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I. " On Systems of Linear Indeterminate Equations and Con- 

 gruences." By H. J. STEPHEN SMITH, Esq., M.A., Fellow 

 and Mathematical Lecturer of Balliol College, Oxford. 

 Communicated by Professor J. J. SYLVESTER. Received 

 January 17, 1861. 



(Abstract.) 



The present communication relates to the theory of the solution, 

 in positive and negative integral numbers, of systems of linear inde- 

 terminate equations, having integral coefficients. In connexion with 

 this theory, a solution is also given of certain problems relating to 

 rectangular matrices, composed of integral numbers, which are of 

 frequent use in the higher arithmetic. Of this kind are the two 

 following : 



1 . " Given (in integral numbers) the values of the determinants 

 of any rectangular matrix of given dimensions, to find all the 

 matrices, the constituents of which are integers, and the determinants 

 of which have those given values. 



2. " Given any rectangular matrix, the determinants of which 

 have a given number D for their greatest common divisor, to find 

 all the supplementary matrices, which, with the given matrix, form 

 square matrices, of which the determinant is D." 



A solution of particular, but still very important cases of these 

 two problems, has been already given by M. Hermite. The method 

 by which in this paper their general solution has been obtained, 

 depends on an elementary, but apparently fertile principle in the 

 theory of indeterminate linear systems ; viz. that if m be the index 

 of indeter minuteness of such a system ('. e. the excess of the number 

 of indeterminates above the number of really independent equations), 

 it is always possible to assign a set of m solutions, such that the 

 determinants of the matrix formed by them shall admit of no 

 common divisor but unity. 



Such a set of solutions is termed a, fundamental set, and possesses 

 the characteristic property, that every other solution of the system 

 can be integrally expressed by means of the solutions contained in 

 it. A set of independent solutions is one in which the determinants 

 of the matrix have a finite common divisor, i. e. are not all zero. 



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