 n+1 . Hence the total number for the ?ith order is 



2 n 3 n+1 . For n being 5, the number is 23,328. 



Cor. The number of elementary relationships included in 



1 8 

 the first n orders is -=- (6 n — 1) ; for n being 7, the number is 



greater than one million. 



Laws of Reduction. — I. When the sex-symbols preceding 

 and succeeding c 1_1 are the same, then c 1-1 can reduce to 1 ; 

 and when they are different, it cannot reduce to 1. This 

 depends on the morphological law, that sex in mankind is 

 dioecious. 



II. When the sex-symbols preceding and succeeding c~ 1+1 

 are the same, then c~ 1+1 must reduce to 1 ; and when they 

 are different, it cannot reduce to 1. This depends on physio- 



