82 HISTORY OF THE THEORY OF NUMBERS. [CHAP. II 



and hence are determined by the matrix (0^). The same discussion was 

 given by J. G. Gamier, 200 who remarked that the determination of the A's, 

 B's, - - is facilitated by the use of determinants. 



J. Struve 201 reduced the solution of (1) to an equation in 2 variables. 



V. Bouniakowsky 202 discussed the solution of one or more indeterminate 

 equations, chiefly of linear type. 



G. Bianchi 203 treated three linear equations in x, y, z, u, solving by 

 determinants for x, y, z as linear functions of u and determining by inspec- 

 tion what positive integral values, if any, may be given to u such that the 

 expressions for x, y, z become integers. 



C. A. W. Berkhan 204 noted that if (1) have positive integral solutions, 

 the x's are in arithmetical progression, with the common difference g h. 



I. Heger 205 considered a system of homogeneous equations 



(2) knXi + ' + kim+nX m+ n = (l = 1, , 7l) , 



with integral coefficients. Let Xn be the numerically least value 4= of Xi 

 in all possible sets of integral solutions, and let Xn, , x\ m+n be one such 

 set. Their products by 1 give a set of solutions. The only possible Xj.*s 

 are multiples of x^. In (2) set 



Xi = &ni, Xi = xii + x\ (i = 2, -, m + n). 

 Then 



kiiX 2 + - - + k im+n x' m+n = (i = 1, , n). 



As before, x' 2 = x 22 ^, where x 22 is the numerically least value 4= of x' 2 in 

 all sets of integral solutions. Let x 22) , x 2 m + n be such a set. Proceeding 

 in this manner, we get 



X m = 



If the determinant of the coefficients of a; m+ i, , x m+n in (2) is not zero, 

 those variables are definite linear functions of Xi, , x m , whence 



x m +j = Xim+j^i + + x mm+ m (j 1, - - , n), 



where the x { m+i may be taken integral. Giving arbitrary integral values to 

 1, , m, we obtain all integral solutions of (2). 



For n non-homogeneous equations in m variables, n < m, let all the 

 determinants D of order n of the matrix of coefficients have the g.c.d. /; 



200 Cours d' Analyse Algdbrique, ed. 2, Paris, 1814, 67-79. 

 1 Erlauterung einer Regel fiir unbest. Aufgaben . . ., Altona, 1819. 

 Bull. phys. math. acad. sc. St. P<5tersbourg, 6, 1848, 196. 

 3 Memorie di Mat. e Fig. Soc. Italiana Sc., Modena, 24, II, 1850, 280-9. 



Lehrbuch der Unbest. Analytik, Halle, I, 1855, 46-53. 



204 Denkschriften Akad. Wiss. Wien (Math. Nat.), 14, II, 1858, 1-122. Extract in Sitzungs- 

 bcr. Akad. Wiss. Wien (Math.), 21, 1856, 550-60. 



