At the end of my previous paper on the Algebra of Relationship (Proc. Roy. Soc. Edinb., vol. x. p. 224), I promised to return to the investigation of the subject, as it seemed capable of further extension. This anticipation has proved correct, and I have now the honour of bringing before the Society the further developments I have made.

In this investigation we consider a particular class of objects, and that class is in its widest extent 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 deduced from the universal properties of mankind. In our consideration of this universe of mankind, we restrict our attention to the classes into which it is divided by qualities depending on ties of anterior or of posterior relationship, that is, of consanguinity or of affinity.