r 



RECENT ADVANCES IN SCIENCE 7 



Lebesgue integral is not. He gives an extension of the Stieltjes 

 integral modelled on the Lebesgue extension of the Riemann 

 integral, as well as the Fr£chet generalisation of the Lebesgue 

 and Stieltjes integral so as to apply to a class of general ele- 

 ments. The last section gives the definition of the Hellinger 

 integral and also the generalisations of this due to Radon and 

 E. H. Moore. 



W. H. Young (Proc. Roy. Soc. A, 191 6, 92, 353-6) proves that, 

 in a large class of cases, an oscillating succession of functions 

 can be shown to contain converging sequences of functions. 



L. Tonelli (Ann. di Mat. 191 6, 25, 275-316) extends results 

 obtained by him in 1910 on polynomials approximating to 

 functions of many real variables (Rev. sent. 18 [2], 95). 



G. Pal (Math, es phys. lapok, Budapest, 191 5, 24, 243-7) 

 gives certain generalisations of the theorem of Weierstrass on 

 the representation of continuous functions by series of poly- 

 nomials. 



O. Szasz (ibid. 191 6, 25, 157-77) seeks conditions that cer- 

 tain linear combinations of given functions may serve as bases 

 for approximation to any continuous function. 



L. Fejer (Math. 6s term. ert. Budapest, 191 6, 34, 209-29) 

 occupies himself with the approximation to a function that 

 we obtain by drawing through n points of its curve a parabgla 

 of the (2M-i)th degree in which the tangent at each one of the 

 points spoken of is parallel to the axis of abscissae. 



In the short account in Science Progress (191 8, 12, 367) 

 of a paper by R. L. Borger, it was mentioned that the funda- 

 mental theorems of Cauchy's theory of functions were deduced 

 from a theorem proved by G. Kowalewski in his book Die 

 komplexen Verander lichen und ihre Funktionen (Leipzig and 

 Berlin, 191 1, p. 187). This theorem is, that if the real func- 

 tions of two variables, u and v, are " properly " differentiable 

 in a rectangle in the rt^-plane, and also Bu/8y = 8v/8y, they 

 are derivatives of a function w with respect to x and y. But 

 this theorem is proved by Kowalewski by integration : in fact 

 w is J (udx + vdy) taken along a path in the rectangle up to a 

 point in the rectangle, and of course that means that we have 

 to prove a theorem exactly analogous to Cauchy's and by the 

 same method. Further, " proper " differentiability is a simple 

 translation of the existence of the derivative of f(z), where z 

 is a complex variable and f=u-\-iv (Kowalewski, p. 199). 



