INDETERMINATE QUARTIC EQUATIONS 81 

 Thus we have the identity 



(z 4 -4i/ 4 ) 3 ] 



4i/ 4 ) 2 ] 4 . (5) 



For example, x=y=l gives, on omission of the common 

 factor 2, 



353 4 =272 4 + 135 4 +300 4 + 150 4 + 180 4 +90 4 . 

 Again, if for shortness we put X for # 4 +4?/ 4 and Y for 

 x* 4i/ 4 , then equation (4) is 



X 8 - Y*=2(2xy)*(x*+16y*)(X*+ 7 4 ). 

 Multiplying each side of this equation by X 8 + Y 8 it becomes 



and if we multiply each side of this again by X 16 + Y 16 , it 

 becomes 



and in general we have 



*+ Y 2r+ *) (6) 



Now the right-hand side of (6), omitting the factor 2 (2xy)\ 

 consists of the product of r+2 factors each of which is the sum 

 of two biquadrates, and therefore the right-hand side of (6) 

 is equal to twice the sum of 2 r+2 biquadrates, which we may 

 call x l9 x 2 , x 3 , . . ., x 2r+2 . Hence bX 2r , c=Y 2r is an alge- 

 braic solution of (2), for the case 7i=2 r+2 -f 2, giving, on sub- 

 stitution in (3), the identity 



Thus r=l gives 

 [X 8 + 



which is an algebraical solution of 



P 4 =P 1 4 +P 2 4 + . . . +P 10 4 . 



L 



