INACCURACY OF SOME LOGARITHMIC FORMULA. 177 



value must, therefore, be expressible by f (a? f ~* a), where f~^ a is a determined 



value of f~' a. Moreover, by [9], the expressions f~' a, and 2 iV + f~'a, are 

 co-extensive. Now a having two values which differ only in sign, let one of 



them = f (<rf~^ a) ; then (since f^ = — 1) the other will = fir .f (.rf"' a) or 



(see [11]) f (t + «■ f~' a). The supposition is that f (ir + a? f~' a) = one of the 



values of a' or f {ip(2 i^r + f ~'a)}. Hence, by [9], one of the quantities 



2 i ir + T + ^ f "' a must = one of the quantities x {2 i v -\- i~^ a). 



2 ? '+ 1 



Hence x must = one of the quantities • . , a formula comprising all 

 rational fractions, which, in their lowest terms, have even denominators. 



§ 3. f-^ (dh) z=zi-U-\- f-' h [Note G.]. 



By [1 1], f (f-* 6 + f-' h) = f f-' 6 . f f-i hordh. 

 Hence, r^6h = rU + r^h. q. e. d. 



^ 4. On the Neperian logarithms of positive numbers. 

 Developing by [7], it appears thatf — ^— 1 = 1 + 1...+ 



X • ^ • • 



the Neperian base. 



Hence, by [9], f~' e = 2 i ff — ^— l. 



Hence, by [22] , the Neperian logarithms of k^ are expressed by 



. n 



Ts e 



f~^F 



[25] 



2i9r— j/ —I 



Now, by § 3, f-' i^. = f-' rTK. + f-^ K^ 



Hence r ^ K^ = f"^ ^^ - f"^ ^ 



And, K' being positive, (in the formulae of this paper capital letters will be 

 used to denote real quantities) 1 — . , ^^ or , " t^^ and 1 — , ,^„, or — ^ ~^l 

 must evidently both lie between 1 and — 1 . 



2 K' x " 2 ** 



Hence it is plain that (l — fqrK^J and (l - Ylfw) ^^^^ ^°*^ approach 

 indefinitely to 0, as n increases without limit. 



MDCCCXXIX. 2 a 



