84 HISTORY OF THE THEORY OP NUMBERS. [CHAP. II 



G. Frobenius 210 proved the following generalization of the theorem of 

 Heger : 205 Several non-homogeneous linear equations have integral solutions 

 if and only if the rank I and the g.c.d. of the Z-rowed determinants of the 

 matrix of the coefficients of the unknowns are the same as for the augmented 

 matrix obtained by annexing a column formed by the constant terms. 

 Again, sets of integral solutions of m independent linear homogeneous 

 equations in n unknowns (n > m) form a fundamental system if and only 

 if the (n m) -rowed determinants formed from them have no common 

 divisor. He discussed (pp. 194-202) the equivalence under linear trans- 

 formation of determinant db 1 of two systems of m linear forms in n vari- 

 ables; on this subject, see Smith, 207 G. Eisenstein, 211 and G. Frobenius. 212 



Ch. Meray 213 considered a system of m linear forms 



(3) <j)i = cax + &#++ jiV (i = 1, , m) 



in n > m unknowns. Multiplication of this system by the matrix 



V-m 



is defined to be the operation of forming the system of m forms 



If we multiply the latter system by a second matrix, we get a system which 

 can be derived from (3) by multiplication by the product of the two 

 matrices. Given a system of m forms (3) with integral coefficients, the 

 w-rowed determinants of whose matrix of coefficients are not all zero and 

 have the g.c.d. d, we can assign a matrix (4) of rational elements of deter- 

 minant 1/d, and a linear substitution on n variables with integral coefficients 

 of determinant unity, such that after multiplication by the matrix and 

 transformation by the substitution, we obtain a system of forms x it 

 =b x 2 , , d= x m . Then the system <; + fc t - = (i = 1, , m) have 

 integral solutions if and only if the m-rowed determinants of the coefficients 

 of the 0's have for their g.c.d. a number d dividing all the m-rowed deter- 

 minants obtained from the preceding determinants by replacing the elements 

 of an arbitrary column by the /c's [Heger 205 ]. When the equations have a 

 set of integral solutions , , \f/ } all sets of integral solutions are given 

 without duplication by 



X = Xi ' ' Xn- m O n -m '' V = 



where the 0's are arbitrary integers and the coefficients of any 9j satisfy 

 the system 0< = (i = 1, , m). 



A. Cayley 214 suggested that, to solve a system of linear homogeneous 



Jour, fur Math., 86, 1878, 171-3. Cf. Kronecker. 218 



211 Berichtc Akad. Wiss. Berlin, 1852, 350. 



212 Jour, fur Math., 88, 1879, 96-116. 



213 Annales sc. de l'6colc normale sup., (2), 12, 1883, 89-104; Comptes Rendus Paris, 94, 



1882, 1167. 

 2 Quar. Jour. Math., 19, 1883, 38-40; Coll. Math. Papers, XII, 19-21. 



