172 
MR. GRAVES ON A RECTIFICATION OF THE 
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 
(rrf) -}• M 
-0 
— a 
+ o. 
+ 
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 x is a rational fraction, a x 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 : +V e 
and —sje, whereas 
1 + 2 ---+ i,2.. .n W ”* 
represents the value + sj e 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* = — ^/e, ^ is a Neperian logarithm of 
— *Je, but yet, if in [4] — ,J e be substituted for y, and e for a, the resulting 
formula, viz. 
\/ — 1(2 i + 1 ) tt + 1 V e 
le 
or V' — 1 (2* + 1 )tt + £ 
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 6 = cos 6 + V — 1 sin 9 [6] 
it will be useful to possess two preliminaries ; 
1 st, a development of f 0 ; 
2nd, a development of f -1 0 ; 
as it will appear that upon the form of these developments depend the desired 
ones of y and x. 
(By f -1 0 is to be understood, according to the notation of Mr. Herschel, 
every such quantity q , that f q = 6). 
