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



To get Binet's 58 solution, set m = l, n = 2 +3b 2 , a, f3 = a^3b. By (14), 



H. Kuhne 74 expressed the preceding solution in terms of three inde- 

 pendent parameters by replacing a by 3pr, by 3qr, m by p z +pq+q 2 , 

 n by 3r 2 , whence (15) is satisfied identically. Thus 



x = 3spr 9r 4 , y = 3sqr + 9r 4 , z = 9pr 3 s 2 , u = 9gr 3 + s 2 , 

 where s^p^+pq+q*, satisfy (10) identically. Not only do any p, q, r 

 lead to a solution , /3, m, n of (15), but conversely, by multiplying them by 

 a common factor, we can make n/3 a square, necessarily r 2 , and then 



D. Mirimanoff 75 wrote (10), with u = l, in the form 



(x -l)(x- co) (x - co 2 ) +?/ 3 = z 3 . 



Set y = u(x u>)+v(x co 2 ), s = Mco 2 (z co)+yco(z co 2 ), and divide by 

 (x co) (x co 2 ) . We get 



Dz = l+3(co 2 -l)w 2 +3(co-l)w 2 y, D = 1+3(1 -co)w 2 +3(l-co 2 )w 2 y. 



Hence we get all solutions (except rr = co, co 2 ) by giving all values to u, v. 

 Real solutions result if and only if u+v, co 2 w+coy, cow+co 2 y are real, i. e., 

 if u and v are conjugate. Writing fe, a, ab for these three sums, we 

 obtain Hermite's solution (12). 



A. Holm 76 derived (2) by the tangent method. Set x = X+B, y=Y-D 

 and take Y = XB Z JD*. Then X = or 35 3 /( 3 - D 3 ) . The latter gives (2) . 



H. Kiihne 77 discussed diophantine equations such that the n variables 

 are expressible rationally in n 1 parameters. His 74 solution of (10) is an 

 example of the method. 



P. F. Teilhet 78 remarked that the solution by Werebrusow 72 is not the 

 general one and stated that all solutions of (10) with 4(x u} =3(z y} are 

 obtained by equating the two expressions 



/21m 2 -n 2 =Fl6mn\ 3 



f 

 V 



or by equating the two 



/ 

 V 



_ 

 ~ 



where m, n are both even or both odd. 

 A. Ge"rardin 79 derived (2) from 



x-B_y--Dy+D- 



y+D~ x*+Bx+B' 2 

 by setting x = B+mh, y = h D, and equating to zero the constant term of 

 the quadratic for h. Thus m = D*/B 2 , h = 3 3 >/( 3 -> 3 ). Similarly for (1). 



7< Archiv Math. Phys., (3), 4, 1903, 180. Cf. Fujiwara. 85 



76 Nouv. Ann. Math., (4), 3, 1903, 17-21. 



78 Proc. Edinburgh Math. Soc., 22, 1903^t, 43. 



77 Math. Naturwiss. Blatter, 1, 1904, 16-20, 29-33, 45-58. Cf. Kuhne 169 , Ch. XXIII. 



78 L'interme'diaire des math., 11, 1904, 31. 



79 Sphinx-Oedipe, 1900-7, 90-93, (52); I'interme'diaire des math., 16, 1909, 85. 



