434 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xm 



F. Ferrari 168 noted that the solution of xl+AiX\-\ ----- \-A n xl=x z n +\ 

 reduces to the solution of *2x\ = xl + i. 



O. Degel 169 noted that, if x, y, z, s are homogeneous coordinates, the 

 surface 



x" 1 + W7/ 2 + nz 2 + 2ayz + 2bxy + 2cxz = s 2 

 can be represented on a plane (Clebsch, Jour, fur Math., 65, 1866, 380) by 



Take 4 = as the plane and set ps = <? in the initial equation. We get a 

 rationally. Hence px, , ps are expressed as homogeneous quadratic 

 functions of b 2 , 3 . By the same method he 170 treated 



z 2 +?/ 2 +z 2 - 2yz - 2zx - 2xy = s 2 

 and found 



px=(u-\-2}(u-\-v), py = uv, pz = 2u, ps = 2vu z . 

 Several writers (pp. 164-6) gave solutions. 



A. Gerardin 171 found m from (l+wa) 2 +(lH-mb) 2 (we) 2 = 2, whence 



He noted (p. 22) the identity 



Ge*rardin 172 gave solutions of cases of 



x~+2(by+cz')x+my*+2ayz+nz 2 = D. 



O. Degel 173 stated that all solutions of llx- = y--3z--w-+2u-+2s z +Wt- 

 are given by px = A+2aB, py A -\-2bB, pz = A+2cB, pw- A +2dB, 

 pu = A-{-2eB, ps = A-\-2fB, pt = A, where a, -,f are distinct and 4=0, and 

 A = lla 2 -6 2 +3c 2 +d 2 -2e 2 -2/ 2 , B=lla+l)-3c-d+2e+2f. 



'V. G. Tariste" 174 noted that, if <x 1} -, a n is one set of integral solu- 

 tions of Wii+ -\-m n xl=0, all solutions are given by 



{ n ,2 n ,1 



a k J2 m i a i +2a' k ^miaia'i }, 

 t=i =i J 



where the a' are any rational numbers and M is such that the :c's are 

 integers. Gerardin (pp. 136-7) remarked that this result follows by taking 

 Xi = ai j rma'i (i = l, , k). 



L. Aubry 175 discussed the integral solutions of x\y\-\ ----- \-x n y n = 0' 



W. Mantel 176 treated xy+xz+yz = N. 



For z 2 +?/ 2 = 2a 2 +2& 2 , see papers 83-87 of Ch. V. 



On S(z-+Zi) =gr, see Bachet la of Ch. VIII. On 2x2-2y* = g, see Tano 207 

 of Ch. XII. On x 2 +3y 2 = u?+3v z , see papers 201 and 211 of Ch. XXII. 



1M Suppl. al Periodico di Mat., 11, 1908, 129-131. 



169 L'interm6diaire des math., 15, 1908, 151-2. 



170 Ibid., 16, 1909, 167. 



171 Sphinx-Oedipe, 6, 1911, 74-5. 



172 L'intermddiaire des math., 18, 1911, 202-3. 



173 Ibid., 20, 1913, 226. 



174 Ibid., 21, 1914, 49. 



176 Ibid., 23, 1916, 133-4. 



178 Wiskundige Opgaven, 11, 1914, 448-90. 



