An extension of descartes' rule of signs

Math. Ann. 70 (1911), p. 223. Oeuvres, tome 1, p. 22–25. L. c. Oeuvres, tome 1, p. 22–25. We have here slightly changed Laguerre's notation. Since the present article has been set in type, a paper by M. Fekete and J. Pólya has appeared (“Über ein Problem von Laguerre”, Rend. Circ. Mat. Pal. 34, p. 89), which discusses related formulas; and announces that (1+x)k is a Cartesian multiplier for sufficiently large values ofk. See Bôcher ‘Introduction to Higher Algebra’, p. 44 where, however, the columns of the augmented matrix are differently ordered. Cf. Meissner's evaluation ofR (loc. cit., note, p. 226, 227), which in his notation is the ‘Minimalzahln’. If we putk 1=R+1,k 2=R+2, we have the solution of Hurwitz for the correspondingf 2(x) of Theorem I.