CHAP, xiii] SOLUTION OF ax 2 +bxy+cy 2 = dz 2 . 405 



given equation by I and note that the product of fx 2 +gxy-\-hy 2 by I is of 

 that same form. 



C. Gill 34 solved x 2 -y 2 = bc by setting x+y = b cot A/2. Next, 



x 2 +axy+by 2 =z 2 

 is satisfied by 



z-\-x = y cot A/2, z x = (ax+by] tan A/2. 



Eliminate z. The resulting equation gives xfy, whence 



y = t (sin A+a sin 2 A/2), x = t (cos 2 A {2- b sin 2 A /2). 

 Take = m 2 + n 2 , sin A = 2mw/. Then 



x = m 2 bn 2 , y = 2mn-{-an 2 , z = m 2 +amn+bn 2 . 



G. L. Dirichlet 35 proved that Az z +2Bzy+Cy 2 = x 2 is solvable in integers, 

 with x prime to 2Z), if the left member is a form of determinant D of the 

 principal genus. 



J. Neuberg 36 noted that x 2 xy+y 2 =z 2 holds if 



x = 2pq-q 2 } y = p*-q*, z = p 2 -pq+q 2 . 



T. Pepin 37 gave special methods to obtain a particular solution of 

 ax 2 +2bxy-\-cy 2 = z 2 . Given one solution x = a, y = P, 2 = 7, to find all, 

 eliminate D = b 2 ac between 



az 2 = (ax+byY-Dy 2 , ay 2 = (aa+bpy-Dp 2 , 



and write p/q for the irreducible fraction equal to (f3zjy}/(^xay). 

 Hence 



q(pz-vy}=p(@x-ay}, p((3z+yy}=q(a(3x+aay+2b(3y}. 



Conversely, these imply the initial quadratic equation. Hence px, /j.y, pz 

 equal quadratic functions of p, q. It is shown that ju is a factor of 2D(3 2 . 



A. Desboves 38 noted that by specializing his 159 formulas we find that the 

 complete solution in integers of X 2 -\-bY 2 -{-dXY Z 2 is 



X = q 2 -bp 2 , Y = dp 2 +2pq, Z = q 2 +bp 2 +dpq, 



where (as below) is to be inserted before the second members. For the 

 case d = 0, the ordinary method is to factor Z 2 X 2 and get 



X = aq 2 -$p 2 , Y = 2pq, Z = aq 2 +pp~ (b = aft. 



For each pair of factors a, /3 of b, the latter equations give all the solutions. 

 It is inexact to say with A. M. Legendre 39 and others that the general 

 solution includes as many particular formulas as there are ways to decom- 

 pose b into two relatively prime factors. The complete solution in integers 

 of X 2 +Y 2 = cZ z for c = m 2 +n 2 (the only solvable case in view of a theorem 



34 Application of the angular analysis to the solution of indeter. problems of the second 



degree, New York, 1848, 15-17. 



35 Zahlentheorie, 155, 158, 1863; ed. 2, 1871; ed. 3, 1879; ed. 4, 1894. 



36 Nouv. Corresp. Math., 1, 1874-5, 197-8. Cf. papers 112a, 124, 125 of Ch. V, and 72 of 



Ch. IV. Cf. J. Bertrand, Traite" ele"m. d'algebre, 1851, 222-4. 



37 Atti Accad. Pont. Nuovi Lincei, 32, 1878-9, 89-97. 



38 Nouv. Ann. Math., (2), 18, 1879, 269; proofs, (3), 5, 1886, 226-33. 



39 Theorie des nombres, ed. 2, 1808, 29. 



