8 On the Theory of 



1331. " The escapement thus drawn will be a dead escapement, 

 because it is formed of portions of circles concentric to the point 

 A (396) but," #c. 



To illustrate as much of the above as refers to the irregularity of the friction 

 upon the pallets, suppose Fig. I. Plate If. the pendulum at rest, and per- 

 pendicular upon the line X W, bisecting the angle YXZ, (which angle is 

 equal to the angles B X P and DXQof lead of the pallets,) and the lines 

 A B and C D the inclined planes of the pallets ; the dotted lines E E and G H 

 will represent the lines of the inclined planes, when the pendulum subtending 

 the line X Z is led to the extremity of the lead one way ; and the dotted lines 

 I K and L M the inclined planes when the pendulum subtending the line X Y, is 

 at the extremity of the lead in the other dissection, (it cannot be too often 

 repeated that the angle of lead must not be compounded with the angle of 

 vibration,) and the points N and O, where a supposed circle circumscribing 

 the points of the teeth of the wheel intersects the inclined planes A B and 

 C D, are the points upon one of which (determined by the pallet from which 

 the wheel has last escaped) the wheel will be in contact with the pallet, when 

 the pendulum having advanced half the angle of lead subtends the perpen- 

 dicular line X W. Now it is visible that the portions N A and O D of the lines 

 A B and C D, which are the parts of those lines the teeth of the wheel act 

 upon when the pendulum ascends, are greater than the lines N B and O C, 

 which are the portions the teeth of the wheel act upon when the pendulum 

 descends ; consequently, the velocity with which the wheel advances is not 

 equal during equal portions of the lead of both pallets. 



By altering the shape of the inclined planes of the pallets from straight 

 lines to portions of circles, the advance of the wheel may be made nearly pro- 

 portional to the advance of the pendulum. Suppose, Fig. 1. Plate II., the 

 pendulum to subtend the perpendicular line XW, and consequently to have 

 vibrated half the angle it is led, and the inclined plane of the pallet IE, in- 

 stead of being the straight line D C, to be a portion D T C of a circle, passing 

 through the three points D, S, (where the radius It V bisects the arc M G of a 

 supposed circle circumscribing the wheel,) and C ; the consequence resulting 

 from giving this shape to the pallets will be, that the wheel will have advanced 

 to the point S half its total advance, and have acted upon very nearly half the 

 surface of the pallet, when the pendulum has vibrated half the angle it is led ; 

 for the portion C S of the circular face of the pallet upon which the wheel 

 has acted during its advance from the point G to the point S, is less than the. 

 portion S D of the pallet upon which it must act during its advance from S to 

 D, by the quantity S T ; which difference between the arcs C S and S 1) is very 

 triflin" when compared with the difference between the straight lines C O and 

 O V, which form the inclined plane C D. The 



