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 a 4 by X, equation (3) shows that every 

 rational quantity X is expressible rationally in the form 

 ? 4 R* S*, in an infinity of ways, viz. : 



X= 



2(c 8 -6 8 ) 2&c(c 8 -& 8 ) 



r2c 4 Z+6 4 (c 8 -& 8 )"] 4 

 I' 



2(c 8 -& 8 ) J 26c(c-6 8 ) 



n=n 



(ii) We may replace x* in (3) by S n 4 , whence we have 

 [6c 3 (c 8 -6 8 + 22z, 4 )] 4 



' +[26c(c 8 -6 8 )] 4 (a; 1 4 +a: 2 4 + . . . +z n 4 ) (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 n=2 and putting x^b 2 , x z =bc, (4) becomes 



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



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

 [&c 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 



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

 or, as it may be written 



3, (5) 



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. 



