CHAP, xxi] HOMOGENEOUS CUBIC EQUATION F(x, y, z)=0. 589 



and obtaining stL = 0, where L is a linear function of s, t, which is zero for 



s = a0(a'. 6', c') + &*(', V, c')+ct(a', V, c), 

 t = a'<j>(a, b, c)+b'x(a, b, c)+c r t(a, b, c). 

 Then the resulting third set of solutions of 7^ = is 

 (5) x : y : z = as to! :bs tb f : cs tc'. 



By (3) for A =5 = 1, C= -a 3 -V, K = Q, c = l, we see that 

 z = a(a 3 +26 3 ), ?/= -6(6 3 +2a 3 ), z = a 3 -b 3 



satisfy z 3 +?/ 3 = (a 3 +6 3 )z 3 [Prestet 181 ]. 



For geometrical interpretations of Cauchy's results, see Lucas. 296 



A. M. Legendre- 88 deduced from one solution of x*+ay 3 = bz 3 the second 



solution 



X = x(x 3 +2ay 3 ), Y = -y(2x*+ay ? '), Z = z(x 3 -ay 3 ). 



Given X, Y, Z, the determination of x, y, z depends on a quartic equation. 

 J. J. Sylvester 289 stated that (4) can be transformed into 



A'u 3 +B'v 3 +C'w 3 +Kuvw (A'B'C' = ABC), 



where uvw is a factor of z, provided (i) the ratio of two of the coefficients 

 A, B, C is a cube, (ii) the " determinant " 27ABC+K 3 has no positive 

 prime factor 6Z+1, and (iii) if 2 m and 2 n are the highest powers of 2 dividing 

 ABC and K, respectively, then either m is of the form 3k 1 or, if not, m 

 exceeds 3n. If a, (3, y give one solution of (4) and if we set 



x = F 2 G+G 2 H+H 2 F-3FGH, 



y = FG 2 +GH 2 +HF 2 -3FGH, 



then x 3 +y 3 +ABCz 3 +Kxyz = Q. For the case A = B = 1, C a prime, and 

 21C-\-K 3 positive and not divisible by a prime 6&+1, he 290 gave a process 

 to obtain all integral solutions of (4) from one initial solution P = (e, g, i) . 

 The process is to apply to P repetitions of transformation (6) and the trans- 

 formation, depending also upon P, from one system I, m, n to the system 



X = 3gm(gl em) -\-3Cin(il en) -\-K(gil 2 e 2 lm) , 

 /i = 3Cin(imgl)-\-3el(emgl) +K(eim 2 g-lm), 

 v = 3el(en il) + %gm(gn irri) -\-K(egri* i~lm) , 



or to the system obtained by interchanging e and g. 



Sylvester 291 stated that F=x*+y 3 -}-z 3 +Qxyz = Q is not solvable in in- 

 tegers; likewise for 2F = 27nxyz when 27n 2 8rc+4 is a prime; and for 

 4F = 27nxyz when 27 n 2 - 36n+ 16 is a prime. Set M 3 - 27 A = A 3 A r where A t 

 has no cubic factor. If A! is even and contains no factor of the form 

 / 2 +3gr 2 , and if A is a prime, x 3 +y s -{-Az 3 = Mxyz has no integral solution 



288 Th6orie des nombres, ed. 3, 2, 1830, 113-7; Maser's transl., 2, 1893, 110-4. 



289 London, Edinburgh, Dublin Phil. Mag., 31, 1847, 189-191, 293-6 for corrected theorems; 



Coll. Math. Papers, 1, 1904, 107-13. 



290 Phil. Mag., 31, 1847, 467-471; Coll. Math. Papers, I, 114-8. 



191 Annali di Sc. Mat. e Fis., 7, 1856, 398-400; Math. Papers, II, 63-4. 



