CHAP, ii] SYSTEM OF LINEAR EQUATIONS. 85 



equations in the unknowns A, B, , we first equate to zero as many 

 unknowns (say A, , E) as possible such that there exists a solution 

 with F 4= 0; we may take F = 1 and have a solution " beginning with 

 F = 1." Next, set F = in the initial equations and equate to zero as 

 many of the earlier unknowns (say A, B, C) as possible such that there 

 exists a solution with D 4= 0; we may take D = 1 and have a solution be- 

 ginning with D = 1 and having F = 0. The third step might lead to a 

 solution with A = I, D = F = 0. Then we have a system of three 

 standard solutions. 



E. de Jonquieres 215 discussed the equations, arising in Cremona trans- 

 formations, 



n 1 



e*i = 3(n - 1), 2 i z ai = n z - 1. 



G. Chrystal 216 proved that if x f , y', z' form a particular set of solutions of 

 ax + by + cz = d, a'x + b'y + c'z = d', 



and if is the g.c.d. of the determinants (6c'), (ca'), (a&') while M is an 

 arbitrary integer, all solutions are given by 



x = x' + (bc')u/, y = y' + (ca')u/e, z = z' + (ab')u/c. 



T. J. Stieltjes 217 gave an exposition of the results by H. J. S. Smith 207 . 



L. Kronecker 218 gave a simple proof by induction of the theorem due 

 to Frobenius 210 that every n-rowed square matrix with integral elements 

 can be reduced by elementary transformations (interchange of rows or 

 columns and simultaneous change of sign of one row or column, and the 

 addition of one row or column to another) to a matrix in which every 

 element outside the diagonal is zero while every element =}= in the diagonal 

 is positive and a divisor of the following element. A matrix has a single 

 such reduced form. 



P. Bachmann 219 gave a detailed account of the theory of systems of 

 linear forms, equations and congruences. For a summary account, see 

 Encyclopedic des Sc. Math., tome I, vol. 3, 76-89. 



J. H. Grace and A. Young 220 gave a simple proof that any system of 

 linear homogeneous equations with integral coefficients has only a finite 

 number of irreducible solutions in integers ^ 0, a solution being called 

 irreducible if not the sum of two solutions in smaller integers ^ 0. 



J. Konig 221 treated, from the standpoint of modular systems, systems of 

 linear equations and congruences whose coefficients are polynomials in 

 assigned variables. 



216 Giornale di Mat., 24, 1886, 1; Comptes Rendus Paris, 101, 1885, 720, 857, 921. * Pam- 

 phlet, Mode de solution d'une question d'analyse indeterminee . . . theorie des trans- 

 formations de Cremona, Paris, 1885. 



216 Algebra, 2, 1889, 449; ed. 2, vol. 2, 1900, 477-8. 



217 Annales Fac. Sc. Toulouse, 4, 1890, final paper, pp. 49-103. 



218 Jour, fur Math., 107, 1891, 135-6. 



219 Arith. der Quadratischen Formen, 1898, 288-370. 



220 Algebra of Invariants, 1903, 102-7. 



221 Einleitung . . . Algebraischen Groszen, Leipzig, 1903, 347460. 



