CHAP. II] SYSTEM OF LINEAR EQUATIONS. 83 



consider the determinants K in which the constant terms appear in one 

 column, and let F be the g.c.d. of the D's and K's. There exist integral 

 solutions if and only if/ = F; while f/F is the least common denominator 

 of all sets of fractional solutions. Cf. Smith 207 and Frobenius. 210 



V. A. Lebesgue 206 would select, if possible, two equations ax\ = F(x 2 , 

 , x n ) and CL'XI = FI(X Z , -, x n ) from the system of linear equations such 

 that a, a' are relatively prime. Determine r, s, p, q so that ar a's = 1, 

 ap a'q = 0. Then Xi = rF sFi, pF qFi =. 0, whence the system is 

 reduced to the former and equations in x 2 , , x n only. To solve 

 ax + by = cz, a'x + b'y = c't, where the g.c.d. of a, b, c is unity, we may 

 set z = Du, where D = aa + b(3 is the g.c.d. of a, 6. Thus x = can + bv/D, 

 y = cfiu av/D. Then the second equation becomes Au + Bv = c't, 

 which may be treated as was the first. Given a system of m linear equa- 

 tions in ra + n unknowns in which an m-rowed minor D is not zero, we get 

 Dxi = fi(y\, - - , y n ], i = 1, , m. It remains to solve the congruences 

 fi = (mod D), which can be treated by the method for linear equations. 



H. J. S. Smith 207 proved that if the excess of the number of unknowns 

 above the number of linearly independent equations is m, we can assign m 

 sets of integral solutions (called a fundamental system of sets of solutions) 

 such that the determinants of the matrix formed by them admit no com- 

 mon divisor > 1. Every* set of integral solutions of the equations can be 

 expressed linearly and with integral multipliers in terms of the fundamen- 

 tal system. By use of this concept he proved the theorem of Heger: 205 A 

 system of linear equations is or is not solvable in integers according as the 

 g.c.d. of the determinants of the matrix of the coefficients is or is not equal 

 to the g.c.d. for the augmented matrix obtained by annexing a column com- 

 posed of the constant terms (cf. Frobenius 210 ). Use is made of the im- 

 portant elementary divisors. 



H. Weber 208 considered the system of equations 



hi = miffn + + rfipffpi + X t - (i 1, , p) 



with integral coefficients o-ji of determinant 5. If 5 ={= we obtain every 

 set of integers hi, , h p and each 5 P-1 times if we take all possible combina- 

 tions of integers for m b , m p and let Xi, , X p run independently 

 through a complete set of residues modulo 5. If 5 = 0, we can apply to the 

 m's such a substitution of determinant 1 that the matrix (<ry;) is trans- 

 formed into one with columns of zeros at the right. Then by a linear sub- 

 stitution on hi, " , h p of determinant 1, we get a matrix having zeros 

 except in the g-rowed minor in the upper left-hand corner. 



E. d'Ovidio 209 treated algebraically a system of n r independent 

 linear homogeneous equations in n unknowns and the conditions that it 

 have the same <*> r solutions as a second such system. 



206 Exercices d'analyse num^rique, Paris, 1859, 66-75. 



207 Phil. Trans. London, 151, 1861, 293-326; abstr. in Proc. Roy. Soc., 11, 1861, 87-9. Coll. 



Math. Papers, I, 367-409. 



208 Jour, fur Math., 74, 1872, 81. 



209 Atti R. Accad. Sc. Torino, 12, 1876-7, 334-350. 



