PKOFESSOB FINLAY ON IMPOSSIBLE EQUATIONS. 237 



whole question turns on the new sense in which it has 

 been proposed to use the term impossible. 

 To illustrate this point, let the equation 

 X -{- V(4a; -f- 1) =: 5 

 be proposed. Clearing this equation of surds, and solving 

 it by the ordinary process, the roots are found to be 2 and 

 12. Now, if 2 be substituted for x in the proposed equation, 

 we obtain 



2 + V9 = 5, 

 which is obviously true; but if 12 be substituted for «, we 

 get 



12 + V49 rr 5, 



which is evidently false, provided that the radical be 

 restricted to a positive signification, or to its arithmetical 

 value. On these grounds, it has been proposed to call 2 

 a possible root of the proposed equation, while 12 has been 

 designated as an impossible root. Thus we see, that an 

 impossible root of an irrational equation is one which does 

 not satisfy the equation when the radical which it involves is 

 restricted to a positive signification. 



In extending the theory to imaginary roots, a difficulty 

 occurs as to the positive signification of an expression of the 

 form V (« — b V — 1). For if 



(a _ /3 V— 1? z=: a — b V— 1, 

 we shall also have 



(— a-j./3V— 1)^ = « — *>/— 1; 



so that V ip, — b V — 1) may be equal either to -|- a — j3 V — 1> 

 or to — a-f-i^V — Ij and it is not immediately evident 

 which of these is to be taken as its positive signification. 

 Now, in extending any algebraical rule to a case not 

 originally contemplated, the extension must invariably be 

 so framed that the new rule may include the original one 

 as a particular case. According to this principle we must 

 evidently take +a — /3 V — 1 as the positive signification of 



