RELATI\-. 3-1 



resemble that of the operation or relation usually shni 



particular, it will be well that the relation expressed by -< should involve the 



Bed by th Same *i n. In 



conception of one member being in the other; addition, that of taking together 

 multiplication, that of one factor's being taken relatively to the other (as we writ. 

 3 X 2 for a triplet of pairs, and Dtp for the derivative of 9); and involution, that of 

 the base being taken for every unit of the exponent. 



2. In the second place, it is desirable that, in certain general circune-tanet de- 

 terminate numbers should be capable of being substituted for the letters operated 



upon, and that when so substituted the equations should hold good when ini 1 

 preted in accordance with the ordinary definitions of the -igns, so that arithmetical 



o 



ebra should be included under the notation employed as a special case of 



For this end, there ought to be a number known or unknown, which is appropri- 

 ately substituted in certain cases, for each one of, at least, some el « of letters. 



3. In the third place, it is almost essential to the applicability of the signs for 

 addition and multiplication, that a zero and a unity should he possible. By a zero 



I mean a term such that 



x -(7 = x , 



whatever the signification of x; and by a unity a term for which the corresponding 



general formula 



x1 = x 



holds good. On the other hand, there ought to be no term a such that a* = x, 



independently of the value of x. 



4. It will also be a strong motive for the adoption of an algebraic notation, if 



other formulae which hold good in arithmetic, such as 



xi,y z = {*,y) z , 



Yx = x y 



x* = x , 

 a;0 = 0, 



1 



continue to hold good; if, for instance, the conception of a differential is possible, 



and Taylor'., Theorem holds, and 6 or (1 + 01 pl»J an important part .„ .ho v 



tem, if there should be a term having the properties of 9 (::.!. fV.M, or propert.es 

 similar to those of space should otherwise be brought ont by the notat.on, or ff there 

 should be an absurd expression having the properties and ON. of J or the square 

 root of the negative. 



