THE SUMS OF PERIODIC SERIES. 539 



It is a principle of great importance in these investigations, that a function of two independent 

 variables which becomes indeterminate for particular values of the variables may have different 

 limiting values according to the order in which we suppose the variables to assume their particular 

 values, or according to the nature of the arbitrary relation which we conceive imposed on them as 

 they approach those values together. 



I would here make one remark on the subject of consistency. We may speak of the sum of an 

 infinite series which is not convergent, if we define it to mean the limit of the sum of a convergent 

 series of which the first n terms become in the limit the same as those of the divergent series. 

 According to this definition, it appears quite conceivable that the same divergent series should have 

 a different sum according as it is regarded as the limit of one convergent series or of another. If 

 however we are careful in the same investigation always to regard the same divergent series, and 

 the series derived from it, as the limits of the same convergent series and the series derived from it, 

 it does not appear possible to fall into error, assuming of course that we always reason correctly. 

 For example, we may employ the series (.3), and the series derived from it by differentiation, &c., 

 without fear, provided we always regard these series when divergent, or only accidentally convergent, 

 as the limits of the particular convergent series formed by multiplying their w"" terms by g^. 



6. We may now consider the convergency of the series (3), in order to find whether we may 

 employ it directly, or whether we must regard it as the limit of (5). 



By integrating by parts in the w"* term of (3), we have 



2 /•„, ,, . n-KX "^ ft 's '^'T*'' 

 - / f{x ) sin ax = / (i; ) cos 



Suppose that/(«) does not necessarily vanish at the limits x = and x = a, and that it alters 

 discontinuously any finite number of times between those limits, passing abruptly from M, to N, 

 when X increases through a,, from M, to iV, when x increases through a,, and so on. Then, if 

 we put S for the sign of summation referring to the discontinuous values of f(x'), on taking the 

 integrals in (9) from x = to ,r = a, we shall get for the part of the integral corresponding to the 

 first term at the right-hand side of the equation 



^Ino) -(-rf(a) + S(N-M) cos -"""} (10). 



nw [ a J 



It is easily seen that the last two terms in (9) will give a part of the integral taken from to a, 

 which is numerically inferior to — , where Z is a constant properly chosen. As far as regards 



the.se terms therefore the series (3) will be essentially convergent, and its sum will therefore be 

 the limit of the sum of (5). 



Hence, in examining the convergency or divergency of the series (3), we have only got to 



TlTTX 



consider the part of the coefficient of sin of which (10) is the expression. The terms /(O), 



f(a) in this expression may be included under the sign .V if we put for the first a = 0, M = o, 

 N = f(0), and for the second a = a, M =/(«)' .^=0. We have thus got a set of series to con- 

 sider of which the type is 



- (iV - 3/) 2 - cos sin (II). 



It n II a * ' 



