Modern Machine Shop Practice, by Joshua Rose, M.E

Modern Machine-Shop Practice is presented to American mechanics as a complete guide to the operations of the best equipped and best managed workshops, and to the care and management of engines and boilers.

The materials have been gathered in part from the author’s experience of thirty-one years as a practical mechanic; and in part from the many skilled workmen and eminent mechanics and engineers who have generously aided in its preparation. Grateful acknowledgment is here made to all who have contributed information about improved machines and details of new methods.

The object of the work is practical instruction, and it has been written throughout from the point of view, not of theory, but of approved practice. The language is that of the workshop. The mathematical problems and tables are in simple arithmetical terms, and involve no algebra or higher mathematics. The method of treatment is strictly progressive, following the successive steps necessary to becoming an intelligent and skilled mechanic.

The work is designed to form a complete manual of reference for all who handle tools or operate machinery of any kind, and treats exhaustively of the following general topics: I. The construction and use of machinery for making machines and tools; II. The construction and use of work-holding appliances and tools used in machines for working metal or wood; III. The construction and use of hand tools for working metal or wood; IV. The construction and management of steam engines and boilers. The reader is referred to the Table of Contents for a view of the multitude of special topics considered.

The work will also be found to give numerous details of practice never before in print, and known hitherto only to their originators, and aims to be useful as well to master-workmen as to apprentices, and to owners and managers of manufacturing establishments equally with their employees, whether machinists, draughtsmen, wood-workers, engineers, or operators of special machines.

The illustrations, over three thousand in number, are taken from modern practice; they represent the machines, tools, appliances and methods now used in the leading manufactories of the world, and the typical steam engines and boilers of American manufacture.

The new Pronouncing and Defining Dictionary at the end of the work, aims to include all the technical words and phrases of the machine shop, both those of recent origin and many old terms that have never before appeared in a vocabulary of this kind.

The wide range of subjects treated, their convenient arrangement and thorough illustration, with the exhaustive Table of Contents of each volume and the full Analytical Index to both, will, the author hopes, make the work serve as a fairly complete ready reference library and manual of self-instruction for all practical mechanics, and will lighten, while making more profitable, the labor of his fellow-workmen.

---

[I-vii]

Table of
contents
for
Volume II.

CHAPTER I.
THE TEETH OF GEAR-WHEELS.
PAGE
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
CHAPTER II.
THE TEETH OF GEAR-WHEELS (Continued).
Worm and Worm-Wheel, or wheel and tangent screw			28 to 31
	General description of		28
	Qualifications of		28
	The wear of		28
Worm-Wheel Teeth, the sliding motion of			28
	When straight have contact on the centres only of the tooth sides		28
	That envelop a part of the worm circumference		28
	The location of the pitch line of the worm		28
	The proper number of teeth in the worm-wheel		29
	Locating the pitch line of the worm so as to insure durability		29
	Rule for finding the best location for the pitch line of the worm		29
	Increasing the face of the worm to obtain a smoother action		29
Worms, to work with a square thread			29
Worm-Wheels, applications of			30
Gear-Wheels with involute teeth			31 to 34
Gear Teeth. Generating the involute curve			31
	Templates for marking the involute curve		32
Involute Teeth, the advantages of			34
Gear Teeth, Pratt and Whitney’s machine for cutting templates for			35
CHAPTER III.
THE TEETH OF GEAR-WHEELS (Continued).
Gear Teeth, revolving cutters for			37
	Pantagraph engine for dressing the cutters for		38
	Numbers of cutters used for a train of wheels		39
Gear-Wheel Teeth. Table of equidistant value of cutters			41
	Depth of, in the Brown and Sharpe system		42
	Cutting the teeth of worm-wheels		42
	Finding the angle of the cutter for cutting worm-wheels		43
	The construction of templates for rolling the tooth curves		43
	Rolling the curves for gear teeth		43
	Forms of templates for gear teeth		44
	Pivoted arms for tooth templates		44
	Marking the curves by hand		45
	Former or Template of the Corliss bevel gear-wheel engine or cutting machine		45
	The use of extra circles in marking the curves with compasses		46
	Finding the face curves by geometrical constructions		47
	The Willis odontograph for finding the radius for striking the curves by hand		47
	The method of using the Willis odontograph		48
	Professor Robinson’s odontograph		49
	Method of using Professor Robinson’s odontograph		49
	Application of Professor Robinson’s odontograph for trains of gearing		51
	Tabular values and setting numbers for Professor Robinson’s odontograph		51
	Walker’s patent wheel scale for marking the curves of cast teeth		51
	The amount of side clearance in cast teeth		53
	Filleting the roots of epicycloidal teeth with radial flanks		53
	Scale of tooth proportions given by Professor Willis		54
	The construction of a pattern for a spur-wheel that is to be cast with the teeth on		54
	Template for planing the tooth to shape		54
	Method of marking the curves on teeth that are to be glued on		55
	Method of getting out the teeth of		56
	Spacing the teeth on the wheel rim		56
	Methods of accurately spacing the pattern when it has an even number of teeth		58
	Method of spacing the wheel rim when it has an odd number of teeth		58
Gear-Wheels, Bevel Pinion, drawings for			59
	Getting out the body for a bevel-wheel		59
	Template for marking the division lines on the face of the wheel		59
	Marking the lines of the division on the wheel		60
Gear-Wheels, Pinion, with dovetail teeth			60
	Testing the angle of bevel-wheels while in the lathe		60
Gear-Wheels, Skew Bevel. Finding the line of contact			61
	Marking the inclination of the teeth		61
Gear-Wheels, Bevel, drawing for built up			61
Gear-Wheels, Worm, or endless screw			62
	Constructing a pattern from which the worm is to be cast		62
	Tools for cutting the worm in a lathe		62
	Cutting the teeth by hand		62
Gear-Wheels, Mortise or cogged			63
	Methods of fastening cogs		63
	Methods of getting out cogs for		63
Gear-Wheel Teeth, calculating the strength of epicycloidal			64
	Factors of safety for		64
	Tredgold’s rule for calculating the strength of		65
	Cut, calculating the strength of		65
Gear-Wheel Teeth. The strength of cogs			66
	The thickness of cogs		66
	The durability of cogs		66
	Table for calculating the strength of different kinds of		67
	The contact of cast teeth		67
	Table for determining the relation between pitch diameter, pitch, and number of teeth in gear-wheels		68
	Examples of the use of the above table		68
	With stepped teeth		69
	Angular or helical teeth		69
	End thrust of angular teeth		69
	Herring-bone angular teeth		69
	For transmitting motion at a right angle by means of angular or helical teeth		69
	Cutting helical teeth in the lathe		69
	For wheels whose shaft axes are neither parallel nor meeting		70
	Elliptical		70
	Elliptical, marking the pitch lines of		70
	Elliptical, drawing the teeth curves of		73
	For variable motion		74
	Form of worm to give a period of rest		74
	Various applications of		74
Gear-Wheels, arrangement of, for periodically reversing the direction of motion			75
	Watt’s sun and planet motion		75
	Arrangements for the rapid multiplication of motion		75
	Arrangement of, for the steering gear of steam fire-engines		75
	Various forms of mangle gearing		79
Gear-Wheel and Rack, for reciprocating motion			77
Friction Wheels.			77
	The material for		77
	Paper		78
	For the feed motion of machines		78
	The unequal wear upon grooved		79
	Form of, for relieving the journals of strain		79
Cams, for irregular motion			80
	Finding the pitch line of		80
	Finding the working face of		80
	The effect the diameter roller has upon the motion produced by a cam		80
	Demonstration of the different motion produced by different diameters of rollers upon the same cam		80
	Diagram of motion produced from the same cam with different diameters of rollers		81
	Return or backing		82
	Methods of finding the shape of return or backing		82
Cam Motion, for an engine slide valve without steam lap			83
	For a slide valve with steam lap		83
Groove Cams, proper construction of			84
	The wear of		84
	Brady’s improved groove cam with rolling motion and adjustment for wear		84
CHAPTER IV.
SCREW-THREADS.
	Screw Threads, the various forms of		85
	The pitch of		85
	Self-locking		85
	The Whitworth		86
	The United States standard		86
	The Common V		86
	The requirements of		86
	Tools for cutting		87
	Variation of pitch from hardening		87
	The wear of thread-cutting tools		88
	Methods of producing		88
	Alteration of shape of, from the wear of the tools they are cut by		89
Screw Thread Cutting Tools. The wear of the tap and the die			89
	Improved form of chaser to equalize the wear		90
	Form of, to eliminate the effects of the wear in altering the fit		90
	Originating standard angles for		91
	Standard micrometer gauge for the United States standard screw thread		91
	Standard plug and collar gauges for		91
	Producing gauges for		92
	Table of United States standard for bolts and nuts		93
	Table of standard for the V-thread		93
	United States standard for gas and steam pipes		93
	Taper for standard pipe threads		95
	Tables of the pitches and diameters at root of thread, of the Whitworth thread		95
	Table of Whitworth’s screw threads for gas, water, and hydraulic piping		96
	Whitworth’s standard gauges for watch and instrument makers		96
	Screw-cutting hand tools		96
Thread-Cutting Tools. American and English forms of stocks and dies			97
	Adjustable or jamb dies		98
	The friction of jamb dies		98
	The sizes of hobs that should be used on jamb dies		99
	Cutting right or left-hand thread with either single, double, or treble threads with the same dies		99
	Hobs for hobbing or threading dies		100
	Various forms of stocks with dies adjustable to take up the wear		101
	Dies for gas and steam pipes		101
Thread-Cutting Tool Taps. The general forms of taps			102
	Reducing the friction of		102
	Giving clearance to		102
	The friction of taper		103
	Improved forms of		103
	Professor J. E. Sweet’s form of tap		104
	Adjustable standard		104
	The various shapes of flutes employed on taps		105
	The number of flutes a tap should have		105
	Demonstration that a tap should have four cutting edges rather than three		106
	The position of the square or driving end, with relation to the cutting edges		106
	Taper taps for blacksmiths		106
	Collapsing taps for use in tapping machines		107
	Collapsing tap for use in a screw machine		107
	The alteration of pitch that occurs in hardening		108
	Gauging the pitch after the hardening		108
	Correcting the errors of pitch caused by the hardening		109
	For lead		109
	Elliptical in cross section		109
	For very straight holes		109
	Tap wrenches solid and adjustable		110
Thread-Cutting. Tapping			110
	Appliances for tapping standard work		111
CHAPTER V.
FASTENING DEVICES.
Bolts, classification of, from the shapes of their heads			112
	Classification of, from the shapes of their bodies		112
	Countersunk		112
	Holes for, classification of		112
	For foundations, various forms of		113
	Hook bolts		113
	The United States standard for finished bolts and nuts		113
	The United States standard for rough bolts and nuts, or black bolts		114
	The Whitworth standard for bolts and nuts		114
Screws			114
Studs			115
Set Screws			115
Bolts for quick removal			116
	That do not pass through the work		117
	That self-lock in grooves and are readily removable		117
	Heads and their bedding		117
Nuts, the forms of, when they are to be steam tight			118
	Various forms of		118
	Jamb nuts and lock nuts		119
Differential Threads for locking purposes			119
	For fine adjustments		119
Nuts, taking up the wear of			120
	Securing devices		120
	Securing by taper pins		121
	Securing by cotters		121
	Securing by notched plates		121
Pins. Securing for exact adjustments			121
	And double eyes fitting		121
	Fixed		122
	Working		122
Bolts, removing corroded			122
Nuts, removing corroded			122
Washers, standard sizes of			122
Wrench, the proper angles of			123
	Box		124
	Monkey		125
	Adjustable, various forms of		125
	Sockets		125
	Novel for carriage bolts		125
	Pin		126
		Improved form of	126
Keys, the various kinds of			126
	The bearing surfaces of		126
Set Screws, application of, to hubs or bosses			127
Keys, with set-screws			127
	The draught of		127
Feathers, and their applications			127
Keys, for parallel rods			128
Taper Pins, proper position of, for locking purposes			128
	Improved method of fitting		128
CHAPTER VI.
THE LATHE.
Lathe, the importance and advantages of			129
	Classification of lathes		129
	Foot		130
	Methods of designating the sizes of		130
	Bench		130
	Power		130
	Hand		130
	Slide Rest for		131
		American form of, their advantages and disadvantages	132
		English forms of	132
		For spherical work	132
		Methods of taking up lost motion of	133
Engine Lathe, general construction of			133
	The construction of the shears of		134
	Construction of the headstock		134
	Construction of the bearings		134
	Construction of the back gear		135
	Means of giving motion to the feed spindle		135
	Construction of the tailstock		135
	Method of rapidly securing and releasing the tailstock		136
Lathe Tailstock, setting over for turning tapers			136
Engine Lathe, construction of carriage			137
	Feed motion for carriage or saddle		137
Lathe Apron, Construction of the feed traverse			138
	Construction of the cross-feed motion		138
Engine Lathe, lead screw and change wheels of			139
	Feed spindle and lead screw bearings		139
	Swing frame for lead screw		139
	Lead screw nuts		140
	With compound slide rest		140
	Construction of compound slide rest		141
	Advantages of compound slide rest		141
	For taper turning		142
	Taper-turning attachments		142
	With compound duplex slide rest		143
	Detachable slide rest		143
	Three-tool slide rest for turning shafting		143
	With flat saddle for chucking work on		143
The Sellers Lathe			143
	Construction of the headstock and treble gear		144
	Construction of the tailstock and method of keeping it in line		145
	Construction of the carriage and slide rest		145
	Methods of engaging and disengaging the feed motions		146
Car Axle Lathe, with central driving motion and two slide rests			147
	The feed motions of		148
Self-Acting Lathe, English form of			148
Pattern Maker’s Lathe			148
	Brake for cone pulley		149
	With wooden bed		149
	Slide rest for		149
Chucking Lathe, English			149
	Feed motions of		150
Pulley Lathe			150
Gap or Break Lathe			151
Extension Lathe			151
Wheel Lathe			151
Chucking Lathe for boring purposes			152
Lathe for turning crank axles			152
	Construction of the headstock		153
	Construction of the feed motions		154
	For turning crank, Arrangements of the slide rests		154
	Application of the slide rest to a crank		155
CHAPTER VII.
DETAILS IN LATHE CONSTRUCTION.
Live Spindle of a lathe, the fit of			157
	With coned journals		157
	Methods of taking up the end motion of		158
	Arranging the swing frame for the change gears		158
	Taking up the wear of the back bearing		158
	The wear of the front bearing of		158
The Taper for the live centre			159
Methods of removing the lathe centres			159
Tapers for the live centres			159
Methods of removing the dead centre			159
Driving Cone, arranging the steps of			159
	Requirements of proportioning the steps of		159
	Rules for proportioning the diameters of the steps of, when the two pulleys are exactly alike and are connected by an open belt		159 to 161
		When the two pulleys are unlike	161 to 164
Back Gear, methods of throwing in and out			165
Conveying motion to the lead screw			165
Attaching the swing frame			166
Feed Gear. Arrangement for cutting worm threads or tangent screws			167
Feed Motion for reversing the direction of tool traverse in screw cutting			168
	For lathe aprons		168
Slide Rest, weighted elevated			168
	Double tool holder for		169
	Gibbed elevating		169
Examples of feed motions			170
Feed Regulators for screw cutting			171
	The star feed		172
Ratchet Feeds			173
Tool Holding devices, the various kinds of			173
Tool Rest swiveling			174
Tool Holder for compound slide rests			174
	For octagon boring tools		175
Lathe Lead and Feed Screws			175
	Lead screws, supporting, long		176
	Position of the feed nut		177
	Form of threads of lead screws		177
	The effect the form of thread has in causing the nut to lock properly or improperly		177
	Example of a lead screw with a pitch of three threads per inch		177
	Example of a lead screw with five threads per inch		178
	Example with a lead screw of five threads per inch		179
	Device for correcting the errors of pitch of		179
Table for finding the change wheels for screw cutting when the teeth in the change wheels advance by four			180
	For finding the change wheels when the teeth in the wheels advance by six		180
	Constructing a table to cut fractional threads on any lathe		181
	Finding the change wheels necessary to enable the lathe to cut threads of any given pitches		181
	Finding the change wheels necessary to cut fractional pitches		181
Determining the pitches of the teeth for change wheels			182
Lathe Shears or beds			182
	Advantages and disadvantages of, with raised V-guide-ways		182
	Examples of various forms of		183
Lathe Shears with one V and one flat side			183
	Methods of ribbing		184
	The arrangement of the legs of		184
Lathe Tailblock			185
	With rapid spindle motion		185
	With rapid fastenings and releasing devices		185
	The wear of the spindles of		185
	Spindles, the various methods of locking		186
	Testing, various methods of		187
CHAPTER VIII.
SPECIAL FORMS OF THE LATHE.
Watchmaker’s Lathes			188
	Construction of the headstock		188
	Construction of chucks for		188
	Expanding chucks for		188
	Contracting chucks for		188
	Construction of the tailblock		189
	Open spindle tailstocks for		189
	Filing fixture for		189
	Fixture for wheel and pinion cutting		189
	Jewelers’ rest for		189
Watch Manufacturers’ Lathe			190
	Special chucks for		190
	Pump centre rest		190
Lathe, hand			191
	Screw slotting		192
	With variable speed for facing purposes		192
	Cutting-off machine		193
	Grinding Lathes		193
		With elevating rest	194
		Universal	195
		Special chucks for	196
	The Morton Poole calender roll grinding lathe		196
	The construction of the bed and carriages		197
		Principles of action of the carriages	197, 198
		Construction of the emery-wheel arbors and the driving motion	198, 199
		The advantages of	199
		The method of driving the roll	200
		Construction of the headstock	200
		The transverse motion	200
The Brown and Sharpe Screw Machine, or screw-making lathe			200
	Threading tools for		203
	Examples of the use of		203
The Secor Screw Machine, construction of the headstock			204
	The chuck		205
	The feed gear		205
	The turret		205
	The cross slide		205
	The stop motions		206
Pratt and Whitney’s Screw Machine			206
	Parkhurst’s wire feed, construction of the headstock, chuck and feed motion		207
	Box tools for		208
	Applications of box tools		208
	Threading tool for		208
	Cutting-off tool for		208
Special Lathe for wood working			208
	The construction of the carriage and reducing knife		209
	Construction of the various feed motions		209
	Construction of the tailstock		209
Lathes for irregular forms			210
	Axe-handle		210
	Back knife gauge		210
	Special, for pulley turning		211
Boring and Turning mill or lathe			211
	Construction of the feed motions		213
	Construction of the framing and means of grinding the lathe		214
	Construction of the vertical feed motions		215
The Morton Poole roll turning lathe			215
	Construction of the slide rest		216
	The tools for		216
Special Lathes for brass work			216, 217
Boring Lathe with traversing spindle			218
	For engine cylinders		219
	Cylinder, with facing slide rests		219
	With double heads and facing rests		220
Lathe for turning Wheel hubs			221
CHAPTER IX.
DRIVING WORK IN THE LATHE.
Drivers, carriers, dogs, or clamps, and their defects			222
	Lathe clamps		222
	Equalizing drivers		223
	The Clements driver		223
	Driver and face plate for screw cutting		223
	Forms of, for bolt heads		224
	Adjustable, for bolt heads		224
	For threaded work		225
	For steady rest work		225
	For cored work		225
	For wood		225
Centres for hollow work			226
	For taper work		226
Lathe Mandrels, or arbors			227
	Drivers for		227
	For tubular work		227
	Expanding mandrels		227
	With expanding cones		228
	With expanding pieces		228
	Expanding, for large work		228
	For threaded work		228
	For nuts, various forms of		229
	For eccentric work		229
Centring devices for crank axles			230
The Steady Rest or back rest			231
	Steady rest, improved form of		232
	Cone chuck		232
	Steady rest for square and taper work		233
	The cat head		233
	Clamps for		233
	Follower rests		234
Chucks and Chucking			234
	Simple forms of chucks		234
	Adjustable chucks for true work		235
	Two-jawed chucks		236
	Box body chucks		237
	Reversible jawed chucks		237
	Three and four-jawed chucks		237
	Combination chucks		237
	The wear of scroll chuck threads		237
	Universal chucks		238
	The wear of chucks		240
	Special forms of chucks		241
	Expanding chucks for ring-work		241
	Cement chuck		241
	Chucks for wood-working lathes		242
Lathe Face Plates			243
	Face plates, errors in, and their effects		243
	Work-holding straps		244
	Face plate, clamping work on		245
		Forms of clamps for	245
	Examples of chucking work on		246, 247
	For wood work		247
Special Lathe Chuck for cranks			248
Face Plate Work, examples of			249
	Errors in chucking		250
	Movable dogs for		250
	The angle plate		251
		Applications of	251
	Angle plate chucking, examples of		251
	Cross-head chucking		251-253
CHAPTER X.
CUTTING TOOLS FOR LATHES.
Principles governing the shapes of lathe tools			254
Diamond-pointed, or front tool			254
Principles governing use of tools			254
	Front rake and clearance of front tools		254
	Influence of the height of a tool upon its clearance and keenness		255
	Tools with side rake in various directions		256
	The effect of side rake		256
	The angle of clearance in lathe tools		257
	Variation of clearance from different rates of feed and diameters of work		257
Round-nosed tools			258
Utmost Duty of cutting tools			258
	Judging the quantity of the tool from the shape of its cutting		259
Square-nosed tools			260
	The height of lathe tools		260
	Side tools for lathe work		261
	Cutting-off or grooving tools		262
	Facing tools or knife tools		262
	Spring tools		263
Brass Work, front tools for			264
	Side tools for		264
Threading tools			264
	Internal threading tools		264
	The length of threading tools		265
	The level of threading tools		265
	Gauges for threading tools		266
	Setting threading tools		266
	Circular threading tools		267
	Threading tool holders		267
Chasers			268
	Chaser holders		268
	Setting chasers		268
Square Threads, clearance of tools for			269
	Diameter at the roots of threads		269
	Cutting coarse pitch square threads		269
	Dies for finishing square threads		269
Tool Holders for outside work			270
	For circular cutters		272
	Swiveled		273
	Combined tool holders and cutting-off tools		273
Power Required to drive cutting tools			273
CHAPTER XI.
DRILLING AND BORING IN THE LATHE.
The Twist Drill			274
	Twist drill holders		274
	The diametral clearance of twist drills		274
	The front rake of twist drills		275
	The variable clearance on twist drills as usually ground		275
	Demonstration of the common error in grinding twist drills		276
	The effects of improper grinding upon twist drills		276
	Table of speeds and feeds for twist drills		277
	Grinding twist drills by hand		279
	Twist drills for wood work		279
Tailstock Chucks for drilled work			279
Flat Drills for lathe work			280
	Holders for lathe work		281
Half-round bit or pod auger			281
	With front rake for wrought iron or steel		281
	With adjustable cutter		281
	For very true work		281
Chucking Reamer			281
	The number of teeth for reamers		282
	Spacing the teeth of reamers		282
	Spiral teeth for reamers		282
	Grinding the teeth of reamers		282
	Various positions of emery-wheel in grinding reamers		282
	Chucking reamers for true work		283
	Shell reamers		283
	Arbor for shell reamers		283
	Rose-bit or rose reamers		283
	Shell rose reamers		284
	Adjustable reamers		284
	Stepped reamers for taper work		285
	Half-round reamers		285
	Reamers for rifle barrels		285
Boring Tools for lathe work			285
	Countersinks		285
	Shapes of lathe boring tools		285
	Boring tools for brass work		286
	The spring of boring tools		286
	Boring tools for small work		287
	Boring tool holders		287
Boring Devices for Lathes			288
Boring Heads			288
Boring Bars			289
	Boring bar cutters		289
	Three versus four cutters for boring bars		290
	Boring bars with fixed heads		290
		With sliding heads	290
		Bar cutters, the shapes of	291
	Boring head with nut feed		291
	Boring bars for taper work, various forms of		292
	Boring double-coned work		293
	Boring bar, centres for		293
Cutting Speeds and feeds for wrought iron			294
	Examples of speeds taken from practice		295
CHAPTER XII.
EXAMPLES IN LATHE WORK.
Technical Terms used in the work			296
Lathe Centres			296
	Devices for truing		297
	Tools for testing the truth of, for fine work		298
	Shapes of, for light and heavy work		299
Centre Drilling, attachment for lathes			300
	The error induced by straightening work after		300
	Machine		300
	Combined centre-drill and countersink		300
	Countersink with adjustable drill		300
	Centring square		300
	Centre-punch		300
	Centre-punch guide		301
	Centring work with the scribing block		301
	Finding the centre of very rough work		301
	Centre-drill chuck		302
	The proper form of countersink for lathe work		302
	Countersinks for lathe work		302
	Various forms of square centres		303
	The advantage of the square centre for countersinking		303
	Novel form of countersink for hardened work		303
	Chucks for centre-drilling and countersinking		303
	Recentring turned work		304
Straightening Work. Straightening machine for bar iron			304
	Hand device for straightening lathe work		305
	Chuck for straightening wire		305
Cutting Rods into small pieces of exact length, tools for			305
Roughing cuts, the change of shape of work that occurs from removing the surface by			306
	Feeds for		306
	Rates of feed for		307
Finishing Work, the position of the tool for			307
	Finishing cast-iron with water		307
	Specks in finished cast-iron work		307
	Scrapers for finishing cast-iron work		307
	Method of polishing lathe work		308
	Filing lathe work		308
	The use of emery paper on lathe work		308
	The direction of tool feed in finishing long work		309
	Forms of laps for finishing gauges or other cylindrical lathe work		310
	Forms of laps for finishing internal work		311
	Grinding and polishing clamps for lathe work		311
	Burnishing lathe work		311
Taper Work, turning			312
	The wear of the centres of		312
	Setting over the tailstock to turn		312
	Gauge for setting over		313
	Fitting		313
	Grinding		313
	The order of procedure in turning		313
	The influence of the height of the tool in producing true		314
Special Forms. Curved work			314, 315
	Standard gauges for taper work		316
	Methods of turning an eccentric		317
	Turning a cylinder cover		318
	Turning pulleys		318
	Chucking device for pulleys		318
Cutting Screws in the lathe			319
	The arrangement of the change gears		319
	The intermediate wheels		319
	The compounded gears		320
	Finding the change wheels to cut a given thread		320
	Finding the change wheels for a lathe whose gears are compounded		321
	Finding the change gears for cutting fractional pitches		321
	To find what pitch of thread the wheels already on the lathe will cut		322
	Cutting left-hand threads		322
	Cutting double threads		322
	Cutting screws whose pitches are given in the terms of the metric system		322
	Cutting threads on taper work		323
	Errors in cutting threads on taper work		324
CHAPTER XIII.
EXAMPLES IN LATHE WORK (Continued).
Ball Turning with tubular saw			325
	With a single tooth on the end of a revolving tube		325
	With a removable tool on an arbor		325
	Tool holder with worm feed		325
	By hand		325
Cams, cutting in the lathe			326
	Improved method of originating cams in the lathe		326
	Motions for turning cams in the lathe		326, 327
	Application of cam motions to special work		327
	Cam chuck for irregular work		328
Milling or knurling tool			328
	Improved forms of		328
Winding Spiral Springs in the lathe			329
Hand Turning			330
	The heel tool		330
	The graver and its applications		330, 331
	Hand side tools		331
	Hand round-nosed tools for iron		331
	Hand finishing tool		331
Hand Tools, for roughing out brass work			332
	Various forms and applications of scrapers		332, 333
	Clockmakers’ hand tool for special or standard work		334
	Screw cutting with hand tools		334
		Outside and inside chasers	334
		Hobs and their uses	335
		The application of chasers, and errors that may arise from the position in which they are presented to the work	336
		Errors commonly made in cutting up inside chasers	337
		V-tool for starting outside threads	337
		Starting outside threads	338
		Cutting taper threads	338
	Wood turning hand tools		338
	The gauge and how to use it		338
	The chisel and its use		339
	The skew chisel and how to use it		339
	Wood turners’ boring tools for lathe work		340
CHAPTER XIV.
MEASURING MACHINES, TOOLS AND DEVICES.
Standards of Measurements, in various countries			341
	Use of, by sight and by the sense of feeling		341
	Variations in standard gauges		341
	The necessity for accurate standards		341
	The Rogers Bond standard measuring machine		342
		Details of construction of	343, 344
		The principle of construction of	344
		The methods of using	345
	The Whitworth measuring machine		345
	The Betts Machine Company’s measuring machine		346
	Professor Sweet’s measuring machine		347
	Measuring machine for sheet metal		348
Circle, division of the			348
	Troughton’s method of dividing the circle		348, 349
	Ramsden’s dividing engine		349
		The construction of	350, 351
	Pratt and Whitney’s dividing device		352
		Practical application of	353
	Index wheel, method of originating, by R. Hoe & Co.		353
		Application of the index wheel (Hoe & Co.’s system)	353
Classification of the measuring tools used by workmen			354
Micrometer Caliper and its principle of construction			354, 355
Gauges. Standard plug and collar gauges			356
	Methods of comparing standard plug and collar gauges		356
	The effects of variations of temperature upon standard gauges		356
	Plug and collar gauges for taper work		357
	The Baldwin standards for taper bolts		359
	Workmen’s gauges for lathe work		359
Calipers, outside, the various forms of			360
	Inside calipers		360
	Calipers with locking devices		360
	Spring calipers		360
	The methods of holding and using		361, 362
	Keyway calipers		363
	The advantages of calipers		363
Fitting. The four kinds of fit in machine work			363
	The influence of the diameter of the work in limiting the application of standard gauges		363
	The wear of tools and its influence upon the application of the standard gauge system		364
	The influence of the smoothness of the surface upon the allowance to be made for drilling or hydraulic fits		365
	Examples of allowance for hydraulic fits		365
	Parallel holes and taper plugs for hydraulic fits		365
Fitting. Practicable methods of testing the fit of axle brasses forced in by hydraulic pressure			366
	Shrinkage or contraction fits		366
		Allowances for	366
		Gauge for	367
		The shrinkage system at the Royal Gun Factory at Woolwich	367
		Experiments by Thomas Wrightson upon the shrinkage of iron under repeated heatings and coolings	368 to 374
		Shrinking work, to refit it	374, 375
CHAPTER XV.
MEASURING TOOLS.
End Measurements of large lathe work			376
	Template gauges for		376
	Trammels or Trains		377
	Adjustable gauges for		377
Compasses—Dividers			377
	Compass calipers		378
Key Seating rule			378
Surface Gauge			378
	Pattern makers’ pipe gauge		379
Squares. The try square			379
	The T square		379
	Various methods of testing squares		379, 380
	Bevel squares		380
Bevel Protractors			380
Hexagon Gauge			381
Straight Edge and its applications			381, 382
	Winding strips and their application		382
Surface Plate or planimeter			383
Templates for curves			384
Wire Gauges, notch			384
	Standard gauges for wire, &c.		384, 386
	Gauge for music wire		386
	Brown and Sharpe wire gauge		387
	Birmingham wire gauge for rolled shell silver and gold		387
	Sheet iron gauge, Russian		387
	Galvanized iron		387
	Belgian sheet zinc		387
	American sheet zinc		387
Rifle Bore gauge			387
Strength of Wire, Kirkaldy’s experiments			387, 388
CHAPTER XVI.
SHAPING AND PLANING MACHINES.
General description of a shaping machine			389
Construction of swivel head			389
	Slide		390
	Vice chuck		390
	Feed motion		390
Hand shaping machine			392
Quick Return Motion, Whitworth’s			392
Vice Chucks, the principles of construction of plain, for planing machine			392
	The proper methods of chucking work in		393
	Holding taper work in		394
	Various forms of		394
	Swiveling		395
	Rapid motion		396
	For vice work		396
Centres for shaping machines			397
Traveling Head in shaping machine			397
Planer Shapers or shaping machines, having a tappet motion for reversing the direction of motion			398, 399
Quick Return Motion shaping machines, link			399
	The Whitworth		400
	Comparisons of the link motion and Whitworth		401
Simple Crank, investigating the motion of			401
Planing Machines, or planer			402
	The various motions of		402, 403
	The table driving gear		404
	Planing machine with double heads		404
	Rotary planing machine		405
CHAPTER XVII.
PLANING MACHINERY.
The Sellers planing machine			406
	The belt shifting mechanism		406, 407
	The automatic feed motions		408
Sliding Head			408
Cross Bar			409
Slides of Planers, the various forms of construction of			410
Wear of the Slides of planer heads, various methods of taking up the			410
Swivel Heads			411
Tool Aprons			411
Swivel Tool-holding devices for planers			411
Planer Heads, graduations of			412
	Safety devices for		413
	Feed motions for		414
	V-guideways for		414
	Flat guideways for		415
	Oiling devices for		415
Planing Machine Tables			415
	Slots and holes in planing machine tables		416
	Forms of bolts for planer tables		417
	Supplementary tables for planer tables		417
	Angle plates for planer tables		418
	Chucking devices for planer tables		418
Planer Centres			418
Planer Chucks			419
	For spiral grooved work		419
	For curved work		420
	Chucking machine beds on planer tables		420
	For large planing machines		422
	Chucking the halves of large pulleys on a planer		423
Gauges for planing V-guideways in machine beds			421
	Planing guideways in machine beds		422
	Gauge for planer tools		424
Planer Tools, the shapes of			424
	For coarse finishing feeds		424
	The clearance of		424
	For slotted work		424
Planer Tool Holder, with tool post			425
	Various applications of		425
	Simple and advantageous form of		426
	Examples of application of		426
CHAPTER XVIII.
DRILLING MACHINES.
Drilling Machines. General description of a power drilling machine			428
	Lever feed		428
	With automatic and quick return feed motions		428
	Improved, with simple belt and uniform motion, two series of rates of automatic feed, and guide for boring bar		429, 430
	Radial		430, 431
	For boiler shells		436
	Cotter or keyway		438
	Drilling Machine, three-spindle		434
		Four-spindle	434
Drilling and Boring machine			431
	Feed motion of		432
Combined Drilling Machine and lathe			433
Boring Machine, horizontal			433
	For car wheels		438
	For pulleys		438
Quartering Machine			434
Drilling and Turning Machine for boiler makers			435
	Feed motions of		436
CHAPTER XIX.
DRILLS AND CUTTERS FOR DRILLING MACHINES.
Jigs or Fixtures for drilling machines			439
	Limits of error in		439
	Examples of, for simple work, as for links, &c.		440
	Considerations in designing		440
	For drilling engine cylinders		440 to 441
	For cutting out steam ports		441
Drills and Cutters for drilling machines			442
	Table of sizes of twist drills, and their shanks		442
	Flat drills for drilling machines		442
	Errors in grinding flat drills		443
	The tit-drill		443
	The lip drill		443
	Cotter or keyway drills		446
Drilling holes true to location with flat drills			444
	Drilling hard metal		444
	Table of sizes of tapping holes		445
Drill Shanks and sockets			445
	Improved form of drill shank		446
	Square shanked drills and their disadvantages		446
Drill Chucks			446
Stocks and Cutters for drilling machines			447
	Tube plate cutters		448
Stocks and Cutters. Adjustable stock and cutter			448
	Facing tool with reamer pin		449
	Counterbores for drilling machines		449
	Drill and counterbore for wood work		449
	Facing and countersink cutters		449
	Device for drilling square holes		450
	Device for drilling taper holes in a drilling machine		451
CHAPTER XX.
HAND-DRILLING AND BORING TOOLS, AND DEVICES.
The Brad-awl			452
Bits. The gimlet bit			452
	The German bit		452
	The nail bit		452
	The spoon bit		452
	The nose bit		453
	The auger bit		453
	Cook’s auger bit		453
	Principles governing the shapes of the cutting edges of auger bits		453
	Auger bit for boring end grain wood		453
	The centre bit		454
	The expanding bit		454
Drills. Drill for stone			454
	The fiddle drill		455
	The fiddle drill with feeding device		455
	Drill with cord and spring motion		455
	Drill stock with spiral grooves		455
	Drill brace		455
	Drill brace with ratchet motion		456
	Universal joint for drill brace		456
	Drill brace with multiplying gear and ratchet motion		456
	Breast drill with double gear		456
	Drilling levers for blacksmiths		457
	Drill cranks		457
	Ratchet brace		457
	Flexible shaft for driving drills		458
	Drilling device for lock work		459
	Hand drilling machine		459
Slotting Machine			459
	Sectional view of		460
	Tool holders		460, 461
	Tools		461, 462
CHAPTER XXI.
THREAD-CUTTING MACHINERY AND BROACHING PRESS.
Pipe Threading, die stock for, by hand			463
	Die stock for, by power		463
	Pipe threading machines, general construction of		463
Bolt Threading hand machine			464
	With revolving head		465
	Power threading machine		465
	With automatic stop motion		466
		Construction of the head	466
		Construction of the chasers	466
		Bolt threading machine with back gear	467
		Single rapid bolt threading machine	467
		Double rapid bolt threading machine	467
		Construction of the heads of the rapid machines	468
	Bolt threading machinery, the Acme		468
		Construction of the head of	468 to 470
		Capacity of	470
Cutting Edges for taps, the number of			471
	Examples when three and when four cutting edges are used, and the results upon bolts that are not round		471, 472
	Demonstration that four cutting edges are correct for bar iron		472
Positions of Dies, or chasers in the heads of bolt cutting machine			473
Dies, methods of hobbing, to avoid undue friction			473
	The construction of, for bolt threading machines		473
	Method of avoiding friction in thread cutting		474
	Hob for threading		474
	Cutting speeds for threading		474
Nut Tapping machine			475
	Automatic socket for		475
	Rotary		475
	Three-spindle		475
Pipe Threading Machine			475 to 477
Tapping Machine for steam pipe fittings			478
Broaching Press			478
	Principles of broaching		478
	Examples in the construction of broaches		479

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[I-xiv]

List of
plates
Vol. II.

Facing
Frontispiece.		MODERN LOCOMOTIVE ENGINE.	Title Page
Plate	I.	TEMPLATE-CUTTING MACHINES FOR GEAR TEETH.	34
„	II.	FORMS OF SCREW THREADS.	85
„	III.	MEASURING AND GAUGING SCREW THREADS.	93
„	IV.	END-ADJUSTMENT AND LOCKING DEVICES.	120
„	V.	EXAMPLES IN LATHE CONSTRUCTION.	148
„	VI.	CHUCKING LATHES.	150
„	VII.	TOOL-HOLDING AND ADJUSTING APPLIANCES.	174
„	VIII.	WATCHMAKER’S LATHE.	188
„	IX.	DETAILS OF WATCHMAKER’S LATHE.	188
„	X.	EXAMPLES OF SCREW MACHINES.	200
„	XI.	ROLL-TURNING LATHE.	215
„	XII.	EXAMPLES IN ANGLE-PLATE CHUCKING.	252
„	XIII.	METHODS OF BALL-TURNING.	325
„	XIV.	STANDARD MEASURING MACHINES.	341
„	XV.	DIVIDING ENGINE AND MICROMETER.	354
„	XVI.	SHAPING MACHINES AND TABLE-SWIVELING DEVICES.	398
„	XVII.	EXAMPLES OF PLANING MACHINES.	404
„	XVIII.	EXAMPLES IN PLANING WORK.	422
„	XIX.	LIGHT DRILLING MACHINES.	428
„	XX.	HEAVY DRILLING MACHINES.	430
„	XXI.	EXAMPLES IN BORING MACHINERY.	434
„	XXII.	BOILER-DRILLING MACHINERY.	436
„	XXIII.	NUT-TAPPING MACHINERY.	475

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[I-1]

A wheel that is provided with teeth to mesh, engage, or gear with similar teeth upon another wheel, so that the motion of one may be imparted to the other, is called, in general terms, a gear-wheel.

When the teeth are arranged to be parallel to the wheel-axis, as in Fig. 1, the wheel is termed a spur-wheel. In the figure, a represents the axial line or axis of the wheel or of its shaft, to which the teeth are parallel while spaced equidistant around the rim, or face, as it is termed, of the wheel.

When the wheel has its teeth arranged at an angle to the shaft, as in Fig. 2, it is termed a bevel-wheel, or bevel gear; but when this angle is one of 45°, as in Fig. 3, as it must be if the pair of wheels are of the same diameter, so as to make the revolutions of their shafts equal, then the wheel is called a mitre-wheel. When the teeth are arranged upon the radial or side face of the wheel, as in Fig. 4, it is termed a crown-wheel. The smallest wheel of a pair, or of a train or set of gear-wheels, is termed the pinion; and when the teeth are composed of rungs, as in Fig. 5, it is termed a lantern, trundle, or wallower; and each cylindrical piece serving as a tooth is termed a stave, spindle, or round, and by some a leaf.

An annular or internal gear-wheel is one in which the faces of the teeth are within and the flanks without, or outside the pitch-circle, as in Fig. 6; hence the pinion p operates within the wheel.

When the teeth of a wheel are inserted in mortises or slots provided in the wheel-rim, it is termed a mortised-wheel, or a cogged-wheel, and the teeth are termed cogs.

When the teeth are arranged along a plane surface or straight line, as in Fig. 7, the toothed plane is termed a rack, and the wheel is termed a pinion.

A wheel that is driven by a revolving screw, or worm as it is termed, is called a worm-wheel, the arrangement of a worm and worm-wheel being shown in Fig. 8. The screw or worm is sometimes also called an endless screw, because its action upon the wheel does not come to an end as it does when it is revolved in one continuous direction and actuates a nut. So also, since the worm is tangent to the wheel, the arrangement is sometimes called a wheel and tangent screw.

The diameter of a gear-wheel is always taken at the pitch circle, unless otherwise specially stated as “diameter over all,” “diameter of addendum,” or “diameter at root of teeth,” &c., &c.

When the teeth of wheels engage to the proper distance, which is when the pitch circles meet, they are said to be in gear, or geared together. It is obvious that if two wheels are to be geared together their teeth must be the same distance apart, or the same pitch, as it is called.

The designations of the various parts or surfaces of a tooth of a gear-wheel are represented in Fig. 9, in which the surface a is the face of the tooth, while the dimension f is the width of face of the wheel, when its size is referred to. b is the flank or distance from the pitch line to the root of the tooth, and c the point. h is the space, or the distance from the side of one tooth to the nearest side of the next tooth, the width of space being measured on the pitch circle p p. e is the depth of the tooth, and g its thickness, the latter also being measured on the pitch circle p p. When spoken of with reference to a tooth, p p is called the pitch line, but when the whole wheel is referred to it becomes the pitch circle.

The points c and the surface h are true to the wheel axis.

The teeth are designated for measurement by the pitch; the height or depth above and below pitch line; and the thickness.

The pitch, however, may be measured in two ways, to wit, around the pitch circle a, in Fig. 10, which is called the arc or circular pitch, and across b, which is termed the chord pitch.

In proportion as the diameter of a wheel (having a given pitch) is increased, or as the pitch of the teeth is made finer (on a wheel of a given diameter) the arc and chord pitches more nearly coincide in length. In the practical operations of marking out the teeth, however, the arc pitch is not necessarily referred to, for if the diameter of the pitch circle be made correct for the required number of teeth having the necessary arc pitch, and the wheel be accurately divided off into the requisite number of divisions with compasses set to the chord pitch, or by means of an index plate, then the arc pitch must necessarily be correct, although not referred to, save in determining the diameter of the wheel at the pitch circle.

The difference between the width of a space and the thickness of the tooth (both being measured on the pitch circle or pitch line) is termed the clearance or side clearance, which is necessary to prevent the teeth of one wheel from becoming locked in the spaces of the other. The amount of clearance is, when the teeth are cut to shape in a machine, made just sufficient to prevent contact on one side of the teeth when they are in proper gear (the pitch circles meeting in the line of centres). But when the teeth are cast upon the wheel the clearance is increased to allow for the slight inequalities of tooth shape that is incidental to casting them. The amount of clearance given is varied to suit the method employed to mould the wheels, as will be explained hereafter.

The line of centres is an imaginary line from the centre or axis of one wheel to the axis of the other when the two are in gear; hence each tooth is most deeply engaged, in the space of the other wheel, when it is on the line of centres.

There are three methods of designating the sizes of gear-wheels. First, by their diameters at the pitch circle or pitch diameter and the number of teeth they contain; second, by the number of teeth in the wheel and the pitch of the teeth; and third, by a system known as diametral pitch.

The first is objectionable because it involves a calculation to find the pitch of the teeth; furthermore, if this calculation be made by dividing the circumference of the pitch circle by the number of teeth in the wheel, the result gives the arc pitch, which cannot be measured correctly by a lineal measuring rule, especially if the wheel be a small one having but few teeth, or of coarse pitch, as, in that case, the arc pitch very sensibly differs from the chord pitch, and a second calculation may become necessary to find the chord pitch from the arc pitch.

The second method (the number and pitch of the teeth) possesses the disadvantage that it is necessary to state whether the pitch is the arc or the chord pitch.

If the arc pitch is given it is difficult to measure as before, while if the chord pitch is given it possesses the disadvantage that the diameters of the wheels will not be exactly proportional to the numbers of teeth in the respective wheels. For instance, a wheel with 20 teeth of 2 inch chord pitch is not exactly half the diameter of one of 40 teeth and 2 inch chord pitch.

To find the chord pitch of a wheel take 180 (= half the degrees in a circle) and divide it by the number of teeth in the wheel. In a table of natural sines find the sine for the number so found, which multiply by 2, and then by the radius of the wheel in inches.

Example.—What is the chord pitch of a wheel having 12 teeth and a diameter (at pitch circle) of 8 inches? Here 180 ÷ 12 = 15; (sine of 15 is .25881). Then .25881 × 2 = .51762 × 4 (= radius of wheel) = 2.07048 inches = chord pitch.

Degrees.	Sine.	Degrees.	Sine.	Degrees.	Sine.
1	.01745	16	.27563	31	.51503
2	.03489	17	.29237	32	.52991
3	.05233	18	.30901	33	.54463
4	.06975	19	.32556	34	.55919
5	.08715	20	.34202	35	.57357
6	.10452	21	.35836	36	.58778
7	.12186	22	.37460	37	.60181
8	.13917	23	.39073	38	.61566
9	.15643	24	.40673	39	.62932
10	.17364	25	.42261	40	.64278
11	.19080	26	.43837	41	.65605
12	.20791	27	.45399	42	.66913
13	.22495	28	.46947	43	.68199
14	.24192	29	.48480	44	.69465
15	.25881	30	.50000	45	.70710

The principle upon which diametral pitch is based is as follows:—

The diameter of the wheel at the pitch circle is supposed to be divided into as many equal parts or divisions as there are teeth in the wheel, and the length of one of these parts is the diametral pitch. The relationship which the diametral bears to the arc pitch is the same as the diameter to the circumference, hence a diametral pitch which measures 1 inch will accord with an arc pitch of 3.1416; and it becomes evident that, for all arc pitches of less than 3.1416 inches, the corresponding diametral pitch must be expressed in fractions of an inch, as ¹⁄₂, ¹⁄₃, ¹⁄₄, and so on, increasing the denominator until the fraction becomes so small that an arc with which it accords is too fine to be of practical service. The numerators of these fractions being 1, in each case, they are in practice discarded, the denominators only being used, so that, instead of saying diametral pitches of ¹⁄₂, ¹⁄₃, or ¹⁄₄, we say diametral pitches of 2, 3, or 4, meaning that there are 2, 3, or 4 teeth on the wheel for every inch in the diameter of the pitch circle.

Suppose now we are given a diametral pitch of 2. To obtain the corresponding arc pitch we divide 3.1416 (the relation of the circumference to the diameter) by 2 (the diametral pitch), and 3.1416 ÷ 2 = 1.57 = the arc pitch in inches and decimal parts of an inch. The reason of this is plain, because, an arc pitch of 3.1416 inches being represented by a diametral pitch of 1, a diametral pitch of ¹⁄₂ (or 2 as it is called) will be one half of 3.1416. The advantage of discarding the numerator is, then, that we avoid the use of fractions and are readily enabled to find any arc pitch from a given diametral pitch.

Examples.—Given a 5 diametral pitch; what is the arc pitch? First (using the full fraction ¹⁄₅) we have ¹⁄₅ × 3.1416 = .628 = the arc pitch. Second (discarding the numerator), we have 3.1416 ÷ 5 = .628 = arc pitch. If we are given an arc pitch to find a corresponding diametral pitch we again simply divide 3.1416 by the given arc pitch.

Example.—What is the diametral pitch of a wheel whose arc pitch is 1¹⁄₂ inches? Here 3.1416 ÷ 1.5 = 2.09 = diametral pitch. The reason of this is also plain, for since the arc pitch is to the diametral pitch as the circumference is to the diameter we have: as 3.1416 is to 1, so is 1.5 to the required diametral pitch; then 3.1416 × 1 ÷ 1.5 = 2.09 = the required diametral pitch.

To find the number of teeth contained in a wheel when the diameter and diametral pitch is given, multiply the diameter in inches by the diametral pitch. The product is the answer. Thus, how many teeth in a wheel 36 inches diameter and of 3 diametral pitch? Here 36 × 3 = 108 = the number of teeth sought. Or, per contra, a wheel of 36 inches diameter has 108 teeth. What is the diametral pitch? 108 ÷ 36 = 3 = the diametral pitch. Thus it will be seen that, for determining the relative sizes of wheels, this system is excellent from its simplicity. It also possesses the advantage that, by adding two parts of the diametral pitch to the pitch diameter, the outside diameter of the wheel or the diameter of the addendum is obtained. For instance, a wheel containing 30 teeth of 10 pitch would be 3 inches diameter on the pitch circle and 3²⁄₁₀ outside or total diameter.

Again, a wheel having 40 teeth of 8 diametral pitch would have a pitch circle diameter of 5 inches, because 40 ÷ 8 = 5, and its full diameter would be 5¹⁄₄ inches, because the diametral pitch is ¹⁄₈, and this multiplied by 2 gives ¹⁄₄, which added to the pitch circle diameter of 5 inches makes 5¹⁄₄ inches, which is therefore the diameter of the addendum, or, in other words, the full diameter of the wheel.

Suppose now that a pair of wheels require to have pitch circles of 5 and 8 inches diameter respectively, and that the arc pitch requires to be, say, as near as may be ⁴⁄₁₀ inch; to find a suitable pitch and the number of teeth by the diametral pitch system we proceed as follows:

In the following table are given various arc pitches, and the corresponding diametral pitch.

Diametral Pitch.		Arc Pitch.		Arc Pitch.		Diametral Pitch.
				Inch.
2		1	.57	1	.75	1	.79
2	.25	1	.39	1	.5	2	.09
2	.5	1	.25	1	.4375	2	.18
2	.75	1	.14	1	.375	2	.28
3		1	.04	1	.3125	2	.39
3	.5		.890	1	.25	2	.51
4			.785	1	.1875	2	.65
5			.628	1	.125	2	.79
6			.523	1	.0625	2	.96
7			.448	1	.0000	3	.14
8			.392	0	.9375	3	.35
9			.350	0	.875	3	.59
10			.314	0	.8125	3	.86
11			.280	0	.75	4	.19
12			.261	0	.6875	4	.57
14			.224	0	.625	5	.03
16			.196	0	.5625	5	.58
18			.174	0	.5	6	.28
20			.157	0	.4375	7	.18
22			.143	0	.375	8	.38
24			.130	0	.3125	10	.00
26			.120	0	.25	12	.56

From this table we find that the nearest diametral pitch that will correspond to an arc pitch of ⁴⁄₁₀ inch is a diametral pitch of 8, which equals an arc pitch of .392, hence we multiply the pitch circles (5 and 8,) by 8, and obtain 40 and 64 as the number of teeth, the arc pitch being .392 of an inch. To find the number of teeth and pitch by the arc pitch and circumference of the pitch circle, we should require to find the circumference of the pitch circle, and divide this by the nearest arc pitch that would divide the circumference without leaving a remainder, which would entail more calculating than by the diametral pitch system.

The designation of pitch by the diametral pitch system is, however, not applied in practice to coarse pitches, nor to gears in which the teeth are cast upon the wheels, pattern makers generally preferring to make the pitch to some measurement that accords with the divisions of the ordinary measuring rule.

Of two gear-wheels that which impels the other is termed the driver, and that which receives motion from the other is termed the driven wheel or follower; hence in a single pair of wheels in gear together, one is the driver and the other the driven wheel or follower. But if there are three wheels in gear together, the middle one will be the follower when spoken of with reference to the first or prime mover, and the driver, when mentioned with reference to the third wheel, which will be a follower. A series of more than two wheels in gear together is termed a train of wheels or of gearing. When the wheels in a train are in gear continuously, so that each wheel, save the first and last, both receives and imparts motion, it is a simple train, the first wheel being the driver, and the last the follower, the others being termed intermediate wheels. Each of these intermediates is a follower with reference to the wheel that drives it, and a driver to the one that it drives. But the velocity of all the wheels in the train is the same in fact per second (or in a given space of time), although the revolutions in that space of time may vary; hence a simple train of wheels transmits motion without influencing its velocity. To alter the velocity (which is always taken at a point on the pitch circle) the gearing must be compounded, as in Fig. 11, in which a, b, c, e are four wheels in gear, b and c being compounded, that is, so held together on the shaft d that both make an equal number of revolutions in a given time. Hence the velocity of c will be less than that of b in proportion as the diameter, circumference, radius, or number of teeth in c, varies from the diameter, radius, circumference, or number of teeth (all the wheels being supposed to have teeth of the same pitch) in b, although the rotations of b and c are equal. It is most convenient, and therefore usual, to take the number of teeth, but if the teeth on c (and therefore those on e also) were of different pitch from those on b, the radius or diameters of the wheels must be taken instead of the pitch, when the velocities of the various wheels are to be computed. It is obvious that the compounded pair of wheels will diminish the velocity when the driver of the compounded pair (as c in the figure) is of less radius than the follower b, and conversely that the velocity will be increased when the driver is of greater radius than the follower of the compound pair.

The diameter of the addendum or outer circle of a wheel has no influence upon the velocity of the wheel. Suppose, for example, that we have a pair of wheels of 3 inch arc or circular pitch, and containing 20 teeth, the driver of the two making one revolution per minute. Suppose the driven wheel to have fast upon its shaft a pulley whose diameter is one foot, and that a weight is suspended from a line or cord wound around this pulley, then (not taking the thickness of the line into account) each rotation of the driven wheel would raise the weight 3.1416 feet (that being the circumference of the pulley). Now suppose that the addendum circle of either of the wheels were cut off down to the pitch circle, and that they were again set in motion, then each rotation of the driven wheel would still raise the weight 3.1416 feet as before.

It is obvious, however, that the addendum circle must be sufficiently larger than the pitch circle to enable at least one pair of teeth to be in continuous contact; that is to say, it is obvious that contact between any two teeth must not cease before contact between the next two has taken place, for otherwise the motion would not be conveyed continuously. The diameter of the pitch circle cannot be obtained from that of the addendum circle unless the pitch of the teeth and the proportion of the pitch allowed for the addendum be known. But if these be known the diameter of the pitch circle may be obtained by subtracting from that of the addendum circle twice the amount allowed for the addendum of the tooth.

Example.—A wheel has 19 teeth of 3 inch arc pitch; the addendum of the tooth or teeth equals ³⁄₁₀ of the pitch, and its addendum circle measures 19.943 inches; what is the diameter of the pitch circle? Here the addendum on each side of the wheel equals (³⁄₁₀ of 3 inches) = .9 inches, hence the .9 must be multiplied by 2 for the two sides of the wheel, thus, .9 × 2 = 1.8. Then, diameter of addendum circle 19.943 inches less 1.8 inches = 18.143 inches, which is the diameter of the pitch circle.

Proof.—Number of teeth = 19, arc pitch 3, hence 19 × 3 = 57 inches, which, divided by 3.1416 (the proportion of the circumference to the diameter) = 18.143 inches.

If the distance between the centres of a pair of wheels that are in gear be divided into two parts whose lengths are in the same proportion one to the other as are the numbers of teeth in the wheels, then these two parts will represent the radius of the pitch circles of the respective wheels. Thus, suppose one wheel to contain 100 and the other 50 teeth, and that the distance between their centres is 18 inches, then the pitch radius or pitch diameter of one will be twice that of the other, because one contains twice as many teeth as the other. In this case the radius of pitch circle for the large wheel will be 12 inches, and that for the small one 6 inches, because 12 added to 6 makes 18, which is the distance between the wheel centres, and 12 is in the same proportion to 6 that 100 is to 50.

A simple rule whereby to find the radius of the pitch circles of a pair of wheels is as follows:—

Rule.—Divide number of teeth in the large wheel by the number in the small one, and to the sum so obtained add 1. Take this amount and divide it into the distance between the centres of the wheels, and the result will be the radius of the smallest wheel. To obtain the radius of the largest wheel subtract the radius of the smallest wheel from the distance between the wheel centres.

Example.—Of a pair of wheels, one has 100 and the other 50 teeth, the distance between their centres is 18 inches; what is the pitch radius of each wheel?

Here 100 ÷ 50 = 2, and 2 + 1 = 3. Then 18 ÷ 3 = 6, hence the pitch radius of the small wheel is 6 inches. Then 18 - 6 = 12 = pitch radius of large wheel.

Example 2.—Of a pair of wheels one has 40 and the other 90 teeth. The distance between the wheel centres is 32¹⁄₂ inches; what are the radii of the respective pitch circles? 90 ÷ 40 = 2.25 and 2.25 + 1 = 3.25. Then 32.5 ÷ 3.25 = 10 = pitch radius of small wheel, and 32.5 - 10 = 22.5, which is the pitch radius of the large wheel.

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.

In Fig. 12, let a, b, and c represent the pitch circles of three gears of which a and b are in gear, while c is compounded with b; let e be the shaft of a, and g that for b and c. Let a be 60 inches, b = 30 inches, and c = 40 inches in diameter. Now suppose that shaft e suspends from its perimeter a weight of 50 lbs., the shaft being 4 inches in diameter. Then this weight will be at a leverage of 2 inches from the centre of e and the 50 must be multiplied by 2, making 100 lbs. at the centre of e. Then at the perimeter of a this 100 will become one-thirtieth of one hundred, because from the centre to the perimeter of a is 30. One-thirtieth of 100 is 3³³⁄₁₀₀ lbs., which will be the force exerted by a on the perimeter of b. Now from the perimeter of b to its centre (or in other words its radius) is 15 inches, hence the 3³³⁄₁₀₀ lbs. at its perimeter will become fifteen times as much at the centre g of b, and 3³³⁄₁₀₀ × 15 = 49⁹⁵⁄₁₀₀ lbs. From the centre g to the perimeter of c being 20 inches, the 49⁹⁵⁄₁₀₀ lbs. at the centre will be only one-twentieth of that amount at the perimeter of c, hence 49⁹⁵⁄₁₀₀ ÷ 20 = 2⁴⁹⁄₁₀₀ lbs., which is the amount of force at the perimeter of c.

Here we have treated the wheels as simple levers, dividing the weight by the length of the levers in all cases where it is transmitted from the shaft to the perimeter, and multiplying it by the length of the lever when it is transmitted from the perimeter of the wheel to the centre of the shaft. The precise same result will be reached if we take the diameter of the wheels or the number of the teeth, providing the pitch of the teeth on all the wheels is alike.

Suppose, for example, that a has 60 teeth, b has 30 teeth, and c has 40 teeth, all being of the same pitch. Suppose the 50 lb. weight be suspended as before, and that the circumference of the shaft be equal to that of a pinion having 4 teeth of the same pitch as the wheels. Then the 50 multiplied by the 4 becomes 200, which divided by 60 (the number of teeth on a) becomes 3³³⁄₁₀₀, which multiplied by 30 (the number of teeth on b) becomes 99⁹⁰⁄₁₀₀, which divided by 40 (the number of teeth on c) becomes 2⁴⁹⁄₁₀₀ lbs. as before.

It may now be explained why the shaft was taken as equal to a pinion having 4 teeth. Its diameter was taken as 4 inches and the wheel diameter was taken as being 60 inches, and it was supposed to contain 60 teeth, hence there was 1 tooth to each inch of diameter, and the 4 inches diameter of shaft was therefore equal to a pinion having 4 teeth. From this we may perceive the philosophy of the rule that to obtain the revolutions of wheels we multiply the given revolutions by the teeth in the driving wheels and divide by the teeth in the driven wheels.

Suppose that a (Fig. 13) makes 1 revolution per minute, how many will c make, a having 60 teeth, b 30 teeth, and c 40 teeth? In this case we have but one driving wheel a, and one driven wheel b, the driver having 60 teeth, the driven 30, hence 60 ÷ 30 = 2, equals revolutions of b and also of c, the two latter being on the same shaft.

It will be observed then that the revolutions are in the same proportion as the numbers of the teeth or the radii of the wheels, or what is the same thing, in the same proportion as their diameters. The number of teeth, however, is usually taken as being easier obtained than the diameter of the pitch circles, and easier to calculate, because the teeth will be represented by a whole number, whereas the diameter, radius, or circumference, will generally contain fractions.

Suppose that the 4 wheels in Fig. 14 have the respective numbers of teeth marked beside them, and that the upper one having 40 teeth makes 60 revolutions per minute, then we may obtain the revolutions of the others as follows:—

Revolu- tions.		Teeth in first driver.		Teeth in first driven.		Teeth in second driver.		Teeth in second driven.
60	×	40	÷	60	×	20	÷	120	=	666⁄100

and a remainder of the reciprocating decimals. We may now prove this by reversing the question, thus. Suppose the 120 wheel to make 6⁶⁶⁄₁₀₀ revolutions per minute, how many will the 40 wheel make?

Revolu- tions.		Teeth in first driver.		Teeth in first driven.		Teeth in second driver.		Teeth in second driven.
6.66	×	120	÷	20	×	60	÷	40	=	5999⁄100	=

revolutions of the 40 wheel, the discrepancy of ¹⁄₁₀₀ being due to the 6.66 leaving a remainder and not therefore being absolutely correct.

That the amount of power transmitted by gearing, whether compounded or not, is equal throughout every wheel in the train, may be shown as follows:—

Referring again to Fig. 10, it has been shown that with a 50 lb. weight suspended from a 4 inch shaft e, there would be 30³³⁄₁₀₀ lbs. at the perimeter of a. Now suppose a rotation be made, then the 50 lb. weight would fall a distance equal to the circumference of the shaft, which is (3.1416 × 4 = 12⁵⁶⁄₁₀₀) 12⁵⁶⁄₁₀₀ inches. Now the circumference of the wheel is (60 dia. × 3.1416 = 188⁴⁹⁄₁₀₀ cir.) 188⁴⁹⁄₁₀₀ inches, which is the distance through which the 3³³⁄₁₀₀ lbs. would move during one rotation of a. Now 3.33 lbs. moving through 188.49 inches represents the same amount of power as does 50 lbs. moving through a distance of 12.56 inches, as may be found by converting the two into inch lbs. (that is to say, into the number of inches moved by 1 lb.), bearing in mind that there will be a slight discrepancy due to the fact that the fractions .33 in the one case, and .56 in the other are not quite correct. Thus:

188.49	inches	×	3.33	lbs.	=	627	.67	inch	lbs.,	and
12.56	„	×	50	„	=	628		„	„

Taking the next wheels in Fig. 12, it has been shown that the 3.33 lbs. delivered from a to the perimeter of b, becomes 2.49 lbs. at the perimeter of c, and it has also been shown that c makes two revolutions to one of a, and its diameter being 40 inches, the distance this 2.49 lbs. will move through in one revolution of a will therefore be equal to twice its circumference, which is (40 dia. × 3.1416 = 125.666 cir., and 125.666 × 2 = 251.332) 251.332 inches. Now 2.49 lbs. moving through 251.332 gives when brought to inch lbs. 627.67 inch lbs., thus 251.332 × 2.49 = 627.67. Hence the amount of power remains constant, but is altered in form, merely being converted from a heavy weight moving a short distance, into a lighter one moving a distance exactly as much greater as the weight or force is lessened or lighter.

Gear-wheels therefore form a convenient method of either simply transmitting motion or power, as when the wheels are all of equal diameter, or of transmitting it and simultaneously varying its velocity of motion, as when the wheels are compounded either to reduce or increase the speed or velocity in feet per second of the prime mover or first driver of the train or pair, as the case may be.

In considering the action of gear-teeth, however, it sometimes is more convenient to denote their motion by the number of degrees of angle they move through during a certain portion of a revolution, and to refer to their relative velocities in terms of the ratio or proportion existing between their velocities. The first of these is termed the angular velocity, or the number of degrees of angle the wheel moves through during a given period, while the second is termed the velocity ratio of the pair of wheels. Let it be supposed that two wheels of equal diameter have contact at their perimeters so that one drives the other by friction without any slip, then the velocity of a point on the perimeter of one will equal that of a point on the other. Thus in Fig. 15 let a and b represent the pitch circles of two wheels, and c an imaginary line joining the axes of the two wheels and termed the line of centres. Now the point of contact of the two wheels will be on the line of centres as at d, and if a point or dot be marked at d and motion be imparted from a to b, then when each wheel has made a quarter revolution the dot on a will have arrived at e while that on b will have arrived at f. As each wheel has moved through one quarter revolution, it has moved through 90° of angle, because in the whole circle there is 360°, one quarter of which is 90°, hence instead of saying that the wheels have each moved through one quarter of a revolution we may say they have moved through an angle of 90°, or, in other words, their angular velocity has, during this period, been 90°. And as both wheels have moved through an equal number of degrees of angle their velocity ratio or proportion of velocity has been equal.

Obviously then the angular velocity of a wheel represents a portion of a revolution irrespective of the diameter of the wheel, while the velocity ratio represents the diameter of one in proportion to that of the other irrespective of the actual diameter of either of them.

Now suppose that in Fig. 16 a is a wheel of twice the diameter of b; that the two are free to revolve about their fixed centres, but that there is frictional contact between their perimeters at the line of centres sufficient to cause the motion of one to be imparted to the other without slip or lost motion, and that a point be marked on both wheels at the point of contact d. Now let motion be communicated to a until the mark that was made at d has moved one-eighth of a revolution and it will have moved through an eighth of a circle, or 45°. But during this motion the mark on b will have moved a quarter of a revolution, or through an angle of 90° (which is one quarter of the 360° that there are in the whole circle). The angular velocities of the two are, therefore, in the same ratio as their diameters, or two to one, and the velocity ratio is also two to one. The angular velocity of each is therefore the number of degrees of angle that it moves through in a certain portion of a revolution, or during the period that the other wheel of the pair makes a certain portion of a revolution, while the velocity ratio is the proportion existing between the velocity of one wheel and that of the other; hence if the diameter of one only of the wheels be changed, its angular velocity will be changed and the velocity ratio of the pair will be changed. The velocity ratio may be obtained by dividing either the radius, pitch, diameter, or number of teeth of one wheel into that of the other.

Conversely, if a given velocity ratio is to be obtained, the radius, diameter, or number of teeth of the driver must bear the same relation to the radius, diameter, or number of teeth of the follower, as the velocity of the follower is desired to bear to that of the driver.

If a pair of wheels have an equal number of teeth, the same pairs of teeth will come into action at every revolution; but if of two wheels one is twice as large as the other, each tooth on the small wheel will come into action twice during each revolution of the large one, and will work during each successive revolution with the same two teeth on the large wheel; and an application of the principle of the hunting tooth is sometimes employed in clocks to prevent the overwinding of their springs, the device being shown in Fig. 17, which is from “Willis’ Principles of Mechanism.”

For this purpose the winding arbor c has a pinion a of 19 teeth fixed to it close to the front plate. A pinion b of 18 teeth is mounted on a stud so as to be in gear with the former. A radial plate c d is fixed to the face of the upper wheel a, and a similar plate f e to the lower wheel b. These plates terminate outward in semicircular noses d, e, so proportioned as to cause their extremities to abut against each other, as shown in the figure, when the motion given to the upper arbor by the winding has brought them into the position of contact. The clock being now wound up, the winding arbor and wheel a will begin to turn in the opposite direction. When its first complete rotation is effected the wheel b will have gained one tooth distance from the line of centres, so as to place the stop d in advance of e and thus avoid a contact with e, which would stop the motion. As each turn of the upper wheel increases the distance of the stops, it follows from the principle of the hunting cog, that after eighteen revolutions of a and nineteen of b the stops will come together again and the clock be prevented from running down too far. The winding key being applied, the upper wheel a will be rotated in the opposite direction, and the winding repeated as above.

Thus the teeth on one wheel will wear to imbed one upon the other. On the other hand the teeth of the two wheels may be of such numbers that those on one wheel will not fall into gear with the same teeth on the other except at intervals, and thus an inequality on any one tooth is subjected to correction by all the teeth in the other wheel. When a tooth is added to the number of teeth on a wheel to effect this purpose it is termed a hunting cog, or hunting tooth, because if one wheel have a tooth less, then any two teeth which meet in the first revolution are distant, one tooth in the second, two teeth in the third, three in the fourth, and so on. The odd tooth is on this account termed a hunting tooth.

It is obvious then that the shape or form to be given to the teeth must, to obtain correct results, be such that the motion of the driver will be communicated to the follower with the velocity due to the relative diameters of the wheels at the pitch circles, and since the teeth move in the arc of a circle it is also obvious that the sides of the teeth, which are the only parts that come into contact, must be of same curve. The nature of this curve must be such that the teeth shall possess the strength necessary to transmit the required amount of power, shall possess ample wearing surface, shall be as easily produced as possible for all the varying conditions, shall give as many teeth in constant contact as possible, and shall, as far as possible, exert a pressure in a direction to rotate the wheels without inducing undue wear upon the journals of the shafts upon which the wheels rotate. In cases, however, in which some of these requirements must be partly sacrificed to increase the value of the others, or of some of the others, to suit the special circumstances under which the wheels are to operate, the selection is left to the judgment of the designer, and the considerations which should influence his determinations will appear hereafter.

Modern practice has accepted the curve known in general terms as the cycloid, as that best filling all the requirements of wheel teeth, and this curve is employed to produce two distinct forms of teeth, epicycloidal and involute. In epicycloidal teeth the curve forming the face of the tooth is designated an epicycloid, and that forming the flank an hypocycloid. An epicycloid may be traced or generated, as it is termed, by a point in the circumference of a circle that rolls without slip upon the circumference of another circle. Thus, in Fig. 18, a and b represent two wooden wheels, a having a pencil at p, to serve as a tracing or marking point. Now, if the wheels are laid upon a sheet of paper and while holding b in a fixed position, roll a in contact with b and let the tracing point touch the paper, the point p will trace the curve c c. Suppose now the diameter of the base circle b to be infinitely large, a portion of its circumference may be represented by a straight line, and the curve traced by a point on the circumference of the generating circle as it rolls along the base line b is termed a cycloid. Thus, in Fig. 19, b is the base line, a the rolling wheel or generating circle, and c c the cycloidal curve traced or marked by the point d when a is rolled along b. If now we suppose the base line b to represent the pitch line of a rack, it will be obvious that part of the cycloid at one end is suitable for the face on one side of the tooth, and a part at the other end is suitable for the face of the other side of the tooth.

A hypocycloid is a curve traced or generated by a point on the circumference of a circle rolling within and in contact (without slip) with another circle. Thus, in Fig. 20, a represents a wheel in contact with the internal circumference of b, and a point on its circumference will trace the two curves, c c, both curves starting from the same point, the upper having been traced by rolling the generating circle or wheel a in one direction and the lower curve by rolling it in the opposite direction.

To demonstrate that by the epicycloidal and hypocycloidal curves, forming the faces and flanks of what are known as epicycloidal teeth, motion may be communicated from one wheel to another with as much uniformity as by frictional contact of their circumferential surfaces, let a, b, in Fig. 21, represent two plain wheel disks at liberty to revolve about their fixed centres, and let c c represent a margin of stiff white paper attached to the face of b so as to revolve with it. Now suppose that a and b are in close contact at their perimeters at the point g, and that there is no slip, and that rotary motion commenced when the point e (where as tracing point a pencil is attached), in conjunction with the point f, formed the point of contact of the two wheels, and continued until the points e and f had arrived at their respective positions as shown in the figure; the pencil at e will have traced upon the margin of white paper the portion of an epicycloid denoted by the curve e f; and as the movement of the two wheels a, b, took place by reason of the contact of their circumferences, it is evident that the length of the arc e g must be equal to that of the arc g f, and that the motion of a (supposing it to be the driver) would be communicated uniformly to b.

Now suppose that the wheels had been rotated in the opposite direction and the same form of curve would be produced, but it would run in the opposite direction, and these two curves may be utilized to form teeth, as in Fig. 22, the points on the wheel a working against the curved sides of the teeth on b.

To render such a pair of wheels useful in practice, all that is necessary is to diminish the teeth on b without altering the nature of the curves, and increase the diameter of the points on a, making them into rungs or pins, thus forming the wheels into what is termed a wheel and lantern, which are illustrated in Fig. 23.

a represents the pinion (or lantern), and b the wheel, and c, c, the primitive teeth reduced in thickness to receive the pins on a. This reduction we may make by setting a pair of compasses to the radius of the rung and describing half-circles at the bottom of the spaces in b. We may then set a pair of compasses to the curve of c, and mark off the faces of the teeth of b to meet the half-circles at the pitch line, and reduce the teeth heights so as to leave the points of the proper thickness; having in this operation maintained the same epicycloidal curves, but brought them closer together and made them shorter. It is obvious, however, that such a method of communicating rotary motion is unsuited to the transmission of much power; because of the weakness of, and small amount of wearing surface on, the points or rungs in a.

In place of points or rungs we may have radial lines, these lines, representing the surfaces of ribs, set equidistant on the radial face of the pinion, as in Fig. 24. To determine the epicycloidal curves for the faces of teeth to work with these radial lines, we may take a generating circle c, of half the diameter of a, and cause it to roll in contact with the internal circumference of a, and a tracing point fixed in the circumference of c will draw the radial lines shown upon a. The circumstances will not be altered if we suppose the three circles, a, b, c, to be movable about their fixed centres, and let their centres be in a straight line; and if, under these circumstances, we suppose rotation to be imparted to the three circles, through frictional contact of their perimeters, a tracing point on the circumference of c would trace the epicycloids shown upon b and the radial lines shown upon a, evidencing the capability of one to impart uniform rotary motion to the other.

To render the radial lines capable of use we must let them be the surfaces of lugs or projections on the face of the wheel, as shown in Fig. 25 at d, e, &c., or the faces of notches cut in the wheel as at f, g, h, &c., the metal between f and g forming a tooth j, having flanks only. The wheel b has the curves of each tooth brought closer together to give room for the reception of the teeth upon a. We have here a pair of gears that possess sufficient strength and are capable of working correctly in either direction.

But the form of tooth on one wheel is conformed simply to suit those on the other, hence, neither two of the wheels a, nor would two of b, work correctly together.

They may be qualified to do so, however, by simply adding to the tops of the teeth on a, teeth of the form of those on b, and adding to those on b, and within the pitch circle, teeth corresponding to those on a, as in Fig. 26, where at k′ and j′ teeth are provided on b corresponding to j and k on a, while on a there are added teeth o′, n′, corresponding to o, n, on b, with the result that two wheels such as a or two such as b would work correctly together, either being the driver or either the follower, and rotation may occur in either direction. In this operation we have simply added faces to the teeth on a, and flanks to those on b, the curves being generated or obtained by rolling the generating, or curve marking, circle c upon the pitch circles p and p′. Thus, for the flanks of the teeth of a, c is rolled upon, and within the pitch circle p of a; while for the face curves of the same teeth c is rolled upon, but without or outside of p. Similarly for the teeth of wheel b the generating circle c is rolled within p′ for the flanks and without for the faces. With the curves rolled or produced with the same diameter of generating circle the wheels will work correctly together, no matter what their relative diameter may be, as will be shown hereafter.

In this demonstration, however, the curves for the faces of the teeth being produced by an operation distinct from that employed to produce the flank curves, it is not clearly seen that the curves for the flanks of one wheel are the proper curves to insure a uniform velocity to the other. This, however, may be made clear as follows:—

In Fig. 27 let a a and b b represent the pitch circles of two wheels of equal diameters, and therefore having the same number of teeth. On the left, the wheels are shown with the teeth in, while on the right-hand side of the line of centres a b, the wheels are shown blank; a a is the pitch line of one wheel, and b b that for the other. Now suppose that both wheels are capable of being rotated on their shafts, whose centres will of course be on the line a b, and suppose a third disk, q, be also capable of rotation upon its centre, c, which is also on the line a b. Let these three wheels have sufficient contact at their perimeters at the point n, that if one be rotated it will rotate both the others (by friction) without any slip or lost motion, and of course all three will rotate at an equal velocity. Suppose that there is fixed to wheel q a pencil whose point is at n. If then rotation be given to a a in the direction of the arrow s, all three wheels will rotate in that direction as denoted by their respective arrows s.

Assume, then, that rotation of the three has occurred until the pencil point at n has arrived at the point m, and during this period of rotation the point n will recede from the line of centres a b, and will also recede from the arcs or lines of the two pitch circles a a, b b. The pencil point being capable of marking its path, it will be found on reaching m to have marked inside the pitch circle b b the curve denoted by the full line m x, and simultaneously with this curve it has marked another curve outside of a a, as denoted by the dotted line y m. These two curves being marked by the pencil point at the same time and extending from y to m, and x also to m. They are prolonged respectively to p and to k for clearness of illustration only.

The rotation of the three wheels being continued, when the pencil point has arrived at o it will have continued the same curves as shown at o f, and o g, curve o f being the same as m x placed in a new position, and o g being the same as m y, but placed in a new position. Now since both these curves (o f and o g) were marked by the one pencil point, and at the same time, it follows that at every point in its course that point must have touched both curves at once. Now the pencil point having moved around the arc of the circle q from n to m, it is obvious that the two curves must always be in contact, or coincide with each other, at some point in the path of the pencil or describing point, or, in other words, the curves will always touch each other at some point on the curve of q, and between n and o. Thus when the pencil has arrived at m, curve m y touches curve k x at the point m, while when the pencil had arrived at point o, the curves o f and o g will touch at o. Now the pitch circles a a and b b, and the describing circle q, having had constant and uniform velocity while the traced curves had constant contact at some point in their lengths, it is evident that if instead of being mere lines, m y was the face of a tooth on a a, and m x was the flank of a tooth on b b, the same uniform motion may be transmitted from a a, to b b, by pressing the tooth face m y against the tooth flank m x. Let it now be noted that the curve y m corresponds to the face of a tooth, as say the face e of a tooth on a a, and that curve x m corresponds to the flank of a tooth on b b, as say to the flank f, short portions only of the curves being used for those flanks. If the direction of rotation of the three wheels was reversed, the same shape of curves would be produced, but they would lie in an opposite direction, and would, therefore, be suitable for the other sides of the teeth. In this case, the contact of tooth upon tooth will be on the other side of the line of centres, as at some point between n and q.

In this illustration the diameter of the rolling or describing circle q, being less than the radius of the wheels a a or b b, the flanks of the teeth are curves, and the two wheels being of the same diameter, the teeth on the two are of the same shape. But the principles governing the proper formation of the curve remain the same whatever be the conditions. Thus in Fig. 28 are segments of a pair of wheels of equal diameter, but the describing, rolling, or curve-generating circle is equal in diameter to the radius of the wheels. Motion is supposed to have occurred in the direction of the arrows, and the tracing point to have moved from n to m. During this motion it will have marked a curve y m, a portion of the y end serving for the face of a tooth on one wheel, and also the line k x, a continuation of which serves for the flank of a tooth on the other wheel. In Fig. 29 the pitch circles only of the wheels are marked, a a being twice the diameter of b b, and the curve-generating circle being equal in diameter to the radius of wheel b b. Motion is assumed to have occurred until the pencil point, starting from n, had arrived at o, marking curves suitable for the face of the teeth on one wheel and for the flanks of the other as before, and the contact of tooth upon tooth still, at every point in the path of the teeth, occurring at some point of the arc n o. Thus when the point had proceeded as far as point m it will have marked the curve y and the radial line x, and when the point had arrived at o, it will have prolonged m y into o g and x into o f, while in either position the point is marking both lines. The velocities of the wheels remain the same notwithstanding their different diameters, for the arc n g must obviously (if the wheels rotate without slip by friction of their surfaces while the curves are traced) be equal in length to the arc n f or the arc n o.

In Fig. 30 a a and b b are the pitch circles of two wheels as before, and c c the pitch circle of an annular or internal gear, and d is the rolling or describing circle. When the describing point arrived at m, it will have marked the curve y for the face of a tooth on a a, the curve x for the flank of a tooth on b b, and the curve e for the face of a tooth on the internal wheel c c. Motion being continued m y will be prolonged to o g, while simultaneously x will be extended into o f and e into h v, the velocity of all the wheels being uniform and equal. Thus the arcs n v, n f, and n g, are of equal length.

In Fig. 31 is shown the case of a rack and pinion; a a is the pitch line of the rack, b b that of the pinion, a b at a right angle to a a, the line of centres, and d the generating circle. The wheel and rack are shown with teeth n on one side simply for clearness of illustration. The pencil point n will, on arriving at m, have traced the flank curve x and the curve y for the face of the rack teeth.

It has been supposed that the three circles rotated together by the frictional contact of their perimeters on the line of centres, but the circumstances will remain the same if the wheels remain at rest while the generating or describing circle is rolled around them. Thus in Fig. 32 are two segments of wheels as before, c representing the centre of a tooth on a a, and d representing the centre of a tooth on b b. Now suppose that a generating or rolling circle be placed with its pencil point at e, and that it then be rolled around a a until it had reached the position marked 1, then it will have marked the curve from e to n, a part of this curve serving for the face of tooth c. Now let the rolling circle be placed within the pitch circle a a and its pencil point n be set to e, then, on being rolled to position 2, it will have marked the flank of tooth c. For the other wheel suppose the rolling wheel or circle to have started from f and rolled to the line of centres as in the cut, it will have traced the curve forming the face of the tooth d. For the flank of d the rolling circle or wheel is placed within b b, its tracing point set at f on the pitch circle, and on being rolled to position 3 it will have marked the flank curve. The curves thus produced will be precisely the same as those produced by rotating all three wheels about their axes, as in our previous demonstrations.

The curves both for the faces and for the flanks thus obtained will vary in their curvature with every variation in either the diameter of the generating circle or of the base or pitch circle of the wheel. Thus it will be observable to the eye that the face curve of tooth c is more curved than that of d, and also that the flank curve of d is more spread at the root than is that for c, which has in this case resulted from the difference between the diameter of the wheels a a and b b. But the curves obtained by a given diameter of rolling circle on a given diameter of pitch circle will be correct for any pitch of teeth that can be used upon wheels having that diameter of pitch circle. Thus, suppose we have a curve obtained by rolling a wheel of 20 inches circumference on a pitch circle of 40 inches circumference—now a wheel of 40 inches in circumference may contain 20 teeth of 2 inch arc pitch, or 10 teeth of 4 inch arc pitch, or 8 teeth of 5 inch arc pitch, and the curve may be used for either of those pitches.

If we trace the path of contact of each tooth, from the moment it takes until it leaves contact with a tooth upon the other wheel, we shall find that contact begins at the point where the flank of the tooth on the wheel that drives or imparts motion to the other wheel, meets the face of the tooth on the driven wheel, which will always be where the point of the driven tooth cuts or meets the generating or rolling circle of the driving tooth. Thus in Fig. 33 are represented segments of two spur-wheels marked respectively the driver and the driven, their generating circles being marked at g and g′, and x x representing the line of centres. Tooth a is shown in the position in which it commences its contact with tooth b at b. Secondly, we shall find that as these two teeth approach the line of centres x, the point of contact between them moves or takes place along the thickened arc or curve c x, or along the path of the generating circle g.

Thus we may suppose tooth d to be another position of tooth a, the contact being at f, and as motion was continued the contact would pass along the thickened curve until it arrived at the line of centres x. Now since the teeth have during this path of contact approached the line of centres, this part of the whole arc of action or of the path of contact is termed the arc of approach. After the two teeth have passed the line of centres x, the path of contact of the teeth will be along the dotted arc from x to l, and as the teeth are during this period of motion receding from x this part of the contact path is termed the arc of recess.

That contact of the teeth would not occur earlier than at c nor later than at l, is shown by the dotted teeth sides; thus a and b would not touch when in the position denoted by the dotted teeth, nor would teeth i and k if in the position denoted by their dotted lines.

If we examine further into this path of contact we find that throughout its whole path the face of the tooth of one wheel has contact with the flank only of the tooth of the other wheel, and also that the flank only of the driving-wheel tooth has contact before the tooth reaches the line of centres, while the face of only the driving tooth has contact after the tooth has passed the line of centres.

Thus the flanks of tooth a and of tooth d are in driving contact with the faces of teeth b and e, while the face of tooth h is in contact with the flank of tooth i.

These conditions will always exist, whatever be the diameters of the wheels, their number of teeth or the diameter of the generating circle. That is to say, in fully developed epicycloidal teeth, no matter which of two wheels is the driver or which the driven wheel, contact on the teeth of the driver will always be on the tooth flank during the arc of approach and on the tooth face during the arc of recess; while on the driven wheel contact during the arc of approach will be on the tooth face only, and during the arc of recess on the tooth flank only, it being borne in mind that the arcs of approach and recess are reversed in location if the direction of revolution be reversed. Thus if the direction of wheel motion was opposite to that denoted by the arrows in Fig. 33 then the arc of approach would be from m to x, and the arc of recess from x to n.

It is laid down by Professor Willis that the motion of a pair of gear-wheels is smoother in cases where the path of contact begins at the line of centres, or, in other words, when there is no arc of approach; and this action may be secured by giving to the driven wheel flanks only, as in Fig. 34, in which the driver has fully developed teeth, while the teeth on the driven have no faces.

In this case, supposing the wheels to revolve in the direction of arrow p, the contact will begin at the line of centres x, move or pass along the thickened arc and end at b, and there will be contact during the arc of recess only. Similarly, if the direction of motion be reversed as denoted by arrow q, the driver will begin contact at x, and cease contact at h, having, as before, contact during the arc of recess only.

But if the wheel w were the driver and v the driven, then these conditions would be exactly reversed. Thus, suppose this to be the case and the direction of motion be as denoted by arrow p, the contact would occur during the arc of approach, from h to x, ceasing at x.

Or if w were the driver, and the direction of motion was as denoted by q, then, again, the path of contact would be during the arc of approach only, beginning at b and ceasing at x, as denoted by the thickened arc b x.

The action of the teeth will in either case serve to give a theoretically perfect motion so far as uniformity of velocity is concerned, or, in other words, the motion of the driver will be transmitted with perfect uniformity to the driven wheel. It will be observed, however, that by the removal of the faces of the teeth, there are a less number of teeth in contact at each instant of time; thus, in Fig. 33 there is driving contact at three points, c, f, and j, while in Fig. 34 there is driving contact at two points only. From the fact that the faces of the teeth work with the flanks only, and that one side only of the teeth comes into action, it becomes apparent that each tooth may have curves formed by four different diameters of rolling or generating circles and yet work correctly, no matter which wheel be the driver, or which the driven wheel or follower, or in which direction motion occurs. Thus in Fig. 35, suppose wheel v to be the driver, having motion in the direction of arrow p, then faces a on the teeth of v will work with flanks b of the teeth on w, and so long as the curves for these faces and flanks are obtained with the same diameter of rolling circle, the action of the teeth will be correct, no matter what the shapes of the other parts of the teeth. Now suppose that v still being the driver, motion occurs in the other direction as denoted by q, then the faces c of the teeth on v will drive the flanks c of the teeth on w, and the motion will again be correct, providing that the same diameter (whatever it may be) of rolling circle be used for these faces and flanks, irrespective, of course, of what diameter of rolling circle is used for any other of the teeth curves. Now suppose that w is the driver, motion occurring in the direction of p, then faces e will drive flanks f, and the motion will be correct as before if the curves e and f are produced with the same diameter of rolling circle. Finally, let w be the driving wheel and motion occur in the direction of q, and faces g will drive flanks h, and yet another diameter of rolling circle may be used for these faces and flanks. Here then it is shown that four different diameters of rolling circles may be used upon a pair of wheels, giving teeth-forms that will fill all the requirements so far as correctly transmitting motion is concerned. In the case of a pair of wheels having an equal number of teeth, so that each tooth on one wheel will always fall into gear with the same tooth on the other wheel, every tooth may have its individual curves differing from all the others, providing that the corresponding teeth on the other wheel are formed to match them by using the same size of rolling circle for each flank and face that work together.

It is obvious, however, that such teeth would involve a great deal of labor in their formation and would possess no advantage, hence they are not employed. It is not unusual, however, in a pair of wheels that are to gear together and that are not intended to interchange with other wheels, to use such sizes as will give to for the face of the teeth on the largest wheel of the pair and for the flanks of the teeth of the smallest wheel, a generating circle equal in diameter to the radius of the smallest wheel, and for the faces of the teeth of the small wheel and the flanks of the teeth of the large one, a generating circle whose diameter equals the radius of the large wheel.

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.

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.

In determining the diameter of a generating circle for a set or train of wheels, we have the consideration that the smaller the diameter of the generating circle in proportion to that of the pitch circle the more the teeth are spread at the roots, and this creates a pressure tending to thrust the wheels apart, thus causing the axle journals to wear. In Fig. 39, for example, a a is the line of centres, and the contact of the curves at b c would cause a thrust in the direction of the arrows d, e. This thrust would exist throughout the whole path of contact save at the point f, on the line of centres. This thrust is reduced in proportion as the diameter of the generating circle is increased; thus in Fig. 40, is represented a pair of pinions of 12 teeth and 3 inch pitch, and c being the driver, there is contact at e, and at g, and e being a radial line, there is obviously a minimum of thrust.

What is known as the Willis system for interchangeable gearing, consists of using for every pitch of the teeth a generating circle whose diameter is equal to the radius of a pinion having 12 teeth, hence the pinion will in each pitch have radial flanks, and the roots of the teeth will be more spread as the number of teeth in the wheel is increased. Twelve teeth is the least number that it is considered practicable to use; hence it is obvious that under this system all wheels of the same pitch will work correctly together.

Unless the faces of the teeth and the flanks with which they work are curves produced from the same size of generating circle, the velocity of the teeth will not be uniform. Obviously the revolutions of the wheels will be proportionate to their numbers of teeth; hence in a pair of wheels having an equal number of teeth, the revolutions will per force be equal, but the driver will not impart uniform motion to the driven wheel, but each tooth will during the path of contact move irregularly.

The velocity of a pair of wheels will be uniform at each instant of time, if a line normal to the surfaces of the curves at their point of contact passes through the point of contact of the pitch circles on the line of centres of the wheels. Thus in Fig. 41, the line a a is tangent to the teeth curves where they touch, and d at a right angle to a a, and meets it at the point of the tooth curves, hence it is normal to the point of contact, and as it meets the pitch circles on the line of centres the velocity of the wheels will be uniform.

The amount of rolling motion of the teeth one upon the other while passing through the path of contact, will be a minimum when the tooth curves are correctly formed according to the rules given. But furthermore the sliding motion will be increased in proportion as the diameter of the generating circle is increased, and the number of teeth in contact will be increased because the arc, or path, of contact is longer as the generating circle is made larger.

Thus in Fig. 42 is a pair of wheels whose tooth curves are from a generating circle equal to the radius of the wheels, hence the flanks are radial. The teeth are made of unusual depth to keep the lines in the engraving clear. Suppose v to be the driver, w the driven wheel or follower, and the direction of motion as at p, contact upon tooth a will begin at c, and while a is passing to the line of centres the path of contact will pass along the thickened line to x. During this time the whole length of face from c to r will have had contact with the length of flank from c to n, and it follows that the length of face on a that rolled on c n can only equal the length of c n, and that the amount of sliding motion must be represented by the length of r n on a, and the amount of rolling motion by the length n c. Again, during the arc of recess (marked by dots) the length of flank that will have had contact is the depth from s to ls, and over this depth the full length of tooth face on wheel v will have swept, and as l s equals c n, the amount of rolling and of sliding motion during the arc of recess is equal to that during the arc of approach, and the action is in both cases partly a rolling and partly a sliding one. The two wheels are here shown of the same diameter, and therefore contain an equal number of teeth, hence the arcs of approach and of recess are equal in length, which will not be the case when one wheel contains more teeth than the other. Thus in Fig. 43, let a represent a segment of a pinion, and b a segment of a spur-wheel, both segments being blank with their pitch circles, the tooth height and depth being marked by arcs of circles. Let c and d represent the generating circles shown in the two respective positions on the line of centres. Let pinion a be the driver moving in the direction of p, and the arc of approach will be from e to x along the thickened arc, while the arc of recess will be as denoted by the dotted arc from x to f. The distance e x being greater than distance x f, therefore the arc of approach is longer than that of recess.

But suppose b to be the driver and the reverse will be the case, the arc of approach will begin at g and end at x, while the arc of recess will begin at x and end at h, the latter being farther from the line of centres than g is. It will be found also that, one wheel being larger than the other, the amount of sliding and rolling contact is different for the two wheels, and that the flanks of the teeth on the larger wheel b, have contact along a greater portion of their depths than do the flanks of those on the smaller, as is shown by the dotted arc i being farther from the pitch circle than the dotted arc j is, these two dotted arcs representing the paths of the lowest points of flank contact, points f and g, marking the initial lowest contact for the two directions of revolution.

Thus it appears that there is more sliding action upon the teeth of the smaller than upon those of the larger wheel, and this is a condition that will always exist.

In Fig. 44 is represented portion of a pair of wheels corresponding to those shown in Fig. 42, except that in this case the diameter of the generating circle is reduced to one quarter that of the pitch diameter of the wheels. v is the driver in the direction the teeth of v that will have contact is c n, which, the wheels, being of equal diameter, will remain the same whichever wheel be the driver, and in whatever direction motion occurs. The amount of rolling motion is, therefore, c n, and that of sliding is the difference between the distance c n and the length of the tooth face.

If now we examine the distance c n in Fig. 42, we find that reducing the diameter of generating circle in Fig. 44 has increased the depth of flank that has contact, and therefore increased the rolling motion of the tooth face along the flank, and correspondingly diminished the sliding action of the tooth contact. But at the same time we have diminished the number of teeth in contact. Thus in Fig. 42 there are three teeth in driving contact, while in Fig. 44 there are but two, viz., d and e.

In an article by Professor Robinson, attention is called to the fact that if the teeth of wheels are not formed to have correct curves when new, they cannot be improved by wear; and this will be clearly perceived from the preceding remarks upon the amount of rolling and sliding contact. It will also readily appear that the nearer the diameter of the generating to that of the base circle the more the teeth wear out of correct shape; hence, in a train of gearing in which the generating circle equals the radius of the pinion, the pinion will wear out of shape the quickest, and the largest wheel the least; because not only does each tooth on the pinion more frequently come into action on account of its increased revolutions, but furthermore the length of flank that has contact is less, while the amount of sliding action is greater. In Fig. 45, for example, are a wheel and pinion, the latter having radial flanks and the pinion being the driver, the arc of approach is the thickened arc from c to the line of centres, while the arc of recess is denoted by the dotted arc. As contact on the pinion flank begins at point c and ends at the line of centres, the total depth of flank that suffers wear from the contact is that from c to n; and as the whole length of the wheel tooth face sweeps over this depth c n, the pinion flanks must wear faster than the wheel faces, and the pinion flanks will wear underneath, as denoted by the dotted curve on the flanks of tooth w. In the case of the wheel, contact on its tooth flanks begins at the line of centres and ends at l, hence that flank can only wear between point l and the pitch line l; and as the whole length of pinion face sweeps on this short length l s, the pinion flank will wear most, the wear being in the direction of the dotted arc on the left-hand side v of the tooth. Now the pinion flank depth c n, being less than the wheel flank depth s l, and the same length of tooth face sweeping (during the path of contact) over both, obviously the pinion tooth will wear the most, while both will, as the wear proceeds, lose their proper flank curve. In Fig. 46 the generating arcs, g and g′, and the wheel are the same, but the pinion is larger. As a result the acting length c n, of pinion flank is increased, as is also the acting length s l, of wheel flank; hence, the flanks of both wheels would wear better, and also better preserve their correct and original shapes.

It has been shown, when referring to Figs. 42 and 44, when treating of the amount of sliding and of rolling motion, that the smaller the diameter of rolling circle in proportion to that of pitch circle, the longer the acting length of flank and the more the amount of rolling motion; and it follows that the teeth would also preserve their original and true shape better. But the wear of the teeth, and the alteration of tooth form by reason of that wear, will, in any event, be greater upon the pinion than upon the wheel, and can only be equal when the two wheels are of equal diameter, in which case the tooth curves will be alike on both wheels, and the acting depths of flank will be equal, as shown in Fig. 47, the flanks being radial, and the acting depths of flank being shown at j k. In Fig. 48 is shown a pair of wheels with a generating circle, g and g′, of one quarter the diameter of the base circle or pitch diameter, and the acting length of flank is shown at l m. The wear of the teeth would, therefore, in this latter case, cause it in time to assume the form shown in Fig. 49. But it is to be noted that while the acting depth of flank has been increased the arcs of contact have been diminished, and that in Fig. 47 there are two teeth in contact, while in Fig. 48 there is but one, hence the pressure upon each tooth is less in proportion as the diameter of the generating circle is increased. If a train of wheels are to be constructed, or if the wheels are to be capable of interchanging with other combinations of wheels of the same pitch, the diameter of the generating circle must be equal to the smallest wheel or pinion, which is, under the Willis system, a pinion of 12 teeth; under the Pratt and Whitney, and Brown and Sharpe systems, a pinion of 15 teeth.

But if a pair or a particular train of gears are to be constructed, then a diameter of generating circle may be selected that is considered most suitable to the particular conditions; as, for example, it may be equal to the radius of the smallest wheel giving it radial flanks, or less than that radius giving parallel or spread flanks. But in any event, in order to transmit continuous motion, the diameter of generating circle must be such as to give arcs of action that are equal to the pitch, so that each pair of teeth will come into action before the preceding pair have gone out of action.

It may now be pointed out that the degrees of angle that the teeth move through always exceeds the number of degrees of angle contained in the paths of contact, or, in other words, exceeds the degrees contained in the arcs of approach and recess combined.

In Fig. 50, for example, are a wheel a and pinion b, the teeth on the wheel being extended to a point. Suppose that the wheel a is the driver, and contact will begin between the two teeth d and f on the dotted arc. Now suppose tooth d to have moved to position c, and f will have been moved to position h. The degrees of angle the pinion has been moved through are therefore denoted by i, whereas the degrees of angle the arcs of contact contain are therefore denoted by j.

The degrees of angle that the wheel a has moved through are obviously denoted by e, because the point of tooth d has during the arcs of contact moved from position d to position c. The degrees of angle contained in its path of contact are denoted by k, and are less than e, hence, in the case of teeth terminating in a point as tooth d, the excess of angle of action over path of contact is as many degrees as are contained in one-half the thickness of the tooth, while when the points of the teeth are cut off, the excess is the number of degrees contained in the distance between the corner and the side of the tooth as marked on a tooth at p.

With a given diameter of pitch circle and pitch diameter of wheel, the length of the arc of contact will be influenced by the height of the addendum from the pitch circle, because, as has been shown, the arcs of approach and of recess, respectively, begin and end on the addendum circle.

If the height of the addendum on the follower be reduced, the arc of approach will be reduced, while the arc of recess will not be altered; and if the follower have no addendum, contact between the teeth will occur on the arc of recess only, which gives a smoother motion, because the action of the driver is that of dragging rather than that of pushing the follower. In this case, however, the arc of recess must, to produce continuous motion, be at least equal to the pitch.

It is obvious, however, that the follower having no addendum would, if acting as a driver to a third wheel, as in a train of wheels, act on its follower, or the fourth wheel of the train, on the arc of approach only; hence it follows that the addendum might be reduced to diminish, or dispensed with to eliminate action, on the arc of approach in the follower of a pair of wheels only, and not in the case of a train of wheels.

To make this clear to the reader it may be necessary to refer again to Fig. 33 or 34, from which it will be seen that the action of the teeth of the driver on the follower during the arc of approach is produced by the flanks of the driver on the faces of the follower. But if there are no such faces there can be no such contact.

On the arc of recess, however, the faces of the driver act on the flanks of the follower, hence the absence of faces on the follower is of no import.

From these considerations it also appears that by giving to the driver an increase of addendum the arc of recess may be increased without affecting the arc of approach. But the height of addendum in machinists’ practice is made a constant proportion of the pitch, so that the wheel may be used indiscriminately, as circumstances may require, as either a driver or a follower, the arcs of approach and of recess being equal. The height of addendum, however, is an element in determining the number of teeth in contact, and upon small pinions this is of importance.

In Fig. 51, for example, is shown a section of two pinions of equal diameters, and it will be observed that if the full line a determined the height of the addendum there would be contact either at c or b only (according to the direction in which the motion took place).

With the addendum extended to the dotted circle, contact would be just avoided, while with the addendum extended to d there would be contact either at e or at f, according to which direction the wheel had motion.

This, by dividing the strain over two teeth instead of placing it all upon one tooth, not only doubles the strength for driving capacity, but decreases the wear by giving more area of bearing surface at each instant of time, although not increasing that area in proportion to the number of teeth contained in the wheel.

In wheels of larger diameter, short teeth are more permissible, because there are more teeth in contact, the number increasing with the diameters of the wheels. It is to be observed, however, that from having radial flanks, the smallest wheel is always the weakest, and that from making the most revolutions in a given time, it suffers the most from wear, and hence requires the greatest attainable number of teeth in constant contact at each period of time, as well as the largest possible area of bearing or wearing surface on the teeth.

It is true that increasing the “depth of tooth to pitch line” increases the whole length of tooth, and, therefore, weakens it; but this is far more than compensated for by distributing the strain over a greater number of teeth. This is in practice accomplished, when circumstances will permit, by making the pitch finer, giving to a wheel, of a given diameter, a greater number of teeth.

When the wheels are required to transmit motion rather than power (as in the case of clock wheels), to move as frictionless as possible, and to place a minimum of thrust on the journals of the shafts of the wheels, the generating circle may be made nearly as large as the diameter of the pitch circle, producing teeth of the form shown in Fig. 52. But the minimum of friction is attained when the two flanks for the tooth are drawn into one common hypocycloid, as in Fig. 53. The difference between the form of tooth shown in Fig. 52 and that shown in Fig. 53, is merely due to an increase in the diameter of the generating circle for the latter. It will be observed that in these forms the acting length of flank diminishes in proportion as the diameter of the generating circle is increased, the ultimate diameter of generating circle being as large as the pitch circles.

[1]This form is undesirable in that there is contact on one side only (on the arc of approach) of the line of centres, but the flanks of the teeth may be so modified as to give contact on the arc of recess also, by forming the flanks as shown in Fig. 54, the flanks, or rather the parts within the pitch circles, being nearly half circles, and the parts without with peculiarly formed faces, as shown in the figure. The pitch circles must still be regarded as the rolling circles rolling upon each other. Suppose b a tracing point on b, then as b rolls on a it will describe the epicycloid a b. A parallel line c d will work at a constant distance as at c d from a b, and this distance may be the radius of that part of d that is within the pitch line, the same process being applied to the teeth on both wheels. Each tooth is thus composed of a spur based upon a half cylinder.

Comparing Figs. 53 and 54, we see that the bases in 53 are flattest, and that the contact of faces upon them must range nearer the pitch line than in 54. Hence, 53 presents a more favorable obliquity of the line of direction of the pressures of tooth upon tooth. In seeking a still more favorable direction by going outside for the point of contact, we see by simply recalling the method of generating the tooth curves, that tooth contacts outside the pitch lines have no possible existence; and hence, Fig. 53 may be regarded as representing that form of toothed gear which will operate with less friction than any other known form.

This statement is intended to cover fixed teeth only, and not that complicated form of the trundle wheel in which the cylinder teeth are friction rollers. No doubt such would run still easier, even with their necessary one-sided contacts. Also, the statement is supposed to be confined to such forms of teeth as have good practical contacts at and near the line of centres.

Bevel-gear wheels are employed to transmit motion from one shaft to another when the axis of one is at an angle to that of the other. Thus in Fig. 55 is shown a pair of bevel-wheels to transmit motion from shafts at a right angle. In bevel-wheels all the lines of the teeth, both at the tops or points of the teeth, at the bottoms of the spaces, and on the sides of the teeth, radiate from the centre e, where the axes of the two shafts would meet if produced. Hence the depth, thickness, and height of the tooth decreases as the point e is approached from the diameter of the wheel, which is always measured on the pitch circle at the largest end of the cone, or in other words, at the largest pitch diameter.

The principles governing the practical construction of the curves for the teeth of the bevel-wheels may be explained as follows:—

In Fig. 56 let f and g represent two shafts, rotating about their respective axes; and having cones whose greatest diameters are at a and b, and whose points are at e. The diameter a being equal to that of b their circumferences will be equal, and the angular and velocity ratios will therefore be equal.

Let c and d represent two circles about the respective cones, being equidistant from e, and therefore of equal diameters and circumferences, and it is obvious that at every point in the length of each cone the velocity will be equal to a point upon the other so long as both points are equidistant from the points of intersection of the axes of the two shafts; hence if one cone drive the other by frictional contact of surfaces, both shafts will be rotated at an equal speed of rotation, or if one cone be fixed and the other moved around it, the contact of the surfaces will be a rolling contact throughout. The line of contact between the two cones will be a straight line, radiating at all times from the point e. If such, however, is not the case, then the contact will no longer be a rolling one. Thus, in Fig. 57 the diameters or circumferences at a and b being equal, the surfaces would roll upon each other, but on account of the line of contact not radiating from e (which is the common centre of motion for the two shafts) the circumference c is less than that of d, rendering a rolling contact impossible.

We have supposed that the diameters of the cones be equal, but the conditions will remain the same when their diameters are unequal; thus, in Fig. 58 the circumference of a is twice that of b, hence the latter will make two rotations to one of the former, and the contact will still be a rolling one. Similarly the circumference of d is one half that of c, hence d will also make two rotations to one of c, and the contact will also be a rolling one; a condition which will always exist independent of the diameters of the wheels so long as the angles of the faces, or wheels, or (what is the same thing, the line of contact between the two,) radiates from the point e, which is located where the axes of the shafts would meet.

The principles governing the forms of the cones on which the teeth are to be located thus being explained, we may now consider the curves of the teeth. Suppose that in Fig. 59 the cone a is fixed, and that the cone whose axis is f be rotated upon it in the direction of the arrow. Then let a point be fixed in any part of the circumference of b (say at d), and it is evident that the path of this point will be as b rolls around the axis f, and at the same time around a from the centre of motion, e. The curve so generated or described by the point d will be a spherical epicycloid. In this case the exterior of one cone has rolled upon the coned surface of the other; but suppose it rolls upon the interior, as around the walls of a conical recess in a solid body; then a point in its circumference would describe a curve known as the spherical hypocycloid; both curves agreeing (except in their spherical property) to the epicycloid and hypocycloid of the spur-wheel. But this spherical property renders it very difficult indeed to practically delineate or mark the curves by rolling contact, and on account of this difficulty Tredgold devised a method of construction whereby the curves may be produced sufficiently accurate for all practical purposes, as follows:—

In Fig. 60 let a a represent the axis of one shaft, and b the axis of the other, the axes of the two meeting at w. Mark e, representing the diameter of one wheel, and f that of the other (both lines representing the pitch circles of the respective wheels). Draw the line g g passing through the point w, and the point t, where the pitch circles e, f meet, and g g will be the line of contact between the cones. From w as a centre, draw on each side of g g dotted lines as p, representing the height of the teeth above and below the pitch line g g. At a right angle to g g mark the line j k, and from the junction of this line with axis b (as at q) as a centre, mark the arc a, which will represent the pitch circle for the large diameter of pinion d; mark also the arc b for the addendum and c for the roots of the teeth, so that from b to c will represent the height of the tooth at that end.

Similarly from p, as a centre, mark (for the large diameter of wheel c,) the pitch circle g, root circle h, and addendum i. On these arcs mark the curves in the same manner as for spur-wheels. To obtain these arcs for the small diameters of the wheels, draw m m parallel to j k. Set the compasses to the radius r l, and from p, as a centre, draw the pitch circle k. To obtain the depth for the tooth, draw the dotted line p, meeting the circle h, and the point w. A similar line from circle i to w will show the height of the addendum, or extreme diameter; and mark the tooth curves on k, l, m, in the same manner as for a spur-wheel.

Similarly for the pitch circle of the small end of the pinion teeth, set the compasses to the radius s l, and from q as a centre, mark the pitch circle d, outside of d mark e for the height of the addendum and inside of d mark f for the roots of the teeth at that end. The distance between the dotted lines (as p) represents the full height of the teeth, hence h meets line p, being the root of tooth for the large wheel, and to give clearance, the point of the pinion teeth is marked below, thus arc b does not meet h or p. Having obtained these arcs the curves are rolled as for a spur-wheel.

A tooth thus marked out is shown at x, and from its curves between b c, a template for the large diameter of the pinion tooth may be made, while from the tooth curves between the arcs e f, a template for the smallest tooth diameter of the pinion can be made.

Similarly for the wheel c the outer end curves are marked on the lines g, h, i, and those for the inner end on the lines k, l, m.

Internal or annular gear-wheels have their tooth curves formed by rolling the generating circle upon the pitch circle or base circle, upon the same general principle as external or spur-wheels. But the tooth of the annular wheel corresponds with the space in the spur-wheel, as is shown in Fig. 61, in which curve a forms the flank of a tooth on a spur-wheel p, and the face of a tooth on the annular wheel w. It is obvious then that the generating circle is rolled within the pitch circle for the face of the wheel and without for its flank, or the reverse of the process for spur-wheels. But in the case of internal or annular wheels the path of contact of tooth upon tooth with a pinion having a given number of teeth increases in proportion as the number of teeth in the wheel is diminished, which is also the reverse of what occurs in spur-wheels; as will readily be perceived when it is considered that if in an internal wheel the pinion have as many teeth as the wheel the contact would exist around the whole pitch circles of the wheel and pinion and the two would rotate together without any motion of tooth upon tooth. Obviously then we have, in the case of internal wheels, a consideration as to what is the greatest number (as well as what is the least number) of teeth a pinion may contain to work with a given wheel, whereas in spur-wheels the reverse is again the case, the consideration being how few teeth the wheel may contain to work with a given pinion. Now it is found that although the curves of the teeth in internal wheels and pinions may be rolled according to the principles already laid down for spur-wheels, yet cases may arise in which internal gears will not work under conditions in which spur-wheels would work, because the internal wheels will not engage together. Thus, in Fig. 62, is a pinion of 12 teeth and a wheel of 22 teeth, a generating circle having a diameter equal to the radius of the pinion having been used for all the tooth curves of both wheel and pinion. It will be observed that teeth a, b, and c clearly overlap teeth d, e, and f, and would therefore prevent the wheels from engaging to the requisite depth. This may of course be remedied by taking the faces off the pinion, as in Fig. 63, and thus confining the arc of contact to an arc of recess if the pinion drives, or an arc of approach if the wheel drives; or the number of teeth in the pinion may be reduced, or that in the wheel increased; either of which may be carried out to a degree sufficient to enable the teeth to engage and not interfere one with the other. In Fig. 64 the number of teeth in the pinion p is reduced from 12 to 6, the wheel w having 22 as before, and it will be observed that the teeth engage and properly clear each other.

By the introduction into the figure of a segment of a spur-wheel also having 22 teeth and placed on the other side of the pinion, it is shown that the path of contact is greater, and therefore the angle of action is greater, in internal than in spur gearing. Thus suppose the pinion to drive in the direction of the arrows and the thickened arcs a b will be the arcs of approach, a measuring longer than b. The dotted arcs c d represent the arcs of receding contact and c is found longer than d, the angles of action being 66° for the spur-wheels and 72° for the annular wheel.

On referring again to Fig. 62 it will be observed that it is the faces of the teeth on the two wheels that interfere and will prevent them from engaging, hence it will readily occur to the mind that it is possible to form the curves of the pinion faces correct to work with the faces of the wheel teeth as well as with the flanks; or it is possible to form the wheel faces with curves that will work correctly with the faces, as well as with the flanks of the pinion teeth, which will therefore increase the angle of action, and Professor McCord has shown in an article in the London Engineering how to accomplish this in a simple and yet exceedingly ingenious manner which may be described as follows:—

It is required to find a describing circle that will roll the curves for the flanks of the pinion and the faces of the wheels, and also a describing circle for the flanks of the wheel and the faces of the pinion; the curve for the wheel faces to work correctly with the faces as well as with the flanks of the pinion, and the curve for the pinion faces to work correctly with both the flanks and faces of the internal wheel.

In Fig. 65 let p represent the pitch circle of an annular or internal wheel whose centre is at a, and q the pitch circle of a pinion whose centre is at b, and let r be a describing circle whose centre is at c, and which is to be used to roll all the curves for the teeth. For the flanks of the annular wheel we may roll r within p, while for the faces of the wheel we may roll r outside of p, but in the case of the pinion we cannot roll r within q, because r is larger than q, hence we must find some other rolling circle of less diameter than r, and that can be used in its stead (the radius of r always being greater than the radius of the axis of the wheel and pinion for reasons that will appear presently). Suppose then that in Fig. 66 we have a ring whose bore r corresponds in diameter to the intermediate describing circle r, Fig. 65 and that q represents the pinion. Then we may roll r around and in contact with the pinion q, and a tracing point in r will trace the curve m n o, giving a curve a portion of which may be used for the faces of the pinion. But suppose that instead of rolling the intermediate describing circle r around p, we roll the circle t around p, and it will trace precisely the same curve m n o; hence for the faces of the pinion we have found a rolling circle t which is a perfect substitute for the intermediate circle q, and which it will always be, no matter what the diameters of the pinion and of the intermediate describing circle may be, providing that the diameter of t is equal to the difference between the diameters of the pinion and that of the intermediate describing circle as in the figure. If now we use this describing circle to roll the flanks of the annular wheel as well as the faces of the pinion, these faces and flanks will obviously work correctly together. Since this describing circle is rolled on the outside of the pinion and on the outside of the annular wheel we may distinguish it as the exterior describing circle.

Now instead of rolling the intermediate describing circle r within the annular wheel p for the face curves of the teeth upon p, we may find some other circle that will give the same curve and be small enough to be rolled within the pinion q for its teeth flanks. Thus in Fig. 67 p represents the pitch circle of the annular wheel and r the intermediate circle, and if r be rolled within p, a point on the circumference of r will trace the curve v w. But if we take the circle s, having a diameter equal to the difference between the diameter of r and that of p, and roll it within p, a point in its circumference will trace the same curve v w; hence s is a perfect substitute for r, and a portion of the curve v w may be used for the faces of the teeth on the annular wheel. The circle s being used for the pinion flanks, the wheel faces and pinion flanks will work correctly together, and as the circle s is rolled within the pinion for its flanks and within the wheel for its faces, it may be distinguished as the interior describing circle.

To prove the correctness of the construction it may be noted that with the particular diameter of intermediate describing circle used in Fig. 65, the interior and exterior describing circles are of equal diameters; hence, as the same diameter of describing circle is used for all the faces and flanks of the pair of wheels they will obviously work correctly together, in accordance with the rules laid down for spur gearing. The radius of s in Fig. 69 is equal to the radius of the annular wheel, less the radius of the intermediate circle, or the radius from a to c. The radius of the exterior describing circle t is the radius of the intermediate circle less the radius of the pinion, or radius c b in the figure.

Now the diameter of the intermediate circle may be determined at will, but cannot exceed that of the annular wheel or be less than the pinion. But having been selected between these two limits the interior and exterior describing circles derived from it give teeth that not only engage properly and avoid the interference shown in Fig. 62, but that will also have an additional arc of action during the recess, as is shown in Fig. 68, which represents the wheel and pinion shown in Fig. 62, but produced by means of the interior and exterior describing circles. Supposing the pinion to be the driver the arc of approach will be along the thickened arc of the interior describing circle, while during the arc of recess there will be an arc of contact along the dotted portion of the exterior describing circle as in ordinary gearing. But in addition there will be an arc of recess along the dotted portion of the intermediate circle r, which arc is due to the faces of the pinion acting upon the faces as well as upon the flanks of the wheel teeth. It is obvious from this that as soon as a tooth passes the line of centres it will, during a certain period, have two points of contact, one on the arc of the exterior describing circle, and another along the arc of r, this period continuing until the addendum circle of the pinion crosses the dotted arc of the exterior describing circle at z.

The diameters of the interior and exterior describing circles obviously depend upon the diameter of the intermediate circle, and as this may, as already stated, be selected, within certain limits, at will, it is evident that the relative diameters of the interior and exterior describing circles will vary in proportion, the interior becoming smaller and the exterior larger, while from the very mode of construction the radius of the two will equal that of the axes of the wheel and pinion. Thus in Fig. 69 the radii of s, t, equal a b, or the line of centres, and their diameters, therefore, equal the radius of the annular wheel, as is shown by dotting them in at the upper half of the figure. But after their diameters have been determined by this construction either of them may be decreased in diameter and the teeth of the wheels will clear (and not interfere as in Fig. 62), but the action will be the same as in ordinary gear, or in other words there will be no arc of action on the circle r. But s cannot be increased without correspondingly decreasing t, nor can t be increased without correspondingly decreasing s.

Fig. 70 shows the same pair of gears as in Fig. 68 (the wheel having 22 and the pinion 12 teeth), the diameter of the intermediate circle having been enlarged to decrease the diameter of s and increase that of t, and as these are left of the diameter derived from the construction there is receding action along r from the line of centres to t.

In Fig. 71 are represented a wheel and pinion, the pinion having but four teeth less than the wheel, and a tooth, j, being shown in position in which it has contact at two places. Thus at k it is in contact with the flank of a tooth on the annular wheel, while at l it is in contact with the face of the same tooth.

As the faces of the teeth on the wheel do not have contact higher than point t, it is obvious that instead of having them ³⁄₁₀ of the pitch as at the bottom of the figure, we may cut off the portion x without diminishing the arc of contact, leaving them formed as at the top of the figure. These faces being thus reduced in height we may correspondingly reduce the depth of flank on the pinion by filling in the portion g, leaving the teeth formed as at the top of the pinion. The teeth faces of the wheel being thus reduced we may, by using a sufficiently large intermediate circle, obtain interior and exterior describing circles that will form teeth that will permit of the pinion having but one tooth less than the wheel, or that will form a wheel having but one tooth more than the pinion.

The limits to the diameter of the intermediate describing circle are as follows: in Fig. 72 it is made equal in diameter to the pitch diameter of the pinion, hence b will represent the centre of the intermediate circle as well as of the pinion, and the pitch circle of the pinion will also represent the intermediate circle r. To obtain the radius for the interior describing circle we subtract the radius of the intermediate circle from the radius of the annular wheel, which gives a p, hence the pitch circle of the pinion also represents the interior circle r. But when we come to obtain the radius for the exterior describing circle (t), by subtracting the radius of the pinion from that of the intermediate circle, we find that the two being equal give o for the radius of (t), hence there could be no flanks on the pinion.

Now suppose that the intermediate circle be made equal in diameter to the pitch circle of the annular wheel, and we may obtain the radius for the exterior describing circle t; by subtracting the radius of the pinion from that of the intermediate circle, we shall obtain the radius a b; hence the radius of (t) will equal that of the pinion. But when we come to obtain the radius for the interior describing circle by subtracting the radius of the intermediate circle from that of the annular wheel, we find these two to be equal, hence there would be no interior describing circle, and, therefore, no faces to the pinion.

The action of the teeth in internal wheels is less a sliding and more a rolling one than that in any other form of toothed gearing. This may be shown as follows: In Fig. 73 let a a represent the pitch circle of an external pinion, and b b that of an internal one, and p p the pitch circle of an external wheel for a a or an internal one for b b, the point of contact at the line of centres being at c, and the direction of rotation p p being as denoted by the arrow; the two pinions being driven, we suppose a point at c, on the pitch circle p p, to be coincident with a point on each of the two pinions at the line of centres. If p p be rotated so as to bring this point to the position denoted by d, the point on the external pinion having moved to e, while that on the internal pinion has moved to f, both having moved through an arc equal to c d, then the distance from e to d being greater than from d to f, more sliding motion must have accompanied the contact of the teeth at the point e than at the point f; and the difference in the length of the arc e d and that of f d, may be taken to represent the excess of sliding action for the teeth on e; for whatever, under any given condition, the amount of sliding contact may be, it will be in the proportion of the length of e d to that of f d. Presuming, then, that the amount of power transmitted be equal for the two pinions, and the friction of all other things being equal—being in proportion to the space passed (or in this case slid) over—it is obvious that the internal pinion has the least friction.

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[I-28]

In Fig. 74 are shown a worm and worm gear partly in section on the line of centres. The worm or tangent screw w is simply one long tooth wound around a cylinder, and its form may be determined by the rules laid down for a rack and pinion, the tangent screw or worm being considered as a rack and the wheel as an ordinary spur-wheel.

Worm gearing is employed for transmitting motion at a right angle, while greatly reducing the motion. Thus one rotation of the screw will rotate the wheel to the amount of the pitch of its teeth only. Worm gearing possesses the qualification that, unless of very coarse pitch, the worm locks the wheel in any position in which the two may come to a state of rest, while at the same time the excess of movement of the worm over that of the wheel enables the movement of the latter, through a very minute portion of a revolution. And it is evident that, when the plane of rotation of the worm is at a right angle to that of the wheel, the contact of the teeth is wholly a sliding one. The wear of the worm is greater than that of the wheel, because its teeth are in continuous contact, whereas the wheel teeth are in contact only when passing through the angle of action. It may be noted, however, that each tooth upon the worm is longer than the teeth on the wheel in proportion as the circumference of the worm is to the length of wheel tooth.

If the teeth of the wheel are straight and are set at an angle equal to the angle of the worm thread to its axis, as in Fig. 75, p p representing the pitch line of the worm, c d the line of centres, and d the worm axis, the contact of tooth upon tooth will be at the centre only of the sides of the wheel teeth. It is generally preferred, however, to have the wheel teeth curved to envelop a part of the circumference of the worm, and thus increase the line of contact of tooth upon tooth, and thereby provide more ample wearing surface.

In this case the form of the teeth upon the worm wheel varies at every point in its length as the line of centres is departed from. Thus in Fig. 76 is shown an end view of a worm and a worm gear in section, c d being the line of centres, and it will be readily perceived that the shape of the teeth if taken on the line e f, will differ from that on the line of centres c d; hence the form of the wheel teeth must, if contact is to occur along the full length of the tooth, be conformed to fit to the worm, which may be done by taking a series of section of the worm thread at varying distances from, and parallel to, the line of centres and joining the wheel teeth to the shape so obtained. But if the teeth of the wheel are to be cut to shape, then obviously a worm may be provided with teeth (by serrating it along its length) and mounted in position upon the wheel so as to cut the teeth of the wheel to shape as the worm rotates. The pitch line of the wheel teeth, whether they be straight and are disposed at an angle as in Fig. 75, or curved as in Fig. 76, is at a right angle to the line of centres c d, or in other words in the plane of g h, in Fig. 76. This is evident because the pitch line must be parallel to the wheel axis, being at an equal radius from that axis, and therefore having an equal velocity of rotation at every point in the length of the pitch line of the wheel tooth.

If we multiply the number of teeth by their pitch to obtain the circumference of the pitch circle we shall obtain the circumference due to the radius of g h, from the wheel axis, and so long as g h is parallel to the wheel axis we shall by this means obtain the same diameter of pitch circle, so long as we measure it on a line parallel to the line of centres c d. The pitch of the worm is the same at whatever point in the tooth depth it may be measured, because the teeth curves are parallel one to the other, thus in Fig. 77 the pitch measures are equal at m, n, or o.

But the action of the worm and wheel will nevertheless not be correct unless the pitch line from which the curves were rolled coincides with the pitch line of the wheel on the line of centres, for although, if the pitch lines do not so coincide, the worm will at each revolution move the pitch line of the wheel through a distance equal to the pitch of the worm, yet the motion of the wheel will not be uniform because, supposing the two pitch lines not to meet, the faces of the pinion teeth will act against those of the wheel, as shown in Fig. 78, instead of against their flanks, and as the faces are not formed to work correctly together the motion will be irregular.

The diameter of the worm is usually made equal to four times the pitch of the teeth, and if the teeth are curved as in figure 76 they are made to envelop not more than 30° of the worm.

The number of teeth in the wheel should not be less than thirty, a double worm being employed when a quicker ratio of wheel to worm motion is required.

When the teeth of the wheel are curved to partly envelop the worm circumference it has been found, from experiments made by Robert Briggs, that the worm and the wheel will be more durable, and will work with greatly diminished friction, if the pitch line of the worm be located to increase the length of face and diminish that of the flank, which will decrease the length of face and increase the length of flank on the wheel, as is shown in Fig. 79; the location for the pitch line of the worm being determined as follows:—

The full radius of the worm is made equal to twice the pitch of its teeth, and the total depth of its teeth is made equal to .65 of its pitch. The pitch line is then drawn at a radius of 1.606 of the pitch from the worm axis. The pitch line is thus determined in Fig. 76, with the result that the area of tooth face and of worm surface is equalized on the two sides of the pitch line in the figure. In addition to this, however, it may be observed that by thus locating the pitch line the arcs both of approach and of recess are altered. Thus in Fig. 80 is represented the same worm and wheel as in Fig. 79, but the pitch lines are here laid down as in ordinary gearing. In the two figures the arcs of approach are marked by the thickened part of the generating circle, while the arcs of recess are denoted by the dotted arc on the generating circle, and it is shown that increasing the worm face, as in Fig. 79, increases the arc of recess, while diminishing the worm flank diminishes the arc of approach, and the action of the worm is smoother because the worm exerts more pulling than pushing action, it being noted that the action of the worm on the wheel is a pushing one before reaching, and a pulling one after passing, the line of centres.

It may here be shown that a worm-wheel may be made to work correctly with a square thread. Suppose, for example, that the diameter of the generating circle be supposed to be infinite, and the sides of the thread may be accepted as rolled by the circle. On the wheel we roll a straight line, which gives a cycloidal curve suitable to work with the square thread. But the action will be confined to the points of the teeth, as is shown in Fig. 81, and also to the arc of approach. This is the same thing as taking the faces off the worm and filling in the flanks of the wheel. Obviously, then, we may reverse the process and give the worm faces only, and the wheel, flanks only, using such size of generating circle as will make the spaces of the wheel parallel in their depths and rolling the same generating circle upon the pitch line of the worm to obtain its face curve. This would enable the teeth on the wheel to be cut by a square-threaded tap, and would confine the contact of tooth upon tooth to the recess.

The diameter of generating circle used to roll the curves for a worm and worm-wheel should in all cases be larger than the radius of the worm-wheel, so that the flanks of the wheel teeth may be at least as thick at the root as they are at the pitch circle.

To find the diameter of a wheel, driven by a tangent-screw, which is required to make one revolution for a given number of turns of the screw, it is obvious, in the first place, that when the screw is single-threaded, the number of teeth in the wheel must be equal to the number of turns of the screw. Consequently, the pitch being also given, the radius of the wheel will be found by multiplying the pitch by the number of turns of the screw during one turn of the wheel, and dividing the product by 6.28.

When a wheel pattern is to be made, the first consideration is the determination of the diameter to suit the required speed; the next is the pitch which the teeth ought to have, so that the wheel may be in accordance with the power which it is intended to transmit; the next, the number of the teeth in relation to the pitch and diameter; and, lastly, the proportions of the teeth, the clearance, length, and breadth.

When the amount of power to be transmitted is sufficient to cause excessive wear, or when the velocity is so great as to cause rapid wear, the worm instead of being made parallel in diameter from end to end, is sometimes given a curvature equal to that of the worm-wheel, as is shown in Fig. 82.

The object of this design is to increase the bearing area, and thus, by causing the power transmitted to be spread over a larger area of contact, to diminish the wear. A mechanical means of cutting a worm to the required form for this arrangement is shown in Fig. 83, which is extracted from “Willis’ Principles of Mechanism.” “a is a wheel driven by an endless screw or worm-wheel, b, c is a toothed wheel fixed to the axis of the endless screw b and in gear with another and equal toothed gear d, upon whose axis is mounted the smooth surfaced solid e, which it is desired to cut into Hindley’s[2] endless screw. For this purpose a cutting tooth f is clamped to the face of the wheel a. When the handle attached to the axis of b c is turned round, the wheel a and solid wheel e will revolve with the same relative velocity as a and b, and the tool f will trace upon the surface of the solid e a thread which will correspond to the conditions. For from the very mode of its formation the section of every thread through the axis will point to the centre of the wheel a. The axis of e lies considerably higher than that of b to enable the solid e to clear the wheel a.

“The edges of the section of the solid e along its horizontal centre line exactly fit the segment of the toothed wheel, but if a section be made by a plane parallel to this the teeth will no longer be equally divided as they are in the common screw, and therefore this kind of screw can only be in contact with each tooth along a line corresponding to its middle section. So that the advantage of this form over the common one is not so great as appears at first sight.

“If the inclination of the thread of a screw be very great, one or more intermediate threads may be added, as in Fig. 84, in which case the screw is said to be double or triple according to the number of separate spiral threads that are so placed upon its surface. As every one of these will pass its own wheel-tooth across the line of centres in each revolution of the screw, it follows that as many teeth of the wheel will pass that line during one revolution of the screw as there are threads to the screw. If we suppose the number of these threads to be considerable, for example, equal to those of the wheel teeth, then the screw and wheel may be made exactly alike, as in Fig. 85; which may serve as an example of the disguised forms which some common arrangements may assume.”

In Fig. 86 is shown Hawkins’s worm gearing. The object of this ingenious mechanical device is to transmit motion by means of screw or worm gearing, either by a screw in which the threads are of equal diameter throughout its length, or by a spiral worm, in which the threads are not of equal diameter throughout, but increase in diameter each way from the centre of its length, or about the centre of its length outwardly. Parallel screws are most applicable to this device when rectilinear motions are produced from circular motions of the driver, and spiral worms are applied when a circular motion is given by the driver, and imparted to the driven wheel. The threads of a spiral worm instead of gearing into teeth like those of an ordinary worm-wheel, actuate a series of rollers turning upon studs, which studs are attached to a wheel whose axis is not parallel to that of the worm, but placed at a suitable inclination thereto. When motion is given to the worm then rotation is produced in the roller wheel at a rate proportionable to the pitch of worm and diameter of wheel respectively.

In the arrangement for transmitting rectilinear motion from a screw, rollers may be employed whose axes are inclined to the axis of the driving screw, or else at right angles to or parallel to the same. When separate rollers are employed with inclined axes, or axes at right angles with that of the main driving screw, each thread in gear touches a roller at one part only; but when the rollers are employed with axes parallel to that of the driving screw a succession of grooves are turned in these rollers, into which the threads of the driving screw will be in gear throughout the entire length of the roller. These grooves may be separate and apart from each other, or else form a screw whose pitch is equal to that of the driving screw or some multiple thereof.

In Fig. 86 the spiral worm is made of such a length that the edge of one roller does not cease contact until the edge of the next comes into contact; a wheel carries four rollers which turn on studs, the latter being secured by cottars; the axis of the worm is at right angles with that of the wheel. The edges of the rollers come near together, leaving sufficient space for the thread of the worm to fit between any two contiguous rollers. The pitch line of the screw thread forms an arc of a circle, whose centre coincides with that of the wheel, therefore the thread will always bear fairly against the rollers and maintain rolling contact therewith during the whole of the time each roller is in gear, and by turning the screw in either direction the wheel will rotate.

To prevent end thrust on a worm shaft it may have a right-hand worm a, and a left-hand one c (Fig. 87), driving two wheels b and d which are in gear, and either of which may transmit the power. The thrust of the two worms a and c, being in opposite directions, one neutralizes the other, and it is obvious that as each revolution of the worm shaft moves both wheels to an amount equal to the pitch of the worms, the two wheels b d may, if desirable, be of different diameters.

Involute teeth.—These are teeth having their whole operative surfaces formed of one continuous involute curve. The diameter of the generating circle being supposed as infinite, then a portion of its circumference may be represented by a straight line, such as a in Fig. 88, and if this straight line be made to roll upon the circumference of a circle, as shown, then the curve traced will be involute p. In practice, a piece of flat spring steel, such as a piece of clock spring, is used for tracing involutes. It may be of any length, but at one end it should be filed so as to leave a scribing point that will come close to the base circle or line, and have a short handle, as shown in Fig. 89, in which s represents the piece of spring, having the point p′, and the handle h. The operation is, to make a template for the base circle, rest this template on drawing paper and mark a circle round its edge to represent on the paper the pitch circle, and to then bend the spring around the circle b, holding the point p′ in contact with the drawing paper, securing the other end of the piece of steel, so that it cannot slip upon b, and allowing the steel to unwind from the cylinder or circle b. The point p′ will mark the involute curve p. Another way to mark an involute is to use a piece of twine in place of the spring and a pencil instead of the tracing point; but this is not so accurate, unless, indeed, a piece of wood be laid on the drawing-board and the pencil held firmly against it, so as to steady the pencil point and prevent the variation in the curve that would arise from variation in the vertical position of the pencil.

The flanks being composed of the same curve as the faces of the teeth, it is obvious that the circle from which the tracing point starts, or around which the straight line rolls, must be of less diameter than the pitch circle, or the teeth would have no flanks.

A circle of less diameter than the pitch circle of the wheel is, therefore, introduced, wherefrom to produce the involute curves forming the full side of the tooth.

The depth below pitch line or the length of flank is, therefore, the distance between the pitch circle and the base circle. Now even supposing a straight line to be a portion of the circumference of a circle of infinite diameter or radius, the conditions would here appear to be imperfect, because the generating circle is not rolled upon the pitch circle but upon a circle of lesser diameter. But it can be shown that the requirements of a proper velocity ratio will be met, notwithstanding the employment of the base instead of the pitch circle. Thus, in Fig. 90, let a and b represent the respective centres of the two pitch circles, marked in dotted lines. Draw the base circle for b as e q, which may be of any radius less than that of the pitch circle of b. Draw the straight line q d r touching this base circle at its perimeter and passing through the point of contact on the pitch circles as at d. Draw the circle whose radius is a r forming the base circle for wheel a. Thus the line r p q will meet the perimeters of the two circles while passing through the point of contact d at the line of centres (a condition which the relative diameters of the base circles must always be so proportioned as to attain).

If now we take any point on r q, as p in the figure, as a tracing point, and suppose the radius or distance p q to represent the steel spring shown in Fig. 89, and move the tracing point back to the base circle of b, it will trace the involute e p. Again we may take the tracing point p (supposing the line p r to represent the steel spring), and trace the involute p f, and these two involutes represent each one side of the teeth on the respective wheels.

The line r p q is at a right angle to the curves p e and p f, at their point of contact, and, therefore, fills the conditions referred to in Fig. 41. Now the line r p q denotes the path of contact of tooth upon tooth as the wheels revolve; or, in other words, the point of contact between the side of a tooth on one wheel, and the side of a tooth on the other wheel, will always move along the line q r, or upon a similar line passing through d, but meeting the base circles upon the opposite sides of the line of centres, and since line q r always cuts the line of centres at the point of contact of the pitch circles, the conditions necessary to obtain a correct angular velocity are completely fulfilled. The velocity ratio is, therefore, as the length of b q is to that of a r, or, what is the same thing, as the radius of the base circle of one wheel is to that of the other. It is to be observed that the line q r will vary in its angle to the line of centres a b, according to the diameter of the base circle from which it is struck, and it becomes a consideration as to what is its most desirable angle to produce the least possible amount of thrust tending to separate the wheels, because this thrust (described in Fig. 39) tends to wear the journals and bearings carrying the wheel shafts, and thus to permit the pitch circles to separate. To avoid, as far as possible, this thrust the proportions between the diameters of the base circles d and e, Fig. 91, must be such that the line d e passes through the point of contact on the line of centres, as at c, while the angles of the straight line d e should be as nearly 90° to a radial line, meeting it from the centres of the wheels (as shown in the figure, by the lines b e and d e), as is consistent with the length of d e, which in order to impart continuous motion must at least equal the pitch of the teeth. It is obvious, also, that, to give continuous motion, the length of d e must be more than the pitch in proportion, as the points of the teeth come short of passing through the base circles at d and e, as denoted by the dotted arcs, which should therefore represent the addendum circles. The least possible obliquity, or angle of d e, will be when the construction under any given conditions be made such by trial, that the base circles d and e coincide with the addendum circles on the line of centres, and thus, with a given depth of both beyond, the pitch circle, or addenda as it is termed, will cause the tooth contacts to extend over the greatest attainable length of line between the limits of the addendum circles, thus giving a maximum number of teeth in contact at any instant of time. These conditions are fulfilled in Fig. 92,[3] the addendum on the small wheel being longer than the depth below pitch line, while the faces of the teeth are the narrowest.

In seeking the minimum obliquity or angle of d e in the figure, it is to be observed that the less it is, the nearer the base circle approaches the pitch circle; hence, the shorter the operative length of tooth flank and the greater its wear.

In comparing the merits of involute with those of epicycloidal teeth, the direction of the line of pressure at each point of contact must always be the common perpendicular to the surfaces at the point of contact, and these perpendiculars or normals must pass through the pitch circles on the line of centres, as was shown in Fig. 41, and it follows that a line drawn from c (Fig. 91) to any point of contact, is in the direction of the pressure on the surfaces at that point of contact. In involute teeth, the contact will always be on the line d e (Fig. 92), but in epicycloidal, on the line of the generating circle, when that circle is tangent at the line of centres; hence, the direction of pressure will be a chord of the circle drawn from the pitch circle at the line of centres to the position of contact considered. Comparing involute with radial flanked epicycloidal teeth, let c d a (Fig. 91) represent the rolling circle for the latter, and d c will be the direction of pressure for the contact at d; but for point of contact nearer c, the direction will be much nearer 90°, reaching that angle as the point of contact approaches c. Now, d is the most remote legitimate contact for involute teeth (and considering it so far as epicycloidal struck with a generating circle of infinite diameter), we find that the aggregate directions of the pressures of the teeth upon each other is much nearer perpendicular in epicycloidal, than in involute gearing; hence, the latter exert a greater pressure, tending to force the wheels apart. Hence, the former are, in this respect, preferable.

It is to be observed, however, that in some experiments made by Mr. Hawkins, he states that he found “no tendency to press the wheels apart, which tendency would exist if the angle of the line d e (Fig. 92) deviated more than 20° from the line of centres a b of the two wheels.”

A method commonly employed in practice to strike the curves of involute teeth, is as follows:—

In Fig. 93 let c represent the centre of a wheel, d d the full diameter, p p the pitch circle, and e the circle of the roots of the teeth, while r is a radial line. Divide on r, the distance between the pitch circle and the wheel centre, into four equal parts, by 1, 2, 3, &c. From point or division 2, as a centre, describe the semicircle s, cutting the wheel centre and the pitch circle at its junction with r (as at a). From a, with compasses set to the length of one of the parts, as a 3, describe the arc b, cutting s at f, and f will be the centre from which one side of the tooth may be struck; hence from f as a centre, with the compasses set to the radius a b, mark the curve g. From the centre c strike, through f, a circle t t, and the centres wherefrom to strike all the teeth curves will fall on t t. Thus, to strike the other curve of the tooth, mark off from a the thickness of the tooth on the pitch circle p p, producing the point h. From h as a centre (with the same radius as before,) mark on t t the point i, and from i, as a centre, mark the curve j, forming the other side of the tooth.

In Fig. 94 the process is shown carried out for several teeth. On the pitch circle p p, divisions 1, 2, 3, 4, &c., for the thickness of teeth and the width of the spaces are marked. The compasses are set to the radius by the construction shown in Fig. 93, then from a, the point b on t is marked, and from b the curve c is struck.

In like manner, from d, g, j, the centres e, h, k, wherefrom to strike the respective curves, f, i, l, are obtained.

Then from m the point n, on t t, is marked, giving the centre wherefrom to strike the curve at h m, and from o is obtained the point p, on t t, serving as a centre for the curve e o.

A more simple method of finding point f is to make a sheet metal template, c, as in Fig. 95, its edges being at an angle one to the other of 75° and 30′. One of its edges is marked off in quarters of an inch, as 1, 2, 3, 4, &c. Place one of its edges coincident with the line r, its point touching the pitch circle at the side of a tooth, as at a, and the centre for marking the curve on that side of the tooth will be found on the graduated edge at a distance from a equal to one-fourth the length of r.

The result obtained in this process is precisely the same as that by the construction in Fig. 93, as will be plainly seen, because there are marked on Fig. 93 all the circles by which point f was arrived at in Fig. 95; and line 3, which in Fig. 95 gives the centre wherefrom to strike curve o, is coincident with point f, as is shown in Fig. 95. By marking the graduated edge of c in quarter-inch divisions, as 1, 2, 3, &c., then every division will represent the distance from a for the centre for every inch of wheel radius. Suppose, for example, that a wheel has 3 inches radius, then with the scale c set to the radial line r, the centre therefrom to strike the curve o will be at 3; were the radius of the wheel 4 inches, then the scale being set the same as before (one edge coincident with r), the centre for the curve o would be at 4, and arc t would require to meet the edge of c at 4. Having found the radius from the centre of the wheel of point f for one tooth, we may mark circle t, cutting point f, and mark off all the teeth by setting one point of the compasses (set to radius a f) on one side of the tooth and marking on circle t the centre wherefrom to mark the curve (as o), continuing the process all around the wheel and on both sides of the tooth.

This operation of finding the location for the centre wherefrom to strike the tooth curves, must be performed separately for each wheel, because the distance or radius of the tooth curves varies with the radius of each wheel.

In Fig. 96 this template is shown with all the lines necessary to set it, those shown in Fig. 95 to show the identity of its results with those given in Fig. 93 being omitted.

The principles involved in the construction of a rack to work correctly with a wheel or pinion, having involute teeth, are as in Fig. 97, in which the pitch circle is shown by a dotted circle and the base circle by a full line circle. Now the diameter of the base circle has been shown to be arbitrary, but being assumed the radius b q will be determined (since it extends from the centre b to the point of contact of d q, with the base circle); b d is a straight line from the centre b of the pinion to the pitch line of the rack, and (whatever the angle of q d to b d) the sides of the rack teeth must be straight lines inclined to the pitch line of the rack at an angle equal to that of b d q.

Involute teeth possess four great advantages—1st, they are thickest at the roots, where they should be to have a maximum of strength, which is of great importance in pinions transmitting much power; 2nd, the action of the teeth will remain practically perfect, even though the wheels are spread apart so that the pitch circles do not meet on the line of centres; 3rd, they are much easier to mark, and truth in the marking is easier attained; and 4th, they are much easier to cut, because the full depth of the teeth can, on spur-wheels, in all cases be cut with one revolving cutter, and at one passage of the cutter, if there is sufficient power to drive it, which is not the case with epicycloidal teeth whenever the flank space is wider below than it is at the pitch circle. On account of the first-named advantage, they are largely employed upon small gears, having their teeth cut true in a gear-cutting machine; while on account of the second advantage, interchangeable wheels, which are merely required to transmit motion, may be put in gear without a fine adjustment of the pitch circle, in which case the wear of the teeth will not prove destructive to the curves of the teeth. Another advantage is, that a greater number of teeth of equal strength may be given to a wheel than in the epicycloidal form, for with the latter the space must at least equal the thickness of the tooth, while in involute the space may be considerably less in width than the tooth, both measured, of course, at the pitch circle. There are also more teeth in contact at the same time; hence, the strain is distributed over more teeth.

These advantages assume increased value from the following considerations.

In a train of epicycloidal gearing in which the pinion or smallest wheel has radial flanks, the flanks of the teeth will become spread as the diameters of the wheels in the train increase. Coincident with spread at the roots is the thrust shown with reference to Fig. 39, hence under the most favorable conditions the wear on the journals of the wheel axles and the bearings containing them will take place, and the pitch circles will separate. Now so soon as this separation takes place, the motion of the wheels will not be as uniformly equal as when the pitch circles were in contact on the line of centres, because the conditions under which the tooth curves, necessary to produce a uniform velocity of motion, were formed, will have become altered, and the value of those curves to produce constant regularity of motion will have become impaired in proportion as the pitch circles have separated.

In a single pair of epicycloidal wheels in which the flanks of the teeth are radial, the conditions are more favorable, but in this case the pinion teeth will be weaker than if of involute form, while the wear of the journals and bearings (which will take place to some extent) will have the injurious effect already stated, whereas in involute teeth, as has been noted, the separation of the pitch circles does not affect the uniformity of the motion or the correct working of the teeth.

If the teeth of wheels are to be cut to shape in a gear-cutting machine, either the cutters employed determine from their shapes the shapes or curves of the teeth, or else the cutting tool is so guided to the work that the curves are determined by the operations of the machine. In either case nothing is left to the machine operator but to select the proper tools and set them, and the work in proper position in the machine. But when the teeth are to be cast upon the wheel the pattern wherefrom the wheel is to be moulded must have the teeth proportioned and shaped to proper curve and form.

Wheels that require to run without noise or jar, and to have uniformity of motion, must be finished in gear-cutting machines, because it is impracticable to cast true wheels.

When the teeth are to be cast upon the wheels the pattern-maker makes templates of the tooth curves (by some one of the methods to be hereafter described), and carefully cuts the teeth to shape. But the production of these templates is a tedious and costly operation, and one which is very liable to error unless much experience has been had. The Pratt and Whitney Company have, however, produced a machine that will produce templates of far greater accuracy than can be made by hand work. These templates are in metal, and for epicycloidal teeth from 15 to a rack, and having a diametral pitch ranging from 1¹⁄₂ to 32.

The principles of action of the machine are that a segment of a ring (representing a portion of the pitch circle of the wheel for whose teeth a template is to be produced) is fixed to the frame of the machine. Upon this ring rolls a disk representing the rolling, generating, or describing circle, this disk being carried by a frame mounted upon an arm representing the radius of the wheel, and therefore pivoted at a point central to the ring. The describing disk is rolled upon the ring describing the epicycloidal curve, and by suitable mechanical devices this curve is cut upon a piece of steel, thus producing a template by actually rolling the generating upon the base circle, and the rolling motion being produced by positive mechanical motion, there cannot possibly be any slip, hence the curves so produced are true epicycloids.

VOL. I.	TEMPLATE‑CUTTING MACHINES FOR GEAR TEETH.		PLATE I.
Fig. 98.
Fig. 99.
Fig. 100.		Fig. 101.

The general construction of the machine is shown in the side view, Fig. 98 (Plate I.), and top view, Fig. 99 (Plate I.), details of construction being shown in Figs. 100, 101 (Plate I.), 102, 103, 104, 105, and 106. a a is the segment of a ring whose outer edge represents a part of the pitch circle. b is a disk representing the rolling or generating circle carried by the frame c, which is attached to a rod pivoted at d. The axis of pivot d represents the axis of the base circle or pitch circle of the wheel, and d is adjustable along the rod to suit the radius of a a, or what is the same thing, to equal the radius of the wheel for whose teeth a template is to be produced.

When the frame c is moved its centre or axis of motion is therefore at d and its path of motion is around the circumference of a a, upon the edge of which it rolls. To prevent b from slipping instead of rolling upon a a, a flexible steel ribbon is fastened at one end upon a a, passes around the edge of a a and thence around the circumference of b, where its other end is fastened; due allowance for the thickness of this ribbon being made in adjusting the radii of a a and of b.

e′ is a tubular pivot or stud fixed on the centre line of pivots e and d, and distant from the edge of a a to the same amount that e is. These two studs e and e′ carry two worm-wheels f and f′ in Fig. 102, which stand above a and b, so that the axis of the worm g is vertically over the common tangent of the pitch and describing circles.

The relative positions of these and other parts will be most clearly seen by a study of the vertical section, Fig. 102.[4] The worm g is supported in bearings secured to the carrier c and is driven by another small worm turned by the pulley i, as seen in Fig. 101 (Plate I.); the driving cord, passing through suitable guiding pulleys, is kept at uniform tension by a weight, however c moves; this is shown in Figs. 98 and 99 (Plate I.).

Upon the same studs, in a plane still higher than the worm-wheels turn the two disks h, h′, Figs. 103, 104, 105. The diameters of these are equal, and precisely the same as those of the describing circles which they represent, with due allowance, again, for the thickness of a steel ribbon, by which these also are connected. It will be understood that each of these disks is secured to the worm-wheel below it, and the outer one of these, to the disk b, so that as the worm g turns, h and h′ are rotated in opposite directions, the motion of h being identical with that of b; this last is a rolling one upon the edge of a, the carrier c with all its attached mechanism moving around d at the same time. Ultimately, then, the motions of h, h′, are those of two equal describing circles rolling in external and internal contact with a fixed pitch circle.

In the edge of each disk a semicircular recess is formed, into which is accurately fitted a cylinder j, provided with flanges, between which the disks fit so as to prevent end play. This cylinder is perforated for the passage of the steel ribbon, the sides of the opening, as shown in Fig. 103, having the same curvature as the rims of the disks. Thus when these recesses are opposite each other, as in Fig. 104, the cylinder j fills them both, and the tendency of the steel ribbon is to carry it along with h when c moves to one side of this position, as in Fig. 105, and along with h′ when c moves to the other side, as in Fig. 103.

This action is made positively certain by means of the hooks k, k′, which catch into recesses formed in the upper flange of j, as seen in Fig. 104. The spindles, with which these hooks turn, extend through the hollow studs, and the coiled springs attached to their lower ends, as seen in Fig. 102, urge the hooks in the directions of their points; their motions being limited by stops o, o′, fixed, not in the disks h, h′, but in projecting collars on the upper ends of the tubular studs. The action will be readily traced by comparing Fig. 104 with Fig. 105; as c goes to the left, the hook k′ is left behind, but the other one, k, cannot escape from its engagement with the flange of j; which, accordingly, is carried along with h by the combined action of the hook and the steel ribbon.

On the top of the upper flange of j, is secured a bracket, carrying the bearing of a vertical spindle l, whose centre line is a prolongation of that of j itself. This spindle is driven by the spur-wheel n, keyed on its upper end, through a flexible train of gearing seen in Fig. 99; at its lower end it carries a small milling cutter m, which shapes the edge of the template t, Fig. 105, firmly clamped to the framing.

When the machine is in operation, a heavy weight, seen in Fig. 98 (Plate I.), acts to move c about the pivot d, being attached to the carrier by a cord guided by suitably arranged pulleys; this keeps the cutter m up to its work, while the spindle l is independently driven, and the duty left for the worm g to perform is merely that of controlling the motions of the cutter by the means above described, and regulating their speed.

The centre line of the cutter is thus automatically compelled to travel in the path r s, Fig. 105, composed of an epicycloid and a hypocycloid if a a be the segment of a circle as here shown; or of two cycloids, if a a be a straight bar. The radius of the cutter being constant, the edge of the template t is cut to an outline also composed of two curves; since the radius m is small, this outline closely resembles r s, but particular attention is called to the fact that it is not identical with it, nor yet composed of truly epicycloidal curves of any generation whatever: the result of which will be subsequently explained.

With a given pitch every additional tooth increases the diameter of the wheel, and changes the form of the epicycloid; so that it would appear necessary to have as many different cutters, as there are wheels to be made, of any one pitch.

But the proportional increment, and the actual change of form, due to the addition of one tooth, becomes less as the wheel becomes larger; and the alteration in the outline soon becomes imperceptible. Going still farther, we can presently add more teeth without producing a sensible variation in the contour. That is to say, several wheels can be cut with the same cutter, without introducing a perceptible error. It is obvious that this variation in the form is least near the pitch circle, which is the only part of the epicycloid made use of; and Prof. Willis many years ago deduced theoretically, what has since been abundantly proved by practice, that instead of an infinite number of cutters, 24 are sufficient of one pitch, for making all wheels, from one with 12 teeth up to a rack.

Accordingly, in using the epicycloidal milling engine, for forming the template, segments of pitch circles are provided of the following diameters (in inches):

12,	16,	20,	27,	43,	100,
13,	17,	21,	30,	50,	150,
14,	18,	23,	34,	60,	300.
15,	19,	25,	38,	75,

 

In Fig. 106, the edge t t is shaped by the cutter t t, whose centre travels in the path r s, therefore these two lines are at a constant normal distance from each other. Let a roller p, of any reasonable diameter, be run along t t, its centre will trace the line u v, which is at a constant normal distance from t t, and therefore from r s. Let the normal distance between u v and r s be the radius of another milling cutter n, having the same axis as the roller p, and carried by it, but in a different plane as shown in the side view; then whatever n cuts will have r s for its contour, if it lie upon the same side of the cutter as the template.

The diameter of the disks which act as describing circles is 7¹⁄₂ inches, and that of the milling cutter which shapes the edge of the template is ¹⁄₈ of an inch.

Now if we make a set of 1-pitch wheels with the diameters above given, the smallest will have twelve teeth, and the one with fifteen teeth will have radial flanks. The curves will be the same whatever the pitch; but as shown in Fig. 106, the blank should be adjusted in the epicycloidal engine, so that its lower edge shall be ¹⁄₁₆th of an inch (the radius of the cutter m) above the bottom of the space; also its relation to the side of the proposed tooth should be as here shown. As previously explained, the depth of the space depends upon the pitch. In the system adopted by the Pratt & Whitney Company, the whole height of the tooth is 2¹⁄₈ times the diametral pitch, the projection outside the pitch circle being just equal to the pitch, so that diameter of blank = diameter of pitch circle + 2 × diametral pitch.

We have now to show how, from a single set of what may be called 1-pitch templates, complete sets of cutters of the true epicycloidal contour may be made of the same or any less pitch.

Now if t t be a 1-pitch template as above mentioned, it is clear that n will correctly shape a cutting edge of a gear cutter for a 1-pitch wheel. The same figure, reduced to half size, would correctly represent the formation of a cutter for a 2-pitch wheel of the same number of teeth; if to quarter size, that of a cutter for a 4-pitch wheel, and so on.

But since the actual size and curvature of the contour thus determined depend upon the dimensions and motion of the cutter n, it will be seen that the same result will practically be accomplished, if these only be reduced; the size of the template, the diameter and the path of the roller remaining unchanged.

The nature of the mechanism by which this is effected in the Pratt & Whitney system of producing epicycloidal cutters will be hereafter explained in connection with cutters.

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