348 SCIENCE PROGRESS 



equilibrium in a plane containing fixed particles repelling with 

 forces inversely proportional to the distance to prove that if all 

 the roots of a polynomial / (x) = o lie on or within any closed 

 polygon then all the roots of /'(x) = o lie within that polygon. 

 J. L. Walsh {Trans. Amer. Math. Soc, 22, 1921, loi ; Comptes 

 Rendus, 172, 1921, 662) has further developed this method and 

 applied it to locate the roots of the Jacobian of two binary forms. 



L. E. Dickson {Trans. Amer. Math. Soc, 22, 1921, 167; 

 Amer. Journ. Alath., 43, 192 1, 102) investigates completely 

 which general homogeneous polynomials can be expressed as 

 determinants with linear elements. Speaking geometrically 

 his result is that the equation to every plane curve, every 

 quadric surface, and a sufficiently general cubic surface can be 

 expressed in this way, but that no other general surface or 

 variety in higher space can be so expressed. Incidentally he 

 points out that the proof given by Jessop {Quartic Surfaces^ 

 p. 161) of the number of disposable constants in the equation to 

 a " determinant " quartic surface is not valid ; a similar 

 argument applied to a binary form would lead to incorrect 

 conclusions. 



L. E. Dickson {C.R., 172, 1921, 636) also investigates the 

 theory of triples of polynomials in n variables with a composition 

 theorem, viz. such that we have 



f{Xi,X2, . . • Xn) (f) {^i, ^2) • • • ^n) =F{^l>^2f ' • • ^n), 



where the X^, X^, • • • Xn are bilinear in the Xi, x^, . . . Xn, and 

 the gj, ^2 • • • ?«• 



E. T. Bell {Amer. Math. Monthly, 28, 1921, 258) proves that 

 if p is an odd prime not dividing 4''-i , then the BernouUian 

 number B2pr has its numerator divisible by p, the particular 

 case in which r = i having been discovered by J. C. Adams. 



H. Cramer {Arkiv for Mat., 15, 1921, No. 5) establishes a 

 number of theorems concerning prime numbers, in particular 

 concerning the order of the difference between two consecutive 

 primes and concerning Landau's question whether there is 

 always at least one prime between w* and (n + I)^ 



Viggo Brun {Proc. Camb. Phil. Soc., 20, 1921, 299) obtains 

 approximate formulae for the function [x], the number of 

 integers not exceeding x, and for tt {x), the number of primes 

 not exceeding x. 



E. Landau {Gottingen Nachrichten, 192 1, 88) gives a shorter 

 version of Hardy and Littlewood's solution of Waring's Problem. 



J. Liouville published in 1 858-65 a series of eighteen articles 

 in which he stated results which express equalities between 

 sums of values of general arithmetical functions when the 

 arguments of the functions involve the divisors of two numbers 

 whose sum is given. An account of these formulae, with refer- 



