CHAP, xix] xy-\-a, xz+a, yz+a ALL SQUARES. 513 



N. Vernon (p. 302) equated the first and second functions (5) to the 

 squares of (r 2 -z?/)/(2r), and (s 2 - xz) /(2s). Then x+y = (r 2 +x-?/)/(2r), etc., 

 which give x, y in terms of z. Then the third function becomes a quartic 

 in z which is made a square as usual. 



D. S. Hart 68 noted that x-+xy-\-y~= D if x = m i ri i , y = 2mn+n z . 

 Thenz 2 +Z2+z 2 =D if z = x(2pq+q^l(p*-q 2 ). Takew = 2,w = l, p = r+%q. 

 Then y-+yz+z- = D if r = 7g/4, p = 90/4. Hence an answer is 195, 325, 264. 



A. Martin and A. B. Evans 69 took x~+axy+y- = (mx y)~ to get x/y. 

 Then x 2 +axz-\-z- and y*+ayz-{-z 2 are made squares by known methods. 



Several writers 70 made the functions (5) squares. R. F. Davis 71 noted 

 the solutions 7, 8, - 15 and 435, 4669, 1656. 



N. G. S. Aiyar 72 solved x-+xy j ry- = c i , etc., by geometry, algebra and 

 trigonometry, without attention to rational values. 



A. Gerardin 73 assumed that a solution of a 2 +aj8+/3 2 = A 2 is known and 

 sought a solution of 



by setting B = x+u or B = axpfq, or x = t a (3, obtaining a quartic 

 function of t which is made a square in three ways. There is found a solu- 

 tion in positive integers by functions of the sixth degree. 

 E. Turriere 74 considered the system 



Ax 2 +Bxy+Cy~= D, Dy 2 +Eyz+Fz 2 = D, Gz 2 +Hzx+Ix 2 = D, 



under the assumption that each has a set of rational solutions, say x , yo 

 for the first. Solving the first with yy = Z(xx ), where Z is a parameter, 

 we get x and y rationally in terms of Z. Similarly, z/y is rational in 

 X=(zZi)/(yyi), and x/z in Y=(xx^/(yyz)- The condition that 

 the product of the values of yfx, zfy, xfz be unity is of the sixth degree in 

 X, Y, Z. The problem is thus reduced to finding the rational points on a 

 certain sextic surface. 



M. Rignaux, 74a to treat the last system, would use a solution x = x 0) y = yo 

 of the first equation, where x , yo are quadratic functions of two param- 

 eters m, n; likewise a solution x = Xi, z = 2i of the third equation in terms of 

 parameters p, q. Hence take x = XtiXi, y = y Xi, z = z Q Xi. The given second 

 equation becomes a quartic in m, n and is solvable in known special cases. 



xy+a, xz+a, yz+a ALL SQUARES. 



Diophantus, III, 12, 13 and IV, 20, asked for three numbers such that 

 the product of any two increased by a given number a shall be a square. 

 For a = 12, he found 2, 2, 1/8; for a =10, complicated fractions; for 

 o = l, x, x+2, 4x+4. In V, 27, the numbers themselves are to be squares. 



68 Math. Quest. Educ. Times, 20, 1874, 59-60. 



69 Ibid., 21, 1874,45-6. 



70 The Math. Visitor, 1, 1880, 105-6, 129-30; Amer. Math. Monthly, 1, 1894, 208 for (4). 



71 Math. Quest. Educ. Times, 11, 1907, 25. 



72 Jour, of Indian Math. Club, 2, 1910, 24-25. 



73 Nouv. Ann. Math., (4), 16, 1916, 62-74. 



74 Ibid., (4), 18, 1918, 43-49. For such a system, see Ch. V, p. 223. 

 74a L'intermediaire des math., 25, 1918, 132-3. 



34 



