214 MR GREGORY ON THE REAL NATURE OF SYMBOLICAL ALGEBRA. 



also the index must be of the same class, — a Ihnitation which I do not remember 

 to have seen any where introduced. Therefore the binomial theorem does not ap- 

 ply to such expressions as (1 + a) °° or (1 + af"" ; and, though it does apply to 



- d 



(1 + af'', since both a and j- are distributive and commutative operations, it does 



— d 



not apply to (l +/(x)Y\ as/(«) and -j- are not relatively commutative. 



Closely connected with the binomial theorem is the exponential theorem, and 

 the same remai-ks will apply equally to both. So that, in order that the relation 



X' 



may subsist, it is necessary, and it suffices, that cc should be a distributive and 

 commutative function. On this depends the propriety of the abbreviated notation 

 for Taylor's theorem 



fix + h) = i"-^'f{x). 



Properly speaking, however, the symbol f ought not to be used, as it implies an 

 aritlunetical relation, and instead, we ought to employ the more general symbol 

 of log— ^ But this depends on the existence of a class of operations on which I 

 may say a few words. 



IV. If we define a class of operations by the law 



/W +/(y) =/(.^y). 



we see that, when x and p are numbers, the operation is identical with the arith- 

 metical logai'ithm. But when x and y are any thing else, the function will have 

 a different meaning. But so long as they are distributive and commutative func- 

 tions, the general theorems such as 



log (1 + .r) = X— - + - — &c. 



being proved solely from laws we have laid down, are true of all sjTnbols subject 

 to those laws. It happens that we are not generally able to assign any known 

 operation to which the series is equivalent when x is any thing but a number, and 

 we therefore say that log (1 -i- x) is an abbreviated expression for the series 



,x — — + — — &c. But there may be distinct meanings for such expressions as 



log (l + T-) or log (4-^ , as there are for £*d^, that is log (^7-) • In the 



d 

 case of another operation, a, we know that lt)g (1 + ^) = ^ . 



