INDETERMINATE QUARTIC EQUATIONS 89 

 Hence we have as our final trial equations 



+\2(x*-y*)\*[\2xy(x 2 +y 2 )>f+\2xy(x*-y*)\*+ (z 4 -*, 4 ) 4 ] (5) 



=[(z 4 +4y 4 )Ma 

 (* 4 + 4/ 4 )j 4 + J4 



(7) 



/ 4 ) 4 - (a; 4 -4t/ 4 ) 4 ] 4 

 + j(2(a; 8 - 16/ 8 ) 4 [(z 8 +4a;y+ 16*/ 8 ) 4 + (8z V) 4 + (8*y ) 4 ] (8) 

 Of these the first two are immediately to be rejected since 

 they imply (see 7, equation (5), infra) 



and [( 



+J2(a; 8 - 16i/ 8 )j 4 (a; 16 + 224V+ 256/ 16 ) 2 

 respectively, equations which are known to be impossible. 1 



The remaining two agree in giving, the former when x=2 

 and y=I, the latter when x=y, 



(5 4 +3 4 ) 4 =(5 4 -3 4 ) 4 +(30) 4 (21 4 +2 4 +8 4 ) 

 or on removal of the common factor 2 from the roots 



353 4 =272 4 +315 4 +30 4 +120 4 , 



a result which direct calculation will verify. Neither of the 

 equations (7) or (8) however seems to yield any more solutions 

 for other values of x and y ; and they must therefore be 

 regarded as, at the best, only more or less likely approxima- 

 tions to an algebraical solution. 



N.B. Hence collecting the results of 2, 3, and 6 we have 



353 4 = 315 4 + 272 4 + 120 4 + 30 4 

 =300 4 + 272 4 + 180 4 + 150 4 + 135 4 +90 4 

 =272 4 +252 4 +234 4 + 198 4 + 189 4 + 130 4 +36 4 +30 4 

 =300 4 + 272 4 + 180 4 + 150 4 + 135 4 + 72 4 + 72 4 + 54 4 + 36 4 + 36 4 ; etc. 



1 Euler, Elements of Algebra, I.e. 

 51 



