CHAP. XXI] TWO EQUAL SUMS OF TWO CUBES. 557 



the determinant being y\-\-y'l if n = 3, and y\ if n = 2. Proceeding as had 

 Hermite 62 and considering the intersections of (13) with the general line 

 in space of n dimensions 



it is shown that the coordinates of any point on (13) are expressed rationally 

 as functions of n 1 parameters a\, , o n _i: 



where a = 0, 61= a n _i, bi = a;_i a n -\ (i = 2, - - , n 1), 

 A=f(ai, , a n _i), B=f(bi, -,&_! 

 V. Schlegel 69 treated a\+al+al = al by setting 



0,4+03 m(a\ a 2 ) = -- . 



These become, for ai+2 = ^, Oi a 2 = y, a4-\-a 3 = u, a^ a 3 = v, 



x = m z v, m z (x 2 i/ 2 ) +it 2 v z = 4(p- q~) , 



u-\-my = n(pq}, umy = -- -. 



n 



The last two give u, y, the second of the four becomes 



mx-\-v 



mxv pq 



Equate each member to r. We thus get x and v in terms of p, q, r, m. 

 By x = m?v, 



(p-q)(m 3 -l) ' 



For any m, we can choose pq to make r rational; then the a* are rational. 

 A. Martin 70 gave Vieta's derivation of (1) with B = r, D= s, and with 



C. Moreau 71 gave the ten numbers < 100,000 which are sums of two 

 positive cubes in two ways. 



A. S. Werebrusow 72 gave the formula 



where M-+MN+ N 2 = 3u<j>t, co 3 = l [Teilhet 78 ]. 



K. Schwering 73 stated that the general solution of (10) is 



(14) x = ma n 2 , y= mfi+ri 2 , z = nam 2 , u 

 where 



(15) o: 2 +a/3+/3 2 



69 El Progreso Mat,, 4, 1894, 169-171. 



70 Math. Magazine, 2, 1895, 153-4; Amer. Math. Monthly, 9, 1902, 79. 



71 L'intermediaire des math., 5, 1898, 66 [253; 4, 1897, 286], 



72 Ibid., 9, 1902, 164-5; 11, 1904, 96, 289. Math. Soc. Moscow, 25, 1905, 417-437. 



73 Archiv Math. Phys., (3), 2, 1902, 280-4. 



