adius
of the arc A F, the flanks of the teeth in that region will be radial.
We have, therefore, chosen a circle whose diameter, A B, is
three-fourths of A P, as shown, so that the teeth, even at the ends of
the wheels, will be broader at the base than on the pitch line. This
circle ought strictly to roll upon the true elliptical curve; and
assuming, as usual, the tracing-point upon the circumference, the
generated curves would vary slightly from true epicycloids, and no two
of those used in the same quadrant of the ellipse would be exactly
alike. Were it possible to divide the ellipse accurately, there would be
no difficulty in laying out these curves; but having substituted the
circular arcs, we must now roll the generating circle upon these as
bases, thus forming true epicycloidal teeth, of which those lying upon
the same approximating arc will be exactly alike. Should the junction of
two of these arcs fall within the breadth of a tooth, as at D, evidently
both the face and the flank on one side of that tooth will be different
from those on the other side; should the junction coincide with the edge
of a tooth, which is very nearly the case at F, then the face on that
side will be the epicycloid belonging to one of the arcs, its flank a
hypocycloid belonging to the other; and it is possible that either the
face or the flank on one side should be generated by the rolling of the
describing circle partly on one arc, partly on the one adjacent, which,
upon a large scale, and where the best results are aimed at, may make a
sensible change in the form of the curve.

The convenience of the constructions given in Figure 252 is nowhere
more apparent than in the drawing of the epicycloids, when, as in the
case in hand the base and generating circles may be of incommensurable
diameters; for which reason we have, in Figure 254, shown its
application in connection with the most rapid and accurate mode yet
known of describing those curves. Let C be the centre of the base
circle; B, that of the rolling one; A, the point of contact. Divide the
semi-circumference of B into six equal parts at 1, 2, 3, etc.; draw the
common tangent at A, upon which rectify the arc A 2 by process No. 1;
then by process No. 2 set out an equal arc A 2 on the base circle, and
stepping it off three times to the right and left, bisect these spaces,
thus making subdivisions on the base circle equal in length to those on
the rolling one. Take in succession as r