( 560 ) 



n » 



J5„ =: i ^ a'm bn-m + è -^ («'"i ^m+n — ^w «'m-|-n) i^^J) 



O 1 



and 



2 T'^ . . 

 Bn = — I /(tt>) sin no) [b\ sin u) -f- 6', sin 2a> + . . ] day 



o 



00 ft' 



— -^ ~z~ V^m — n <^m-|-»)j 

 1 ^ 



or 



5„ = ^ ^ 6'„, a„_,„ -\- ^ :2 {am b',n+n — b'm Clm+n) {IV) 



1 1 



2. If we suppose 



ƒ' (.r;) = 131^ + 31, cos ^ + ÜJ, cos 2x -{- ... 

 =z ^j sin ar -}-(£, sin 2ir -f- •• • 

 the four preceding equations give immediately, by putting y (x) =f(x) 



n CO 



2I„ = ^ ^ a,„ an—m -\- ^ «m «m+n (1) 



1 



n 00 



2I„ = - i ^ 6,„ &„_„, + ^' 6,„ J,„4.„ (2) 



I 1 



n 00 



^•„ z= i ^ a,n bn—m + i -S" (a„j 6,„_)-„ — b,n ttmJ^n) (3) 



1 



3. From the four equations of Art. 1, the beautiful theorem of 

 Parseval may be easily deduced. For^ supposing that for all the 

 values on the circumference of the circle modz = l, we have 



è «0 + «1 ^ 4- «i,^* + • • • = <P (^) 



Z 



it is evident, if we assume in succession z = 6'^ and z = g— '"'^ , that 

 F, {oi) -\-iF,{io) = (p {é'-) G, («>) - i G, (w) =r tp (gtw) 

 F^{(M) — iF^{oi) — (p{e-i''>) (?^ (to) + I Ö, (to) = ifj (e-'"). 

 Multiplying these equations and adding the results we obtain 

 2 [F,(a>) (?, (a>) -f F, (to) G^, {io)] = <^(e'-) tf? («'-) + <P («-''") t|> («-'") 

 where 



F^ (tt>) =z i'", = i a, + «1 c<^« ct> + «1 cos 2a> + .. . 

 (?, (ü>) = G, = -I a'„ 4- a\ cos a> + a', cos 2a> + .. . 

 F^ (to) := /-', =: a, sin to + «a **^ 2to -|- . . . • 



(?, (to) = 6r, == a'j sin to -}- a', sin 2to -[- • • • 



