ON THE THEORY OP NUMBERS. 319 



Gauss (see arts. 20 and 104 of this Report, or Dirichlet, Crelle, vol. xxi. pp. 141, 



2oi'ir 2fttir / 2ai7r\ 



142). From these values of Se ^ and Se ^ we infer that 2n^a7— e '^ / 



/ 2bin \ 



and 2n\a7— e ^ / are two quantities of the form Y+if'^Z »^\, and Y—ii^^Z \/\, 

 Y and Z denoting integral functions of x with integral coefficients ; i. e. 

 that 4X=Y^ — (— 1)'^XZ-. From this equation, which is a generalization 

 of that obtained by Gauss for the case when \ is a prime (Disq. Arith. 

 art. 357), we can deduce a solution of the equation T^— XY^=4. In the 



( 2aiV \ 

 x—e ^ )=Y+ii^^Z >^\, let us first write i for x, and then —i 

 for i, and let us denote by X^ Y^, Z^, X_j, Y_^, Z_{ the values which X, 



Y, and Z acquire when i and — i are written for x. We thus find, denoting 

 the number of numbers less than X and prime to it by X', 



4n.V2-e ^ ) [-i-e ^ j=2^'+2ncos"i^4 + XJ 



= [YiY_i+XZiZ_,] + >/\l^"ZiYi-+r'-'z.iYi2 ; 

 or, writing 



T for i[Y,Y_,+XZ,Z_..], Ufori[i'^%Y_i+i-'^'Z_iY,], 



and observing that XiX_j=l, 



i(T+U VX)=2^'n cos=^^+^\ T^-\U^=4., 



where it is easily seen that T and U are integral numbers. When fx is even, 

 we may obtain a solution of the equation more simply by writing -j-1 or — 1 

 for X. (See the notes of Jacobi and Dirichlet already referred to.) 



It is to be observed, however, that the solution obtained by these methods 

 is not in general the least solution. Its ordinal place in the series of solutions 

 depends (as we shall hereafter see) on the number of classes of forms of det. D. 



97. Solution of the General Indeterminate Equation of the second degree. — 

 The solution of the indeterminate equation ax"^ + 2bxy-\-cy'^ + 2dx + ley -\-f=-0 

 depends on the problem of the representation of a given number by a qua- 

 dratic form. We confine ourselves to the case which presents the greatest 

 complexity, that in which 6"— ac=D is a positive and not square number. 

 The methods of solution contained in Euler's Memoirs relating to it (see 

 Comment. Arith. vol. i. pp. 4, 297, 549, 570, vol. ii. p. 263 ; and the Algebra, 

 vol. ii. cap. vi.) are incomplete in several respects: first, because Euler 

 always assumes that a single solution is known, and only proposes to deduce 

 all the solutions from it; secondly, because it is not possible, from a given 

 solution, to deduce any other solutions than those which belong to the same 

 set with the given solution, whereas the equation may admit of solutions be- 

 longing to different sets ; and lastly, because he gives no method for distin- 

 guishing between the integral and fractional values contained in the formulae 

 by which x and y are expressed. The first complete solution of the problem 

 was given by Lagrange in his Memoir " Sur la solution des Problemes Ind6- 

 terniines du second degre" (Hist, de I'Academie de Berlin for 1767, vol. 

 xxiii. p. 165-311). But the following method of solution, which is different 

 in some respects and much simpler, will be found in a subsequent memoir, 

 "Nouvelle methode pour resoudre les problemes iud6termines en nombres 

 entiers" (Hist, de I'Academie de Berlin for 1767, vol. xxiv. p. 181) ; and in the 

 Additions to Euler's Algebra (paragraph 7). If we multiply by aD and 

 writep iovax + by + d, g for (b^ —ac) y + (bd—a€),M tor (bd—aey —(b^—ac) 

 (d^— a/"), the given equation becomes g'^~Dp^=M. Confining ourselves 



