AND OTHER SURD EQUATIONS. 219 



but (6) was immediately derived from (3) by writing for n 

 its numerical value, and yet if in (4) for n we substitute its 

 value, and obliterate all even powers of unity, we get 



which does not agree with (8.) The fact is, the expression 

 following the negative sign in the right-hand member of 

 this equation being necessarily positive, the equation itself 

 is a clear violation of our symbolical conventions, and is 

 therefore inadmissible. This sufficiently explains the 

 reason why certain expressions in terms of «, seem to satisfy 

 (and algebraically do strictly satisfy) certain irrational equa- 

 tions, which are nevertheless impossible. Thus, the equation 

 (a) is easily shown to be impossible (see Art. 3, cor. 3), 

 and yet it is strictly satisfied by a: rz n^. Further discussion 

 of this part of the subject I leave until I come to speak 

 of impossible eapressiont, with which, as will be seen, it is' 

 closely and intimately connected. 



It has been demonstrated that the root of (/Sj) in terms 

 of n, may be at once deduced from that of (aj) by writing 

 flj n'^, bi Ji^, for tti hi respectively. We thus get 



which verifies the solution (9). The invariable possi- 

 bility of equation (/3i), and all the other conclusions which 

 have been established with regard to that equation and its 

 congener (a^), are immediately deducible from the above 

 solution. 



Of course (9), like equation (1), from which it is derived, 

 may be exhibited in a variety of forms ; these, however, it 

 is needless to develop. 



Adapting the foregoing results to the original equations, 

 we b *»ve the following conclusions : — 



