88 



The theory of independent and fundamental sets of solutions in some 

 respects resembles that of independent and fundamental systems of 

 units in Lejeune Dirichlet's celebrated generalization of the solution 

 of the Pellian equation. 



By the aid of the same principle of fundamental sets, the follow- 

 ing criterion is obtained for the resolubility or irresolubility of inde- 

 terminate linear systems. 



" A linear system is or is not resoluble in integral numbers, accord- 

 ing as the greatest common divisor of the determinants of the 

 matrix of the system is or is not equal to the corresponding greatest 

 common divisor of its augmented matrix." 



[The matrix of a linear system of equations is, of course, the 

 rectangular matrix formed by the coefficients of the indeterminates ; 

 the augmented matrix is the matrix derived from that matrix, by 

 adding to it a vertical column composed of the absolute terms of the 

 equations.] 



A system of linear congruences may, of course, be regarded as a 

 system of linear indeterminate equations of a particular form ; and 

 the criterion for its resolubility or irresolubility is implicitly con- 

 tained in that just given for any indeterminate system. But this 

 criterion may be expressed in a form in which its relation to the 

 modulus is very clearly seen. 



Let 



A a #i+ Af, 2 # 2 + . . . +A;, n # M = Ai, w+ i,mod M,*=l, 2,3, ... n 

 represent a system of congruences ; let us denote by v Vn-it 

 V p Vo> the greatest common divisors of the determinant, first minors, 

 &c., of the matrix of the system [so that, in fact, V M is the deter- 

 minant itself, vi the greatest common divisor of the coefficients 

 Ay, and Vo = l]j ^7 D n , T)n-i> DI, Do the corresponding 

 numbers for the augmented matrix ; let also ^ and di respectively 



represent the greatest common divisors of M with -^-, and of M 



V*-i 



with T ; and put 



Then the necessary and sufficient condition for the resolubility of 

 the system is 



