INDETERMINATE QUARTIC EQUATIONS 73 

 Equation (4) is then satisfied by taking 





- c)[{X(a 8 - <P)x* + ^(fe 2 ~ e 2 )ay M/ s - <?)(f z +fc + c 2 ) - r(c +/){X(o 3 - 



2[{X(a - d)*! 3 + ft(6 - eJ-ivVt/ 3 - c 3 ) 2 (/ 2 +fc + c 2 ) - {X(a 3 - d 3 )x, + /x{& 3 - e*)x s }] 



on substituting for x 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 2 =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, jx, 

 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 



*i 4 + v-QS=*Q\*+ v.Q'S+v(2P)'. (1) 



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

 becomes 



Xa 3 6+ Xa6 3 = (xc 3 d+ [Lcd 3 + 2va*x* (2) 



If now we take d=Xa& 3 /[xc 3 , (2) is satisfied by taking 



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



so that d=X 2 & 4 ((z 2 c 8 -X 2 6 8 )/2 (A ^c 11 a: 4 . 



Hence, omitting the common denominator, and replacing 

 x throughout by #/c 2 , we have as a solution of (1) 



- X 2 & 8 ) - 2[x 3 ^c 3 a: 4 , Q' 2 =2^ v c i x*+ X 2 6 4 ( {J .*c 8 - X 2 & 8 ) 



2P==2X{x6c((x 2 c 8 -X 2 6 8 )a;. 

 Thus we have the identity 



+(ji[2 ( x 3 vc 4 a; 4 -X 2 6 4 ((x 2 c 8 -X 2 6 8 )] 4 

 4 +(ji[2fx 3 ^c 4 a; 4 +X 2 6 4 ((x 2 c 8 -X 2 6 8 )] 4 

 4 (4) 



