CHAP, xxi] TERNARY CUBIC FORM MADE A CONSTANT. 593 



numerical multiples of the invariants S and T of F. If Si = bri+c{, f is a 

 square for , = c , f = -6. Thus (Mordell 162 of Ch. XXII) if we find 176 all 

 rational solutions of 



we can deduce all the rational solutions of F = Q. The method is applied 

 in detail to the canonical cubic x*-{-y*-{-z*-\-6mxyz = Q. 



W. H. L. Janssen van Raay 311 solved ylz-\-z/x-{-xly = 3 in integers by 

 reducing it to a 3 +6 3 +c 3 = 3a6c. 



A. Hurwitz 312 proved (p. 226) that a curve (4) with integral coefficients 

 has either no rational point or an infinity of rational points if A, B, C are 

 not zero and relatively prime in pairs, while no one of them is divisible by 

 a square of a prime, and at most one of them is 1. Next, if A = B = 1, 

 C=J= 1, and C is not divisible by a square of a prime, the curve has either 

 1, 2 or an infinity of rational points. Finally, tf A = B = C=1, K^l, 3, 

 5, the curve has 3 or an infinity of rational points. There is a discussion 

 of cubic curves without a double point (genus 1), the coefficients of whose 

 equation belong to an algebraic field. A rational point is one whose coor- 

 dinates are proportional to three numbers of the field. By use of an elliptic 

 parameter, there are found all complete sets of a finite number of rational 

 points, such that the line joining any two (distinct or identical) meets the 

 curve in a point of the set. The most general cubic curves with exactly 

 one or exactly four rational points are determined. Cf. Levi. 307 



M. Weill, 3120 starting with one solution a, b, c of Ax z -\-By 3 -\-Cz 3 = 0, wrote 

 x = a+\8, y = b-\-\'d, z = c+5, and equated to zero the coefficient AXa 2 

 +BX'6 2 +Cc 2 of 35, and hence found 5 rationally, thus obtaining the second 

 solution (3) due to Cauchy. Given two sets of solutions a, b, c and a', b', c', 

 he wrote x = a-{-da', etc., found 5 rationally, and obtained Desboves' 301 

 special case of Cauchy's (5). 



TERNARY CUBIC FORM MADE A CONSTANT. 



J. L. Lagrange 163 determined cubic forms F(x, y, z) whose product by 

 F(X, Y, Z) is of that form. Cf. Libri 64 - 65 of Ch. XXV. 



G. L. Dirichlet 313 employed the roots a, /3, 7 of a cubic equation with 

 integral coefficients and without rational roots. Let F(x, y, z] denote the 

 product of x+ay+o^z by the similar functions of ft and 7. First, let a 

 single root a be real. If T, U, V form a fundamental solution of 

 F(T, U, V) = 1, and X, Y, Z form one solution of F(x, y, z) = m, an infinite 

 set of solutions of the latter is given by the development of 



One solution of any set can be found by a finite number of trials. But if all 

 three roots are real, it is stated that there exist two fundamental solutions 

 from which all can be found by multiplication and powering. 



311 Wiskundige Opgaven, 12, 1915, 206-8. 



312 Vierteljarhschrift d. Naturfor. Gesell. Zurich, 62, 1917, 207-29. 



312a Nouv. Ann. Math., (4), 17, 1917, 47-51. 



313 Bericht Akad. Wiea. Berlin, 1841, 280-5; Werke, I, 625-32. 



39 



