1904.] 



The General Theory of Integration. 



445 



increase in strength of colour, which was generally observed in the 

 light transmitted through these films when the plane of polarisation 

 of obliquely incident light was changed from that of incidence to a 

 perpendicular position is accounted for. 



In Part III some evidence is brought to show that the allotropic 

 silvers obtained by Carey Lea" 55 " are particular cases of the media which 

 have been considered in the second part. 



"The General Theory of Integration." By W. H. Young, ScJD., 

 St. Peter's College, Cambridge. Communicated by Dr. E. W. 

 Hobson, F.RS. Received April 23,— Eead May 19, 1904. 



(Abstract.) 



The paper begins with a recapitulation of the well-known definitions 

 of integration and of upper and lower integration (integral par exces, 

 par defaut; oberes, unteres Integral). The theorem on which the 

 Darboux definition of upper (lower) integration is founded is stated 

 and proved in the following form : — 



Given any small positive quantity e±, we can determine a positive 

 quantity e, such that, if the fundamental segment S be divided up 

 in any manner into a finite number of intervals, then, provided 

 only the length of each interval is less than e, the upper summa- 

 tion of any function over these intervals differs by less than e\ 

 from a definite limiting value (the upper integral). 



Next follows a discussion as to whether it is admissible to adopt a 

 more general mode of division of the fundamental segment than that 

 used by Eiemann, Darboux and other writers, when forming summa- 

 tions (upper, lower summations), defining as limit, the integral (upper, 

 lower integral), of a function over the fundamental segment. It is 

 shown by examples first that the restriction as to the finiteness of the 

 number of intervals into which the fundamental segment is divided 

 cannot be removed without limitations ; but that it can be removed, 

 provided the content of the intervals is always equal to that of the 

 fundamental segment. Secondly it is shown that the error introduced 

 by taking the summation over an infinite number of intervals whose 

 content is less than that of the fundamental segment, is not in general 

 corrected by adding to the summation the content of the points 

 external to the intervals multiplied by corresponding value (upper, 

 lower limit) of the function. Similarly it is shown that the more 



* ' Araer. Journ. of Science,' 1886. 

 VOL. LXXIII. 2 I 



