RECENT ADVANCES IN SCIENCE 367 



also the extension of these integrals so as to embrace the case 

 where the convergence is not absolute will not have the effect 

 of removing the discontinuity of the integral, even if we adopt 

 the postulates formulated by Harnack and E. H. Moore, sup- 

 posing that these conditions are modified so as to take account 

 of the work of Lebesgue and the results of the theory of integra- 

 tion with respect to discontinuous functions. Young, confining 

 himself for simplicity to the case in which the function with 

 respect to which we integrate is the variable itself, considers 

 certain generalisations of the usual processes in dealing with 

 functions which possess Harnack points, so that a Lebesgue 

 integral does not exist. With these generalisations it is to be 

 noticed that the process of integration by parts is allowable 

 and the second theorem of the mean is valid. 



Young (Proc. Roy. Soc. 191 7, A, 93, 28-41) gives a short 

 account of some of the formulae which are fundamental in the 

 modern theory of multiple integrals, integration being with 

 respect to a function of bounded variation of two variables. 



B. H. Camp (Amer.Journ. Math. 191 7, 39, 31 1-34) discusses 

 multiple integrals over infinite fields, and shows that, for 

 functions whose integrals over finite fields exist, the three 

 definitions of J. Pierpont (1906), C. de la Vallee-Poussin, and 

 G. H. Hardy (1902) are equivalent when any one of them 

 exists absolutely. Thus Hardy's concept is more general, since 

 it alone of the three does not presuppose absolute existence. 

 Camp makes certain applications to Fourier's double and 

 quadruple integrals. 



W. F. Osgood (Bull. Amer. Math. Soc. 191 7, 23, 4°4) com- 

 municates a theorem on the singular points of analytic trans- 

 formations which will shortly be published elsewhere. 



R. L. Borger (Bull. Amer. Math. Soc. 191 7, 23, 287-90) 

 proves a theorem about two real functions of the real variables 

 x and y which satisfy conditions immediately suggested by 

 Riemann's conditions of the analytic character of a function 

 of a complex variable. Borger 's theorem is proved without 

 integration and by means of a theorem of Kowalewski, and an 

 immediate consequence of it is that, if any function of a complex 

 variable possesses a finite derivative at each point of a simply 

 connected closed region, then this derivative is continuous, all 

 the derivatives of the function exist, and the function is develop- 

 able in a power series. 



