16 On the Theory of 



G D, (equidistant each to each from the point 1), where the circles 

 forming the circular rests intersect the circle circumscribing the 

 points of the teeth of the wheel, the pendulum is led a quantity 

 equal to each of those angles at each vibration. For, supposing 

 the wheel advancing, at the moment the tooth 2 has reached the 

 extremity of the inclined plane of the pallet A A, the point S of the 

 pallet will have reached the point H, and the point T of the pallet 

 BB will have reached the point G, ready to receive the tooth 1, 

 which at that instant will drop upon it ; and when, by the action 

 of the tooth 1 upon the pallet BB, it is returned to its former 

 place, the point F of the pallet AA will be ready to receive the 

 tooth 3, which will at that instant drop upon it. 



To render the above demonstration as apparent as possible, the 

 wheel has been drawn with only six teeth, because, supposing a 

 wheel even of the size of the wheel in the figure, with thirty teeth, 

 (the number for a second pendulum), in which case the pallet 

 would be only one-fifth of their present thickness, the portion of the 

 circle the chord would subtend would be so short, and conse- 

 quently the space between the chord prolonged and the tangent so 

 small, that the distance between the points I and W would be 

 much less apparent. In the case of a scape wheel of the size 

 usually employed in seconds pendulum clocks, in laying down the 

 lines for the purpose of determining the distance between the two 

 centers, it will be sufficient, unless when very great accuracy is 

 required, to draw two lines, bisecting the points which mark the 

 thickness of the pallets upon the circle of the wheel, and upon 

 those lines at the points where they intersect the circle of the wheel, 

 to raise two perpendiculars and to take the point where these per- 

 pendiculars meet, as the center of motion of the pallets. 



Were the center of motion of the pallets placed higher or lower 

 than the proper place, as above determined ; See Fig. 2 and 3, 

 Plate III, in both which the angle of lead is drawn the same as in 

 Fig. 1. In the one case, the action of the tooth of the wheel upon 

 the inclined plane of the pallet AA, See Fig. 2, (here the center of 

 motion of the pallets is raised), would lead the pendulum an angle 

 less than the angle of lead O I Q, as drawn, by a quantity equal to 



