In mechanics, circular motion is transmitted by means of wheels, drums, or pulleys; and accordingly as the driving and driven are of equal or unequal diameters, so are equal or unequal velocities produced. Hence the principle on which the following rules are founded.

1. When time is not taken into Account.

Rule. - Divide the greater diameter, or number of teeth, by the lesser diameter or number of teeth; and the quotient is the number of revolutions the lesser will make, for one of the greater.

Example. - How many revolutions will a pinion of 20 teeth make, for 1 of a wheel with 125 ?

125 ÷ 20 = 6.25 or 6¼ revolutions.

To find the number of revolutions of the last, to one of the first, in a train of wheels and pinions.

Rule. - Divide the product of all the teeth in the driving by the product of all the teeth in the driven; and the quotient equal the ratio of velocity required.

Example 1. - Required the ratio of velocity of the last, to 1 of the first, in the following train of wheels and pinions; viz., pinions driving-the first of which contains 10 teeth, the second 15, and third 18. Wheels driven first, 15 teeth, second, 25, and third, 32.

10 X 15 X 18 / 15 X 20 X 32 = .225 of a revolution the wheel will make to one of the pinion.

EXample 2. - A wheel of 42 teeth giving motion to one of 12, on which shaft is a pulley of 21 inches diameter driving one of 6; required the number of revolutions of the last pulley to one of the first wheel.

42 X 21 / 12 X 6 = 12.25 or 12¼ revolutions.

NotE. - Where Increase or decrease of velocity is required to be communicated by wheel-work, it has been demonstrated that the number of teeth on each pinion should not be less than 1 to 6 of its wheel, unless there he some oilier important reson for a higher ratio.

2. When Time must be regarded.

Rule. - Multiply the diameter or number of teeth in the driver, by its velocity in any given time, and divide the product by the required velocity of the driven; the quotient equal the number of teeth or diameter of the driven, to produce the velocity required.

Example 1. - If a wheel, containing 84 teeth, makes 20 revolutions per minute, how many must another contain, to work in contact, and make 60 revolutions in the same time ?

81 x 20 ÷ 60 = 28 teeth.

Example 2. - From a shaft making 45 revolutions per minute, and with a pinion 9 inches diameter at the pitch line, I wish to transmit motion at 15 revolutions per minute; what, at the pitch line, must be the diameter of the wheel ?

45 X 9 ÷ 5 = 27 inches.

Example 3. - Required the diameter of a pulley to make 16 revolutions in the same time as one of 24 inches making 36. 24 X 36 ÷ = 54 inches.

The distance between the centres and velocities of two wheels being given, to find their proper diameters.

Rule. - Divide the greatest velocity by the least; the quotient is the ratio of diameter the wheels must bear to each other.

Hence, divide the distance between the centres by the ratio + 1; the quotient equal the radius of the smaller wheel; and subtract the radius thus obtained from the distance between the centres; the remainder equal the radius of the other.

Example. - The distance of two shafts from centre to centre is 50 inches, and the velocity of the one 25 revolutions per minute, the other is to make 80 in the same time; the proper diameters of the wheels at the pitch lines are required.

80 ÷ 25 = 3.2, ratio of velocity, and 50 ÷ 3.2 + 1 = 11.9 the radius of the smaller wheel; then 50 - 11.9 = 38.1, radius of larger; their diameters are 11.9 X 2 = 23.8 and 38.1 X 2 = 76.2 inches.

To obtain or diminish an accumulated velocity by means of wheels, pinions, or wheels, pinions, and pulleys, it is necessary that a proportional ratio of velocity should exist, and which is thus attained: multiply the given and required velocities together; and the square root of the product is the mean or proportionate velocity.

Example. - Let the given velocity of a wheel containing 54 teeth equal 16 revolutions per minute, and the given diameter of an inter-mediate pulley equal 25 inches, to obtain a velocity of 81 revolutions in a machine; required the number of teeth in the intermediate wheel and diameter of the last pulley.

√81 X 16 = 36 mean velocity.

54 X 16 ÷ 36 = 21 teeth and 25 x 36 ÷ 81 = 11.1 inches, diam. of pulley,

To determine the proportion of wheels for screw-cutting by a Lathe.

In a lathe properly adapted, screws to any degree of pitch, or number of threads in a given length, may he cut by means of a leading screw of any given pitch, accompanied with change wheels and pinions; coarse pitches being effected generally by means of one wheel and one pinion with a carrier, or intermediate wheel, which cause no variation or change of motion to take place. Hence the following

RulE. - Divide the number of threads in a given length of the screw which is to be cut, by the number of threads in the same length of the leading screw attached to the lathe; and the quotient is the ratio that the wheel on the end of the screw must bear to that on the end of the lathe spindle.

Example. - Let it be required to cut a screw with 5 threads in an inch, the leading screw being of ½ inch pitch, or containing 2 threads in an inch; what must be the ratio of wheels applied?

5 ÷ 2 = 2.5, the ratio they must bear to each other. Then suppose a pinion of 40 teeth be fixed upon for the spindle, -40 X 2.5 = 100 teeth for the wheel on the end of the screw.

But screws of a greater degree of fineness than about 8 threads in an inch are more conveniently cut by an additional wheel and pinion, because of the proper degree of velocity being more effectively attained; and these, on account of revolving upon a stud, are commonly designated the stud-wheels, or stud-wheel and pinion; but the mode of calculation and ratio of screw are the same as in the preceding rule. Hence, all that is further necessary is to fix upon any:5 wheels at pleasure, as those for the spindle and stud-wheels; then multiply the number of teeth in the spindle-wheel by the ratio of the screw, and by the number of teeth in that wheel or pinion which is in contact with the wheel on the end of the screw; divide the product by the stud-wheel in contact with the spindle-wheel; and the quotient is the number of teeth required in the wheel on the end of the leading screw.