CHAP, xxi] BINARY CUBIC FORM MADE A SQUARE. 571 



A. Desboves 165 gave for a = 6 = this result with v replaced by v/2. 

 He 166 reduced ax 3 -\-by 3 = cz~ to Lagrange's 161 case by multiplication by o 2 c 3 . 

 H. Brocard 167 noted that x 3 -\-(2a-\-l}(x 1) = ?/ 2 has the special solution 



R. F. Davis 168 made So; 3 Sz+16 the square of px 2 +x 4:, obtaining a 

 quadratic for x with rational roots if 8p 3 8p+16=D. Hence solutions 

 like p Q, dbl, 2 lead to new solutions x. 



G. de Rocquigny 169 proposed for solution a; 3 zdbl =y~. H. Brocard 170 

 noted that for the upper sign it has solutions x = Q, 1, 3, 5. E. B. Escott 171 

 noted that for the lower sign it is impossible as shown by use of modulus 3. 



L. C. Walker 172 reproduced Lagrange's 163 work, applying it to x 3 +ay 3 = z z . 



The least positive integral solution 173 of x 3 66y 3 = D has a: = 25. 



L. Aubry 174 found restrictions on possible solutions of x 3 -}-x 2 -\-2x-}-l D. 



A. Gerardin 175 assumed that XQ, y , z is a known solution of 



ax 3 + bx 2 y + cxy 1 + dy 3 = 2 2 



and took x = x -\-mf, y = y Q -\-mg, z = z +mh. There results a quadratic 

 equation Am z -\-Bm-i-C = 0. He took in turn 



4 = 0, = 0, C = 0, 2 -4AC=D. 

 L. J. Mordell 176 wrote the proposed cubic in the form 

 (2) g* = 4h 3 -g 2 ha 2 -gza 3 , 



which is the syzygy connecting the seminvariants a, 



and g = a?d b 3 -\-3bh of the quartic 



/= ax*+b 



Given integral solutions of (2) in which a is odd and prime to h, we can 

 find integers a, , e such that / has the invariants g 2 and gr 3 , and b is 

 prime to a. Conversely, every such quartic yields a solution of (2) with 

 a odd and prime to h. Hence to find all solutions (with y odd and prime 

 to x) of 

 (3) 2 2 = 4z 3 -0 2 :n/ 2 -032/ 3 , 



take a representative / of each class of binary quartics with the invariants 

 gr 2 , </3 ; apply to / a suitable linear substitution ( p q r ,} of determinant unity 

 to obtain a quartic /' having a' odd and prime to b' ; then x = h', y = a', 



166 Comptes Rendus Paris, 87, 1878, 161. 



166 Nouv. Ann. Math., (2), 18, 1879, 398. 



167 Nouv. Corresp. Math., 3, 1877, 23-24. 



168 Proc. Edinb. Math. Soc., 13, 1894-5, 179-80. 



169 L'intermediaire des math., 9, 1902, 203. 

 "Ibid., 10, 1903, 131. 



171 Ibid., 132. 



172 Amer. Math. Monthly, 10, 1903, 49-50. 



173 Math. Quest. Educ. Times, (2), 14, 1908, 29. 



174 L'intermediaire des math., 18, 1911, 276-7. 



175 Sphinx-Oedipe, 8, 1913, 161. 



176 Quar. Jour. Math., 45, 1913-4, 170-186. 



