222 MR. R. HARLEV ON IMPOSSIBLE 



From what has been already done, it will have appeared 

 evident that the chief value of w, as an element of operation, 

 is this — that it enables us to discover certain expressions 

 for Xy which, in every circumstance, seemingly satisfy the 

 proposed equation. The consideration of these expressions 

 will readily enable us further to determine the relation 

 that must subsist among the several co-efficients of ar, in the 

 given equation, in order to that equation being possible. 

 This latter object, however, may be more easily effected, 

 as we have seen, by simpler means. 



I have spoken of the roots of x, in terms of n, as satis- 

 fying the proposed irrational equation only in appearance. 

 I proceed to explain my meaning. 



In the last article it was shown, that when b is negative, 

 the equation 



X 4- V2a x-\-h =z 

 1 11 I 



is impossible ; that is, it has no root whatever. And yet we 

 have also shown, that the expression 



a n^ -\-n s/c^ n^ -f- 6, 

 111 



being substituted for x, seems to satisfy the equation in 



1 



every circumstance. How are these conclusions to be har- 

 fhonized ? If we recur to the verification of the solution 

 (1), we shall be furnished with a satisfactory explanation 

 of this difficulty. We there find that the substitution of 

 the above root in the expression »^2a a; -}- 6, gives 



V2a X -\- b ^H a n •\' Va* W + b. 

 Ill 1 1 I 



Now it has been before remarked, that when b is negative, 



the right-hand member of this equation is negative also ; 

 but the left-hand member is always positive; hence, when 

 b is negative, the above expresses an impossible relation; 



viz., the equality of two quantities, one of which is positive, 

 and the other negative. The root, therefore, which.p sents 



