to A'B', the stroke of the engine, and that again to AB, the throw of the
crank. The highest position of P will be that shown in the figure,
determined by placing the crank vertically, as OC. At that instant the
motions of C and C' are horizontal, and being inclined to CC' they must
be equal. In other words, the motion is one of translation, and the
radius of curvature at P is infinite.

To find the center of curvature at D, assume the crank pin A to have a
velocity A_a_. Then, since the rod is at that instant turning about the
farther end A', we will have D_d_ for the motion of D. The instantaneous
axis of the connecting rod is found by drawing perpendiculars to the
directions of the simultaneous motions of its two ends, and it therefore
falls at A', in the present position. But the perpendicular to the motion
of the crank pin is the line of the crank itself, and consequently is
revolving about O with an angular velocity represented by AO_a_. The
motion of A' is in the direction A'B', but its velocity at the instant is
zero. Hence, drawing a vertical line at A', limited by the prolongation
of _a_O, we have A'_a_' for the motion of the instantaneous axis.
Therefore, by drawing _a_'_d_, cutting the normal at _x_, we determine
D_x_, the radius of curvature.

Placing the crank in the opposite position OB, we find by a construction
precisely similar to the above, the radius of curvature E_z_ at the other
extremity of the axis of the curve. It will at once be seen that E_z_ is
less than D_x_, and that since the normal at P is vertical and infinite,
the evolute of DPE will consist of two branches _x_N, _z_M, to which the
vertical normal PL is a common asymptote. These two branches will not be
similar, nor is the curve itself symmetrical with respect to PL or to any
transverse line; all of which peculiarities characterize it as something
quite different from the ellipse.

[Illustration: FIG. 23.]

[Illustration: FIG. 24.]

[Illustration: FIG. 25.]

Moreover, in Fig. 22, the locus of the instantaneous axis of the trammel
bar (of which the part EH corresponds to the connecting rod, when a crank
OH is added to the elliptograph there discussed) was found to be a
circle. But in the present case this locus is very different. Beginning
at A', the instantaneous axis moves downward and to the right, as the
crank travels from A in the direction of the arrow, until it becomes
vertical, when the axis will be found upon C'R, at an in