328 MEMOIRS OF THE AMERICAN ACADEMY. 



A subjacent sign of infinity may indicate that the correlate is indeterminate, so that 



I 



oo 



will denote a lover of something. We shall have some confirmation of this presently. 

 If the last subjacent number is a one it may be omitted. Thus we shall have 



I 



1 





^11=51=5. 



This enables us to retain our former expressions Iw ,5 oh , etc. 



The associative principle does not hold in this counting of factors. Because it 

 does not hold, these subjacent numbers are frequently inconvenient in practice, and 

 I therefore use also another mode of showing where the correlate of a term is to be 

 found. This is by means of the marks of reference, f J || § % which are placed sub- 

 jacent to the relative term and before and above the Correlate. Thus, giver of a 

 horse to a lover of a woman may be written 



The asterisk I use exclusively to refer to the last correlate of the last relative of the 

 algebraic term. 



Now, considering the order of multiplication to be : — a term, a correlate of it, a 

 correlate of that correlate, etc., — there is no violation of the associative principle. 

 The only violations of it in this mode of notation are that in thus passing from rel- 

 ative to correlate, we skip about among the factors in an irregular manner, and that 

 we cannot substitute in such an expression as^roh a single letter for oh. I would 



iggest that such 



may be found useful in treating other cases of 



associative multiplication. By comparing this with what was said above concerning 

 functional multiplication, it appears that multiplication by a conjugative term is func- 

 tional, and that the letter denoting such a term is a symbol of operation. I am 

 therefore using two alphabets, the Greek and Madisonian, where only one was ne- 

 cessary. But it is convenient to use both. 



Thus far, we have considered the multiplication of relative terms only. Since our 

 conception of multiplication is the application of a relation, we can only multiply 

 absolute terms by considering them as relatives. Now the absolute term "man" is 

 really exactly equivalent to the relative term « man that is — ," and so with any 

 other. I shall write a comma after any absolute term to show that it is so regarded 

 as a relative term. Then man that is black will be written 



m,b . 



