INACCURACY OF SOME LOGARITHMIC FORMULAE. 
179 
Now let cbs 
-i 
R 
/ R s + S 3 
(characterized, to distinguish it from any of the 
-l 
R 
l) be the arc, when radius = 1, in the first 
other values of cos 
v R + o' 
positive or negative semicircle, according as S is positive or negative, whose 
R R 
cosine = ; (as ~ > pJ , ^ always lies between 1 and — 1, it is evident 
/ R 3 + S 3 
that such an arc cbs 
/ R- + S 3 
R 
premised, will its sine = 
/ R 3 + S 5 
S 
is always assignable) then, by what has been 
Hence 
V R- + S 3 
f cos 
-1 
R 
_ R + v'-i S 
/R 3 +S 3 /R 3 + S 3 
Again, let 1 / R 3 + S 3 designate the tabular Neperian logarithm of / R 3 + S 3 ; 
then, by [27], will 
Hence 
—l 
f cos 
f (cos ~ 1 
f(_ _i 1 /R 8 + S 3 ) = /R 3 + S 3 
R 
/ R 3 + S 3 
R 
VR* + S 3 
= . f ( — — 1 1 /R s + S 3 ) or (see [11] ) 
- /-ll /R 3 + S 2 ) = R + -/-IS 
R 
Hence, by [9] , 
f 1 (R + / — lS)orf -1 fi = 2?7r + cos 1 ^ — / — 1 1 / R 3 + S 3 [28] 
in which expression the real and imaginary parts of f -1 0 are separated. 
Corollary. 1 may remark that from the ambiguity of d cos -1 6, which 
= + — Q 2 d &, the arcs in odd positive and even negative semicircles whose 
cosines = 6, a quantity between 1 and — 1 , will be found on development to 
be represented by 
2! '”+y- 
l 3 . 3 9 ... (2 72 — l) 3 2« + l 
1 . 2 ... 2 «. 2 » + 1 . 
Similarly, the arcs in odd negative and even positive semicircles whose 
cosines = 0, are represented by 
l 3 . 3 3 . . . (2 ft — l) 9 2n+l 
As 
27V-- + 
+ 
1.2 
2 n . 2 ft + 1 . 
I 3 . 3 2 . . . (2ft — l) 3 2n + l 
1. 2. ..2ft. 2ft + 1. 
2 A 2 
