446 Wisconsm Academy of Sciences, Arts, aiid Letters. 



Then if we give in turn to t all values from — nc to + nc, instead 

 of e~^^* we have the series 



e—vcziJf-e—(nc—l}zi-\- -{-c'^-\- -l-e("c— l)2t -{-enczi 



= C—nczi[l-\~ezi-\-e'^i-\- .... -[-e(2/ic— l)2^-}-e2nc2i) 

 l_g(2nc+l)2i 



C—nczi 



l—e^i 



2n^+l_, 

 = e — nczi , 



g ^ smf— .^T— 2) sin(— .;-'-r) 



2 . sin \z £ia^2 



e^' 



Similarly it is found that 



sin( 1 z) sinf = z) 



9^1(2) = r-l^ . <Pz'^) = 



sin^2 sini^^; 



etc. By these substiuUtions the integral shown above is changed 

 into 



'2k^ [from t = — eiot^=-\-c\ 



2^ • -^ I L_ . ■ , dz 



—It {siny^z) 



or, if we put nz = Xj into 



sin o-jX.sin n:o3: sin n-^ a: sm crc 



. dx 



(sin-^y ^ 



