RECENT ADVANCES IN SCIENCE 347 



his proof of 191 5 of the illegitimacy of " classes which contain 

 themselves as members," and replies to certain criticisms. 



A. Rosenthal (Gott. Nachr. 191 6, 305-21) makes some 

 contributions to Caratheodory's theory of measurability, and 

 M. Leau (Compt. rend. 191 7, 165, 141-4) writes on the measure 

 of linear aggregates. W. Sierpinski (ibid. 164, 882-4) discusses 

 some problems which imply non-measurable functions, gives 

 (ibid. 993-4) an extension of the notion of density of an aggre- 

 gate, and (Ann. di Mat. 191 7, 26, 131-50) finds the necessary 

 and sufficient conditions that an aggregate of points should be 

 an " arc." 



E. Guillaume (Rev. de Metaphys. 191 8, 25, 285-323) con- 

 siders the theory of Lorentz as a given logical construction, and 

 tries to introduce into it a parameter which plays the part of 

 universal time (cf. his papers in Arch, de Geneve, 191 7, 43, 

 89-1 12, 185-98). L. de la Rive (ibid. 281-4) gives a geometrical 

 construction of the equations of relativity. R. Bricard (Nouv. 

 Ann. 191 7, 17, 201-22) gives an elementary sketch of the 

 principle of relativity in space of one dimension. L. Amaduzzi 

 (Scientia, 191 8, 24, 239-43, 321-6) gives a short, simple, and 

 excellent account of the principle of relativity. 



Theory of Numbers and Algebra. — G. H. Hardy and S. 

 Ramanujan (Proc. Lond. Math. Soc. 191 8, 16, xxii ; 17, 75-115) 

 develop their application (cf. Science Progress, 191 8, 13, 5) 

 to the principal problems of the theory of partitions of the 

 analytic methods which have proved fruitful in the theory of 

 the distribution of primes and allied subjects. 



G. Julia (Compt. rend. 191 7, 164, 352-5, 484-6, 571-4, 

 619-22, 910-13, 991-3) continues his researches on arithmetical 

 binary forms (cf. Science Progress, 191 8, 13, 180). 



H. H. Mitchell (Trans. Amer. Math. Soc. 191 8, 19, 119-26) 

 gives a proof that certain ideals in a cyclotomic realm are 

 principal ideals. 



M. Amsler (Compt. rend. 191 7, 165, 102-5) considers the 

 development of a quadratic irrational in a continued fraction. 



O. Szasz (Journ. fur Math. 147, 132-60) solves a general 

 problem relating to the convergence of continued fractions. 



G. Humbert (Journ. de Math. 191 6, (7) 2, 79-103, 104-54, 

 1 5 5-67) solves some problems on Hermite's (1854) method of 

 approximation to a given irrational number. L. R. Ford 

 (Trans. Amer. Math. Soc. 191 8, 19, 1-42) also studies Hermite's 



