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) = 0 are given, and I propose to 
consider the more general question, viz., When two roots real or 
imaginary of the equation f{z) — 0 whose coefficients are possibly 
imaginary are given, do any corresponding limitations exist for 
the roots of f'\z) = 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 /' (, z ) = 0 within a circle whose centre is the 
middle point of AB and whose radius is ^AB cot - , where n is the 
degree of the polynomial f(z). 
(2) If all the roots of f(z) = 0 lie inside a given oval curve 
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) = 0 be the positions of equal centres of force attracting 
according to the law of the inverse distance, then the roots of 
f (z)— 0 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 potygon 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) = 0 
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 £ such that the point at infinity 
becomes 0 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 
