CHAP. XIX] PKODUCTS BY Twos PLUS LINEAR FUNCTIONS SQUARES. 521 



s = 2abg, r = (a? c 2 )g, q = 2acg, p = (d 2 a 2 )g, n = 2adg. Then 



</Q 2 +6 2 )(a 2 +c 2 ) ^(a 2 +d 2 )(a 2 +6 2 ) _g(a 2 +c 2 )(a 2 +d 2 } 



a 2 +d 2 a 2 +c 2 a 2 +6 2 



To satisfy (1), take a=f-+fh+h z , b=f 2 -h 2 , c = 2fh+h~, d=f-+2fh. For 

 four numbers, see Euler. 106 



J. Cunliffe 108 made xy-\-z, etc., squares by taking yx = 2n, z = ri 2 . 

 Then xy-\-z = (x-{-n) 2 , while xz +?/ and yz+x are linear functions of x and 

 may be equated to squares. S. Jones took y = x l, z = x 4. "J. B." 

 took y = t 2 x v 2 , z = v 2 x, whence xy+z = t 2 x 2 . Then xz-\-y = (vx r}' 2 gives x. 

 From yz+x= D, we get a quartic in r which is solved as usual. 



W. Wright 109 took xy a = p 2 , yz b = q 2 and made p 2 -}-a and q*+b 

 squares. Then xz c= D if D/?/ 2 c= D, which is easily satisfied. 



Cunliffe 110 took xy+z = A z , xz+y = B 2 . Thus (y+z)(x+l}=A 2 +B 2 . 

 Hence set y+z = a 2 +b 2 , x+l=c 2 +d 2 , A=ac+bd, B = ad-bc. Also, 

 (y-z)(x-l)=A 2 -B 2 . Hence take y-z = A-B, x-l = A+B. By the 

 two values of x, we get 6 in terms of a, c, d. To get integral values of 

 6, equate the denominator c d to unity. 



D. S. Hart 111 made xy+z, etc., and xy+x+y, etc., all squares by taking 



E. N. Barisien 112 treated the system 



xzy = t 2 , (z+a)xy = u?, ( 

 Subtract the first from the other two. Thus 



ax = U? - 1 2 , bx = v 2 - 1 2 , av 2 - bu 2 = (a - 6) t z . 

 Set v = t+h, u = t+l. Discarding the denominator 2ha 2lb, we have 

 t = bl z -atf, u = U 2 +ah 2 -2alh, v = ah 2 +U 2 -2blh, x = 4lh(h-l)(bl-afi). 



Then y, z can be found from Az y = B. Set B = Ap+r, z = q+p; then 

 y = Aq-r. [Take g= -ah/I, f= -Z 2 /a]. Then 



x = 4f 2 g(a+g}(b+g), u=f(g 2 +2ag+ab}, v=f(g 2 +2bg+ab], 

 t=f(g 2 -ab}, z = g, 2/=/ 2 {3^ 4 -hV(a+6)+6a^ 2 -a 2 6 2 }. 



He 113 elsewhere merely stated the latter solution. 



V. G. Tariste" 114 treated the case a = 1, 6 = 2 of the last problem. Then 



t 2 = 2u 2 , whose general solution is u = \(A 2 +B 2 ); v, t = \(A 2 -B 2 2AB). 



Several writers 115 made xy+z, yz+x, xz+y all squares (Diophantus 

 III, 14). 



E. Bahier, 85 pp. 208-212, made xy v, xz v,yz v squares the sum of two 

 of which equals the third. 



108 The Gentleman's Math. Companion, London, 3, No. 17, 1814, 463-6. 



109 Ibid., 467-8. 



110 Ibid., 5, No. 27, 1824, 349-53. 



111 Math. Quest. Educ. Times, 28, 1878, 67-8. 



112 Sphinx-Oedipe, 1907-8, 180-1. 



113 Mathesis, (3), 9, 1909, 154-5. 



114 L'intermediaire des math., 19, 1912, 38-9. 



115 Zeitschr. Math. Phys., Hist.-lit. Abt., 37, 1892, 138; Math. Quest. Educ. Times, 25, 1914, 



40, 102-4; Amer. Math. Monthly, 24, 1917, 88-89, 294. 



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