84 ON THE ALGEBRAICAL SOLUTION OF 

 4. QUESTION 3. Solve the equation 



In the identity 

 put =(a; 4 -|-2/ 4 -f-2 4 )/8 and we obtain 



Let us now choose x, y, and z so that 



# 4 +2/ 4 +z 4 =2v 4 . 

 To do this, since we have identically 



it will be sufficient to take 



x=2ab+b 2 , y=a?-b 2 , z=a*+2ab, 

 for since this makes x 2 +xy+y 2 =(a 2 +ab+b 2 ) 2 , we shall have 1 



(2ab+ 6 2 )M- ( 2 -6 2 ) 4 + ( 2 + 2a6) 4 =2(a 2 + a6+ & 2 ) 4 . 

 Making these substitutions (2) becomes 



(a2 _ 62)4 



/16. (3) 

 Now since 2 J^' 



(3) may be written 



( 2 +a6+& 2 ) 4 -4] 4 +(8a6+46 2 ) 



This is an algebraical solution of equation (1) and it may 

 be integralised by multiplying each root by (c 2 +cd+d 2 ), or 



1 This solution of the equation 2P 4 = Pj* + P 2 4 + P 3 4 involves the assumption 

 P l = P 2 + P 3 , a restriction which the identity 



2-484813* = 575528 4 + 155873 4 + 116745 4 

 ehowe to be unnecessary. 



