590 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxi 



except when M/A is the square of an integer. Likewise if A is of the 

 form p 3u)1 , where p is a prime. Also without the assumption that A! is 

 even, provided it has no factor / 2 -}-3<7 2 , while A=2 3w1 ; or A/2 is a prime 

 <?i4, and M/9 is an integer; or A/4 is a prime qi2, and M/18 is an 

 integer; or A is a prime and A, B are of the respective forms qn2, qriQ, 

 or qn4:, qn3, or gn3, qn. 



E. Lucas 292 stated Cauchy's results on the cubic (4) as follows: (i) If 

 a, b, c is one set of integral solutions, another set x, y, z is given by 



a o c 



(ii) If a, b, c and a', 6', c' are two distinct sets of solutions, then 

 x y z 



a b c = 0, Aaa'x+BWy + Ccc'z = Q 

 a' b' c' 



give a third set. But (i) and (ii) do not yield all solutions. Lucas 293 had 

 stated as exercises these results without relation to Cauchy. They were 

 verified by Moret-Blanc, 294 and restated by A. Gerardin. 295 



Lucas 296 stated the generalizations to any homogeneous cubic F(x, y, z) 

 = 0. 1. The tangent at a point mi with rational coordinates Xi, y\, Zi, 

 and on F = 0, cuts the cubic at a rational point m, i. e., 



dF dF 6F 



F = 0, X7r-+yj-+zir = V 

 dxi dy\ dzi 



determine x, y, z rationally. The point m is distinct from m\ unless the 

 tangent is parallel to an asymptote or passes through a point of inflexion. 

 2. The secant mim z through two rational points on the cubic cuts the 

 cubic in a rational point (in general distinct from m\, w 2 ). 3. The conic 

 through five rational points on a cubic cuts it in a sixth rational point. 



S. Realis 297 obtained a second solution (quadratic in a, /3, 7) of 

 x 3 -f 2y z +3z 3 = 6xyz from one solution a, /3, 7. 



Realis 298 noted that all integral solutions except x = y=z of 



3 +2/ 3 +s 3 3xys 

 are given by 



z=(a-6) 3 +(a-c) 3 , 2/ = (6-c) 3 -H&-a) 3 , z = (c-a) 3 -{-(c-&) 3 . 

 If a, /3, 7 is one set of solutions of 



A 



another set is g'.ven by 



292 Bull. Bibl. Storia Sc. Mat., 10, 1877, 175; Amer. Jour. Math., 2, 1879, 178. 



293 Nouv. Ann. Math., (2), 14, 1875, 526. 

 * M Ibid. t (2), 20, 1881,201. 



296 Sphinx-Oedipe, 5, 1910, 90. 



298 Nouv. Ann. Math., (2), 17, 1878, 507-9; Amer. Jour. Math., 2, 1879, 180. 



297 Nouv. Corresp. Math., 4, 1878, 34&-S2. 



298 Ibid., 5, 1879, 8-11. 



