432 HISTORY OP THE THEORY OF NUMBERS. [CHAP, xm 



E. Cesaro 157 found various sets of solutions of 



v 2 v(x-\-y+z} +xy+yz+zx = 

 Re"alis stated and Rochetti 158 proved that 



2(xy+yz+zx) - 



has an infinitude of solutions. Solving for z, we are to make xy n-= D , 

 whence n 2 +c 2 is to be expressed as a product of two factors. Or, choose 

 p, q, r, s so that n=prqs; then four solutions are 



x = p z +q z , ?/ = r 2 -f-s 2 , z = (ps) 2 +(gd=r) 2 . 



A. Desboves 159 gave the complete solution in integers of the general 

 homogeneous quadratic equation in n variables, when one solution x, y, 

 is given. Regard mx, my, as the same solution as x, y, . First, 

 letn = 3: 



(5) aX*+bY z +cZ 2 +dXY+eXZ+fYZ = 0. 



Let X = px, Y = py+p, Z = pz+q. Then (5) gives p as a rational function 

 of p, q, x, y, z, so that 



X= - (bp 2 +cq*+fpq)x, 



(6) Y = (dx + by +fz) p 2 cyq" 2 + (ex -f- 2cz) pq, 



Z= bzp 2 + (ex +fy + cz) q* + (dx + 2by} pq. 



This is the general solution of (5), since we can find p, q such that (6) 

 becomes any assigned solution. A convenient modification (pp. 233-5) 

 of the method of Gauss 147 leads to (6). Special cases of (6) have been 

 noted above (Desboves 38 ). 



For any n, set X = px+r, Y = py+p, in the proposed equation 

 F(X, Y, - ) = 0. We get p and then 



X = Mr-Nx, Y = Mp-Ny, Z = Mq-Nz,-->, 



dF dF 



N=F(r,p,q, ), 



. 

 or op 



The results are no more general than those for the case r = 0. 

 A. Meyer 160 gave criteria for the solvability in integers of 



(7) axt+bif+czt+du^O, 



where a, b, c, d are integers not zero without square factors and such that 

 no three have a common factor. Write (a, fe) for the positive g.c.d. of 

 a, 6, and set 



a=(a, 6) (a, c)(a, d)a, b = (b } a)(b, c)(b, d)fi, 

 c=(c, a)(c, b)(c, d)y, d = (d, a) (d, b) (d, c)5. 



Then necessary conditions for the solvability of (7) in integers not all zero 

 are (I) a, , d are not all of the same sign, and (II) (a, c) (a, d) (b, c) (b, d)y8 

 is a quadratic residue of (a, b), with five similar conditions derived by per- 

 muting the letters. Again, (7) is solvable if and only if conditions (I) 



157 Nouv. Corresp. Math., 6, 1880, 273. 



168 Mathesis, (1), 1, 1881, 165. 



169 Nouv. Ann. Math., (3), 3, 1884, 225-39. 



160 Vierteljahrsschrift Naturforsch. Gesell. Zurich, 29, 1884, 209-222. 



