216 MR R. HARLEY ON IMPOSSIBLE 



(3) or (3^) is identical (except in form) with (1) or (P), 

 it immediately follows that (1) or {V) embraces all possible 

 solutions of (a) ; and that, therefore, no loss of generality 



has been sustained by the exclusion of the negative sign 

 from before the introduced radical. Indeed, this is as might 

 have been expected from the logical and consistent charac- 

 ter of the operation: it accords with that exact compre- 

 hensiveness of result, which must ever attach to those 

 mathematical investigations which are conducted with 

 due regard to the entire data of the problem discussed. 



When the foregoing verifications involve the violation of 

 our symbolical conventions, it is clear that the roots indi- 

 cated by the formula (1) or (1^) and (3) or (3^) are rejec- 

 tive. But this can only be the case when the right-hand 

 members of the equations (2) or (2^) and (4) or (4^) are 

 negative (Art. 1). Now, still bearing in mind that the 

 symbol */ is to be interpreted positively, we readily dis- 

 cover, by mere inspection, that the right-hand member of 

 (2) or (2^) is negative only when b is so; but that the 



right-hand member of (3) or (3^) is always negative. We 

 hence conclude that (6) can never strictly satisfy (a), and 



that (5) does so only when h is positive. Combining these 



conclusions with what has been befjre demonstrated in this 

 article, it is immediately seen that (6) is always a root of 

 (/3,) and that (5) is so only when b is negative. 



These important conclusions may be deduced from other 

 and more simple considerations. Thus, resolving (a) or (/3) 



by the common process, we obtain the relations marked (5) 

 and (6.) Now from (5) we get 



4^2ax -4- i rr « c/) V^^ -f- b (7) 5 



111 1 1 1 



And from (6) we get 



^/2ax-\'b r: a + Va" + * (8) 



1111 1 I 



