INDETERMINATE QUARTIC EQUATIONS 61 

 Equation (3) is now identically satisfied by taking 



8(< 4 



say dr=3z 4 (z 4 2 -z 3 2 ) 2 (3z 3 2 +z 4 2 ), 



where rf=a: 1 (z 4 2 + z 3 2 )( 18z 4 2 z 3 2 -z 4 4 -2 3 4 ) 

 Hence rfz 1 =a; 1 ^r+^ 1 =3a; 1 z 4 (z 4 2 -Z3 2 ) 2 (3z 3 2 +z 4 2 ) 



+* 1 z 4 (z 4 2 +z 3 2 )(18z 4 2 z 3 2 -z 4 4 -z 3 4 ) 



and by symmetry 



^2=2^3 



Hence dP 1 =d(z 1 +z 3 )=z 1 [2z 4 (z 4 + 



Thus, finally, we have the algebraic identity 



+ [a; 7 

 on writing x for z 4 and y for z 3 . 



For example x=l, y=2 gives 76 4 +1203 4 =1176 4 +653 4 , 

 and x=l, y=3 gives 133 4 +134 4 =158 4 +59 4 . 



1 This identity ia due to Euler (Commentationes Arithmetical, vol. ii. p. 289), who 

 obtained it by a different method. 



