CHAP, xxi] HOMOGENEOUS CUBIC EQUATION F(x,y,z)=Q. 591 



and values of y, z derived by permuting the triples of letters cyclically. 

 All solutions of x 3 -\-y s -\-z* = x 2 y-\-y-z+z z x are given. 



A. Desboves 299 proved that if x, y, z is one set of solutions of 

 = Q, a second set of solutions is given by 



3 +2By 3 ), Y= -y(2Ax*+By 3 ), Z = z(Ax* - By 3 ) . 



For A I this result is due to Legendre. 288 



J. J. Sylvester 300 called the intersection of the tangent at a point P on a 

 cubic with the cubic the tangential of P. He proved for A = B = C = I 

 that (3) gives the tangential to (4) at the point (a, b, c) and that the point 

 on the cubic collinear with (a, b, c) and (a', b', c') has the coordinates 



(7) 6ca' 2 -6'c'a 2 , ca6' 2 -cV6 2 , abc' 2 -a'b'c z . 



A. Desboves 301 noted that Cauchy's formula (5) becomes, for (4), 

 x = 3BW(ab f - ba f ) +3Ccc'(ac' -ca') -K(a 2 b'c' - a'*bc), 



with similar expressions for y, z. Since a, 6, c and a', b', c' satisfy (4), we 

 can express A, B as linear functions of C, K. Substitute the resulting value 

 of B into x, etc. We get (7). This result, which is simpler than, but 

 equivalent to, Cauchy's (5), had been found otherwise by Sylvester, 300 

 whose published announcement without proof was limited to the case 

 A=B = C=1, and, for K = 0, but A, B, C arbitrary, by Desboves 302 and by 

 P. Sondat. 303 From the fact that (7) satisfy Ax*+By*+Cz* = Q, we have 

 the identity 



(6V 3 - &' 3 c 3 ) (a W - a' 2 6c) 3 + (c V 3 - c' 3 a 3 ) (6 2 aV - 6' 2 ac) 3 



+ (a 3 6 /3 - a' 3 6 3 ) (cV6' - c' 2 a&) 3 = 0. 



This leads to solutions of the system of equations [cf. Bini 438 ] 



or 



Desboves 304 simplified Cauchy's proofs of (2) and (5), gave also a direct 

 proof of (2), and showed that a 2 divides F(0, w, v), etc., a fact seemingly 

 overlooked by Cauchy. Hence we may take x = F(0, w, v)/a~, etc., 

 obtaining polynomials of degree 4 for x, y, z. As new results, he proved 

 that if one solution of F = is given we can reduce its complete solution to 

 that of a biquadratic equation. He sought an F such that the latter is 

 -A 4 +#?? 4 = Cf 2 , where C = A+B, the only biquadratic hitherto solved com- 

 pletely. The resulting F is 



He obtained the solution of/(z, ?/)+cz 3 = 0, with coefficients of special type, 

 given solutions m, n of the cubic f(x, y) = 0. 



A. Holm 305 noted that the tangent to a cubic at a rational point, not an 

 inflexion point, cuts the cubic in a new rational point. In case there is a 



299 Nouv. Ann. Math., (2), 18, 1879, 404. Same by R. Norrie. M 



300 Amer. Jour. Math., 3, 1880, 61-6; Coll. Papers, 3, 1909, 354-7. 



301 Nouv. Ann. Math., (2), 20, 1881, 173-5; (3), 5, 1886, 563-5. 



302 Ibid., (2), 18, 1879, 407-8. 



303 Ibid., (2), 19, 1880, 459. 



304 Ibid., (3), 5, 1886, 545-579. 



<* Proc. Edinburgh Math. Soc., 22, 1903-4, 40. 



