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In Fig (I have developed curve)

314, I have developed curve 1, as a true epicycloidal curve, which practically explains itself. I have then taken circles approximating the shape of this curve and transferred them to Fig. 315, which shows the true epicycloidal curve attached to outline in the tooth numbered 1. Inspecting this shaped tooth, it will be found that in order to make a pointed tooth with an epicycloidal curve, the point extends beyond the arbitrary diameter selected for the diameter of the wheel, so that if the tooth should be made a true epicycloid, it would be flattened at the top. This would not do, however, in that it would be a bad form of tooth and quickly round on the top, destroying in a measure the advantages derived from an epicycloidal curve. My attention was drawn to this matter at first from the fact that in a large number of watches having wheels formed by the same cutter we had different variations or rating at different intervals of the twenty-four hours, although the watches would practically correspond at the beginning and end of the twenty-four hours. Investigating the fact and finding that the escapement was correct and satisfactory--that the mainspring developed the true curve in running down and winding up--I concluded that the trouble lay in the shape of the wheel teeth. Examine the tooth in Fig. 314, lettered D: The shape of the tooth 1 is the correct epicycloid, and the shape of the outline 4 is what the workman termed a "nice looking tooth." Now, if when the point a of the wheel tooth contacted the pinion the rating was correct, and when leaving at b the rating was correct, then at all intermediate points the running of the watch must have been practically incorrect. I then realized the failure of attempting to make a true epicycloidal curve to a tooth. It will be seen by glancing at these again, that the lines are practically parallel, merely beginning the curves lower down, so that while it was an epicycloid, it was an epicycloid secured by reducing the diameter of the wheel or by making the space wider. I then determined on making an approximate curve--that is, a curve the outline of which is marked 5 in tooth--which I found would follow in a measure the even workings of a proper epicycloidal curve. I have exaggerated the differences slightly here, in order to show the lines, but the curve is formed on much the same theory as the epicycloid, and approximates clearly its entire length, except at the extreme point, where it varies sufficient to make a departure at about the time that the next succeeding tooth of the wheel comes into engagement with the next succeeding pinion leaf, thus maintaining in a measure the practical uniformity of the epicycloid and providing a good wearing curve for the face of the tooth. No arbitrary rule of determining this curve can be selected, and it must be largely left to the discretion of the designer. It would need a sheet of paper much larger than a page of this book to draw the outline of an approximate curve for study and analyzing, but it would be well for the designer to draw his epicycloid curve on a very large scale--say one or two hundred diameters--and then curve the top only of his tooth so as to meet in a point, make the templet of a shape desired, and, by a pantograph engine, form his cutter.

Regarding the shape of the pinion: In Fig. 314, I have developed an epicycloid 2 by the same generating circle, and shown in Fig. 315 the outline of a pinion marked 2, that is made from this curve. In case the pinion was a driver, I have shown in Fig. 314 an epicyloid marked 3, developed by a generating circle, the diameter of which is the pitched radius of the wheel, and have made the outline of a pinion tooth in Fig. 315, with the curve, and marked it 3. The inner circle of Fig. 315 in the pinion is the pitched circle--the outer circle is a circle supposed to be the diameter of the pinion, the addendum of which above the pitch circle is 1.50. It will be seen from a glance at Fig. 315 that the outlines of teeth developed by epicycloids 2 and 3 would extend beyond the arbitrary selected diameter and develop an immense amount of engaging friction, as it will be seen that "lead" takes place at 45 degrees before the line of centers, hence no watch would keep any uniform time in which there was an attempt to make the face of the pinion an epicycloid curve. Look at the outline marked 6 of the pinion tooth: We have here a tooth which is only 1.25 addendum; or, in other words, the face of which is a semi-circle the diameter of the thickness of a tooth. This is all that is necessary, in that this factor above the pitch circle is merely a factory of safety. Examine the engaging teeth and pinion, it will be seen that "lead" can only take place slightly before the teeth reach the line of centers.--T. F. Sheridan.