-i^* On Criteria for determiriing Numerical Equations. 



V. Let Q be prime to 3 R. 



Then putting /^+ 3^^ = Q^, the complete solution of the 

 equation immediately preceding is contained in the two sy- 

 stems. 



1st. D = 2/ 3R = 2^. 

 2nd. D = (/±3^) 3R=/+-, 

 and for both systems, 



f±g ^^ = {h±S k \/'^)3. 

 The second system must therefore be rejected, for g evi- 

 dently contains 3, and therefore /= 3 R+^ will contain 3, and 

 therefore D and therefore Q will do the same, contrary to 

 supposition. 



Hence 



.Vr , d / 1 



= \/f+/v/' 



and the three roots of the equation being 



"(A + fA V ^) + (X — /x \/~^6) 



will evidently be all rational, which of course includes (he ne- 

 cessity of their being also integer. 



Again 2°, if we suppose that Q does contain 3, D'^ will con- 

 tain 27, and consequently D will contain y ; and we shall have 



_ 1_ 



"S" --^ V "~ 27' 



--•(ly--(f)- 



Here R being prime to — , it may be shown, as in the last 

 case, that the complete solution is 



consequently 



