RECENT ADVANCES IN SCIENCE 4 35 



work of Dini and Scheefer, occupies a very important place 

 in the theory of functions of a real variable. Maurice Frechet 

 (Compt.Rcnd. 191 5, June 28) gives an extension of the definition 

 of an integral for any aggregate of numbers. 



Paul Appell (ibid. Sept. 27) investigates a second form 

 of theta functions of the fourth degree. W. B. Ford (Proc. Nat. 

 Acad. Sci. 191 5, 1, No. 7) represents arbitrary functions as 

 limits of definite integrals depending on a parameter when the 

 parameter becomes infinite, or by a series of definite integrals. 



In 191 5 F. R. Moulton presented to the same Academy a 

 paper on the solution of an infinite system of differential equa- 

 tions of the analytic type. It is interesting to notice in this 

 connection that, under certain conditions, there is a rigorous 

 solution of the problem of infinitely many bodies. To the same 

 Academy, H. Blumberg presented a paper in which he applied 

 the methods of what Prof. E. H. Moore called " general analysis" 

 in the strikingly original book he published some years ago, to 

 giving a uniform theory for factorisation of various types of 

 expressions. 



On the subject of differential equations, D. Pompieu (Compt. 

 Rend. 191 5, August 30) gives a double solution of Riccati's 

 equation, and Prof. T. Levi-Civita (Atti del R. Institute Veneto, 

 191 5, 74, 907) discusses the reduction of the differential equa- 

 tions of the problem of three bodies. 



There is, of course, an infinity of functions whose values 

 at a finite number of points are the same as those of a 

 given function, but Prof. E. T. Whittaker (Proc. Roy- Soc. 

 Edinb. 191 5, 35, 181) shows that a certain one of these 

 functions is represented by a well-known expansion in the 

 theory of interpolation, and that it can be constructed when 

 any one function of the above infinity is known. The function 

 thus representative of the set of functions mentioned is called 

 the " cardinal function," and the end of Whittaker's paper 

 contains a suggestion for the use of functions analogous to this 

 cardinal function to simplify the theory of functions of a real 

 variable. 



Geometry. — The second volume of the Opera Matematiche di 

 Luigi Cremona was published at Milan in 191 5 and contains 

 principally various papers of a historical nature on modern 

 geometry, the prolegomena to a geometrical theory of surfaces 

 written in 1 866, and sundry papers on gauche curves and surfaces . 



