AND OTHER 8UBD EQUATIONS. 221 



Again, let the equation 



« + Vai + 2 rr 

 be proposed. This agrees in form with (a), and since the 



secondlerm under the radical is positive^ the equation is 

 possible, having one root, viz., 



the root of its congener is 



i4-V| = 2. 

 It is worthy of remark here, that when (a) or (/S) is solved 



by the common process, the positive root belongs to (j8j), 

 and the negative one to (a,) ; and that, when both roots 

 are positive, ^aj) is impossible and (j8i) has two roots. The 

 reason of this is at once obvious, from the following simple 

 considerations — 



Since, by (a), a; rr — */2a a; -|- ^> 



and, by (j8), x zz V^a x -{- b, 

 11 11 1 



Xj in the first instance must be negative, and in the second 

 positive, otherwise we should have to subject it to incom- 

 patible conditions. 



8. Tlie method of solution explained and illustrated in 

 the last article, is evidently capable of application to any 

 class of surd equations whatever, provided only that such 

 equations, when rationalized by the common method, are 

 capable of algebraical resolution. 



Whether we employ n or n" (p being any odd integral 

 number, positive or negative,) for -1, we have seen that the 

 results are virtually identical, and that nothing is gained 

 as to generality by the use of one symbol rather than the 

 other ; while the employment of n has the advantage over 

 that of n" in point of simplicity and convenience. In 

 practice, therefore, it will be found better ta employ n ex- 

 clusively' 



