Chapter I: =the Teeth of Gear-Wheels.=

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=Gear-Wheels.= Spur-wheels, bevel-wheels, mitre-wheels,
crown-wheels, annular or internal wheels 1
Trundle-wheels, rack and pinion-wheel and tangent screw, or
worm and worm-wheel 1
The diameter of the pitch circle of 1
=Gear-Wheel Teeth.= The face, the flank, the depth or height 1
The space, the pitch line, the point, the arc pitch, the chord
pitch, the line of centres 2
Rules for finding the chord pitch from the arc pitch; table of
natural sines; diametral pitch; finding the arc from the
diametral pitch; table of arc and diametral pitches 3
=Gear-Wheels.= The driver and follower, a train of gears 3
Intermediate gears 3
The velocity of compounded wheels 4
Finding the diameters of the pitch circles of 4
Considered as revolving levers 5
Calculating the revolutions of, and power transmitted by 5
The angular velocity of 6
=Gear-Wheels.= Hunting tooth in, stop motion of 7
=Gear-Wheel Teeth.= The requirements and nature of the teeth
curves 7
Cycloidal curves for the faces of; epicycloidal and involute
curves; the hypocycloidal curve; method of forming or
generating the epicycloidal and hypocycloidal curves for the
faces and flanks of gear teeth 8
Applications of the epicycloidal and hypocycloidal curves in
the formation of gear teeth 9
The diameter of the circle for generating the epicycloidal and
hypocycloidal curves; graphical demonstration that the flank
curves are correctly formed to work with the face curves of the
other wheel 10
Graphical demonstration that the curves are correct independent
of either the respective sizes of the wheels, or of the curve
generating circles 11
=Gear-Wheels.= Hand applications of the rolling or
generating circle to mark the tooth curves for a pair of wheels 12
=Gear-Wheel Teeth.= The variation of curve due to different
diameters of wheels or of rolling circles 12
Tracing the path of contact of tooth upon tooth in a pair of
gear-wheels; definition of the "arc of approach;" definition of
the "arc of recess;" demonstration that the flanks of the teeth
on the driver or driving-wheel have contact with the faces of
the driven wheel during the arc of approach, and with the
flanks of the driven wheel during the arc of recess 13
Confining the action of the teeth to one side only of the line
of centres, when motion rather than power is to be conveyed 13
Demonstration that the appearance or symmetry of a tooth has no
significance with regard to its action 14
Finding how many teeth will be in constant action, the diameter
of the wheels, the pitch of the teeth, and the diameter of the
rolling circle being given 15
Example of the variation of tooth form due to variation of
wheel diameter 15
=Gear Teeth.= Variation of shape from using different
diameters of rolling circles 16
Thrust on the wheel shafts caused by different shapes of teeth 16
=Gear-Wheels.= Willis' system of one size of rolling circle for
trains of interchangeable gearing 16
Conditions necessary to obtain a uniform velocity of 16
=Gear Teeth.= The amount of rolling and of sliding motion of 16
The path of the point of contact of 16
The arcs of approaching and of receding contact 16
Lengths of the arcs of approach and of recess 16
The influence of the sizes of the wheels upon the arcs of
contact 17
Influence of the size of the rolling circle upon the amount of
flank contact 18
Demonstration that incorrectly formed teeth cannot correct
themselves by wear 18
The smaller the diameter of the rolling circle, the less the
sliding motion 18
Influence of the size of the rolling upon the number of teeth
in contact in a given pair of wheels 19
Demonstration that the degrees of angle the teeth move through
exceed those of the path of contact, unless the tooth faces
meet in a point 19
Influence of the height of the teeth upon the number of teeth
in contact 20
Increasing the arc of recess without increasing the arc of
approach 20
Wheels for transmitting motion rather than power 21
Clock wheels 21
Forms of teeth having generating or rolling circles, as large
or nearly as large as the diameters of the wheels 21
=Gear-Wheels.= Bevel 21
The principles governing the formation of the teeth of bevel-
wheels 22
Demonstration that the faces of the wheels must be in line with
the point of intersection of the axis of the two shafts 22
=Gear Teeth.= Method of finding the curves of, for bevel
gear 22
=Gear-Wheels.= Internal or annular 23 to 27
Demonstration that the teeth of annular wheels correspond to
the spaces of spur-wheels 23
=Gear-Wheels Internal.= Increase in the length of the path
of contact on spur-wheels of the same diameter, and having the
same diameter of generating or rolling circle 23
Demonstration that the teeth of internal wheels may interfere
when spur-wheels would not do so 23
Methods of avoiding the above interference 23
Comparison of, with spur-wheels 23
The teeth of: demonstration that it is practicable to so form
the teeth faces that they will have contact together as well as
with the flanks of the other wheel 24
Intermediate rolling circle for accomplishing the above result 24
The application of two rolling circles for accomplishing the
above result 24
Demonstration that the result reached by the employment of two
rolling circles of proper diameter is theoretically and
practically perfect 24
Limits of the diameters of the two rolling circles 25
Increase in the arc of contact obtained by using two rolling
circles 25
Demonstration that the above increase is on the arc of recess
or receding contact, and therefore gives a smooth action 25
Demonstration that by using two rolling circles each tooth has
for a certain period two points of contact 25
The laws governing the diameters of the two rolling circles 25
Practical application of two rolling circles 26
Demonstration that by using two rolling circles the pinion may
contain but one tooth less than the wheel 26
The sliding and rolling motion of the teeth of 27