THE SUMS OF PERIODIC SERIES. 571 



dti . . 



whence, — =21' sin nw, where 

 dy 



Y'=nAe''^ - nBe-!' -- f {f(y') - (- i)''F(y')} (e'<»-J'> + e-"'^-"') dy'. 



■K -J f, 



The values of A and B are to be determined by (80) and (Si), which require that 



dY 



—- ± A K = when « = ± /3. 



dy ^ ' 



We thus get 



(n + h) 6-'' J - (n-h) e-^B - -/^ \f(y')-(- iyF(y')]{(.n + h) £"»-"■)+ (w _ h)e-^^-'-''>} dy'=0, 



and the equation derived from this by changing the signs of h and /3 ; whence the values of 

 A and B may be found. We get finally 



M = 2Fsinna', (gi), 



where 



F= - {(« + A) e"? - (« - A) e-"^; -' (e"* + e""") T |( « + /«) e"<^-^) + (« _ A) e'-'^-yn 



T -'o 



{/(y')-(-ir^,(y')}rfy 



' - /■'(e"^-."' - 6-"<^-v')l/,{y') - (- i)"F,(2/')} rfy' 



+ - |(« + A) e"^ + (n - ft) f-"^} -' (e"» - e""^) f ^ {(w + ft) e"'^"*' + (n - ft) e-«(S-v)l 



" ^/'('""""'^ - """'"""'> ^•^=(2'') - (- ^y^^iy')\ H (92). 



47. The two expressions for u given, one by (90), and the other by (gi) and (92), are 

 necessarily equal for values of x and y lying between the limits and it, - fi and /3 respectively. 

 They are also equal for the limiting values y = - fi and y = (i, but not for the limiting values 

 a; = and x = n, since for these values (91) fails; that is to say, in order to find from this series 

 the value of u for .r = or .r = tt, we should have ^rst to sum the series, and fhett put .r = 



or J? = TT. 



The comparison of these expressions leads to two remarkable formula;. In the first place it 

 will be observed that the first and second lines in the right-hand side of (92) are unchanged when 

 y changes sign, while the third and fourth lines change sign with 7. This is obvious with respect 

 to the first and third lines, and may be easily proved with respect to the second and fourth by 

 taking - y' instead of y' for the variable with respect to which the integration is performed, and 

 remembering thai J\(y), F,{y) are unchanged, and /,(?/), F.(y) change sign, when y changes sign. 

 Consequently the part of w corresponding to the first two lines of (92) is equal to the part expressed 

 by the first two in (90), and the j)art corresponding to the last two lines of (!)2) equal to the part 

 expressed by the last two in (90). Hence the ecpiation obtained by ecpiating the two ex- 

 pressions for u splits into two; and each of the new equations will again sj)lit into two in eon- 

 sequence of the independence of the functions /, F, which are arbitrary from y = to // = fi. 

 As far however as anything peculiar in the transformations is concerned, it is evident that we may 

 suppress one of the functions /, F, suppose F, and consider only an element of the integral 



