INACCURACY OF SOME LOGARITHMIC FORMULAE. 
177 
value must, therefore, be expressible by f (x f 1 a), where f 1 a is a determined 
value of f _1 a. Moreover, by [9], the expressions f -1 a, and 2i«r + f ~ l a, are 
co-extensive. Now a having two values which differ only in sign, let one of 
them = f (<rf _1 a) ; then (since f v = — 1) the other will = f?r . f (x f -1 a) or 
(see [11]) f (t + x f _1 a). The supposition is that f (t + < 2 ? f -1 «) = one of the 
values of a or f {x (2 i vr + f _1 a)}. Hence, by [9], one of the quantities 
2 in _|_ r -p x f _1 a must = one of the quantities x (2 i w + f _1 a). 
2 2+1 
Hence x must = one of the quantities • . , a formula comprising all 
Z 2 
rational fractions, which, in their lowest terms, have even denominators. 
§ 3. f _1 {6 h) = f- 1 6 + f" 1 h [Note G.]. 
By [11], f (f- 1 d + f _1 A) = f f- 1 d . f f 1 h or 6h. 
Hence, f _1 6h — f -1 6 + f 1 h. q. e. d. 
§ 4 . On the Neperian logarithms of positive numbers. 
Developing by [ 7 ], it appears that f — — 1 = 1 + 1 ...+ j~cr — ~ 
the Neperian base. 
Hence, by [9], f _1 e = 2 'nr — , s /— 1 . 
Hence, by [ 22 ], the Neperian logarithms of k 2 are expressed by 
f ~ 1 /c 2 
Sin— — 1 
Now, by §3, f~‘ rrx* = f_1 ITir* + f RS - 
Hence f “ 1 K« = f~‘ I ~ 5 - f-' 
. = e 
[25] 
And, K' being positive, (in the formulae of this paper capital letters will be 
used to denote real quantities) 1 — yyy 2 or FTK 3 and 1 ~ 1 ~ + K 3 or “ F+Tc 5 
must evidently both lie between 1 and — 1 . 
2 K 2 \ n 2 n 
Hence it is plain that ( 1 — p + and ( 1 — p+^ 5 ) will both approach 
indefinitely to 0 , as n increases without limit. 
2 A 
MDCCCXXIX. 
