S52 Mb. stokes, on THE CRITICAL VALUES OF 



finite from a; = a - ^ to .r = a + ^', and which shall be such that / F(jv)dw^ i f(.v)dx. 



Suppose that we are considering the series (3). Then, if C„ be the coefficient of sin in the 



expansion of F{x) in a series of the form (3), it is evident that C„ will approach the finite limit A„ 

 when ^ and f' vanish, where A„ = — I f(x) sin dx. But so long as ^ and ^' differ from 



zero the series 2C„ sin is convergent, and has F{x) for its sum, and F(x) becomes equal 



a 



to f{x) when ^ and (T vanish, for anv value of x except a. We might therefore be disposed to 



conclude at once that the series (3) is convergent, and has f(x) for its sum, unless it be for the 



particular value a = ai but this point will require examination, since we might conceive that the 



series (3) became divergent, or if it remained convergent that it had a sum different from /(.r), when 



t and t' were made to vanish before the summation was performed. If we agree not to consider 



the series (3) directly, but only the limit of the series (5) when g becomes 1, it follows at once 



from (7) that for values of «• different from a that limit is the same as in Art. 4. For x = a the 



limit required is that of 1 \f{u. - e) + f(.a + e) \ when € vanishes. If f(x) does not change sign 



as .v passes through a the limit required is therefore positive or negative infinity, according as /(.r) 



is positive or negative ; but if f(x) changes sign in passing through os the limit required may be 



zero, a finite quantity, or infinity. The expression just given for the limit may be proved without 



difficulty. In fact, according to the method of Art. 4, we are led to examine an integral of 



the form 



where ^ is a constant quantity which may be taken as small as we please, and supposed to vanish 

 after h. Now by a known property of integrals the above integral is equal to 



- }/(« - |i) +./"(« + ?•)! f TTri ' '"^^'^^ ?■ ^''^^ between and t 



But / — — -rr , which is equal to tan"' .f , becomes equal to — when h vanishes, and the limit of 



^, when h vanishes must be zero, since it cannot be greater than ^, and ^ may be made to vanish 

 after h. 



22. The same thing may be proved by the method which consists in summing the series 



. Httx . n-irx 11.11, 1 . 1 . 



2 sm sin - to « terms. It we adopt this method, then so long as we are considering a 



a a 



value of X different from a it will be found that the only peculiarity in the investigation is, that the 



quantity under the integral sign in the integrals we have to consider becomes infinite for one value 



of the variable ; and it may be proved just as in Art. 20, that this circumstance has no effect on 



the result. If we are considering the value x — a, it will be found that the integral we shall have 



to consider will be 



I /•? sin i/f f , 2a ^ 2n „ 1 , 



where v is first to be made infinite, and then ^ may be supposed to vanish. If /(a + e) +/(n - f) 

 approaches a finite limit, or zero, when e vanishes, as may be the case it f{x) changes sign in 

 passing through eo , it may be proved, just as in the case in whicli f(x) does not become infinite, 

 jthat the above integral approaches the same limit as ^ {f(a + e) + f(a - e)\. In all cases 



