6 SCIENCE PROGRESS 



continuous function of a well-known type. Systems of lattice- 

 points are of importance in the theories of numbers, binary 

 quadratic forms, algebraic numbers, elliptic functions, modular 

 functions, and crystallography. 



D. Pompeiu {Compt. Rend. 191 6, 163, 430-32) gives a neces- 

 sary and sufficient condition that a given discontinuous func- 

 tion may be a derived function. 



L. Galvani {Rend, del Circ. mat. di Palermo, 19 16, 41, 103- 

 34) considers " convex " functions of one and of two real 

 variables which are defined in any aggregate. 



G. H. Hardy (Mess, of Math. 191 7, 46, 175-82) considers 

 Stieltjes's " problem of moments." It has been suggested by 

 Lebesgue (1909) that it should be possible to prove some at 

 any rate of Stieltjes's results, and in particular the theorem con- 

 cerning the uniqueness of the solution of the problem, by 

 methods independent of the theory of continued fractions. 

 Hardy gives two different proofs of the uniqueness which fulfil 

 this condition. 



T. H. Hildebrandt (Bull. Amer. Math. Soc. 191 7-18, 24, 

 1 13-44, 177-202) gives a paper which is very closely related to 

 the recent paper by G. A. Bliss (see Science Progress, 19 18, 

 12, 544). Many definitions of integration, which are more or 

 less related to the Lebesgue definition, have recently been 

 given, and Hildebrandt discusses some of these definitions 

 and considers their relation to the Lebesgue integral. In the 

 first section he discusses the types of definition of integration 

 which are extensions of the Darboux upper and lower integral 

 method of defining a Riemann integral : W. H. Young's 

 definition is shown to be equivalent to the Lebesgue definition ; 

 and Pierpont's definition is also considered. The second 

 section is devoted to a consideration of the definitions of the 

 integrals of functions which are not integrable according to 

 Lebesgue 's definition, because functions integrable are always 

 absolutely integrable (Jordan, Harnack, Moore, Borel, Den- 

 joy). The third section is devoted to the Stieltjes integral, 

 which has recently come into the foreground on account of 

 the part which it plays in the theory of linear functional opera- 

 tions on continuous functions. Hildebrandt points out that 

 a Stieltjes integral is expressible in terms of a Lebesgue inte- 

 gral of another function and conversely, but that in spite of 

 this the Stieltjes^integral seems to be applicable where the 



