§ 1. In my previous papers on this subject I used the relationship terms, not in a representative but in a class sense; for instance, cA was employed to denote the children of the man A, and U was employed to denote the total assemblage of mankind, or a limited portion of that assemblage. I have found it useful for the purpose on hand to analyse these symbols into their component elements. Let U denote a man representatively, that is, any man, then mankind is appropriately represented by ΣU, where Σ has its ordinary mathematical meaning of taking the sum. Also UA is the appropriate mathematical expression for the man who has the name A, and A standing by itself is to be regarded as a contraction for UA. This notation is useful where, as in the present case, the universe of the investigation is composed of individuals; but since the universe may be continuous in its nature, by taking U to denote the whole, a more general basis is given to the Algebra of Logic, and accordingly I adopted that notation in my work on the subject.