To delineate our entrance pallet after one-half of the engaged tooth has passed the inner angle of the entrance pallet, we proceed, as in former illustrations, to establish the escape-wheel center at A, and from it sweep the arc b, to represent the pitch circle. We next sweep the short arcs p s, to represent the arcs through which the inner and outer angles of the entrance pallet move. Now, to comply with our statement as above, we must draw the tooth as if half of it has passed the arc s.
To do this we draw from A as a center the radial line A j, passing through the point s, said point s being located at the intersection of the arcs s and b. The tooth D is to be shown as if one half of it has passed the point s; and, consequently, if we lay off three degrees on each side of the point s and establish the points d m, we have located on the arc b the angular extent of the tooth to be drawn. To aid in our delineations we draw from the center A the radial lines A d’ and A m’, passing through the points d m. The arc a is next drawn as in former instructions and establishes the length of the addendum of the escape-wheel teeth, the outer angle of our escape-wheel tooth being located at the intersection of the arc a with the radial line A d’.
As shown in Fig. 92, the impulse planes of the tooth D and pallet C are in contact and, consequently, in parallel planes, as mentioned on page 91. It is not an easy matter to determine at exactly what degree of angular motion of the escape wheel such condition takes place; because to determine such relation mathematically requires a knowledge of higher mathematics, which would require more study than most practical men would care to bestow, especially as they would have but very little use for such knowledge except for this problem and a few others in dealing with epicycloidal curves for the teeth of wheels.