89 



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 || represent any square matrix in integral numbers, v &* 

 determinant, v 1> V 2* Vi> Vo t ne greatest common divisors 

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

 unit-matrices ||aj| and ||/3||, of the same dimensions as ||Aj|, and 

 satisfying the equation 



The following result (among many which may be deduced from 

 this transformation of a square matrix) admits of frequent ap- 

 plications : 



" If D be the greatest common divisor of the determinants of the 

 matrix of any system of n independent linear equations ; of the D n 

 sets of values (incongruous mod. D) that may be attributed to 

 the absolute terms of the equations, the system is resoluble for 

 D"- 1 , and irresoluble for D' 1 - 1 ( D-l)." 



As an example of the use that may be made of this result, it is 

 shown, in conclusion, that it supplies an immediate demonstration of 

 a fundamental principle in the general theory of complex integral 

 numbers, composed of the root of any irreducible equation, having 

 its first coefficient unity, and all its coefficients integral ; viz. that 

 the number of incongruous residues, for any modulus, is always 

 represented by the norm of the modulus. A demonstration of this 

 principle has, however, already been given in the ' Quarterly Journal 

 of Pure and Applied Mathematics/ in a paper signed Lanavicensis ; 

 to whom, therefore, the honour of priority in this inquiry is due. 



