78 ON THE ALGEBRAICAL SOLUTION OF 

 Hence we have 



u=$ir + w' = 



, Qx'y' 2 x, 3 + [v'(3u' a - v' 2 ) 2 - 1 8u'y' 2 ].i 2 ar 2 - 2*'(3' z - v 

 x ^ x%r + sc = 



/=x r + y'- - - - - - - , 



ISy' 3 ^ 3 y 



2 v / -v /3 )iv'(3^ /2 -?; /2 ) 2 -18^ /2 j+182/' 4 ]a; 1 3 



-[x'\v'(3u' 2 - v'*) 2 - 18^y 2 |+ 2x V(3u /2 - v' 2 ) 2 ]^ 2 ^ 



' 



In these equations u f , v\ x', y' must have such values as 

 satisfy equation (4). Thus if we take u'=2, t/=l, ^'=3, 2/'=l, 

 we shall have 



u=(9'7x 1 2 -66x l x 2 + 12a: 2 2 )/6o; 1 2 , v=l, 



These give, finally, the identity 



1 2 -66z 1 ay}- 12a; 2 2 )] 4 + [4 

 + [2ic 1 (1007a; 1 3 - 726a: 1 2 n: 2 + 132o; 1 a: 2 2 )] 4 + [2a; 1 (899a: 1 3 



)] 4 (8) 

 The solution u=2, v=l, =3, y=l gives from (1) 



5 4 =3 4 +4 4 +4 4 +2 4 +2 4 



and w=7, v=6, rc=3, 2/=19 gives 



85 4 =84 4 + 38 4 + 22 4 + 16 4 + 13 4 . 

 If in (8) we put 3^=1, o; 2 =2, we derive 



205 4 = 166 4 + 156 4 + 133 4 + 92 4 + 74 4 . 



It is to be remarked that if we have any solution of the 

 form 



P 4 =P 1 4 +P 2 4 + (*+i/) 4 + (x-y 

 =P 1 4 +P 2 4 +2(* 2 +3 2 / 2 ) 2 



