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LXI. On an Improvement in the Analysis of Equations. By 

 J. R. Young, Professor of Mathematics in Belfast College"^. 



IN the analysis of numerical equations our chief difficulties 

 are with those of an even degree, into which equations of 

 an odd degree may always be changed. The following brief 

 sketch of an improved method of discussing such equations 

 will, I think, be acceptable to the readers of the Philosophical 

 Magazine, although 1 reserve minute details, as well as some 

 important extensions, for a future communication. 

 Let 



a;2«+flA-2«-i + i^«-2 + c^2«-3^,,^^_,_;5._0 . . (A) 



be any equation of an even degree with numerical coefficients j 

 then the left-hand member may be decomposed into the fol- 

 lowing pair of conjugate factors, namely, 



and 



«+ia^«-' + ,Y/^Q«^-M^'"-'-f^^'''-^-....-^T» 



and, consequently, if these be separately equated to zero, and 

 either of the roots of the proposed equation substituted for x, 

 one of these two new equations will always be satisfied. It 

 follows, therefore, that when a real root is substituted, the 

 expression under the radical, namely, 



Qa2_^,V,2«-2_c^2n-3_...._yt, . , . . (B) 



must always be positive ; otherwise a real expression, viz. that 

 which precedes the radical, would neutralize an imaginary 

 one. 



Hence, if we perform the partial analysis of the equation 

 (A) by the method of Budan, commonly called the method 

 of Fourier, we may discuss the doubtful intervals, which that 

 analysis leaves for further examination, by an appeal to the 

 inferior polynomial (B). Applying the before- mentioned 

 analysis to this, in reference to the doubtful intervals, we 

 should infer that whenever, throughout any such interval, (B) 

 was plus, the sought roots may be real ; and that when, 

 throughout, the sign was minus, they must be imaginary. 

 When the sign of (B) passed from plus to minus, or from 

 minus to plus, in the interval, the doubt would remain; and 



* Communicated by the Author. 



