. 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.

For all practical purposes it will make no difference whether such
parallelism takes place after eight or nine degrees of angular motion of
the escape wheel subsequent to the locking action. The great point, as
far as practical results go, is to determine if it takes place at or
near the time the escape wheel meets the greatest resistance from the
hairspring. We find by analysis of our drawing that parallelism takes
place about the time when the tooth has three degrees of angular motion
to make, and the pallet lacks about two degrees of angular movement for
the tooth to escape. It is thus evident that the relations, as shown in
our drawing, are in favor of the train or mainspring power over
hairspring resistance as three is to two, while the average is only as
eleven to ten; that is, the escape wheel in its entire effort passes
through eleven degrees of angular motion, while the pallets and fork
move through ten degrees. The student will thus see we have arranged to
give the train-power an advantage where it is most needed to overcome
the opposing influence of the hairspring.

[Illustration: Fig. 92]

As regards the exalted adhesion of the parallel surfaces, we fancy there
is more harm feared than really exists, because, to take the worst view
of the situation, such parallelism only exists for the briefest
duration, in a practical sense, because theoretically these surfaces
never slide on each other as parallel planes. Mathematically
considered, the theoretical plane represented by the impulse face of
the tooth approaches parallelism with the plane represented by the
impulse face of the pallet, arrives at parallelism and instantly passes
away from such parallelism.

TO DRAW A PALLET IN ANY POSITION.

As delineated in Fig. 92, the impulse planes of the tooth and pallet are
in contact; but we have it in our power to delineate the pallet at any
point we choose between the arcs _p s_. To describe and illustrate the
above remark, we say the lines _B e_ an