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

Aida Ammei,148 just after 1807, noted that xl+2xl-\ ----- hnx' = t/2 has the solution

«i = - of + Z) Ta^ xr = 2aiOr, y = j) ja],

r=2 y=l

and that xl +8x2+ 6a?J H ----- hiw(w+l)a£=y2 has the solution

G. Libri149 noted that ax2+ by2 +cz2+d = Dissolvable if a V+ 6V +c'z2 = is, where a', &', CA are any three of a, Z>, c, d. For example, if

we get a solution x = np+g, a/ = rp+s, 3 = wp+$, where p is found rationally in terms of the indeterminates q, s, t. If an2, 6r2, cm2 are relatively prime integers and if no one of a, b, c is divisible by 4, we can assign the value ±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=z2+41w2 — 1132* since F=0 is solvable. Likewise for 23^2+7/2-13s2 and az2+5z2-22/2, where a is a prime =3, 13, 27, 37 (mod 40).

A. Cauchy150 treated the homogeneous equation F(x, y, z)~Q of degree N, with the given set of integral solutions a, 6, c. Let re, y, 0 be another set. The ratios of u, v, w are determined by au-{-bv-i-cw = Q, xu+y+zw^Q. Then

F(wXj wy, —ux—vy^Q, F(wa,wb, —ua—vb)—Q. Set y/x = p, b/a = P. Then

Fl=F(wi wp, -tt-»p)=0, F2=F(w, wP, -u-vP} = 0.

Let <£, Xj >A be the partial derivatives of F(x, y, z) with respect to x, y, z. Then

x<t>+yx+z\I/ = NF(x, y, z), a<f>(a, b, Thus au+bv+cw = 0 is satisfied by (2) u=<t>(a, 6, c)+br— en, v = x+cm—ar,

for m, n, r arbitrary integers. If the latter can be chosen to make F1 = for a rational p(p=f=P), we get the new solution x :y :z = w : wp : —u — v

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

F(x, y, z)=Ax*+Byz+Cz*+Dyz+Ezx+Fxy, note that the condition Fi=F2 now gives p=P or

p= -P+[(Ev+Du)w-~Fw2-2Cuv1[ot, a=Bw*

Replace P by its value b/a and use au+bv+cw=Q, F(a, &, c)=0. Thus ), where p = Cu*-Ewu+Aw~, y~Av2-Fuv+Bu2. Then all

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

149 Memoria sopra la teoria del mimeri, Firenze, 1820, 10-14.

160 Exercices de mathe*matiqiies, Paris, 1826; Oeuvres, (2), VI, 286.