[Use Ex. 1.]

Ex. 4. The pedal of r m — a m cos mS is obtained by changing m into

m/{m + Y).
Ex. 5. The pedal of a conic with respect to its centre is a unicursal
quartic.
Ex. 6. The line joining to a focus of a curve is bisected by a singular
focus of the pedal.
The ordinary foci of the pedal are the feet of the perpendiculars from
on the lines joining the intersections of the curve with the circular
lines through 0.
Ex. 7. Any tangent to the curve of Ex. 1 is

x cos a + y sin a> =/(<»).
The normal is

— xsinu> + y cos a =/' (<u),

and the corresponding normal to the evolute is

X cos a + y sin a = — /" (<a).
The coordinates of the point of contact of the tangent and of the centre
of curvature are

(cos o) ./ — sin <i> ./', sin o> ./ + cos to ./')
and ( - sin a ./' — cos a ./", cos to ./' — sin a ./").
The radius of curvature is /(<») +/" (a).
Ex. 8. Tangents are drawn in any direction to a given curve. Show
that the algebraic sum of the radii of curvature at the points of contact
is zero. Show also that the centroid of the points of eontact coincides
with the centroid of the corresponding centres of curvature, and is
independent of the direction of the tangents.
