8 SCIENCE PROGRESS 



Generalising a theorem of Tchebychef, T. Nagel {Liouville, 

 4, 192 1, 343-56) proves that, if f(x) be any irreducible integral 

 function of degree greater than i , with integral coefficients, and 

 P„ be the greatest prime divisor of /(i) ./(2) . . . /(«), then the 

 limit as n tends to infinity of n{log nYlP^ is zero, where e is 

 any positive number less than i . 



P. J. Heawood (Proc. Lond. Math. Soc, 20, 1921, 233-50) 

 discusses rational approximations to incommensurable real 

 numbers ; and O. Perron (Math. Ann., 83, 192 1, 77-84) im- 

 proves results of Kronecker and Minkowski on the degree of 

 approximation to n real numbers by rational fractions with a 

 common denominator. 



L. E. Dickson (Proc. Lond. Math. Soc, 20, 1921, 225-32) 

 develops an arithmetic of quaternions, taking as integral 

 quaternions those a -h bi + cj + dk in which a, b, c, d are 

 integers ; A. Hurwitz, in his Vorlesungen iiber die Zahlentheorie 

 der Quaternionen, published just before his death, had taken 

 as integral those in which a, b, c, d are either integers or halves 

 of odd integers. 



On the reciprocity formula for algebraic numbers there are 

 papers by L. J. Mordell {Proc. Lond. Math. Soc, 20, 192 1, 289- 

 96) for the quadratic field, and by L, Koschmieder {Math. 

 Ann., 83, 1921, 280-85) for the cubic field, the latter making 

 use of elliptic functions. 



H. Hancock {Liouville, 4, 1921, 327-42) obtains integral 

 solutions of the equation x^ + yi^ = ^ in the quadratic field. 



C. Siegel {Math. Zs., 11, 1921, 246-75) deals with a theorem 

 stated by Hilbert that every number in an algebraic field (pro- 

 vided that neither it nor any of its conjugate numbers are 

 negative real quantities) can be expressed as the sum of the 

 squares of four numbers of the; field. 



E. Landau {Math. Zs., 11, 1921, 317-18) has a note on the 

 uniform convergence of Dirichlet's series ; and K. Vaisala 

 {Acta Dorpat, A. i, 1921) generalises them by considering series 

 ^a^ e-V^ where X„ = r„e^«, and lim r„ = 00. 



Geometry. — In a plane a rigid system can be brought from 

 one position to another by a single rotation about a point ; in 

 three dimensions by a rotation about a line together with a 

 translation along the line. R. A. P. Rogers {Proc Roy. Irish 

 Acad., 36 A, 1922, 60-73) discusses the similar reduction of a 

 displacement in Euclidean space of n dimensions ; this is, of 

 course, a representation of a real, orthogonal linear trans- 

 formation, 



F. Severi {Rend. Lombardo, 54, 1921, 243-54) sketches a 

 method for enumerative questions concerning curves in space 

 of any number of dimensions ; the problem is first solved for 

 a rational curve, and then the general case of genus p is arrived 



