An introduction to the modern theory of equations, by Florian Cajori.

28

THEOItY OF EQUATIONS

coordinates. rTo determine the points on this circle for which
P and Q vanish, we must solve the equations g(t) 0 and
h(t) = O, for the given value of r. But we know by ~ 7 that if
h (t) =0 and g (t) 0 have roots at all, they cannot have more
than 2n. From this it follows that neither P nor Q can be
equal to zero at all points of an area in the plane, for in that
event we could select r such that the circle would pass through
that area, and P and Q would vanish at an infinite number of
points on this circle.
The value of Q may be written,
Q = I(sin no + a sin (n-1) +2 sin( 2)
From this expression it is readily seen that r may be taken
so large that Q has the- same sign as sin nAf on all points of the
circle where sin nsS is numerically larger than some value c,
which may be as small as we please, but not zero. Mark on
the circle the points
0, 7T 27T (2n —1)7T
0, _, -, -, __ -
and designate them, respectively, by 0, 1, 2, **., 2 n-1. Thus,
the circle is divided into 2 n arcs, (01), (12), (23),..., (2 n-1, 0),
in which sin ni f is alternately
+ and -. The figure shows
the division for n=5. In
\12 1, passing from are (01) to arc
sa.~ 12~ A ~ ((12), the function Q, for sufficiently large values of r,
5 11 P 10 O changes from + to-. Since
5 '/ Ir~ ~ by ~ 25, Q is a continuous
function having real values,
in going along the circle from
+ to -, it must at the point
~7 t8~ h1 pass through zero. Similarly, Q must pass through