74 ON THE ALGEBRAICAL SOLUTION OF 

 If in this we replace # 4 by x* y* we obtain 



- X 2 6 4 ( [i 2 c 8 - X 2 6 8 )] 4 + if 2x |i6c( ; tx 2 c 8 - X 2 6 8 )?/] 4 



4 -?/ 4 )+X 2 6 4 

 4 (5) 



Equation (5) is an algebraical solution of the equation 



for all finite values of X, JA, and v. 



Thus, putting c=x=2, b=y=\, equation (5) becomes 



- X 2 (256(A 2 -X 2 )] 4 



(6) 

 For example X=2, |i=l, ^=3 gives 



3 4 + 2- 33 4 + 3- 14 4 = 1 7 4 + 2- 23 4 + 3- 28 4 . 



r=r r~r 



Again, if in (4) we replace vx* by Sv r a; r 4 2v r y r *, we obtain 

 an equation 



4 -Si' r 2/ r 4 )+ X 2 6 4 (jx 2 c 8 -X 2 6 8 )] 4 



*) (7) 

 which is an algebraical solution of the equation 



X 1 P 1 4 +X 2 P 2 4 + . . . +X r+J P? +2 =X 1 P / 1 4 +X 2 P' 2 4 + . . . 



+ X r+2 P r+2 



for all values of r except r=0. 



Fhially, if in (7) we put all the ?/'s]equal to zero, we obtain 



X[6c 3 |X[i((i 2 c 8 -X 2 6 8 )+ 2 [ * 3 2j/ r a; r 4 S] 4 



-X 2 6 4 ([jL 2 c 8 -X 2 6 8 )] 4 

 = X[6c 3 |X (i( [X 2 c 8 - X 2 6 8 ) - 2 [ x 3 2 JV r r 4 j] 4 



X 2 6 8 )] 4 



(8) 



