554 Mr. stokes, ON THE CRITICAL VALUES OF 



where A„ or B„ is supposed given, and want to know, Jirst, whether the series is convergent, 

 secondly, whether if it be convergent it is the direct developement of its sum f{x), and thirdly, 

 whether we may directly employ the formulae already obtained, trusting to the formulas themselves 

 to give notice of the cases to which they do not apply by leading to processes which cannot be 

 effected. 



25. If the series 2.<4„ or 2fi„ is essentially convergent, it is evident <i fortiori that the series 

 (3) or (22) is convergent. 



c c 



li A„= S — cos n-y + C„, or ii B„ = S - sin ny + C„, where 2C„ is essentially convergent, the 

 n n 



given series will be convergent, as is proved in Art. 6. 



In either of these cases let /(a?) be the sum of the given series. Suppose that it is the series 



of sines which we are considering. Let E^ be the coefficient of sin in the direct developement 



of /(.t). Then we have 



. rnrx „„ . WTT.r 

 fLv) = 2^„ sm = S£„ sm ; 



and since both series are convergent, if we multiply by any finite function of x, (p(.v), and integrate, 

 we may first integrate each term, and then sum, instead of first summing and then integrating. 



Taking ^(j;) = sin '- , and integrating from .r = to x = a, we get E„ = A„, so that the given 



series is the direct developement of its sum /(.r). The proof is the same for the series of 

 cosines. 



26. Consider now the more general case in which the series S - ^„ is essentially convergent. 



The reasoning which is about to be ofTered can hardly be regarded as absolutely rigorous ; 

 nevertheless the proposition which it is endeavoured to establish seems worthy of attention. 

 Let M„ be the sum of the first n terms of the given series, and F(n, x) the sum of the first n terms 



of the series 2 - A„ cos . Then we have 



Wit a 



/(M» + m - ?<„) d"' = F(.n + ™i ■'>') - E(n, x) = \//(«, x), suppose (44). 



Now by hypothesis the series "S, - A„ is essentially convergent, and therefore a fortiori the 



series 2 A„ cos is convergent, and therefore >i/(c5 , .r) = 0, whatever be tiie value of m. 



Mir a ^ 



Let the limits of .r in (44) be 31 and x + A.t, and divide by I^x, and we get 



-r- / (u, + „-u„)dx= -^-— : 



and as we have seen the limit of the second side of this equation when we suppose n first to 

 become infinite and then Aa' to vanish is zero. But for general values of x the limit will remain 

 the same if we first suppose A.r to vanish and then n to become infinite; and on this supposition 

 we have 



limit of (m„+„, - «„) =0, for n = M ; 



so that for general values of x the series considered is convergent. 



To illustrate the assumption here made that for general values of x the order in which n and 

 ^x assume their limiting values is immaterial, let \f/(y, x) be a continuous function of x which 



