RECENT ADVANCES IN SCIENCE 619 



the arithmetical-philosophical concept of equivalence and 

 of repetition and on the concepts of number and order (228- 

 34, 235-8 ; cf. A. Nattucci, 282-3), and by F. Palatini on 

 infinitesimals and the actual infinite (127-33). 



Arithmetic and Algebra. — W. J. MacMillan (Amer. Journ. 

 Math. 1916, 38, 387-96) gives a theorem connected with 

 irrational numbers consisting in the fact that the limiting 

 value of the geometric mean of certain positive real numbers 

 is zero or iJ2 t according as a certain number in the function is 

 rational or irrational. 



A. J. Kempner (Trans. Amer. Math. Soc. 1916, 17, 476-82) 

 gives a new form of infinite series which defines transcendental 

 numbers, and which is of great interest by the side of the 

 series of Liouville, E. Maillet, and G. Faber. It is interesting 

 that his theorem shows that the important function which 

 Fredholm introduced in 1891 has transcendental values for an 

 infinite set of real rational values of the argument having the 

 origin as a limiting point. 



E. Noether (Math. Ann. 1915, 77, 103-28), in a paper on 

 the most general regions of whole transcendental numbers, 

 constructs a region of these numbers, which is to have certain 

 properties, by means of Zermelo's theorem on well-ordering 

 aggregates . 



A. Perna (Giorn. di Mat. 191 5, 53, 46-93) continues his 

 paper on the construction of transcendental numbers, and 

 gives an exposition of the investigations of others on the 

 arithmetical properties of transcendental functions. 



H. B. Fine (Proc. Nat. Acad. Sci., Washington, D.C., 19 16, 

 2, No. 9) gives a condition under which Newton's method of 

 approximation for calculating a real root of an equation will 

 with certainty lead to such a root, and gives an analogous 

 investigation for the extension of this method to a system of 

 equations. 



H. Taber (Amer. Journ. Math. 1916, 38, 337-72) gives 

 conditions for the complete reducibility of groups of linear 

 substitutions. 



O. E. Glenn (Trans. Amer. Math. Soc. 1916, 17, 405-17) 

 treats of a transvectant operation between general connexes 

 in which the variables are ternary and contragredient which 

 gives all invariant formations of the simultaneous systems of 

 the connexes. His method bears a closer resemblance to 



