216 MR GREGORY ON THE REAL NATURE OF SYMBOLICAL ALGEBRA. 



is the same as (+)" . The connection between these expressions is so intimate, 

 that, being subject to the same laws, they may be used indifferently the one 

 for the other. This has been the case most particulaxly in the theory of equations. 

 The most general form of the root is usually expressed by a (cos 6 + ( — )* sin «) , 



while the more correct symbolical form would be (+)* a, since the expression 



«« + P, a;"-' + Pj x"-^ + &c. + P„ = 



does not involve any sine or cosine, but may be considered as much a function 

 of + as of X, so that the former symbol may be easily supposed to be involved in 

 the root. Hence, instead of the theorem that every equation must have a root, I 



p 

 would say every equation must have a root of the form (+)* a, p and q being 

 numbers, and a a distributive and commutative function. 



