430 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xm 



solutions of F(x, y, z)=0 are given by axfa = by/(3 = czf'Y, where a, (3, 7 

 have been defined and u, v, w are given by (2). In particular, x = a/a, 

 y = f3/b, z=y/c are solutions. 



To apply this method for Af = 3, we remove the factor p P from FiF z 

 and have a quadratic in p, whose discriminant is to be made a perfect 

 square if new rational solutions exist. To avoid treating this quadratic, 

 Cauchy 287 of Ch. XXI gave a method independent of the above. 



G. Poletti 151 treated the general equation of degree two in three un- 

 knowns. First, for solution in rational numbers, solve for one unknown u 

 in terms of the other two v, w. Since the radical Z is to be rational, a 

 quadratic function of v, w is to be a square Z 2 . Solve the latter for v; 

 a new radical Y is to be rational, whence 



Solving this for w, we see that a radical X is to be rational : 



(F) X 2 = AY*+BZ 2 +C. 



Hence the rational solution of the initial equation is equivalent to that of 

 (F), where A, B, C are given integers. This in turn is evidently equivalent 

 to the solution in relatively prime integers of 



(G) z 2 = A2/ 2 +z 2 +a 2 . 



Set 7r = 2 Ay 2 and call < the g.c.d. of x, y; \f/ that of z, t. The quotient 

 7T i of TT by <V 2 is an integer. Thus the problem reduces to 



(H) x\-Ay\ = ^\ Bzl+Ctl^irrf, 



where Xi=x/4> and y\ = yl<t> are relatively prime, and likewise also z 1} t\. 

 From Legendre's theory of the quadratic forms of divisors of x\ Ayl, 

 we get TTi as a quadratic function of two parameters y', z', and \f/ as one of 

 y'ij z\ ; then by the composition of quadratic forms, we get Xi, y\ as functions 

 of the four parameters y', z', y', z(. To get the linear forms 4A+6i of 

 the divisors n, use Legendre's text. By (H 2 ) these must divide p 2 +-BCV 2 , 

 whose divisors are of certain linear forms 4BC^-\-j3i. Equate each of the 

 latter to 4A+fe* and solve for integers , ^. For each such set of solutions, 

 we can tell by a theorem of Legendre whether or not (H 2 ) is solvable in 

 integers. There is a similar discussion of the solution of (F) in integers. 

 A. Cayley 152 treated the generalization of Euler's 82 equation (5), viz., 



(3) $(x, y) = ax*+!3x+y-{y z -'r 1 y-6 = <}>(a, b). 

 This is a special case of 



(4) (abcfgh) (x'y'z'Y = (abcfgh) (xyz)\ 



where the second member denotes ax 2 -{-by 2 -\-cz- -\-2fyz -\-2gxz -\-2hxy. It is 

 assumed that the latter has a linear automorph (transformation into itself) , 

 which may be taken to be such that z' = z. For z' = z = l, h = 0, (4) becomes 

 (3). We can find a solution of (4) by Hermite's method: set o/ = 2 x, 



151 Memorie Accad. Sc. Torino, 31, 1827, 409-49. Cf. Atti della Societ& Ital. delle Scienze 



residcnte in Modena, Vol. 19. 

 162 Nouv. Ann. Math., 10, 1857, 161-5; Coll. Math. Papers, III, 205-8. 



