240 PROFESSOR FINLAT ON IMPOSSIBLE EQUATIONS. 



III. 



The preceding method may be generalized with the ut- 

 most facility. Let us consider, for instance, the equation 



X-{-WX'z=zO (1), 



where X and X' denote any rational and entire algebraic 

 functions of x. If we assume 



WX'=i/ (2), 



and .-. J^' =: y" (20, 



equation (1) becomes 



X-\-2/ = (10. 



Eliminating x between (!') and (2'), we obtain an equation 

 of the form 



9(y) = o (3), 



where f denotes a rational and entire algebraic function 

 of the quantity ?/, to which it is applied. Now, if the radical 

 in equation (1) be restricted to a positive signification, it is 

 evident from (2) that y must be positive; and therefore the 

 negative roots of (3) must be rejected, as giving impossible 

 values of a;. Consequently, if the number of negative 

 roots in equation (3) be found by means of the theorem of 

 Sturm, the number of impossible roots of (1) may thence 

 be readily ascertained. Thus, if p and q denote the de- 

 grees of the functions X and X' respectively, it is evident 

 from (2') that every negative root in equation (3) will give 

 q impossible roots for equation (1). 



IV. 



The same method may be applied to equations contain- 

 ing any number of radicals. For the sake of clearness, 

 let us first consider the particular equation 



mjs/(ax -^-b) ■-\- nA/(cx -^ d) zzf (1> 



If we assume 



\/(ax -\-b)zzi/, 's/Cpx -}-d) zzz, (2), 



and.', ax -\- b zz i/^, ex -\- d zz i:^ (20, 



equation (1) becomes 



