CHAP, xxii] SINGLE EQUATIONS OF DEGREE FOUR. 663 



Take (2x 2 -\-ax} 2 =(ax 6) 2 . We get x rationally and a condition on o, 6, 

 which is solved for a. Take 6= m?/2, whence a follows rationally. 



A. Desboves 261 gave identities yielding an infinitude of solutions of 

 ax 3 -\-by 3 = cv* for certain values of c. He 262 noted that aX 4 +bY*=cZ 3 for 



X=x(Zax*-5by*), Y = y(5ax*-3by*), Z = ax*+by\ 

 c = 81a?x s - 158a&zy +81&Y, 



and gave long formulas yielding solutions of aX 4 +bY* = cZ* when c is 

 represented by a certain form of degree 20. Further, X 4 Y 4 = cZ 4 is 

 solvable when c is of one of the forms 



xy(x 2 +4y 2 ), z 8 +4?/, 2xy(x 2 -y 2 } (x 4 +y*-6x 2 y*). 

 S. Re"alis 263 gave various quartic equations not having a rational root, as 



(mod 5). 

 Several 264 solved x 3 + y 3 = (x y) 4 . Set x+y = u, xy=z. Then 



Set r 2 = (83) 2 . Hence there are two types of solutions. 



R. W. D. Christie 265 made 12a6c(a+6+c) a square, but not a biquadrate 

 as claimed. A. Gerardin 266 noted that it is a biquadrate for (a, b } c) = (1, 2, 

 6), (3, 4, 9), etc. 



E. Grigorief 267 noted that 11 4 = 12 3 +17 3 +20 3 . P. F. Teilhet 268 gave 

 cases of x* = yl+ yl+yl for z = 3, 10, 17, 20, 29, 36, 43, 55, 62. He 269 noted 

 that 



8 4 = 14 3 +34 2 +14 2 = 12 3 +48 2 +8 2 = three such sums. 

 K. Kommerell 270 gave as the positive integral solutions of 



xyz (x + y z) = t 2 , 



where y\, z\ ,are without square factors, d 2 y\ is relatively prime to e 2 Zi, and 

 T 2 - tyiZi U 2 = a\d 2 y l - e^) 2 . 



A. Hurwitz 271 proved that x s y-\-y 3 z-\-z s x = is impossible since 



u' ! +v 7 +w 7 = 

 is impossible. 



261 Nouv. Ann. Math., (2), 18, 1879, 408. 



Ibid., 440-4. 



Ibid., (3), 2, 1883, 370; 4, 1885, 376, 427-31; Mathesis, 7, 1887, 96; Jour, de math. sp6c., 



1888, 90 (and questions 66, 67). Reprinted, C. A. Laisant's Algebre, 1895, 224-6. 

 264 Zeitschr. Math. Naturw. Unterricht, 20, 1889, 264-5. 



266 Educ. Times, 49, 1896. 



2M Bull. Soc. Philomathique, (10), 3, 1911, 244. 



267 L'intermediaire des math., 9, 1902, 319, 

 Ibid., 10, 1903, 170-1. 



269 Ibid., 11, 1904, 18. 



270 Math. Naturw. Mitteilungen, Stuttgart, (2), 7, 1905, 74-8. Cf. Brehm, 285 Euler 249 ; 



also papers 12, 22 of Ch. V. 



271 Math. Annalen, 65, 1908, 428-30. Generalization, Hurwitz, 212 Ch. XXVI. 



