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.
Fig. 37.
Fig. 38.
When a wheel is intended to be formed to work correctly with any other wheel having the same pitch, or when there are more than two wheels in the train, it is necessary that the same size of generating circle be used for all the faces and all the flanks in the set, and if this be done the wheels will work correctly together, no matter what the number of the teeth in each wheel may be, nor in what way they are interchanged. Thus in [Fig. 37], let a represent the pitch line of a rack, and b and c the pitch circles of two wheels, then the generating circle would be rolled within b, as at 1, for the flank curves, and without it, as at 2, for the face curves of b. It would be rolled without the pitch line, as at 3, for the rack faces, and within it, as at 4, for the rack flanks, and without c, as at 5, for the faces, and within it, as at 6, for flanks of the teeth on c, and all the teeth will work correctly together however they be placed; thus c might receive motion from the rack, and b receive motion from c. Or if any number of different diameters of wheels are used they will all work correctly together and interchange perfectly, with the single condition that the same size of generating circle be used throughout. But the curves of the teeth so formed will not be alike. Thus in [Fig. 38] are shown three teeth, all struck with the same size of generating circle, d being for a wheel of 12 teeth, e for a wheel of 50 teeth, and f a tooth of a rack; teeth e, f, being made wider so as to let the curves show clearly on each side, it being obvious that since the curves are due to the relative sizes of the pitch and generating circles they are equally applicable to any pitch or thickness of teeth on wheels having the same diameters of pitch circle.
