INACCURACY OF SOME LOGARITHMIC FORMULiE. 179 



T> 



Now let chs "^ ^^ (characterized, to distinguish it from any of the 



Other values of cos "' , „, ^) be the arc, when radius = 1, in the first 

 positive or negative semicircle, according as S is positive or negative, whose 

 cosine = .„, ^, ; (as ,„, ^, always lies between 1 and — 1, it is evident - 



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that such an arc c6s ~' . is always assignable) then, by wiiat has been 



V Iv + b 



premised, will its sine = ,-^ — ga - 



Hence 



, _, R _ R -fAA-i^S 

 **=°* ^R^ + S* v^R^ + S* 



Again, let 1 -v/ R« + S* designate the tabular Neperian logarithm of v' R^ + S^ ; 



then, by [27], will 



f(_ ^-ZTi 1 -v/ R^ + S^) = >/R^ + S» 



Hence 



f c6s -' ;7^P^. . f (- -v/^Tl i/R^ + S^) or (see [1 1] ) 



f(c6s -' ^^f^g, - ^^'^l\ ^/WTS') = R+ V~^\ S 

 Hence, by [9] , 



T) 



f-^(R+ \/^S)orf-'() = 2?7r + c6s"'^^^^5=^=^- ^"^l-ZR^+S* [28] 



in which expression the real and imaginary parts of f ~' ^ are separated. 



Corollary. I may remark that from the ambiguity of d cos ~* 6, which 



= + -s/T--^ d 6, the arcs in odd positive and even negative semicircles whose 



cosines = ^, a quantity between 1 and — 1, will be found on development to 



be represented by 



2jwH fl... ^^ — 9 ... 



.2 1.2...2».2ra+ 1. 



Similarly, the arcs in odd negative and even positive semicircles whose 

 cosines = 6, are represented by 



2^ • ^1.2... 2w.2ra+l. 

 As 



£__ l\3\..{2n- If g2" + ' / 



2 •" 1.2...2W.2M + 1. 



2 a2 



