520
ON POLYZOMAL CURVES.
[414
viz., of a circle having its centre on the major axis at a distance = ae sin 6 from the
centre, and its radius = b cos 6. (I notice, in passing, that this gives in practice a
very convenient graphical construction of the ellipse.) It may be remarked that for
6 = ± sin -1 e, the circle becomes
viz., this is the circle of curvature at one or other of the extremities of the major
axis ; as 6 passes from 0 to + sin _1 e we have a series of real circles, which, by their
continued intersection, generate the ellipse; as 6 increases from 6 = ± sin -1 e to i 90°,
the circles continue real, but the consecutive circles no longer intersect in any real
point,—and ultimately for 6=± 90°, the circles become evanescent at the two foci
respectively.
121. In the case q> 1, we have a real representation of
(x — qae) 2 + y 2 + b 2 (q 2 - 1),
as the squared distance of the point (x, y) from a point (X, 0, Z) out of the plane
of the figure, viz., putting this = (x — X) 2 -f y 2 + Z 2 ,
we have
qae — X, Z 2 = b 2 (q 2 — 1),
whence
or, what is the same thing,
CL~ — O“ ir
that is, the locus is the focal hyperbola, viz., a hyperbola in the plane of zx, having
its vertices at the foci, and its foci at the vertices of the ellipse.
122. If instead of the form first considered, we start from the trizomal form
2bz + V« 2 + (y — aeiz) 2 + Va? + (y+ aeiz) 2 = 0,
then we have the zomal or circle of double contact under the form
a? + (y — qaei) 2 = a 2 (1 — q 2 ) ;
or putting herein q — — i tan </>, this is
a? + (y — ae tan <£) 2 = a 2 sec 2 </>;
so that we have the ellipse as the envelope of a variable circle having its centre
on the minor axis of the ellipse, distance from the centre = ae tan </>, and radius
= a sec tf>. This is, in fact, Gergonne’s theorem, according to which the ellipse is
the secondary caustic or orthogonal trajectory of rays issuing from a point and
refracted at a right line into a rarer medium. It is to be remarked that for
tan (f> = + y , the equation of the circle is
