59

tances. It is therefore necessary that those divisions of

the circumference of the one, which indicate the thickness

of the teeth and width of the spaces, should be contained

an exact number of times in the circumference of the

other. Hence it follows that the numbers of the teeth

in the two wheels are proportional to their respective radii

or diameters ; and, on the other hand, the lengths of these

diameters depend on the relative velocities at which the

two wheels are intended to revolve. These are the points

which first demand our attention, and which wre shall

proceed shortly to elucidate.

Prop. I.—If the distance between the parallel axes of

two wheels, and their relative velocities be known, the

diameter of each wheel may be found.

The velocities of wheels gearing together, are in the

inverse ratio of their diameters; or, in other words, the

one wheel revolves with so much the less velocity as its

diameter exceeds that of the other, and vice versa.

Thus, the velocity is doubled, by causing a wheel of a

certain diameter to work into another of half that dia-

meter; and conversely to reduce the velocity by one-half,

it is necessary to employ a wheel of twice the diameter.

This being premised, let us now call A and B the two

axes; suppose them to be situated at the distance of three

feet apart, and let it be required to make the axis A

revolve twice during one revolution of the axis B.

Divide the distance between the axes into three equal

parts ; from the point A, as a centre, with a radius equal

to one of these parts, describe a circle; then, from the

centre B, with a radius equal to the sum of the two

remaining parts, describe another circle, which will touch

the former in a point situated upon the line joining their

centres. These circles indicate the true diameters of the

wheels; they are called the primitive or proportional

circles, or, more frequently, in reference to the teeth, the

pitch circles, of the wheels.

Hence it appears that, in order to the solution of the

general problem, it is only necessary to divide the dis-

tance between the axes into as many equal parts as there

are units in the sum of the numbers expressing the velo-

cities of each wheel; then to take for the radius of the

smaller wheel a number of parts corresponding with

the number which denotes the lower velocity; and con-

versely.

Now, since the circumferences of circles are directly

as their diameters, it is obvious that, in any pair of

wheels gearing together, the number of teeth in each is

also directly proportional to the respective diameters ;

so that, when the number of teeth in the one is known,

that of the other may be found by the following simple

proportion :—

Prop. II.—The number of teeth in the one wheel is to

the number of teeth in the other as the diameter of the

first is to that of the second.

Supposing the number of teeth in both wheels, and the

distance of their centres to be known, their diameters

may be found thus :—

Prop. III.—The sum of the numbers of teeth in the

two wheels is to the distance between their centres as the

number of teeth in either wheel is to its radius.

In the case of a rack gearing with a wheel, the primi-

tive circle becomes a straight line, which, in determining

the form of the teeth, must be drawn touching the pitch-

circle of the wheel. In a trundle, the pitch-circle passes

through the centres of the staves. In the examples of

trundle gearing, which form the subject of Plate XXII., it

will be observed that the diameter of the staves is equal

to the thickness of the teeth of the wheel or rack which

imparts the motion; this circumstance, however, which

is by no means essential, does not frustrate the application

of the principles we shall establish, to other cases.

In a pair of toothed wheels working together, the first

and most important objects to be attained are the reduc-

tion of the friction to a minimum, and the equable

distribution of the strain over the teeth as they come

successively into action. To satisfy these conditions, it

is necessary that the form of the teeth should be such

that the working surface of a tooth of the driving

wheel should roll over, or be developed upon, that of the

driven wheel, without any sliding or rubbing action.

Now, taking as an example the pair of spur-wheels repre-

sented at Fig. 1, Plate XXIII., and supposing the smaller

to be the driver; if a circle, 0 G C, be described with

the radius 0 C of the driven wheel as a diameter, and

if this circle be conceived to roll in the interior of the

primitive circle ABC, the line in which the point C

will travel, will coincide, as we have already demon-

strated, with the radius 0 C. And supposing the circle

0 G C, to roll on the exterior of the primitive circle

C D E of the driving wheel, the same point 0 will de-

scribe an epicycloid, which will, in every possible position,

be a tangent to the straight line 0 C, upon which it will

be developed, or, in other words, will roll without sliding,

since the curve and straight line are generated by one

and the same point in the circle 0 G C. Hence it follows

that the point of contact of any two teeth must always

be in the circle 0 G C.

In the case of a pinion driven by an internal spur

wheel, Fig. 3, Plate XXII., it is obvious that the

same construction which we have above described is

applicable. When a rack is substituted for the driving

wheel, Fig. 2, Plate XXI., the circle A E C, drawn

with a diameter equal to the radius A C of the primitive

circle of the pinion, generates a cycloid, which fulfils, in

this case, the required conditions. But if, on the other

hand, the pinion is the driver, as in Fig. I, Plate XXI.,

the generating circle then becomes a straight line, M N,

which, by moving round the circle BAD, would cause

any point, such as A, to describe an involute A c; this

curve, therefore, will act in the required manner upon

the flank of a tooth of the rack, provided the latter be

perpendicular to the pitch line M N.

These illustrations involve the statement of the theore-

tical principle which determines the form of the teeth of

all wheels working in gear ; but in practice certain modi-

fications are admissible. This observation we conceive it