6 SCIENCE PROGRESS 



D, Jackson himself {Trans. Amer. Math. Soc, 22, 1921, 

 117-28, 158-66) has dealt with the method of least mth 

 powers, in which the trigonometric sum of order n is found, 

 such that the integral of the mth power of the absolute value 

 of f{x) — Tn(x) over the period 27r has the least possible value. 

 The sum always exists and is uniquely determined for my i, 

 and approaches the Tchebychef sum as a limit as ;« tends to 

 infinity. 



Allied to the foregoing is a paper by C. Jordan {Proc. Lond. 

 Math. Soc, 20, 1921, 297-325), which, given values j'^, j'l, . . . y^-i 

 of a function for x^, x-^, . . . x^-x determines a sequence of 

 polynomials (jy^ix), of successive degrees 1,2,... 7n, such that if 



fm{x) = Co + qc^i + . . . + c^^„, 



where the c are independent of m, then ^fc — /^C^i)]^ is a 

 minimum for all values of m. 



E. A. Milne and S. Pollard (Proc. Lond. Math. Soc, 20, 

 192 1, 264-88), using the methods of Lebesgue integration, 

 discuss the maximum errors of certain integrals and sums 

 involving functions whose value is not precisely determined. 



Recent numbers of the Paris Coniptes Rendus contain a 

 good deal of work on integral functions ; T. Varopoulos {C.R., 

 173, 1921, 515, 693, 963) determines functions whose addition 

 to the argument of an integral function does not alter its order 

 of growth ; P. Fatou (ibid., 571-3) investigates integral functions 

 with two distinct multiplication theorems ; and G. Valiron 

 (ibid., 1059-61) gives several properties of the roots of the 

 equation f(z) = Z. 



D. G. Taylor (Proc. Edin. Math. Soc, 39, 1921, 63-7) 

 obtains expressions for the addition of a third of a period to the 

 argument of an elliptic function ; and F. Bowman (Proc Lond. 

 Math. Soc, 20, 1921, 251-63) for the differential coefficient of 

 the complete third Jacobian elliptic integral with regard to the 

 modulus, with examples of its application, 



A pair of complex numbers %, z^ can be regarded as repre- 

 senting a point of a four-dimensional space, and two anal3^tical 

 functions w^ = f\(zx, ^2), ^2 = Ai^i) ^2) give a representation of 

 a region of this space upon another region. Such a representa- 

 tion is not conformal, and K. Reinhardt (Math. Ann., 83, 1921, 

 211-55), investigating it in detail, shows that there is nothing 

 corresponding to Riemann's theorem — i.e. there are certain 

 simple 4-dimensional regions (corresponding to circles in the 

 plane) which cannot be reversibly represented on each other. 



R, Nevanlinna (Oversigt Finska, A, 63, 1920-21, No, 6) 

 obtains upper limits for the coefficients of a power series which 

 gives a conformal representation of a star-region, 



W. Windau (Math. Ann., 83, 1921, 256-79) extends to 



