INDETERMINATE QUARTIC EQUATIONS 61 

 Equation (3) is now identically satisfied by taking 



r= 



say dr=3z 4 (2 4 2 -z 3 2 ) 2 (3z 3 2 -fz 4 2 )> 



where d= 

 Hence =a? 1 - 



and by symmetry 

 cfo 2 =2a? 1 2 8 [ 

 Hence ^P 1 =^(2 1 +2 3 )=^ 1 



Thus, finally, we have the algebraic identity 



x 5 y 2 + 



+ [x 7 + Sxy- 1 7x 

 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 is due to Euler (Commentationes Arithmeticce, vol. ii. p. 289), who 

 obtained it by a different method. 



