AND OTHER SURD EQUATIOXS. 209 



hence either '7f-\' v^^^ 0, 



1 3 



or, J^— -v^f = 0, for all roots of (2). 



1 s 



Cor. 2. In like manner it may be shown, that every 

 value oi a which will satisfy the equation 



will also satisfy either one or other of the equations 



v7+v7+v7=o, 



1 3 3 



v7+ v7— s/f— 0, 



1 2 3 



v7— v7-}- V7=: 0, 



1 % 3 



— 'v7+^7+^7— 0. 



1 3 3 



Cor. 3. If, when a surd equation is rationalized, and 

 all the roots of the resulting equation are obtained, none of 

 these roots are found to satisfy the proposed equation, that 

 equation has no root whatever. For if such an equation 

 had any root, that root would necessarily satisfy (and there- 

 fore be also a root of) the rational equation. 



4. Definition. — An equation which has no root what- 

 ever, is designated as impossible. This definition is pro- 

 posed by Mr. Cockle in the Mechanics' Magazine, Vol. 

 xlix. (p. 365), where it is clearly demonstrated that the 

 very supposition of the existence of such equations involves 

 an arithmetical contradiction. It is needless, therefore, to 

 argue the propriety of the definition, which we adopt with- 

 out any hesitation. 



5. For the purpose of clearly illustrating the preceding 

 principles, let us consider the particular equation 



4 -f Va; — 3 -{- Va; -f 21 iz 0. 



Multiplying hj (4 -{- a/x — 3 — V^; + 21) (1 + Va^ — 3> 

 in order to eliminate radicals, we get 



8 (x — 4) =: ; .-. a; — 4. 



2 E 



