556 Mb. stokes, on THE CRITICAL VALUES OF 



Let 2G„sin = Fix); 



a 



then f{x) -F{,v) = 2«-^C„ sin '^^ (48). 



Now if (b(x) = 2m„, where the series '2u„, 2 — " are both convergent, we may find <p'{x) 



^ dx ' 



by differentiating under the sign of summation. This is evident, since by the theorem referred to 



du 

 in Art. 2 (note), we may find /2 -^ dx by integrating under the sign of summation. Conse- 



dx 



quently we have from (48) 



r-^{x)-F^-^(x) = J=(^Y"l-C„Z'^ (49); 



and since the series 2 - C„ is essentially convergent, the convergency of the series forming the 



right-hand side of (49) cannot become infinitely slow (see Sect. III.), and therefore, the w"' term 

 being a continuous function of x, the sum is also a continuous function of x, and therefore 

 f^ix) — F'^(x) is a function which by Art. 23 can be expanded in a series of sines or cosines. 

 But F'^(x) is also such a function, being in fact a constant, and therefore /^(a?) is a function 

 of the kind considered in Art. 23, which is what is assumed in obtaining the formula (35). 



It may be observed that these results do not require the assumptions of Art. 26 in the case in 

 which the series 2C'„ is essentially convergent, or composed of an essentially convergent series 



c c 



and of a series of the form SiS" - sin ny or "2.8 - cos n-y, according as C„ is the coefficient of a 



n ' n ' 



cosine or of a sine. 



SECTION II. 



Mode of ascertaining the nature of the discontinuity of the integrals which are analogous 

 to the series considered in Section I, and of obtaining the developements of the 

 derivatives of the expanded functions. 



28. Let us consider the following integral, which is analogous to the series in (I), 



y^0(/3)sin/3.rd/3 (50), 



2 ra , 

 where 0(/^) = - /(x) sin fix' dx (51). 



Although the integral (50) may be written as a double integral, 



- P /""/(*') sin /3 J? sin /3 it' rfj3dcr' (52), 



IT Jo •'o 



