540 Royal Society : — 



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 



Ai,i#i+Ai,2# 2 + • . . +Af, B a?»=AM»+i,raod M, i=\, 2,3, ... n 



represent a system of congruences ; let us denote by v«> V»— li • • • 

 Vi) Vo> the greatest common divisors of the determinant, first minors, 

 &c, of the matrix of the system [so that, in fact, v» is the deter- 

 minant itself, Vi the greatest common divisor of the coefficients 

 A i./, and Vo=l]; by D,„ D„_i, . . . Di, Do the corresponding 

 numbers for the augmented matrix ; let also li and d l respectively 



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



Vi-i 



with ^ ; and put 



m = d H Xd n -iX ... Xdi, 

 piXc»-lX ... XC U . . . 



Then the necessary and sufficient condition for the resolubility of 

 the system is 



m=fx ; 



and when this condition is satisfied, the number of solutions is pre- 

 cisely m. 



The demonstration of this result (which seems to exhaust the 

 theory of these systems) is obtained by means of the following 

 theorem : — 



" If ||A|i represent any square matrix in integral numbers, v»» its 

 determinant, v»-i> V«— 2, • . . Vi> V*> the greatest common divisors 

 of its successive orders of minors, it is always possible to assign two 

 unit-matrices ||<i|| and ||/3||, of the same dimensions as ||A||, and 

 satisfying the equation 



