HOW TO FIND THE ROLLER DIAMETER FROM THE LENGTH OF THE FORK.
To illustrate, let us imagine the small circles _A B_, Fig. 54, to
represent the center of a pallet staff and balance staff in the order
named. We divide this space into four equal parts, as shown, and the
third space will represent the point at which the pitch circles of the
fork and roller will intersect, as shown by the arc _a_ and circle _b_.
Now if the length of the radii of these circles stand to each other as
three to one, and the fork vibrates through an arc of ten degrees, the
jewel pin engaging such fork must remain in contact with said fork for
thirty degrees of angular motion of the balance.
[Illustration: Fig. 54]
Or, in other words, the ratio of angular motion of two _mobiles_ acting
on each must be in the same ratio as the length of their radii at the
point of contact. If we desire to give the jewel pin, or, in ordinary
horological phraseology, have a greater arc of roller action, we would
extend the length of fork (say) to the point _c_, which would be
one-fifth of the space between _A_ and _B_, and the ratio of fork to
roller action would be four to one, and ten degrees of fork action would
give forty degrees of angular motion to the roller--and such escapements
have been constructed.
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