14:6 Mr. W. Williams on the 



n = 00, have not been determined. If the function is con- 

 tinuous, and 



i. If the continuity is ordinary continuity ; or 

 ii. If the function has not an infinite number of oscilla- 

 tions ; or 

 iii. If the infinitely numerous oscillations satisfy Lipschitz* $ 

 condition; 



then the term vanishes, and S oo =F(a?). In all other cases 

 the term must be treated as indeterminate. We may, of 

 course, investigate its values for different types of continuous 

 functions, and so widen the limitations of the function F. But 

 we cannot determine the general nature of these limitations 

 because we cannot evaluate the integrals 



jV(*±*)-F^] siD(2 y 1H - 



by known methods of integration until we are provided with 

 conditions other than those involved in the definition of a 

 " continuous function/'- — such other conditions, for example, 

 as i., ii., and iii. above. 



38. It is necessary to remark that a series of the form 



a x sin x + a 2 sin 2a? + . . . + a n sin nx . . . 



may be convergent, and tend to a definite limiting value 

 which we may denote by F(a?) for all values of x, and yet it 

 may be impossible to derive the coefficients by Fourier's 

 method from F(a;) because 'F(aO may not be integrable 

 according to Riemann's definition. Eiemann has given an 

 example of such a series in the paper already mentioned. In 

 a case of this sort, however, since the coefficients are not 

 determined by Fourier's method, the series is not really a 

 Fourier series. For a Fourier series is one in which the 

 coefficients are defined by the definite integrals 



If 77 1 C n . 



a n =— \ ¥(v) cos nv^v, b n = — \ F(v) smnvftv, 



7Tj_ w 7Tj_ n 



and the object of our investigation is to determine the most 

 general conditions under which the series thus defined is 

 convergent. * 



Hence, since it is necessary that the function should be 

 integrable in order that the coefficients to be derived from it 

 may be finite and determinate, we get when the function has 

 no infinite values the following necessary and sufficient con- 

 ditions for the convergency of a Fourier series :— 



