adii the chords A 1, A 2, A 3,
etc., of the describing circle, and with centres 1, 2, 3, etc., on the
base circle, strike arcs either externally or internally, as shown
respectively on the right and left; the curve tangent to the external
arcs is the epicycloid, that tangent to the internal ones the
hypocycloid, forming the face and flank of a tooth for the base circle.

[Illustration: Fig. 255.]

In the diagram, Figure 253, we have shown a part of an ellipse whose
length is ten inches, and breadth six, the figure being half size. In
order to give an idea of the actual appearance of the combination when
complete, we show in Figure 255 the pair in gear, on a scale of three
inches to the foot. The excessive eccentricity was selected merely for
the purpose of illustration. Figure 255 will serve also to call
attention to another serious circumstance, which is, that although the
ellipses are alike, the wheels are not; nor can they be made so if there
be an even number of teeth, for the obvious reason that a tooth upon one
wheel must fit into a space on the other; and since in the first wheel,
Figure 255, we chose to place a tooth at the extremity of each axis, we
must in the second one place there a space instead; because at one time
the major axes must coincide; at another, the minor axes, as in Figure
255. If, then, we use even numbers, the distribution, and even the forms
of the teeth, are not the same in the two wheels of the pair. But this
complication may be avoided by using an odd number of teeth, since,
placing a tooth at one extremity of the major axes, a space will come at
the other.

It is not, however, always necessary to cut teeth all round these
wheels, as will be seen by an examination of Figure 256, C and D being
the fixed centres of the two ellipses in contact at P. Now P must be on
the line C D, whence, considering the free foci, we see that P B is
equal to P C, and P A to P D; and the common tangent at P makes equal
angles with C P and P A, as is also with P B and P D; therefore, C D
being a straight line, A B is also a straight line and equal to C D. If
then the wheels be overhung, that is, fixed on the ends of the shafts
outside the bearings, leaving the outer faces free, the moving foci may
be connected by a rigid link A B, as shown.

[Illustration: Fig. 256.]

This link will then communicate the same motion that would result from
the use of the complete elliptical wheels, and we may therefore
dis