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Proceedings of the Royal Society 
e~ l , and also after the term. Hence, the number of different 
ways in which a distinction of sex can be r times introduced is 
equal to the number of combinations 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 r . Hence, the number of terms for one genus-notion is 
l+(?i+l)2 + (’lii^2 2 + . . . + 2" +1 , 
I * JL 
that is, 3 m+1 . Hence, the total number for the ?ffh order is 2”3 K+1 . 
The number for the 5th order is 23,328. 
Cor. 1. The number of varieties for the nth. order is 2” +1 . 
Cor. 2. The number of elementary relationships included in the 
first n orders is 
5 
For n being 5 the number is 27,990. 
There have been those who have conceived the idea of framing a 
philosophical language (Max Muller, “ Science of Language,” vol. ii. 
lect. 2). The fact that without going beyond relationships involv- 
ing more than seven generations, and without taking into account 
any combination of relationships, we have more than one million 
different elementary relationships in which two persons can be said 
to stand to one another, may serve to give some idea of the 
difficulties inherent in that task. At the same time it shows how 
mathematical analysis can step in where ordinary language fails. 
§ 9. I shall now state briefly the properties of the different kinds 
of symbols required in this analysis, and first of the relationship 
symbols. 
The order in which two fundamental symbols occur in a relation- 
ship is in general essential; that is, the symbols are in general 
non-commutative with one another. Thus cc~ l is not equivalent 
to c~ 1 c. 
A relationship is not altered by varying the mode of association 
of its fundamental symbols. Thus (c 2 )c = c(c 2 ), that is, grand- 
child of child is equivalent to child of grandchild. Again c 2 (c~ 1 ) 
= c(c 1_1 ), that is, grandchild of parent is equivalent to child of 
brother or of sister or of self. 
The symbol c has already been defined (§ 6) in such a manner 
as to satisfy the distributive law, and the symbol y in such a 
