INDETERMINATE CUBIC EQUATIONS 55 



Avhich 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 = x l r + a, u = x^r + c 9 v = x 3 r + d 

 and (2) on expansion and rearrangement becomes 



+ (Zabx^ - x} - 2x*)r* 



+ (Wb-Vx^ - 2c,x, - 4dx 3 )r = 0. 

 Hence, making the coefficient of r vanish by taking 



x 3 = (3a 2 6-6 3 ^ - 

 equation (4) is satisfied by taking 



r = (x? + 2x 2 2 - 

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



2 - 4c(3a*6 - 



+ 4(c 2 



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

 Hence, we find 



2 6 - b 3 ) 3 - Wab^x, 3 - 4c(3a 2 6 - b*)xS 



u = [Sbcd 2 x, 3 + { (3a 2 b - b 3 ) 2 - 24abd 2 }x 1 *x* - 4c(3a 2 6 - 



+ 4(c 2 +2<f)z 2 3 ]/86d:r 1 3 , 



v = [{(3a 2 6 - 6 3 ) 3 - 24abd*(3a?b - b 3 ) + 326dP}z 1 3 

 - 6c{ (3a 2 6 - 6 3 ) 2 - Sabd* }x 2 x 2 + 4(3a 2 6 - 6 3 )(3c 2 



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

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



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

 braical solution 



x=(89x 1 2 -88a: 1 a: 2 - 

 u\ 



t7=! 



Hence we derive the following solutions, on integralising, 



