On Linear Indeterminate Equations and Congruences. 539 



the exhibition of movements that are not only local but progressive 

 in space, it is needless to describe here, because the principles it in- 

 volves are essentially the same as those which are stated above. 



Jan. 1 7. — " On the Homologies of the Eye and of its Parts in the 

 Invertebrata." By J. Braxton Hicks, M.D. Lond., F.L.S. 



Jan. 24. — "On the Calculus of Symbols, with Applications to the 

 Theory of Differential Equations." By W. H. L. Russell, A.B. 



January 31. — Major-General Sabine, R.A., Treasurer and Vice- 

 President in the Chair. 



The following communications were read : — 



"On Systems of Linear Indeterminate Equations and Congru- 

 ences." By H. J. Stephen Smith, Esq., M.A. 



The present communication relates to the theory of the solutio 

 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 indeierminateness of such a system (i. 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 & 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. 

 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- 

 2N2 



