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XVI. — On Impossible EquatioTis. By Professor Finlay. 



Bead February 4, 1851. 



The following paper is intended as a supplement to Mr. 

 Harley's paper on the same subject, read before this 

 Society about a month ago, in which the fundamental 

 principles of the theory were established, and some of the 

 simplest cases of irrational equations were solved, in a very 

 elegant and direct manner. My paper is divided into 

 five paragraphs. The first contains the definition of the 

 new sense of the term " impossible," with some illustrations 

 relative to that definition. The second and third para- 

 graphs contain the discussion of an irrational equation con- 

 taining a single radical of any order. The third and fourth 

 paragraphs contain the discussion of an irrational equation 

 containing two or more radicals of any order. The object of 

 the discussion, in all cases, is to ascertain, a 'priori, the 

 number of impossible roots which the equation contains, 

 and to determine the possible roots exclusively of the im- 

 possible ones. Although the paper is extremely short, I 

 should hope that what it contains is sufficient to show the 

 method of separating the possible from the impossible 

 roots in any irrational equation. 



I. 



An impossible equation is one the roots of which are all 

 impossible. In this definition, the term root, as applied 

 to an equation, is used in the ordinary sense, and the 



