CHAP. XXII] QUARTIC FUNCTION MADE A SQUARE. 641 



A positive v cannot surpass a certain number L which makes every coefficient 

 in (3) positive; hence we have only to try v = Q, 1, , L 1. If v= t, 

 where Q<t<x, let s be the least t for which 4a 2 (+c)>6 2 and substitute 

 s+w for v in (3) ; we get an equation like Ax 2 +Bx-{-4:a 2 e = (s-\-w) 2 with all 

 coefficients positive, whereas x 2 >t 2 =(s-\-w) 2 ; hence the only cases to try 

 are v= 1, , (s 1). Finally, if v= u, Q<u>x, let r be the re- 

 mainder <x on dividing u by x and n the quotient. Set 



By 2 2 >a 2 z 4 , we have b>n and need only try nl, , 6 1. 



C. G. J. Jacobi 148 stated that the analysis of Euler 144 " 5 to find an infinitude 

 of rational values of x, given one, making the quartic f(x) a square is the 

 same as that of Euler's 149 (earlier) solution of the transcendental equation 



f x dx 

 - 



For the latter, Euler used a chain of n equations f(p, q) = 0, f(q, r) = 0, 

 f(r, s) = 0, , where 



f(p, q) = a+2p(p+q)+- Y (p*+q'>)+28pq+2epq(p+q)+tpY 



is symmetrical in p and q, whereas in the diophantine problem Euler's 

 canonical equation Qy z -\-2Py R = Sx 2 -{-Tx-{-U = Q is not symmetric in 

 x } y, as pointed out by L. Schlesinger, 150 who discussed at length Jacobi's 

 above remark. The latter had been discussed by T. Pepin. 151 Jacobi 

 observed that the analysis of the multiplication of elliptic integrals (4) 

 gives an infinitude of rational T/'S for which also V/(?/) is rational, if a rational 

 x makes V/(z) rational, and drew from the theory of abelian integrals the 

 conclusion 152 : If f(x) is of the fifth or sixth degree and if one rational value 

 of x makes V/(z) rational, there exist an infinitude of x's of the form 

 a+6Vc, with a, b } c rational, for which V/Oc) ='+&' Vc, with a', b' rational; 

 and the extension to f(x) of degree 2n-j-l or 2n+2 and x's satisfying an 

 equation of degree n with rational coefficients. J. Ptaszycki 153 remarked 

 that the last theorem follows at once from the representation of a rational 

 function by means of polynomials which enter in the development into a 

 continued fraction of the square root of this function. The generalization 

 of Jacobi's theorem has been considered by J. von Sz. Nagy. 154 



The problem to make a quartic a rational square was proposed in 1865 

 as a prize subject by the Accad. Nuovi Lincei of Rome. 



L. Calzolari 155 wrote a-\-bv-}-cv 2 -}-dv 3 -\-ev* = w 2 in the form 



+c'v 2 +Q 2 , Q=2ev 2 +dv+k, 

 b' = 2be-dk, c' 



14 Jour, fur Math., 13, 1834, 353-5; Werke, II, 51-5. 



149 Institutiones Calculi Integrate, 1, 1763, Ch. 6, Prob. 83, 642. 



160 Jahresber. d. Deutschen Math.-Vereinig., 17, 1908, 63 (with history of /(x) = D). 



161 Atti Accad. Pont. Nuovi Lincei, 30, 1876-7, 224-37. 



162 Cf. Jacobi, Jour, fur Math., 32, 1846, 220; Werke, II, 135; Schwering 23 ' of Ch. XXI. 



163 Jahresber. d. Deutschen Math.-Vereinig., 18, 1909, 1-3. 

 1M Ibid., 4-7. Cf. Nagy 16 of Ch. XXIII. 



165 Giornale di Mat., 7, 1869, 317-50. 



42 



