68 ON THE ALGEBRAICAL SOLUTION OF 



For example c=2, b=x=l gives 



223 4 + 2056 4 =287 4 + 1020 4 + 2024 4 . 

 We may now deduce several results from (3). 

 (i) Replacing x* by X, equation (3) shows that every 

 rational quantity X is expressible rationally in the form 

 -# 4 - 4 , in an infinity of ways, viz. : 



Y _fc 2 (c 8 -6 8 +2Z)1 4 [2c 4 Z-6 4 (c 8 -6 



L 2(c 8 -& 8 ) J *[_ 26c(c 8 -& 8 ) 



rc 2 (c 8 -6 8 -2Z)1 4 _r2c 4 Z+& 4 (c 8 -& 8 )1 4 



L 2(c 8 -6 8 ) J L 26c(c 8 -6 8 ) J ' 



n=n 



(ii) We may replace # 4 in (3) by 2# n 4 , whence we have 



[6c 3 (c 8 -6 8 + 22z n 4 )] 4 + [2c 4 2# n 4 -6 4 (c 8 -& 8 )] 4 

 )] 4 +[2c 4 S^ 4 +6 4 (c 8 -6 8 )] 4 

 (o; 1 4 +a; 2 4 + . . . +x n *) (4) 



Equation (4) is an algebraical solution of the equation 



for all values of m greater than 2. 



Many particular results of some interest are included in (4). 

 Thus taking n2 and putting ic 1 =6 2 , x 2 =bc, (4) becomes 



[6c 3 (c 8 -6 8 +26 4 & 4 +c 4 )] 4 +[2c 4 -6 4 (6 4 +c 4 )-& 4 (c 8 -6 8 )] 4 

 =[6c 3 (c 8 -6 8 -26 4 6 4 +c 4 )] 4 +[2c 4 -& 4 (6 4 +c 4 )+6 4 (c 8 -6 8 )] 4 



+ [26c(6 8 -c 8 )] 4 (6 8 +6 4 c 4 ). 



Hence dividing each root by (6 4 +c 4 ) we have 

 [6c 3 (6 4 +c 4 )] 4 +[6 4 (6 4 +c 4 )] 4 =[6c 3 (c 4 -36 4 )] 4 +[6 4 (3c 4 -6 4 )] 4 



+ [26 3 c(6 4 -c 4 )] 4 +[26 2 c 2 (6 4 -c 4 )] 4 , 

 or, as it may be written 



&11 12 _p^-3^)T 

 ~ 4 * + 



The equations (4) and (5) have an important application 

 to the solution of the problem of finding a number of biquad- 

 rates whose sum is a biquadrate. 1 



1 See the writer's paper, Part III, Quest. 2. 



