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



rational asymptote, the line parallel to it and through a rational point cuts 

 it again in a rational point. 



A. S. Werebrusow 306 obtained solutions of (4) with K = Q from one 

 solution. 



B. Levi 307 considered a cubic equation with rational coefficients which 

 corresponds to a cubic curve C of genus unity (transformable birationally 

 into a straight line) and determined points on C by use of an elliptic param- 

 eter. By a configuration of rational points on C is meant the set of all 

 rational points deduced from one or more rational points by the operations 

 of finding the tangential point to a given point and finding the third inter- 

 section with C of the secant joining two points of the set. There are 

 theorems on the number of points in a finite configuration of such rational 

 points (cf. Hurwitz 312 ). There is a discussion of the cubic 



xz"y(yx)(y kx) = 



into which any cubic with a rational point can be transformed birationally. 

 A. Thue 308 considered Ax 3 +By 3 = Cz z in which x, y, z are relatively prime 

 in pairs and z^y^x>0. We can find integers p, q, r, without a common 

 factor and numerically < V3z, such that px+qy = rz. Hence 



where a, b, c are integers. Hence Aax-\-Bby = Ccz. From this and the 

 former linear relation we get the ratios of x, y, z. He introduced further 

 numbers and deduced many relations with the aim to obtain limits for 

 a, 6, c, etc. 



L. Chanzy 309 applied Lucas' 296 three methods to the equation 



The tangent at (x\, y\) meets the cubic at the point with the ordinate 



i2 



The line joining the known points (x\, y\), (x 2 , 2/2) meets the cubic in the 

 point with the ordinate 



while x 3 follows from (3 #1X2/2 y\} = (y 3 yi}(x 2 Xi}. 



L. J. Mordell 310 considered a ternary cubic form F(x, y, z). Given one 

 set of solutions, we can find a linear unitary substitution which transforms 

 F = into jSi 2 +2S 2 +S 3 = 0, where Sj is a function of degree j of 77, f. 

 Its discriminant f=Sl SiS s is a binary quartic whose invariants are 



808 Matem. Shorn. (Math. Soc. Moscow), 27, 1909, 211-227. 



807 Atti IV Congresso Intcrnaz. Mat., Roma, 2, 1909, 173-7. Supplement to his four papers, 



Atti R. Accad. Sc. Torino, 41, 1906, 739-64; 43, 1908, 99-120, 413-434, 672-681. 

 "Skrifter Videnskapsselsk. Kristiania (Math.), 1, 1911, No. 4, pp. 19-21; 2, 1911, No. 15, 



7 pp. The related No. 20 is considered under Thue 178 of Ch. XXIII. 

 <"> Sphinx-Oedipe, 8, 1913, 166-7. 

 Quar. Jour. Math., 45, 1913-4, 181-6. 



