CHAP. II] REAL LINEAR FORMS, APPROXIMATION. 95 



members are linearly independent, so that r is the ordinary (absolute) rank 

 of the rectangular matrix 



(dik) (i = 1, , P', k = 1, -, q). 



This matrix is said to be of relative rank (or rank of rationality) R if R is 

 the least number such that, by means of a linear substitution on the rows 

 with arbitrary coefficients, the matrix can be transformed into a matrix 

 all but r of whose rows contain only zero elements and all but R rows 

 contain only integral elements. Necessary and sufficient conditions for 

 the approximate solution in integers of our equations are expressed in dif- 

 ferent forms: R of the 's can be given arbitrary values, while the choice 

 of r R of the 's is limited only by certain conditions of rationality, 

 while the remaining p r 's are uniquely determined in terms of the earlier 



rC Q 

 s ^* 



A. Hurwitz 263 proved that if co is irrational there exists an infinitude 

 of pairs of integers x, y for which yfx co | < l/(V5x 2 ). Likewise, 

 y/x co < 1/| V&c 2 } if co is not equivalent to (1 + V5)/2. 



H. Minkowski 264 found by use of lattice points and other geometrical 

 concepts the fundamental theorem that, if /i, -,f n are linear homogeneous 

 functions of Xi, - , x n with any real coefficients whose determinant A is 

 not zero, we can assign integral values not all zero to x i} , x n such that 

 |/| = A/I A | for i = 1, , n. If a\, , a n -\ are real, we can find 

 integers x\, , x n without a common factor and with x n > such that 



a- 



x n 



For n > 1, consider n linear forms /i, -,/ in xi, , x n with a determinant 

 A =}= 0, such that r of the forms have real coefficients and s = (n r)/2 pah's 

 have conjugate imaginary coefficients, and let p be any real number ^ 1. 

 Then integral values not all zero can be assigned to x\, , x n such that 265 



-EI//*<^ 2 ^ n ~ - W /^I A I \ p ' 



n.js 1 ' ' I \T/ |T(1 + l/p)} r 2- 2 '."{r(l + 2/p)}< 



except for p = 1, s = 0, n = 2, when the members may be equal; here F 

 denotes the ordinary gamma function. He obtained (p. 161) Lagrange's 255 

 result on the minimum of x ay. 



A. Hurwitz 266 gave an elegant analytic proof of Minkowski's 264 theorem, 

 and the fact that the inequality sign may be taken in n 1 of the n relations. 



Ch. Hermite 267 remarked that Euclid's g.c.d. process leads to approxi- 



263 Math. Annalen, 39, 1891, 279. This and papers cited on p. 158 of Vol. I of this History 



give approximations by use of Farey series. 



264 Geometrie der Zahlen, 1896, 104-123. Extracts in Math. Papers Chicago Congress, 1896, 



201-7; French transl., Nouv. Ann. Math., (3), 15, 1896, 393^03. 



265 Also in Comptes Rendus Paris, 112, 1891, 209; Werke, I, 261-3. 



266 G6ttingen Nachrichten, 1897, 139. French transl., Nouv. Ann. Math., (3), 17, 1898, 

 64-74. Cf . P. Bachmann, Allgemeine Arith. d. Zahlenkorper, 1905, 335-41; G. Humbert, 

 Annales de la Fac. Sc. Toulouse, (3), 3, 1911, 8-12. 



267 Le Matematiche pure ed applicate, Citta di Castello, 1, 1901, 1-2; Werke, IV, 552-3. 



