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



n = l+gand25+10pg+(p 2 -10)g 2 -2p2 3 +g 4 =(5+pg-2 2 ) 2 . The last holds 

 if 10p = 2S, q= -51/60, whence x = 15/16. Cf. Gerardin. 346 



W. Lenhart 326 discussed S(#J+o;i) = S(i/?+2/i), where i = l, -, n. 

 Assign any values to #,-, yi (i = 3, - -, ri). Then seek numbers (in his 203 

 table of sums of two cubes) t=x\-\-x\, t' = yl+yl, such that 



where f depends on the chosen values of #,-, ?/,, i^3. For n = 2, he found 

 Xi = 5, 0:2 = 6, ?/i = 7, 2/2 = 1. For w>2 he took = ' and found 



(12, 5, 1; 11, 8, 2), (14, 13, 11, 8; 17, 12, 5, 3), 

 (21, 14, 10,4, 1; 20, 17, 5, 3, 2). 



B. Peirce (ibid.) took Xi = diX-\-bi, ?/i = a t --|-& n _j + i and found that the 

 condition gives 



R. Hoppe 327 considered the rational solutions of x 3 +y 3 = x y. Set 

 y = x(lu)l(l+u). Then x and y are rational in u if w/(l-f3w 2 ) = D. 

 If u is a solution, 



u 



= 



is a second solution, etc. The nth such solution is found. 



C. Hermite 328 noted the solution x a(ab c r ) 1 y = a? b-c, z = b(c- ab), 

 u = a?c b z of 

 (1) x-y-\-y 2 z-}-z 2 u-{-u 2 x = 0. 



J. Joffroy 329 stated that a 2 6 3 = 7-10 n is impossible. A. Morel gave an 

 erroneous extension to a 2 6 3 =t=10 Wl + +10" 7 . 



S. Re"alis 330 gave long cubic functions x, y, z, w of a, J3, 7 for which 



Re"alis 331 obtained as solutions of (1) : 



2= - 



as well as formulas of the third and fourth degrees. 



T. Pepin 332 noted that a surface of degree m is osculated at an arbitrary 

 point of a given surface only when there is a positive integer n satisfying 



and proved that 1, 5, 20 are the only integral values <675 of m. E. de 

 Jonquieres 333 used the discriminant of the quadratic in n to show that 



328 Math. Miscellany, New York, 2, 1839, 96-7; Extract, Sphinx-Oedipe, 8, 1913, 93-4. 



327 Zeitschr. Math. Phys., 4, 1859, 359-61. 



328 Nouv. Ann. Math., (2), 6, 1867, 95. 



829 Nouv. Ann. Math., (2), 10, 1871, 95-6, 288. 



830 Nouv. Corresp. Math., 4, 1878, 346-52. 



331 Nouv. Ann. Math., (2), 18, 1879, 301-4. 



332 Jour, de Math., (3), 7, 1881, 71-108. 



338 Atti Accad. Pont. Nuovi Liucei, 37, 1883-4, 183-8. 



