456 SCIENCE PROGRESS 



(not zero) from a to b, the integral from a to b of f(x) is also 

 positive. Further, if Sciolette's demonstration is not possible, 

 f(x) is not integrable. 



One of Weierstrass's fundamental theorems in the theory 

 of analytic functions is that a function-element which can be 

 continued along every path within a simply connected region 

 defines a regular one-valued function for that region. A. 

 Pringsheim (Sitzungsber. der K.B. Akad. zu Munchen, 1915, 

 27 and 58) proves this theorem in a very simple way by means 

 of approximating step-like polygons (Treppenpolygone). 



Prof. W. F. Osgood (Bull. Amer. Math. Soc. 1916, 22, 443) 

 extends the scope of a fundamental theorem on the continua- 

 tion of analytic functions of several variables and simplifies its 

 proof. A case of the theorem was given by Kistler in 1905, 

 and a more general theorem by Hartogs in 1906 ; but both 

 these proofs involve n-ple integrals, whereas that of Osgood 

 makes use of only Cauchy's integral-formula for functions of a 

 single variable. 



Weierstrass stated the theorem that a function of n com- 

 plex variables which is meromorphic at every point of space is 

 rational ; and he understood by " space " the extended space 

 of analysis — the n spheres of the n complex variables. But 

 the theorem is true for other spaces, and Prof. W. F. Osgood 

 (Trans. Amer. Math. Soc. 1916, 17, 333) accordingly lays 

 down a general definition of infinite regions which will include 

 the cases of projective geometry, the geometry of inversion, 

 the geometry of the space of analysis, and so on. 



G. D. Birkoff (Trans. Amer. Math. Soc. 19 16, 17, 386) 

 shows that the classical results of Weierstrass and Mittag- 

 Leffler on the formation of infinite products of functions with 

 assigned singularities admit of a natural extension to infinite 

 products of matrices. The importance of this lies in the fact 

 that, in a large part of the theory of functions of a single com- 

 plex variable — such as for the functions defined by linear 

 difference and differential equations — the matrix of analytic 

 functions rather than the single analytic function must be taken 

 as the fundamental element. 



R. D. Carmichael (Trans. Amer. Math. Soc. 1916, 17, 207) 

 lays the foundations of the theory of a large class of series which 

 include the factorial series and are suitable for the representation 

 of functions which are defined throughout a half-plane and 



