CHAP, xix] QUADRATIC FORMS IN x, y; x, z;y, z MADE SQUARES. 511 



can usually be made squares by values of x, y, z which are rational functions 

 of a parameter k whenever the auxiliary equation ^jp 2 -\-yaq 2 = aftr 2 can be 

 similarly solved. For, if rational functions p\, q if TI of k satisfy the latter, 

 a linear relation p\x-\-q\y-}-riZ = Q implies A = D. Similarly, if rational 

 functions p 2 , #2, r 2 of m satisfy the auxiliary equation, then r z x4-pzy+g.zZ = Q 

 implies B=H. Solving the two linear equations, we obtain x, y, z as 

 quadratic functions of m and k. For these values, C becomes a quartic 

 function of m whose first and last coefficients are squares of functions of k, 

 so that C can be made a square. For a = = 2, 7 = 1, we have the problem 

 of a rational triangle with rational medians. Euler's 46 equations (1) are 

 treated by this method and by a related method. In the second paper he 

 used the method to make px 2 +q-y 2 pz 2 , py 2 -\-q 2 z 2 pw 2 , pz 2 -{-q 2 w 2 px 2 , 

 pw 2 -}-q 2 x 2 py 2 all squares. 



On 2x 2 -\-2y~ z-= D, etc., see triangles with rational medians, Ch. V. 



QUADRATIC FORMS IN x AND y, x AND z, y AND 2, MADE SQUARES. 

 J. Cunliffe 64 found rational numbers x, y, z such that 

 (4) x z xy+y z , x 2 -xz+z 2 , y 2 -yz+z 2 



are squares, by equating the first and second to the squares of 4ax, 

 4bx, whence 



lQa 2 -y 2 16& 2 -z 2 

 x = - 



8a y Sbz 



Equate the denominators. Thus y = 5a 36, z = 5b 3a. Then 



y z -yz+z z =(7a-nb) 2 



if a : b = n 2 49 : I4n 94. J. Whitley equated the first two functions (4) 

 to the squares of x ny and x mz ; hence take 



Set p = 2n l, q = n z l, v = n? n-\-l. Then v~ = p 2 pq-\-q 2 . Equating 

 y 2 yz-{-z 2 = p-'rri i 2pgra 3 +(4g 2 +p# 2p 2 )m 2 -}-(2pq 4g 2 )w+y 2 



to the square of pm z qm-\-v, we get m rationally. 

 To find rational numbers such that 65 



(5) tf+xy+y*, x*+xz+z z , if+yz+z 2 



are squares, equate the first and second to the squares of x-\-y m and 

 x-\-z n. We get x and z in terms of y. The third condition leads to a 

 quartic in y, which is made a square as usual. 



Lowry 050 made a = x 2 +axy+by~, (3 = x 2 + o.ixz +hz 2 , y = y*+a 2 yz+b2Z 2 

 squares. Set r = n(ain-\-2?n),s = m 2 bin 2 , p = u(au+2v), a- = v 2 bu 2 . Take 

 yfx = p/o-, zfx = rfs. Then <x<r 2 (x 2 = (v 2 + auv + bu 2 ) 2 . Similarly, /3 = D . Since 



64 The Gentleman's Math. Companion, London, 3, No. 14, 1811, 310-11. 



66 Ibid., 4, No. 21, 1818, 757-60; J. Cunliffe, Leybourn's Math. Repository, New Ser., 2, 



1809, I, 93-5. Cf. Ch. V. 113 

 65a New Series of Math. Repository (ed., T. Leybourn), 3, 1814, I, 153-164. 



