THE SUMS OF PERIODIC SERIES. 557 



the integration with respect to x must be performed first, because, the integral of sin jiw sin ^x'dfi 

 not being convergent at the limit oc , / sin /3iT? sin /3.r'd/3 would have no meaning. Suppose, 

 however, that instead of (52) we consider the integral, 



- /"°r/(^')e-*^sin/3a7sin/3.r'rf/3d^' (53), 



TT "0 Jo 



where A is a positive constant, and e is the base of the Napierian logarithms. It is easy to see that 

 at least in the case in which the integral (50) is essentially convergent its value is also the limit to 

 which the integral (53) tends when h tends to zero as its limit. It is well known that the limit of (53) 

 when h vanishes is in general /(,i); but when x = the limit is zero; when x = a the limit is i/(«) ; 

 and when /(.r) is discontinuous it is the arithmetic mean of the values of /(.r) for two values of ,i; 

 infinitely little greater and less respectively than the critical value. When ,v > o it is zero, and in 

 all cases it is the same, except as to sign, for negative as for positive values of x. 



We may always speak of the limit of (53), but we cannot speak of the integral (50) till we 

 assure ourselves that it is convergent. Now we get by integration by parts, 



//(•r') sin (ix'dx = - ^/('O cos /3*'' + -y./'i^f') sin (iw - — J fix) sin (ix'dx (54). 



P P P 



When this integral is taken between limits, the first term will furnish a set of terms of the form 



C L 



— cos /3a, where a may be zero, and the last two terms will give a result numerically less than -7, 



where L is a constant properly chosen. Now whether a be zero or not, j cos /3a sin /3.r ~— is 



convergent at the limit co , and moreover its value taken from any finite value of /3 to /3 = eo is 

 the limit to which the integral deduced from it by inserting the factor e"*^ tends when h vanishes. 

 The remaining part of the integral (50) is essentially convergent at the limit 03 . Hence the 

 integral (50) is convergent, and its value for all values of x, both critical and general, is the limit to 

 which the value of the integral (53) tends when A vanishes. 



29. Suppose that we want to find f"(x), knowing nothing about /(.i), at least for general 

 values of x, except that it is the value of the integral (50), and that it is not a function of the 

 class excluded from consideration in Art. 1. We cannot differentiate under the integral sign, 

 because the resulting integral would, usually at least, be divergent at the limit eo . We may 

 however find f"{x) provided we know the values of x for which f{x) and/'(.r) are discontinuous, 

 and the quantities by which f{x) and f(x) are suddenly increased as x increases through each 

 critical value, supposing the extreme values included among those for which f(x) or f(x) is 

 discontinuous, under the same convention as in Art. 6". Let a be any one of the critical values 

 of X ; Q, Q, the quantities by which f{i'),f{,!r) are suddenly increased as x increases through a; 

 S the sign of summation referring to the critical values of x; d)-(/3) the coefficient of sin fix in 

 the direct developement of f"{x) in a definite integral of tlie form (50). Then taking the integrals 

 in (54) between limits, and applying the formula (51) to f"(x), we get 



d),(/3) = - /3^rf)(/3) + - jiSQ cos fta-- SQ, sin /3«. 



TT TT 



We may find 0^(/3) in a similar manner. Wc get thus when n is even 

 (- iytp^ifi) = /3''(/.(/3)--/3''-' .yQco8/3a + - /3''-'5'Q, sin ^a + ... 



TT TT 



+ (- 1)^"^'- 5- (/^., sin /3a (55), 



TT 



Vol.. VIII. Part V. 4C 



