It is obvious, however, that such teeth would involve a great deal of labor in their formation and would possess no advantage, hence they are not employed. It is not unusual, however, in a pair of wheels that are to gear together and that are not intended to interchange with other wheels, to use such sizes as will give to for the face of the teeth on the largest wheel of the pair and for the flanks of the teeth of the smallest wheel, a generating circle equal in diameter to the radius of the smallest wheel, and for the faces of the teeth of the small wheel and the flanks of the teeth of the large one, a generating circle whose diameter equals the radius of the large wheel.
It will now be evident that if we have planned a pair or a train of wheels we may find how many teeth will be in contact for any given pitch, as follows. In Fig. 36 let A, B, and C, represent three blanks for gear-wheels whose addendum circles are M, N and O; P representing the pitch circles, and Q representing the circles for the roots of the teeth. Let X and Y represent the lines of centres, and A, H, I and K the generating or rolling circle, whose centres are on the respective lines of centres--the diameter of the generating circle being equal to the radius of the pinion, as in the Willis system, then, the pinion M being the driver, and the wheels revolving in the direction denoted by the respective arrows, the arc or path of contact for the first pair will be from point D, where the generating circle G crosses circle N to E, where generating circle H crosses the circle M, this path being composed of two arcs of a circle. All that is necessary, therefore, is to set the compasses to the pitch the teeth are to have and step them along these arcs, and the number of steps will be the number of teeth that will be in contact. Similarly, for the second pair contact will begin at R and end at S, and the compasses applied as before (from R to S) along the arc of generating circle I to the line of centres, and thence along the arc of generating circle K to S, will give in the number of steps, the number of teeth that will be in contact. If for any given purpose the number of teeth thus found to be in contact is insufficient; the pitch may be made finer.