78 ON THE ALGEBRAICAL SOLUTION OF 

 Hence we have 



' - 2x'(3u' - 



, _ 



, _ 6 W + MSn'* - 1/ 2 ) 2 



., r ! , = ---- -^t t -x 1 - aia 



18y' 3 o^ s 



/2 - v' 22 - ISu ' 



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

 satisfy equation (4). Thus if we take u'=2, v'=l, x'=3, y'=\, 

 we shall have 



u= (97 x^- 66:^2+ 1 2x 2 z )/Qx 1 z , v=l, 



?/=(953a; 1 3 -981a; 1 2 a; 2 +330a; 1 a; 2 2 -36.T 2 3 )/18a; 1 3 . 

 These give, finally, the identity 

 [(97o; 1 2 -66a; 1 a; 2 + 12a; 2 2 ) 2 + (6a: 1 2 ) 2 ] 4 =[(97a; 1 2 -66^+ 12cc 2 2 ) 2 



-(6V) 2 ] 4 



+[2a; 1 (1007a; 1 3 -726a; 1 2 a; 2 +132a; 1 a; 2 2 )] 4 +[2^ 1 (899a; 1 3 -1236a: 1 2 a; 2 



+528a; 1 a; 2 2 -72a; 2 3 )]* (8) 

 The solution w=2, v=\, x=3, y=l gives from (1) 



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

 and w=7, v=Q, a;=3, y=19 gives 



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

 If in (8) we put ^=1, a; 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 =/Y+P 2 4 + (x 



