212 MR. R. HARLEY ON IMPOSSIBLE 



ascertain the exact number of roots belonging to each. A 

 very little consideration will show, that for any surd equa- 

 tion of a given form to be impossible, a certain determinate 

 relation must obtain among the co-efficients of w; and to 

 discover that relation becomes at once an interesting and 

 important enquiry. To find also a method of solution, 

 equally applicable to all irrational equations, by which the 

 true roots (if any exist) may be exclusively evolved, is 

 plainly a very desirable object. These, then, are the two 

 main purposes of the present paper; how far they are 

 accomplished I shall not pretend to say. 



7. I shall not now attempt to give a general discussion 

 of this subject, but shall confine attention to certain surd 

 equations of a limited degree. To illustrate the method I 

 propose for the attainment of the objects specified in the 

 last article, let us consider the literal equations 



ax -\- \/bx -\- c ■=! d, (a), 



ax — i^bx -\- c zz d, (^). 



These are readily put under the more simple and conve- 

 nient forms 



X -\~ »/2a X 4- b z=: 0, (a), 



1 111 1 



X — ^2ax-\-b ZZ (/S), 



1111 I 



where x zz x — --, a ZZ -^, and b zz —(ac-\- bd). 



_, a Ii: — , and b zz - 



1 a 1 2a^ 1 a 



Equation (jS) may be written thus : — 



X -f V2^(— Ifx-^b (— 1)^ zz ; 

 11 11 



or, substituting n for — 1, 



X -\- »^2a ri? X -f- b n^ ZZL^\ 

 1 111 



which agrees in form with (a). It hence appears that if 



X zzf{a, b) be the solution of (a), the solution of (/3) will 

 111 1 . 1 



be X zzf{a «^ b w^). 

 1 1 I 



