348 SCIENCE PROGRESS 



method, gives the (rational, complex) fractions a geometrical 

 interpretation, and then studies approximations by means of 

 continued fractions. These researches bear some analogy to 

 those of Humbert (i9 l S, l 9 l &) for the case of real numbers. 



R. W. Brink {ibid. 1 86-204) gives a new integral test of the 

 second kind for the convergence and divergence of infinite 

 series. 



H. B. Mitchell (ibid. 43-52) obtains some conclusions as to 

 the position of the imaginary roots of a polynomial from the 

 real roots of its derivative. 



M. Bauer (Jahresber. der D.M.V. 191 6, 25, 294-301) writes 

 on the determination by iteration of the real roots of an algebraic 

 equation. 



G. Polya (Vierteljahrsschr. Zurich, 191 6, 61, 546-8) proves 

 a theorem on algebraic equations with real roots alone. 



G. Darbi (Ann. di Mat. 191 7, 26, 191-7) finds a property of 

 Abelian equations with cyclic groups. 



W. L. Hart (Bull. Amer. Math. Soc. 191 8, 24, 334-5) proves, 

 after a method due to F. Riess, a theorem on infinite systems 

 of linear equations. 



R. D. Carmichael (ibid. 286-96) derives numerous elementary 

 inequalities for the roots of an algebraic equation. The results 

 generalise many known inequalities and contain new ones of 

 interest. 



Louise D. Cummings (ibid. 336-9) draws attention to a 

 neglected paper (1847) of Kirkman, which is of importance in 

 the theory of triad systems. 



A. A. Bennett (ibid. 477-9) derives, by means of quite 

 elementary considerations, the equation of the probability- 

 curve from the sequence of binomial coefficients. 



C. H. Forsyth (ibid. 431-7) discusses his (191 6) interpolation 

 formula for giving what he calls " tangential interpolation of 

 ordinates among areas." 



E. T. Bell (ibid. 376-80) proves some properties of certain 

 remarkable determinants of integers, which follow immediately 

 from simple considerations in the theory of numbers. 



W. H. Metzler (Amer. Math. Monthly, 191 8, 25, 1 1 3-1 5) gives 

 another proof of Pascal's theorem (191 5) on certain determin- 

 ants which are expressible as the sum of two squares. 



Sir T. Muir (Proc. Roy. Soc. Edinburgh, 191 8, 38, 146-53) 

 has a note on Cayley's (1846) construction of an orthogonant. 



