oti Mr. Graves's Theory of Imaginary Logarithms, 403 

 ^ 1 1 



,'. ff=e^.*. loga = K = la-\-2kTt V — I', 



and if «=1, the first of these equations gives e^*'^'^~l = l. 



It has been objected to Mr. Graves's exponential expres- 

 sion for 1, that that expression has innumerable other values 

 besides the value 1 ; and that therefore any one of these has 

 as much claim to the logarithm attributed to 1 as this 1 itself. 



It is true that such innumerable values exist; since, in making 

 the proposed substitution for x in [2.], the development repre- 

 sents J, only when k is chosen equal to ^. But the very same 

 is true as respects the more limited exponential form of Euler ; 

 the development [2.] becomes that of e, only when Jc is chosen 

 equal to the k involved in the exponent of the first member of 

 [3.]. In both cases, values, different from those retained, will 

 be expressed by the exponential, if the k in [2.] be chosen of 

 different value from the /i: in the denominator of o^. What- 

 ever objection therefore be brought against Mr. Graves's ex- 

 ponential, the very same may with equal propriety be brought 

 against that of Euler; and if from any considerations it be 

 overruled in the theory of the latter, the same considerations 

 must be equally cogent in that of the former ; and as Euler's 

 form is only a particular case of that of Graves, the latter is the 

 form to be used when the utmost generality is to be expressed. 



In reality, however, the objection adverted to is untenable 

 when applied to either theory ; for although the exponential 

 truly represents an infinite variety of different values, yet of 

 only one of these values is the exponent itself the general lo- 

 garithm. This is a fact which I believe has not heretofore 

 been insisted upon ; but that it ought to be insisted upon will, 

 I think, appear from the following considerations. 



Taking either of the two preceding exponential expressions 

 for 1, let the other values involved in that expression be p, q, 

 t'f s, &c. Now of no one of these, p for instance, can the pro- 

 posed exponent be called the general logarithm, because par- 

 ticular values of the arbitrary constant in that exponent, if 

 Euler's be employed, or of pairs of values for the two con- 

 stants, if Graves's be employed, can be chosen for which the 

 exponential will fail to produce^. It is unquestionable thatp 

 has a general logarithm ; but it is equally unquestionable, from 

 this fact, that the proposed cannot be it; the only one of the 

 series of values furnished by the exponential that can claim the 

 exponent for its general logarithm is the 1, because this is the 

 only value which the exponential alvcays gives, whatever 

 changes we introduce into the arbitrary constants; the simple 

 circumstance that k=0 renders the exponential incompetent to 



2D2 



