4 SCIENCE PROGRESS 



W. H. Young and (Mrs.) G. C. Young (Compt. Rend. 191 6, 

 163, 509-1 1) consider the " normal " frontier of a region or of 

 a set of points. 



G. Hamel, it may be remembered, decided (1905), on the 

 supposition that the continuum can be well-ordered, that 

 there are discontinuous solutions of the functional equation 

 f(x + y) = fix). + f(y). In 1916 H. Blumberg showed that 

 such solutions are non-measurable, and in 191 7 obtained a 

 generalised result (Bull. Amer. Math. Soc. 191 8, 24, 220). 



An account of A. Palatini's paper (Nuovo Citnento, 191 7) 

 on Einstein's theory of relativity and the motion of the peri- 

 helion of Mercury's orbit is given in Nature, 19 18, 100, 492). 

 Other recent noteworthy papers on the theory of relativity 

 are by A. Einstein (Sitzungsber. der K. Preuss. Akad. der Wiss, 

 zu Berlin, 1916, 1111-16, B. Cabrera (Rev. de la R. Acad, de 

 Cienc. ex. de Madrid, 11 [8 articles], 12, 546-70, 738-52), and 

 F. Kottler (Sitzungsber. der K. Akad. der Wiss. in Wien, 19 16, 

 125 [Ha], 899-919). Other information as to the principle of 

 relativity is given under " Astronomy " in these " Recent 

 Advances." 



Arithmetic, Theory of Numbers, and Algebra. — E. Borel 

 (Compt. Rend. 1916, 163, 596-8) has a note on the approxima- 

 tion of incommensurable numbers by rational numbers, and 

 O. Nicoletti (Rend, del Circ. mat. di Palermo, 191 7, 42, 73-9) 

 translates some geometrical results into analytical language, 

 and thus obtains a class of iterations operating on a complex 

 variable, by which we can approximate to the roots of a 

 quadratic equation. 



G. Rados (Math, es term. ert. Budapest, 191 5, 33, 702-10) 

 treats a question from the theory of congruences of higher 

 degrees, gives (ibid. 758-62) a new derivation of the well- 

 known criterion of the solvability of quadratic binominal con- 

 gruences, and (ibid. 19 16, 34, 62-70) gives an analogue of 

 Wilson's theorem. Rados also (ibid. 641-55) gives a new ex- 

 position of the theory of binominal congruences. 



V. Amato (Rend, del Circ. mat. di Palermo, 191 7, 42, 48-60) 

 gives a resolution, in a quadratic corpus, of binominal con- 

 gruences in which the modulus is a prime ideal number of the 

 second degree. 



L. Grosschmid (Math, es term. ert. Budapest, 191 6, 34, 236- 

 52) has a paper on the distribution of quadratic residues. 



