INDETERMINATE CUBIC EQUATIONS 55 



which is an indeterminate cubic in x, u, v having the particular 

 solution x = a, u = c,v = d. Hence, to find an algebraical solu- 



(3) 



(4) 

 (5) 



tion, we put x = xp + a, u = x 3 r + c, v = x 3 r + d 

 and (2) on expansion and rearrangement becomes 



+ (3a' 2 6 - b 3 x, - 2d 2 - 4dx 3 )r = 0. 

 Hence, making the coefficient of r vanish by taking 



+ 4(c 2 



equation (4) is satisfied by taking 

 r = (x i 2 + 2xi-3abx 1 z )/bx 3 

 = [(3a 2 6 - 6 3 ) 2 - 24a6(f]a; 1 2 - 4c(3a 2 6 - 



SbdFx* 

 on substituting the value of x s given by (5). 



Hence, we find 

 x = [{ (3a 2 6 - 6 3 ) 3 - 16abd*}x* - 4c(3a 2 & - b s )x' z x 



u = [Sbcd 2 x* + { (3a 2 b - 6 3 ) 2 - 



- 4c(3a 2 6 - 



- b 3 ) + 32bd t }x l 3 



+ 4(3a 2 6 - 6 3 )(3c 2 + 2<f )x l x 



lt 



v = [{(3a 2 6 - 6 3 ) 3 - 

 - 6c{ (3a 2 6 - b 3 ) 2 - 



- 8c(c 2 



For example, as a particular solution of (2), we may take 

 a=2, 6=1, c=2, d=l. (7) 



Substituting these values in (6), we obtain, finally, the alge- 

 braical solution 



o;=(89a; 1 2 -88a; 1 a; 2 +24a; 2 2 )/8a; 1 2 , y=l, 



u= (I6x 1 3 + 73a; 1 2 2 -88a; 1 a: 2 2 + 24x 2 3 )/8x 1 3 , 



Hence we derive the following solutions, on integralising, 



