144 Mr. W. Williams on the 



function*. Riemann's condition is as follows : — Consider a 

 function F(#) between a and b. Divide (b — a) into intervals 

 6\ 8 2 . . . £ n , so that (b — a) = (Si + 8 2 -h. . . + S n ). Let D denote 

 the numerical value of the difference between the greatest and 

 least values of F(#) within the interval S 1 ; similarly D 2 for 

 the interval 8 2j &c. Then D n is called the oscillation of the 



f* 

 function within the interval 8 n . In order that | F(#)d# 



may have a determinate value, ^ a 



(S 1 D 1 + S 2 D 2 + S 3 D 3 + ..,+ SJ) n ) 



must tend to the value zero when 8 1 S 2 . . ,S n are diminished 

 without limit, the necessary and sufficient condition for which 

 is that the sum of the intervals within which the oscillations D 

 are greater than a given finite quantity <r, however small, 

 must be infinitely small when the intervals are infinitely 

 small. If the oscillation within an interval 6' taken on either 

 side of a given point is always > a when 8 is diminished 

 without limit, the function is said to be discontinuous at that 

 point, and the point is spoken of as a point of discontinuity ; 

 and, on the other hand, if the oscillation is <a, the point is 

 a point of continuity. If every point within a finite segment 

 is a point of discontinuity, the function is said to be discon- 

 tinuous over that segment, as, for example, a function which 

 has an infinite number of maxima and minima of finite am- 

 plitude over a finite range of points. If within a given segment 

 the points of continuity are finite in number, the segment 

 can be broken up into a finite number of other segments, 

 over which the function is discontinuous. But if between two 

 points there are no segments of discontinuity, there may, 

 nevertheless, be any number whatever, finite or infinite, of 

 points of discontinuity. In the first case the function is 

 not integrable, since the sum of the intervals of discontinuity 

 is finite. In the second case, Hankel, who has investigated 

 this matter with the view of rendering Riem ami's condition 

 less indeterminate in character, has shown that the sum of the 

 intervals of discontinuity cannot be finite f- Hence, the 

 function is, in such a case, integrable, and accordingly, Rie- 

 mann's condition may be more precisely stated as follows : — 

 A function is integrable between a and b if it is finite and de- 



* " Ueber die Darstellbarkeit einer Function durch eine trigono- 

 metrische Reihe ; " Abhandlungen der k. Gesellschaft der Wissenschaften 

 zu Gottingen, vol. xiii. This paper has also been translated, and published 

 in the Bulletin des Sciences Mathematiques, 1873, p. 35. 



t " Untersuchungen ueber die unendlich oft oscillirenden und unste- 

 tigen Functionen j" Tubingen, 1870. 



