ed by the arc, so that for 30 degrees it is reduced to 1/16 of
that amount, that is, to 1/14400. Conversely, let O B be a tangent of
given length; make O F=1/4 O B; then with centre F and radius F B
describe an arc cutting the circle O D G (tangent to O B at O) in the
point D; then O D will be approximately equal to O B, the error being
the same as in the other construction and following the same law.

[Illustration: Fig. 253.]

The extreme simplicity of these two constructions and the facility with
which they may be made with ordinary drawing instruments make them
exceedingly convenient, and they should be more widely known than they
are. Their application to the present problem is shown in Figure 253,
which represents a quadrant of an ellipse, the approximate arcs C D, E,
E F, F A having been determined by trial and error. In order to space
this off, for the positions of the teeth, a tangent is drawn at D, upon
which is constructed the rectification of D C, which is D G, and also
that of D E in the opposite direction, that is, D H, by the process just
explained. Then, drawing the tangent at F, we set off in the same manner
F I = F E, and F K = F A, and then measuring H L = I K, we have finally
G L, equal to the whole quadrant of the ellipse.

[Illustration: Fig. 254.]

Let it now be required to lay out twenty-four teeth upon this ellipse;
that is, six in each quadrant; and for symmetry's sake we will suppose
that the centre of one tooth is to be at A, and that of another at C,
Figure 253. We, therefore, divide L G into six equal parts at the points
1, 2, 3, etc., which will be the centres of the teeth upon the rectified
ellipse. It is practically necessary to make the spaces a little greater
than the teeth; but if the greatest attainable exactness in the
operation of the wheels is aimed at, it is important to observe that
backlash, in elliptical gearing, has an effect quite different from that
resulting in the case of circular wheels. When the pitch-curves are
circles, they are always in contact; and we may, if we choose, make the
tooth only half the breadth of the space, so long as its outline is
correct. When the motion of the driver is reversed, the follower will
stand still until the backlash is taken up, when the motion will go on
with a perfectly constant velocity ratio as before. But in the case of
two elliptical wheels, if the follower stand still while the driver
moves, which must happen when the motion is