INDETERMINATE QUARTIC EQUATIONS 89 

 Hence we have as our final trial equations 



Y 

 (x^-y^] (5) 



4i/ 4 ) 4 ] 4 

 (z 4 -4i/ 4 )j 4 



(6) 



(7) 



4 ] (8) 



Of these the first two are immediately to be rejected since 

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



and [(a; 4 +42/ 4 ) 4 +(a; 4 -42/ 4 ) 4 ] 4 =[(a; 4 +42/ 4 ) 4 -(a; 4 -4i/ 4 ) 4 ] 4 



+ J2(z 8 - lGy*)\*(x le + 22x 8 y 8 + 256i/ 16 ) 2 

 respectively, equations which are known to be impossible. 1 



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

 and /=!, the latter when xy, 



(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. 



^.#. 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, l.c. 

 M 



