14 On the Theory of 



three teeth * ; circumscribe the points of the teeth of the wheel by 

 a supposed circle Y E Z, divide each of the spaces D E and E F 

 between the teeth DE and F into two equal parts at G and H, 

 draw the straight lines D G and F H and prolong them until they 

 meet at I ; the point I will be the proper center of motion for the 

 pallets ; from the center I draw the two portions of circles K L and 

 M N, intersecting the circumscribing circle at the points DG and 

 H F, the circular rests of the pallets will be a portion of these cir- 

 cles ; the inner rest of the smaller, the outer rest of the larger 

 circle. To determine the angle of lead of the pallets, prolong the 

 two lines IHF and IGD to O and P, and from the point I draw 

 the two straight lines I Q and I R, forming the two angles O I Q and 

 RIP equal to one another, and, from the points F and S of inter- 

 section of the sides of the angles by the portions of circles K L and 

 M N, in the one angle, and the points D and T in the other ; 

 draw the straight lines F S and D T these will be the inclined 

 planes or faces of the pallets. 



As this mode of proceeding would be very difficult, not to say 

 impossible in practice, except in the case of large wheels, on ac- 

 count of the little distance, the points that determine the thickness 

 of each pallet are from each other ; the preferable mode is, after 

 having determined the place of the pallets upon the circumference 

 of the wheel as above described, to draw from the center X of the 

 wheel, Fig. 1, Plate III, the two lines X U and X U, bisecting the 

 arcsFH and DG which mark the thickness of the pallets; and 

 from the points V and V where those lines intersect the circle, to 

 raise the two perpendiculars VW and VW, which lines will be 

 tangents to the circle ; and that done to draw the lines F II I and 



* It may be worth while to notice, as a general rule, that a pair of pallets 

 always occupy the space or portion of the circle contained between the num- 

 ber of the teeth they take over anil one more, thus taking over two teeth they 

 require the space contained between three teeth ; were they to take over teu 

 teeth, they would occupy the space contained between eleven. The reason 

 is evident ; in the one case the thickness of the pallets is without the teeth 

 and the other within, and the thickness of each pallet being equal to half a 

 space, the thickness of the two together must be equal to one entire space; 

 no allowance, as before observed, being made for dross. 



