THE SUMS OF PERIODIC SERIES. 549 



and compare these expansions with that given by integration by parts, we shall have 



(34), 



/' (0) = - ^ (o. + £,)' /' («) = - :?^ (o, - E,), 



ia 4 o 



/"(0)= ^AO, + E,), f"(a)^ ^^(0,-E,), 



and so on, the signs of the coefficients being alternately + and — , and the index of— increasing 

 by 2 each time. 



16. The values of /''(O) and /''(a) when f{x) is expanded in a series of sines and /i is odd, 

 or when f(x) is expanded in a series of cosines and fx is even, will be expressed by infinite series. 

 To find these values we should first have to obtain the direct expansion of /**(«), which would be 

 got by differentiating the equation (24) or (30) (n times, expanding A„ or B„ according to powers 



off— , and rejecting the terms which would render the series contained in the iiP' derived equation 



divergent. The reason of this is the same as before. 



17- The direct expansions of the derivatives of /(.r) may be obtained in a similar manner in 

 the cases in which /(.r) itself, or any one of its derivatives is discontinuous. In what follows, 

 a will be taken to denote a value of a: for which /(.») or any one of its derivatives of the 

 orders considered is discontinuous; Q, Q,, ... Q^ will denote the quantities by which/(a), f(<v), ... 

 f^ix) are suddenly increased as x increases through a; S will be used for the sign of summation 

 relative to the different values of a, and will be supposed to include the extreme values and a, 

 under the convention already mentioned in Art. 6. Of course f(x) may be discontinuous for a 

 particular value of x while /''(a;) is continuous, and vice versa. In this case one of the two Q, Q^ 

 will be zero while the other is finite. 



The method of proceeding is precisely the same as before, except that each term such as 



f{.v) cos in the indefinite integral arising from the integration by parts will give rise to a series 



a 



such as - SQ cos in the integral taken between limits. We thus get in the case of the even 



a 



derivatives of f{ps), when /(«) is expanded in a series of sines, 



(-\y A'-,= [ — ] A„ - -.(—] SQ cos +-._ .S-Q, sin — 



\ a / a \ a I a a \ a I a 



2 iriTt 

 a \ a 



A'Q.cos -— - ... + (- ly" .-.SQ,., sin (.W). 



a a a 



In the case of the odd derivatives of /(^r), when /(,i) is expanded in a scries of cosines, we get 



(- 1) 2 ^1;= — W?„ + - ^Qsin - +- .Vy, cos 



^ a I a \ a I a a \ a J a 



tzl 2 . nira , ,., 



+ (-!)» -5«^.,sin-^ (;iO). 



When the several values of a, Q., Q, ... are given, these equations enable us to find the direct 

 expansion of /"(«). Tiie series corresponding to the odd derivatives in the first case and the even 

 in the second might easily be found. 



Vol.. VIII. I'art V. 4l{ 



