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SCIENCE PROGRESS 



RECENT ADVANCES IN SCIENCE 



FUIIE MATHEMATICS. By Dorothy M. Wrinch, Fellow of Girton 

 College, Cambridge, Lecturer at University College, London. 



Prof. Hobson {Proc. Lond. Math. Soc, 2, xviii, part 4, p. 249} 

 has made an important contribution to the theory required 

 for the further development of integral equations. His paper 

 deals with the so-called Hellinger's integral, which arose pri- 

 marily in connection with the theory of quadratic forms involv- 

 ing an infinite number of variables. Helhnger regarded them 

 as a new species of limit, but Hahn reduced them to the ordinary 

 integrals associated with Lebesgue. Prof. Hobson makes a 

 simpler reduction by a more concise and simple method, and 

 extends the whole theory to a wider class of integrals, of which 

 these are, in fact, only special cases. 



In order to define the nature of a Hellinger's integral, we 

 suppose an interval /3-a to be divided into n parts, a to y^, 

 yi to 3/2, and so forth, the last being %_i to /3. If g{y) is a 

 continuous monotone function of y defined for the whole in- 

 terval, and such that g{y) never decreases as y increases, while 

 f{y) is a continuous function defined for the interval, and 

 constant in any part of it in which g{y) is constant, then the sum 



^-"{f{yr)-f(yr-.)y 



giyr) -g{y r-l) 



when it has a finite upper limit, independent of the mode of 

 division of the interval, is called a Hellinger's integral. Hel- 

 hnger proposed that it be denoted symbolically by 



/ 



'[df{y)y 

 dg{y) 



in his paper in Crelle's Journal, vol. cxxxvi (1909). It is clear 

 that a very difficult existence theorem is involved in the 

 definition, and the fundamental result in this connection, derived 

 by the author, is as follows : 



If r:^: is a variable equal to g{y), and if f{y) = F{x) when 

 expressed in terms of this variable^ then the necessary and 

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