142 Mr. W. Williams on the 



Hence, any element of the form 



1 



sinm#d# 

 a _„ ]6 sine' 



must vanish when e = 0, provided F has not an infinite 

 number of oscillations at the point a, for it cannot exceed the 

 value it would have if sin m6 were put equal to 1 all through. 

 The sum of the finite number of elements of this form which 

 occur in the integral $ n at the points a x a 2 . . . is therefore 

 zero* Again, since T?(6 + x) is continuous up to (a—fx-^e) and 

 beyond (a + /u^e) , we can always choose for t a value such that 

 ~F(a±/j,6±t) —¥(a±/jt6) is as small as we please, however small 



27T 



fie may be, t being = or < - -, and n = cc . Hence, by 



(14), the elements p which occur in the neighbourhood of 

 the infinite values of F(0 + x) are infinitely small when n = cc , 

 and therefore, as before, A and C vanish when n = co . If, 

 then, F(0 + a?) is not infinite when = 0, S o0 =F(#), provided 

 the conditions relating to the portion B are fulfilled ; but if 

 ¥{6 + x) is infinite when 0=0, the value of B is go , and 

 therefore S^^co, or the series is divergent, as we should 

 expect. Hence, if the function contains a finite number of 

 infinite values of the above kind, Fourier's series is, ceteris 

 paribus, convergent for all values of x except those corre- 

 sponding to the infinite values, and for these values of x the 

 series is divergent. 



32. If the function F(0 + #) is indeterminate over a finite 

 range of values of x — for example, if it has an infinite number 

 of discontinuities, or maxima and minima of finite amplitude, 

 over that range — the coefficients of the series and therefore 

 S n cannot be determinate. But the function may have an 

 infinite number of discontinuities, or maxima and minima of 

 finite amplitude, or singularities in the neighbourhood of a 

 finite number of points ; for, since the range within which 

 these singularities occur in the neighbourhood of one of 

 these points is infinitely small, and the function is never in- 

 finite, the elements of the integrals which determine the 

 coefficients and S n corresponding to this range must be 

 infinitely small. Hence, since there is only a finite number 

 of such points, the sum of the elements corresponding to them 

 vanishes, so that the values of the integrals are determined 

 only by the continuous portions of the function. Hence, the 

 coefficients of the series are finite and determinate, and S^ 

 tends to a definite limit for all values of x except those corre- 

 sponding to the indeterminate points in the function ; and 



