INDETERMINATE QUARTIC EQUATIONS 59 



PART II 



FOB the purposes of Diophantine Analysis, biquadratic 

 equations may be divided into two classes according as they 

 do or do not admit of representation, by a perfectly general 

 transformation of their variables, as indeterminate cubics. 

 Thus, for example, the equation 



by the perfectly general transformation P-^x+y, P z =uv, 

 P\=xy, P' z =u+v, becomes 



i.e. x 3 y+xy 3 =v?v+uv 3 , 



which is an indeterminate cubic in x and u ; while, on the other 

 hand, the equation 



does not seem capable, by a perfectly general transformation 

 of its variables, of being represented as an indeterminate cubic 

 in any number of variables. From what has been already 

 shown, 1 it is clear that the former class of equations admits of 

 an algebraic solution (at least when the number of variables 

 exceeds 2) by a process universally applicable, and it is to this 

 class the present paper is confined, though the methods of 

 solution will not be restricted to that already given. The 

 biquadratic equations of the second class require special 

 artifices for their solution and a separate paper will be devoted 

 to them. 2 



1 In the writer's paper, Part I. 2 Viz., Part III. 



