2 SCIENCE PROGRESS 



sufficient condition for the existence of the integral {i.e. of the 

 limit defined above) is that F{x) should be the indefinite integral 

 of a function ^(a;) whose square is summable in the interval {a, b) 

 of X, which of course corresponds to the interval (a, /3) of y. 



Moreover, the Lebesgue integral to which the Hellinger limit 

 is equivalent is merely 



r [<l>{x)Ydx 



nJ a 



We have somewhat expanded the statement of the theorem 

 given by Prof. Hobson himself. 



The author proceeds, after establishing this very important 

 result completely, to some generalisations. For example, if 

 fi{y), f-i{y) are two functions with the property of f{y) in the 

 preceding statement, then the sum 



v-» myr) -/lb.-:)] Ulyr) -Uyr-.)] 

 '^' g{yr)~g{yr.i) 



denoted by Hellinger as 



-^ dMy)dMy) 

 dg{y) 



when the limit exists, is such as to possess the necessary limit 



as a Lebesgue integral if <^i and <^a are summable in the interval 

 {a, b), and have Fx and F^ as their indefinite integrals. Much 

 more extensive and fundamental generalisations follow, in- 

 cluding, for instance, types of Hellinger integrals which converge 

 to Lebesgue integrals of the form 



/:': 



•y a 



dx 



where n is any power not restricted to integer values. For 

 these we must refer the reader to Prof. Hobson's paper. The 

 investigation appears likely to be of considerable utility in 

 applied mathematics, where limits of the type defined are of 

 not infrequent occurrence. 



Other papers of interest in the same Proceedings (parts 4, 

 5, 6) are noticed more briefly below : 



H. J, Priestley, part 4, p. 226, publishes a note on the values 

 of n, which make the function 



— 'P-'^ix) 



dx n ^ ' 



