Dr. Macfarlane's Analysis of Relationships, 437 



philosopher has recently published* the first part of a memoir 

 containing a summary of his investigations. I have not taken 

 up the subject of relation in general, but have restricted my- 

 self, in the first place, to a well-defined and important part, 

 namely the relations of men due to consanguinity and affinity. 



The particular class of objects considered in the investiga- 

 tion is in its widest extent the natural class mankind, by which 

 term I mean the entire number of men who have existed, 

 exist, or will exist. The universal properties of the symbols 

 are based on the universal properties of the class. For parti- 

 cular investigations we may limit in any manner the collec- 

 tion of men considered — for example, to the inhabitants of 

 Christendom, or the subjects of Queen Victoria, or the citizens 

 of a given town, or the members of a given household. 



Let U denote any person in the collection of men consi- 

 dered ; then Ua is an appropriate mathematical expression for 

 the person whose name is A, and Ub for the person whose 

 name is B. For the sake of shortness Ua may be written A. 

 Also 2U is an appropriate mathematical expression for all the 

 persons in the collection. 



Let cA be used to denote any child of A ; then — A denotes 



c 



either parent of A. To express a certain child of A, two chil- 

 dren of A, three children of A, &c, we require IcA, 2cA, 3cA, 

 &c. The completeness of a number may be indicated by a dot 

 over the number, as 3cA, the three children of A. When the 

 value of the complete number is not expressed, the expression 

 takes the form 2c A, all the children of A. Subscript num- 

 bers, as in CjA, c 2 A, &c, are the appropriate mathematical 

 symbols for expressing the eldest child of A, the second child 

 of A, &c. 



Since cA denotes any child of A, cc A will denote any child 



of any child of A, hence any grandchild of A. Similarly, c - A 



G 



will denote any child of either parent of A — that is, any brother 



or sister of A, or A himself (or herself). Also -cA will denote 



c 



either parent of any child of A, hence any consort of A or 



A himself (or herself) ; and — A will denote either parent 



c c 



of either parent of A — that is, any grandparent of A. The 



expression -A may be denoted by c _1 A ; then the above are 

 c 



denoted by c 2 A, c 1-1 A, c _1+1 A, c~' 2 A respectively. The ex- 

 * American Journal of Mathematics, yol. iii. p. 15. 



