INDETERMINATE QUARTIC EQUATIONS 73 

 Equation (4) is then satisfied by taking 



_ 



2[{X(a - d) 



on substituting for a; 3 its value given by (5). These values of 

 x 3 and r when substituted in (3) render it an identity and 

 constitute a solution which is clearly algebraical. 



To satisfy equations (3) the solutions d?=a 2 , e 2 =6 2 , / 2 =c 2 

 obviously make r zero and therefore lead to no new result, 

 but we shall presently show how solutions of a different 

 character may be obtained whatever be the values of X, (i, 

 and v. 



3. We shall now show how the equation 



may be solved by a single formula which holds for all values 

 of r except zero. 



Let us first solve the equation 



^l 4 +^2 4 =^ / l 4 +^ / 2 4 +"(2P) 4 . (1) 



Putting Q^a+b, Q 2 =c-d, Q\=a-b, Q' 2 =c+d, P=ax, (1) 

 becomes 



Ia 3 b+lab 3 =u.c 3 d+v.cd 3 +2va*x* (2) 



If now we take d=\ab 3 /\nc 3 , (2) is satisfied by taking 



i.e. a=x6([A 2 c 8 -X 2 6 8 )/2[xVc 8 rc 4 (3) 



so that rf=X 2 6 4 ( ! x 2 c 8 -X 2 6 8 )/2[A 3 j/c 11 a; 4 . 



Hence, omitting the common denominator, and replacing 

 x throughout by xjc 2 , we have as a solution of (1) 



Thus we have the identity 



+!Ji[2i J L 3 rc 4 a; 4 -X 2 & 4 (( A 2 c 8 -X 2 6 8 ) 

 4 +!i[2( J i 3 j/c 4 a; 4 +X 2 6 4 ( [A 2 c 8 -X 2 6 8 



4 (4) 



