644 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxil 



We obtain Weierstrass' normal form 



(6) y 2 = 4s 3 -0 2 s-03 = 4(s-e 1 )(s-e 2 )(s-e 3 ), 



where g 2 , gz are the invariants of/; also 2/ = |/ / (r)v/(s O 2 - Euler 27 of 

 Ch. XV discussed the problem to find s such that s e t - are squares for 

 i = l,2,3 (whence their product gives w 2 /4), but evidently restricted attention 

 to the case in which each et is rational. Haentzschel showed how, from three 

 primitive solutions of (6), to find four infinite sets of solutions by means of 

 Weierstrass ^-function. 



Removing the assumption of a rational root r, but assuming one solution 

 x , ?/o of f=y 2 , he applied a certain linear fractional transformation giving a 

 quartic whose leading coefficient is a square. 



G. Humbert 164 stated that all the methods which have been proposed to 

 deduce rational solutions of ax 4 -}- -\-e = z i from one or more initial solu- 

 tions are identical at bottom, and gave the method in geometrical and 

 analytic form. 



On x*x z y+xYxy*+y 4 = D, see papers 63-66 of Ch. II, Vol. I. On 

 xy(x*y 2 )=Az 2 , see papers 11, 18; also Congruent Numbers in Ch. XVI. 



For other special quartics made squares, see papers 101 of Ch. I; 21, 

 92-4, 96-7, 109, 138-40 of Ch. IV; and 9, 72, 73, 77, 92, 133 of Ch. V; and 

 various papers of Chs. XIV-XX. 



(1) 



A 4 + 4 = C 4 +Z> 4 . 

 L. Euler 165 took A=p+q, D = pq, C = r+s, B = rs and derived 



pg(p 2 +g 2 )=rs(r 2 +s 2 ). 

 Set p = ax, q by, r = kx, s = y. Then 



y z lx z =(k*-a s b)!(ab*-k). 



If k = ab, x = l, then y = a, C=A, B=^rD. Set therefore k = ab(l+z). 

 Then 2/ 2 /z 2 = a 2 Q/(6 2 -l-z) 2 , where 



Let Q be the square of b 2 l+fz-\-gz 2 and choose /, g to make the terms in 

 z, z 2 agree. Thus 



3b 2 -l ^ ^ 



' ' 2 



2 ' 8(& 2 -l) ' b 2 +g 2 



Then x :y = b 2 lz : a(b 2 1+fz+gz 2 ). As examples, Euler took b = 2, 

 6 = 3, and found the solution 



A = 2219449, B= -555617, C = 1584749, Z) = 2061283, 

 and an erroneous 166 one replaced in his next paper by 

 _ A = 12231, 5 = 2903, C = 10381, D = 10203. 



164 L'interm6diaire des math., 25, 1918, 18-20. 



165 Novi Comm. Acad. Petrop., 17, 1772, 64; Comm. Arith., I, 473; Op. Om., (1), III, 211. 

 184 This error waa also noted in Pinterm6diaire des math., 2, 1895, 6, 394; 7, 1900, 86; Mathe- 

 sis, 1889, 241-2. 



