CHAP. II] REAL LINEAR FORMS, APPROXIMATION. 99 



continued fractions, and found necessary and sufficient conditions that a 



given fraction be in Hermite's series of fractions tending towards w. The 



main condition was generalized by E. Cahen. 284 



Humbert 285 gave simple proofs of the theorems by Hurwitz. 263 



M. Fujiwara 286 supplemented Hurwitz 's 263 second theorem as Borel 270 



had the first. 



J. H. Grace 287 proved that if f ^ k ^ 2 and if x/y and x'/y' are two 



consecutive rational approximations to an irrational number 6 such that 



y 



then xy f - x'y = 1 [Hermite 259 for k = V3, Mhikowski 268 for k = 2]. 

 He 288 proved that Minkowski's 268 last result is final, i.e., if k < |, it is 

 possible to choose a and 6 such that there is not an infinitude of integers 

 x for which | y ax b \ < k/\ x |. 



284 Comptes Rendus Paris, 162, 1916, 779-782. 

 286 Jour, de Math., (7), 2, 1916, 155-167. 



286 Tdhoku Math. Jour., 11, 1917, 239-242. Cf. 14, 1918, 109-115. 



287 Proc. London Math. Soc., (2), 17, 1919, 247-258. 



288 Ibid., 316-9. 



