To prove this we may show that the pitch radii of the two wheels are in the same proportion as their numbers of teeth, thus:—
| Proof.— | Radius of small wheel | = | 10 | × | 4 | = | 40 |
| radius of large wheel | = | 22.5 | × | 4 | = | 90.0 |
Suppose now that a pair of wheels are constructed, having respectively 50 and 100 teeth, and that the radii of their true pitch circles are 12 and 6 respectively, but that from wear in their journals or journal bearings this 18 inches (12 + 6 = 18) between centres (or line of centres, as it is termed) has become 183⁄8 inches. Then the acting effective or operative radii of the pitch circles will bear the same proportion to the 183⁄8 as the numbers of teeth in the respective wheels, and will be 12.25 for the large, and 6.125 for the small wheel, instead of 12 and 6, as would be the case were the wheels 18 inches apart. Working this out under the rule given we have 100 ÷ 50 = 2, and 2 + 1 = 3. Then 18.375 ÷ 3 = 6.125 = pitch radius of small wheel, and 18.375 - 6.125 = 12.25 = pitch radius of the large wheel.
The true pitch line of a tooth is the line or point where the face curve joins the flank curve, and it is essential to the transmission of uniform motion that the pitch circles of epicycloidal wheels exactly coincide on the line of centres, but if they do not coincide (as by not meeting or by overlapping each other), then a false pitch circle becomes operative instead of the true one, and the motion of the driven wheel will be unequal at different instants of time, although the revolutions of the wheels will of course be in proportion to the respective numbers of their teeth.
If the pitch circle is not marked on a single wheel and its arc pitch is not known, it is practically a difficult matter to obtain either the arc pitch or diameter of the pitch circle. If the wheel is a new one, and its teeth are of the proper curves, the pitch circle will be shown by the junction of the curves forming the faces with those forming the flanks of the teeth, because that is the location of the pitch circle; but in worn wheels, where from play or looseness between the journals and their bearings, this point of junction becomes rounded, it cannot be defined with certainty.
In wheels of large diameter the arc pitch so nearly coincides with the chord pitch, that if the pitch circle is not marked on the wheel and the arc pitch is not known, the chord pitch is in practice often assumed to represent the arc pitch, and the diameter of the wheel is obtained by multiplying the number of teeth by the chord pitch. This induces no error in wheels of coarse pitches, because those pitches advance by 1⁄4 or 1⁄2 inch at a step, and a pitch measuring about, say, 11⁄4 inch chord pitch, would be known to be 11⁄4 arc pitch, because the difference between the arc and chord pitch would be too minute to cause sensible error. Thus the next coarsest pitch to 1 inch would be 11⁄8, or more often 11⁄4 inch, and the difference between the arc and chord pitch of the smallest wheel would not amount to anything near 1⁄8 inch, hence there would be no liability to mistake a pitch of 11⁄8 for 1 inch or vice versâ. The diameter of wheel that will be large enough to transmit continuous motion is diminished in proportion as the pitch is decreased; in proportion, also, as the wheel diameter is reduced, the difference between the arc and chord pitch increases, and further the steps by which fine pitches advance are more minute (as 1⁄4, 9⁄32, 5⁄16, &c.). From these facts there is much more liability to err in estimating the arc from the measured chord pitch in fine pitches, hence the employment of diametral pitch for small wheels of fine pitches is on this account also very advantageous. In marking out a wheel the chord pitch will be correct if the pitch circle be of correct diameter and be divided off into as many points of equal division (with compasses) as there are to be teeth in the wheel. We may then mark from these points others giving the thickness of the teeth, which will make the spaces also correct. But when the wheel teeth are to be cut in a machine out of solid metal, the mechanism of the machine enables the marking out to be dispensed with, and all that is necessary is to turn the wheel to the required addendum diameter, and mark the pitch circle. The following are rules for the purposes they indicate.
The circumference of a circle is obtained by multiplying its diameter by 3.1416, and the diameter may be obtained by dividing the circumference by 3.1416.
The circumference of the pitch circle divided by the arc pitch gives the number of teeth in the wheel.
The arc pitch multiplied by the number of teeth in the wheel gives the circumference of the pitch circle.
Gear-wheels are simply rotating levers transmitting the power they receive, less the amount of friction necessary to rotate them under the given conditions. All that is accomplished by a simple train of gearing is, as has been said, to vary the number of revolutions, the speed or velocity measured in feet moved through per minute remaining the same for every wheel in the train. But in a compound train of gears the speed in feet per minute, as well as the revolutions, may be varied by means of the compounded pairs of wheels. In either a simple or a compound train of gearing the power remains the same in amount for every wheel in the train, because what is in a compound train lost in velocity is gained in force, or what is gained in velocity is lost in force, the word force being used to convey the idea of strain, pressure, or pull.