82 ON THE ALGEBRAICAL SOLUTION OF 



COR. We may now give algebraical solutions of the 

 equation 



where r=6+2w. 



For we have the identity 1 



M_ L .i^-[^(^-3^)]* + [cV-3 



_[6 3 (6 4 -3c 4 )] 4 + [c 3 (c 4 -36 4 )] 4 + [2&c(& 4 -c 4 )] 4 (6 4 +c 4 ) 



(& 4 +c 4 ) 4 



Put now &=w 1 2 , c=2v 1 2 and this becomes 



+ [8 VV( V- iKWA V+ !K 8 ) 4 (7) 



Again, equation (5) is 

 [X*+ 7 4 ] 4 = [Z 4 - 7 4 ] 4 + [2Z 7 3 ] 4 + [4^Z 2 7] 4 (x 8 + 16?/ 8 ) 



+ [4^Z7 2 ] 4 (^+ ley 8 ) (8) 



If then we put x=u^ t y=2v 1 5 , x 8 + I6y 8 becomes % 24 +2 12 v 1 24 

 which is expressible rationally as the sum of four biquadrates. 

 Making these substitutions in (8) we see that if X ' and 7' 

 denote the new values of X and 7, then [JC /4 -f 7 /4 ] 4 is ex- 

 pressible simultaneously as the sum of 6, 8, or of 10 rational 

 biquadrates which may all be made integral by multiplying 

 each root by (% 8 + ICi^ 8 ). As a matter of fact we have 



V 2 ) 4 



+ 2 1 V 4 ] 



2 )] 4 K 24 + 2 1 V 4 ] 



v 1 12 )( V 2 - 



1 See 3 of Part I. 



