INACCURACY OF SOME LOGARITHMIC FORMULAE. 185 



maintained in the Encyclopedia MetropoHtana, article Algebra, 284. Indeed, when — 2 is admitted to 

 be one of the values of 4*, the extension of the notion " logarithm " must be greatly abridged to deny 

 that, relatively to the base 4, § is a logarithm of — 2. 



Note G. — From this theorem it does not follow that f ~' fl^ = 2 f ~ fl ; an expression that has only 

 half as many values as f~' 3 + f~' fl, which admits the addition of any one value of f ~ 9 to any 

 other. 



This instance is adapted to give notice of a very insidious species of fallacy, whose intrusion, in 

 reasoning on subjects like the present, should be guarded against with vigilance. 



Note H. — As 2 i it comprises exactly the same values as 2 (i — i) it, and serves as well to show- 

 that the integer in the numerator of [26] may be chosen without reference to that in the denominator, 

 it is preferred for briefness and concinnity in a general formula. 



Note I. — The solution of this problem assists in constructing the figure whose equation is 



M. Vincent has inserted in the commencement of the 15th volume of the " Annales de Mathe- 

 matiques," &c. published at Nismes in 1824 and 1825, and edited by M. J. D. Gergonne, an in- 

 genious paper on the construction of some discontinuous transcendental curves. His paper is entitled 

 " Considerations nouvelles sur la nature des courbes logarithmiques et exponentielles. Par M. Vin- 

 cent, Professeur de Mathematiques au College Royal de Reims, ancien eleve, de I'ecole normale." 

 His general principles appear to me to be correct ; but, in my opinion, he has occasionally fallen into 

 error. For instance, he seems to take it for granted when a is positive, that whatever value of a* be 

 considered, da' = la a^ da; whereas, when the V-^ value of n^is considered (see Appendix § 7.) 

 da'''= (V'^ 2 8* + la)a"'da;. 



To obviate some objections to my general theory, I may here observe incidentally that M. Stein, 

 who has occasionally written on the subject of logarithms in the same journal, would introduce a very 

 confused and inconvenient notation by supposing a" to vary its signification according to the form in 

 which the value of x is expressed — by supposing, for instance, that, while o' = a, a7 would = (a*)^ 



or + a. Hence, by the same analogy a "^^ would = a . 1 '^^. According to the usual interpreta- 



tion of a , which I have adopted, and by which it is identical with f (a; f ~' a), a', a^ and a '^'^ have 

 all tlie same signification. 



The following definition of a*, derived from the characteristic property which led to the extension 

 of tlie exponential notation beyond integral exponents, has been suggested to me by my friend Mr. 

 Hamilton, Royal Astronomer of Ireland : 



"a* comprises every successive function pa; of a;, which, independently of a; and y, satisfies the 

 conditions (^x <^y = f {x -\- y) pi = a." 



From this definition does not follow, in all its generality, the equation a'^ a^ = «' + ■'', for the pro- 

 duct of the i"" value of a* (which I would designate by a *) multiplied by the i"" value of a^ is not 

 necessarily among the values of a* "*■ ^ ; a legitimate consequence of the definition of a* is the equa- 

 tion a^ a^ = a^ "*■ ^. 



MDCCCXXIX. 2 B 



