INDETERMINATE QUARTIC EQUATIONS 81 

 Thus we have the identity 



(z 4 -4t/ 4 ) 3 ] 4 



+ [8a*/ 3 (z 4 +4</ 4 ) 2 (a: 4 -4?/ 4 )] 4 + (8ccy 3 (a; 4 +4i/ 4 )(a; 4 -42/ 4 ) 2 ] 4 . (5) 

 For example, x=yl 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 x 4 +4y 4 and Y for 

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



X 8 - 7 8 =2(2zt/)V+16*/ 8 )(Z 4 + 7 4 ). 

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



X 16 - 7 16 =2(2zi/)V+16*/ 8 )(Z 4 + 7 4 )(Z 8 + 7 8 ), 

 and if we multiply each side of this again by X 16 + 7 16 , it 

 becomes 



X 3 *- Y 32 =2(2xy)*(tf+ 16y*)(X*+ 7 4 )(X 8 + 7 8 )(X 16 + 7 16 ), 

 and in general we have 



(X 2r+2 + 7 2r+2 ) (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 lt x 2 , x 3 , . . ., av+2- Hence b=X 2r , c=Y Zr is an alge- 

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

 stitution in (3), the identity 



7 2r+2 ) 

 Thus r=l gives 

 [Z 8 + 7 8 ] 4 = [X s - 7 8 ] 4 + [2X Z 7 6 ] 4 + [IxyX* 7 4 ] 4 



(X*+ 7 4 )(Z 8 + 7 8 ), 

 which is an algebraical solution of 



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



L 



