207 



XV. — On Impossible and certain other Surd Equations. 

 By Robert Harley, Esq. 



Read January 7, 1851. 



1. The ordinary method of resolving a given surd equation 

 proceeds on the assumption, that the symbol of radicality 

 which enters into it may sustain indifferently either a posi- 

 tive or negative interpretation. It almost invariably happens, 

 however, that from the nature of the enquiry whence such 

 an equation originates, the sign of the radical is necessarily 

 restricted to a plus signification ; and that, therefore, every 

 value of " the unknown," which, with this limitation, will 

 not satisfy the given equation, is inadmissible as a root. 



2. Now it is frequently found, that when the symbol V 

 is thus restricted in its signification, all the roots obtained 

 by the ordinary method of solution, are rejective ; that in 

 fact they are foreign roots, belonging to one or more other 

 equations which, when cleared of radicals, produce the very 

 equation that results from the rationalization of the given 

 one. These foreign roots are introduced by the elimina- 

 tion of the symbol of radicality from the proposed equation ; ^ 

 for that elimination is effected either by multiplying by a 

 factor or factors involving " the unknown," or by an evi- 

 dently equivalent process. 



3. That the removal of radicals does not eliminate fi-om 

 the given equation any of its roots, is easily proved. For 

 let / / be any rational functions whatever of ss, connected 



1 2 



by the equation 



V7+ V7 = 0, (1); 



