Schithert^s investigation of Keplev^s Prohtem, 



3 



§ 3. If c be the uumber 2,718 , the hyperbolical losaiithni 



of which is 

 following equations 



shall have by analytical Trigonometry the 



sin 07 



%/ 



'X 



v/-i 



>/ 



v-i 



9 



COS X 



2 



? 



tans: a? 



^/ 



1 





s 



the equation 11; we shall obtain 



c 



%/ 



1 



%/ 



S 



c f %/-'-,! 



c 



v/ 



> 



whence arises c ^^ 



1 





^. 



.(^. 



) 



yW 



If . — is called \. we shall 



v/( 1 +?)— v^C 



) 



.r 



^+1 



multiply 



■> 



V(i+0+>/C 



■i1 



and b\ 



Numerator and Denominator by the 1 



i 



\ 



{l+e)-{l-e) 



e 



(i+eH2v/(i— t=)+Ci-e) i+VCJ 



e2) 



; whence by dividing the 



_ » 



Numerator and Denominator of e^>^~^ by t + i. we get 



iV— 1 



c 



vZ-i 



A 



C 



1 — AC 



C 



V-1 . ^— -^^ 



V-i 



1— A 



c ' >/-^ 



By taking the logarithms of these equal quantities, and pultin 





* ' 



Zc=l, 



'» = £-{- 



^(1-AC— 'V"--^)— /(t 



AC 



v/-) 



If we now make use of the well known series, 



?(i 



oc 



2 



a 



-3 



a- 



n 



3 



• 9 



n 



, we obtain 



i(i 



AC 



i^-1 



A» • 



AC'V 



/-x 



A' 



;, (cV-^_c-'v^-0-4-| (c^^v^-^— c-^V-0+ — •+r (c''^"' 



ftv^-i 



whence by substitutin 



c^v/ 



c 



v/ 





V^-1 



Ssin .1% ue set 



p 



& 



-\ 



