242 PBOFESSOB FINLAY ON IMPOSSIBLE EQUATIONS. 



which will not satisfy the proposed equation, unless the 

 first radical be taken with the positive, and the second with 

 the negative sign. 



V. 



The theory which has just been given for the case of two 

 radicals of the second order, may now be readily extended 

 to the case of any number of radicals of any order. For, 

 if t, u, V, &c., be assumed equal to the several radicals, we 

 shall obtain, in all cases, a series of equations free from 

 radicals; and the number of these equations being always 

 equal to the number of the unknown quantities x, t, u, v, 

 &c., the values of t, u, v, &c., may always be found by the 

 ordinary methods of elimination, at least when the pro- 

 posed equation is numerical. Let ti, m„ v,, &c., be any 

 system of simultaneous values of t, u, v, &c., then it is 

 evident that if <i, u^ Vij &c., be all positive, the correspond- 

 ing value, or values, of a: will be possible ; but if any one 

 of these quantities be negative, the corresponding values of 

 X will be impossible. 



