588 HISTORY OF THE THEORY OF NUMBERS. [CHAP. XXI 



L. Matthiessen 284 discussed (1) in three ways. One way is to multiply 

 (7), the corresponding equation with the right number V s , by z 3 , where 

 t; 3 2 3 = ?/. Thus, for 11 3 +12 3 +13 3 +14 3 = 20 3 , take z = 5, whence y=1000. 



H. Brocard, "E. A. Majol," and F. Ferrari 285 discussed a sum of three 

 consecutive cubes equal to a sum of two squares. 



L. Aubry 286 treated (y-ky+y 3 +(y+k) 3 =3y(y-+2k 2 ) =u\ First, let 

 y = 2a?, ?/ 2 +2& 2 = 6& 2 , u = 6ab. Then 2a 4 = 36 2 -P, which is satisfied if 



a = q 2 - 3p 2 , b = g 4 +4pg 3 + 18p 2 g 2 + 12p 3 +9p 4 , 



k = q* -\- 1 2pq 3 + 18p 2 g 2 + 3Qp 3 q + 9p 4 . 

 Second, let y = 6a 2 , y 2 +2k~ = 26 2 , u = Qab. Then 18a 4 = b 2 - k 2 , which holds if 



(r, s) = (72, 1) or (9, 8). For p = q = l, the second set gives 



23 3 +24 3 +25 3 = 204 2 , 

 which occurs in a manuscript of Lucas'. Or we may set 7/ = 3a 2 or a 2 . 



HOMOGENEOUS CUBIC EQUATION F(x, y, 2)=0. 



A. Cauchy 287 derived a second solution from a given solution a, b, c. 

 Let <f>(x, y, z}, X) ^ De the first partial derivatives of F(x, y, z) with respect 

 to x, y, z, respectively. Then F = f or 



(1) x : y : 2 = as ta : bs tft : cs ty, 



where, if u = <f>(a, b, c), v=\(a, b, c), w = $(a, b, c), the parameters a, /3, 7 

 satisfy ua-\-vf3-\-wy = Q, while 



We may take a, ft, 7 = 0, w, v, w, 0, u; or v, u, 0. In each case one 

 of the terms (1) is very simple. He showed that we may take such a 

 simple value and obtain the following solution 



O TO O 



a-x b-y c-z 



, w, -v) F(-w, 0, u) F(v, -u, 

 These become 



x y z 



(3) a(Bb 3 -Cc 3 ) b(Cc 3 -Aa 3 ) c(Aa 3 -Bb 3 ) 

 for the case 



(4) F=Ax 3 +By 3 +Cz 3 +Kxyz = Q. 



If a, b, c and a', b', c' are two given sets of solutions of F 0, where F 

 is any ternary cubic form, Cauchy obtained a third set by expanding 



F(as-ta', bs - tb', cs - tc'} = 



2M Zeitschr. Math. Naturw. Unterricht, 37, 1906, 190-3. 

 286 L'interm6diaire des math., 15, 1908, 41-43. 



286 Sphinx-Oedipe, 8, 1913, 28-9. Cf. Lucas 88 * of Ch. XXIII. 



287 Exercices de mathdmatiques, Paris, 1826, 233-260; Oeuvres de Cauchy, (2), 6, 1887, 302. 



For a less effective method, see Cauchy 150 of Ch. XIII. 



