CHAP. II] ONE LINEAR EQUATION IN w>3 UNKNOWNS. 75 



Adding, we get the given equation since f n = f. His second method con- 

 sists in treating these equations taken in reverse order, after each is divided 

 by the /,- in the second member. He noted that the method of Euler 144 

 is applicable also to Aa + Bb + Cc = u. 



K. Weihrauch 164 denoted by E(M : N) the integral part obtained on 

 dividing M by N, and by R(M : N) the remainder. Thus [if A! 



(3) AiXi + A 2 X 2 + + A n X n = A 



gives 



x l = E(A : A,} - x 2 E(A 2 : AJ - - - x n E(A n : AJ + t lt 



ti=^- [R(A : AJ - x 2 R(A 2 :AJ - > - x n R(A n : AI)}. 

 AI 



Treating the latter similarly, we get x 2 . Finally, we get a relation between 

 x n -i, x n , whose solution involves a new parameter n _i. Thus 



(4} y- = M 4- fl-,Zi 4- 4- a- d i (i = 1 



V*/ *<- -LrJ- i ~ "-jiH T T^ U-tn l^n 1 ^fr J.j 



in which MI, , M n is a set of solutions of (3), and 



Aian + A 2 a 2 y + + A n a n f = 0' = 1) > w !) 



The condition that (4) shall give all solutions of (3) is 



1 



, ay | = 1 (i = 2, , n; j = 1, , n - 1), 

 -^i 



where the symbol denotes an (n 1) -rowed determinant. 



T. J. Stieltjes 165 reduced a^i + + a n +ix n+ i = u to an equation in 

 one variable. If X = (ai, a 2 ) is the g.c.d. of a\, a 2 , we can find relatively 

 prime integers a, 7 such that aia + a 2 7 = X. Taking /3 = a 2 /X, 

 6 = i/X, we have ad (3y = 1. Set 



/ / / / 



Xi = ax\ + px 2) X 2 = yXi + ox 2 . 

 Then the initial equation is equivalent to 



Similarly, we can feplace the first two new terms by (ai, a 2 , a^x'i, etc., and 

 finally get dd? = u, where d = (ai, , a+i) is the g.c.d. of ai, , a+i. 

 Givuig to a;2, , x'*+i all sets of integral values, we get all solutions of the 

 initial equation if it be solvable, viz., if u be divisible by d. A system of 

 n independent sets of solutions is fundamental (Smith 207 ) if the g.c.d. of 

 the n + 1 n-rowed determinants is unity. 



W. F. Meyer 166 solved (3) by use of recurring series obtained by simplify- 

 ing and extending C. G. J. Jacobi's 167 generalized continued fraction 

 algorithm. 



184 Untersuchungen iiber eine Gl. 1 Gr., Diss. Dorpat, 1869. Zeitschrift Math. Phys., 19, 



1874, 53. 



165 Annales Fac. Sc. Toulouse, 4, 1890, final paper, pp. 38-47. 

 168 Verhand. des ersten Intern. Math.-Kongresses, 1897, Leipzig, 1898, 168-181. 

 167 Jour, fur Math., 69, 1868, 29-64; Werke, VI, 385-426. 



