INDETERMINATE QUARTIC EQUATIONS 71 



=(^-6)(c?+3&)(a 3 -c 3 ) 3 X 2 +3(c 2 -a 2 )(a 3 -c 3 ) 



Hence omitting A and the factor (a c)(6 d) common to 

 XjT+a, x z r+b, x^r+c, x z r+d we have that if 



Xa 4 +n& 4 =Xc 4 +[^ 4 , (3) 



then also 



x[2a(a 2 +oc+c 2 )(a 3 -c 3 ) 2 X 2 -3(6+d!)(a 2 +ac+c 2 )(a 3 -c 8 ) 



= X[2c(a 2 +ac+c 2 )(a 3 -c 3 ) 2 X 2 -3(6+^)(a 2 +ac+c 2 )(a 3 -c 3 )(d 3 -6 3 ) 



that is to say, if one solution of equation (1) be known, another 

 can be found. This second solution will not be algebraical, 

 nor if we attempt to satisfy (3) by putting c 2 =a 2 , d! 2 =6 2 will it 

 be anything but nugatory. If, however, any solution of (3) 

 other than c 2 =a 2 , d 2 =b 2 be used, a second solution, in general 

 distinct from the first, will be found and so on ad infinitum. 

 Thus, for example, to solve the equation 



P 1 4 +3P 2 4 =P' 1 4 +3P' 2 4 , 



starting from the particular case 2 4 +3-3 4 =4 4 +3-l 4 , we derive 

 the following : 



