352 Mr Grace, On the Zeros of a Polynomial. 



The Zeros of a Polynomial. By J. H. Grace, M.A., Peter- 

 house. 



[Received 11 November 1901.] 



(1) Between two real zeros of a polynomial with real coeffi- 

 cients there is at least one real zero of the derived function. 

 I regard this theorem as giving a limitation for the roots of 

 f'(z)=0 when two roots of f(z) = are given, and I propose to 

 consider the more general question, viz., When two roots real or 

 imaginary of the equation f(z) = whose coefficients are possibly 

 imaginary are given, do any corresponding limitations exist for 

 the roots of/'(V) = 0? It will be found that if A] B represent 

 the given roots in the Argand diagram, then there is at least one 

 root of the equation f'{z) =0 within a circle whose centre is the 



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middle point of A B and whose radius is \ AB cot - , where n is the 

 degree of the polynomial f{z). 



(2) If all the roots qff(z) = lie inside a given oval carve 

 in the Argand diagram, then so also do all the roots of f (z) = 0. 



This well-known theorem can be easily established by ele- 

 mentary mechanical considerations. In fact if the roots of 

 f(z) = be the positions of equal centres of force attracting 

 according to the law of the inverse distance, then the roots of 

 f (z) = represent the equilibrium points. Now if all the poles 

 are on one side of a straight line, all the equilibrium points are 

 on that side, because for points on the line or on the opposite side 

 of the line the components of force perpendicular to the line are 

 all in the same direction. Hence allowing the line to envelope 

 an oval curve enclosing all the centres of force, we see that all the 

 equilibrium points are within the oval. The same reasoning 

 applies to any convex polygon enclosing the centres of force, and 

 it applies even in the extreme case in which all the centres lie on 

 the boundary, except that if they all lie on the same straight line 

 then all the equilibrium points lie on that line. 



(3) Suppose now that the oval enclosing the roots of f (z) = 

 is a circle, and consider the effect of inverting with respect to an 

 external point 0. Bearing in mind that this is equivalent to a 

 homographic transformation of z such that the point at infinity 

 becomes and also that the expression f'{z) is the first polar of 

 the point at infinity, we infer from the theorem of (2) that if all 



