An instrument much employed in the best practice to find the radius which will strike an arc of a circle approximating the true epicycloidal curve, _and for finding at the same time_ the location of the centre wherefrom that curve should be struck, is found in the Willis' odontograph. This is, in reality, a scale of centres or radii for different and various diameters of wheels and generating circles. It consists of a scale, shown in Fig. 135, and is formed of a piece of sheet metal, one edge of which is marked or graduated in divisions of one-twentieth of an inch. The edge meeting the graduated edge at O is at angle of 75° to the graduated edge.
On one side of the odontograph is a table (as shown in the cut), for the flanks of the teeth, while on the other is the following table for the faces of the teeth:
TABLE SHOWING THE PLACE OF THE CENTRES UPON THE SCALE.
+------+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ |No. of|1/4|3/8|1/2|5/8|3/4| 1|1- |1- |1- | 2|2- |2- | 3|3- | |Teeth | | | | | | |1/4|1/2|3/4| |1/4|1/2| |1/2| |------+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | 12 | 1| 2| 2| 3| 4| 5| 6| 7| 9| 10| 11| 12| 15| 17| | 15 | ..| ..| 3| ..| ..| ..| 7| 8| 10| 11| 12| 14| 17| 19| | 20 | 2| ..| ..| 4| 5| 6| 8| 9| 11| 12| 14| 15| 18| 21| | 30 | ..| 3| 4| ..| ..| 7| 9| 10| 12| 14| 16| 18| 21| 25| | 40 | ..| ..| ..| ..| 6| 8| ..| 11| 13| 15| 17| 19| 23| 26| | | | | | | | | | | | | | | | | | 60 | ..| ..| ..| 5| ..| ..| 10| 12| 14| 16| 18| 20| 25| 29| | 80 | ..| ..| ..| ..| ..| 9| 11| 13| 15| 17| 19| 21| 26| 30| | 100 | ..| ..| ..| ..| 7| ..| ..| ..| ..| 18| 20| 22| ..| 31| | 150 | ..| ..| 5| 6| ..| ..| ..| 14| 16| 19| 21| 23| 27| 32| |Rack. | ..| 4| ..| ..| ..| 10| 12| 15| 17| 20| 22| 25| 30| 34| +------+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
The method of using the instrument is as follows: In Fig. 136, let C represent the centre, and P the pitch circle of a wheel to contain 30 teeth of 3 inch arc pitch. Draw the radial line L, meeting the pitch circle at A. From A mark on the pitch circle, as at B, a radius equal to the pitch of the teeth, and the thickness of the tooth as A _k_. Draw from B to C the radial line E. Then for the flanks place the slant edge of the odontograph coincident and parallel with E, and let its corners coincide with the pitch circle as shown. In the table headed _centres for the flanks of the teeth_, look down the column of 3 inch pitch, and opposite to the 30 in the column of numbers of teeth, will be found the number 49, which indicates that the centre from which to draw an arc for the flank is at 49 on the graduated edge of the odontograph, as denoted in the cut by _r_. Thus from _r_ to the side _k_ of the tooth is the radius for the compasses, and at _r_, or 49, is the location for the centre to strike the flank curve _f_. For the face curve set the slant edge of the odontograph coincident with the radial line L, and in the table of centres for the faces of teeth, look down the column of 3-inch pitch, and opposite to 30 in the number of teeth column will be found the number 21, indicating that at 21 on the graduated edge of the odontograph, is the location of the centre wherefrom to strike the curve _d_ for the face of the tooth, this location being denoted in the cut at R.