172 MR. GRAVES ON A RECTIFICATION OF THE ^I 



Neperian logarithm of a, which logarithm is a quantity assignable only in the 

 case when a is positive, and may then be found from the development | 



Independently of the circumstance that neither of these formulae for y and x 

 provides for the case when a is negative or impossible, and that neither [3] nor 

 [4] provides for the case when y is impossible, their incompleteness will appear 

 from what follows. 



That [2] is incomplete is prima facie obvious, from the known fact that 

 when a; is a rational fraction, a* has as many values as there are units in the 

 denominator of x reduced to its lowest terms, whereas [2] never exhibits more 

 than one value. 



Thus, e* (e being the Neperian base and 1 e = 1) has two values, viz : +,^e 

 and —^e, whereas 



represents the value + ^ only. 



The imperfection of [3] and [4] arises from the imperfection of [2], of which 

 [3] and [4] are reverted solutions. 



Thus, as one of the values of e* = — /i/e, ^ is a Neperian logarithm of 

 —^e, but yet, if in [4] —^/^ be substituted for y, and e for a, the resulting 



formula, viz. 



^^(2.- + i). + l./7 ^^^_ .^^^^aii^ma-. 

 ie 

 comprises, whatever value be given to i, only imaginary quantities, among 

 which, of course, \ cannot be found. 



For the purpose of developing y and x correctly, adopting the equation 



f fl = cos fl + \^"^^ sin 9 [6] 



it will be useful to possess two preliminaries ; 

 1st, a development of f ^ ; 

 2nd, a development of f ~* tf ; 

 as it will appear that upon the form of these developments depend the desired 

 ones oiy and x. 



(By f ~^ ^ is to be understood, according to the notation of Mr. Herschel, 

 every such quantity q, that f j = 6). 



