CHAP, xiii] QUADRATIC EQUATION IN n^3 UNKNOWNS. 429 



Aida Ammei, 148 just after 1807, noted that z?+2x;H ----- \-nxl = y 2 has 

 the solution 



n n 



Xi - - a\ + X) ra z r , x r = 2a^a r , y = 2 ja*, 



r=2 j=l 



and that xl+3xl+6xl -\ ----- \-%n(n+l}xl = y 2 has the solution 



Ar(r+l) 2 



r=2 " r=l * 



G. Libri 149 noted that ax 2 +by*+cz*+d = Q is solvable if a'x^+b'y 2 +c'z 2 = 

 is, where a', 6', c' are any three of a, 6, c, d. For example, if 



an 2 + &r 2 + cm 2 = 0, 



we get a solution x-np+q, y = rp+s, z = mp-\-t, where p is found rationally 

 in terms of the indeterminates q, s, t. If an 2 , br 2 , cm 2 are relatively prime 

 integers and if no one of a, b, c is divisible by 4, we can assign the value db 1 

 to the denominator of the fraction for p and hence get integral solutions 

 x, y, z. 



Every integer can be expressed in the form F=o; 2 +41w 2 113s 2 since 

 F = is solvable. Likewise for 23z 2 +2/ 2 -13z 2 and ax 2 +5z 2 -2y 2 , where a 

 is a prime -3, 13, 27, 37 (mod 40). 



A. Cauchy 150 treated the homogeneous equation F(x, y, z)=0 of degree 

 N, with the given set of integral solutions a, b, c. Let x, y, z be another set. 

 The ratios of u, v, w are determined by au-\-bv-\-cw = Q, xu-}-yr-\-zw = 0. 

 Then 



F(wx,wy, ux vy)=Q, F(wa, wb, ua vb~)=0. 



Set y/x = p, b/a = P. Then 



F 1 =F(w,wp, uvp)=Q, F 2 =F(w,wP, u vP)=Q. 



Let <f>, x, t be the partial derivatives of F(x, y, z) with respect to x, y, z. 

 Then 



X(f)+yx+z^ = NF(x } y, z), a<t>(a, b, 



Thus au-\-bv-\-cw = is satisfied by 



(2) u = (j>(a, b, c)-\-br en, v = x+cm ar, 



for m, n, r arbitrary integers. If the latter can be chosen to make Fi = F 2 

 for a rational p(p=t=P), we get the new solution x :y : z = w :wp : u vp 

 of F = 0. 



To apply (pp. 292-301) this general method to 



F(x, y, z)=Ax 2 +By 2 +Cz 2 +Dyz+Ezx+Fxy, 

 note that the condition FI = F 2 now gives p = P or 



p= -P+[(Ev+Dii)w-Fw 2 -2Cuv1!a, a=Bw 9 --Dvw+Cv 2 . 



Replace P by its value bfa and use au+bv+cw = Q, F(a, b, c)=0. Thus 

 }a ) } where p = Cu 2 -Ewu+Aw 2 , y = Av 2 -Fuv+Bu 2 . Then all 



" 8 Y. Mikami, Abh. Gesch. Math. Wiss., 30, 1912, 248. See papers 59, 66 of Ch. IX. 



149 Memoria sopra la teoria dei numeri, Firenze, 1820, 10-14. 



160 Exercices de math<matiques, Paris, 1826; Oeuvres, (2), VI, 286. 



