INDETERMINATE CUBIC EQUATIONS 51 

 To make the coefficient of r vanish we may take 



and equation (4) is then satisfied by taking 



Substituting the value of x from (5) in (6) we derive 



_ 



If we now put A = 

 equations (3) take the form 



A - P= 



(7) 



fc 8 ) 8 -^i 8 +a a 8 +3 8 ) (8) 



A P 1 = 

 + 3 

 A - P= 



A P= 



(9) 



As equation (1) is homogeneous, (9) is the integralised 

 form of its algebraical solution and presents the roots as rational 

 functions of five variables x l9 x 2 , x 3 , X, y. ; of the third degree 

 in x l9 x%, # 3 , the ninth in x l9 x 2 , # 3 , [x, and the tenth in x i9 x 2 , 



As a numerical example, x l =x z -=\ = l, o: 3 =(A=2 gives, on 

 removal of the common factor, 3 3 +4 3 +5 3 =6 3 , the lowest 

 solution which exists. 



(ii) Let w=4, so that we have to solve 



(20 



r r W\ 

 I *4 r V / 



Here we may take as our particular solution 



Po^P^X, P 2 =-P 3 = ( ,, P 4 =0 

 Making then the substitutions 



equation (I 7 ) takes the form 

 ) 3 = (x r+ X) 3 + ( 



