Practical Treatise on Milling and Milling Machines/Chapter 4

by Brown & Sharpe Mfg. Co.

chapter iv

Spiral Head—Indexing and Cutting Spirals

The mechanism known as the spiral head constituted one of the fundamental parts of the original universal milling machine. Its primary purpose was that of indexing and rotating work in
Spiral Head conjunction with the movement of the table for cutting flutes in twist drills. The great possibilities it offered in cutting a large range of spirals, and for doing many other jobs, were soon recognized and developed, until it is now used for an endless variety of operations. With it, ordinary indexing to obtain even spacing on the periphery of pieces, as in cutting teeth in cutters, ratchets, clutch gears, gear wheels and flutes in reamers, taps, drills, etc., can be quickly accomplished. Spiral forms of all common leads can be accurately reproduced by its use.

The spiral head and foot-stock are furnished as a part of all universal milling machines and can be applied, with few exceptions, to plain and vertical spindle machines. Used in connection with a vertical spindle milling attachment, on a plain machine, much the same variety of work can be done as on the universal milling machine.

In construction, spiral heads of today embody the same principles as the one on the original universal milling machine, but improvements have made them more solid and convenient to operate. Likewise, improvements have been made in the design and construction of the foot-stock.

Since our spiral head is typical of these mechanisms, a description of its various points may aid in understanding the methods of indexing and cutting spirals. The head itself consists of a hollow, semi-circular casting in which is mounted a spindle that is connected to an index crank through a worm and wheel. Fig. 4 shows the construction of this part. The head casting has dove-tailed bearings at each side that fit the contour of a base plate, which can be clamped to the surface of the table. The alignment of the head with the table longitudinally is provided by means of a tongue on the under side of the base plate that fits a T slot in the table.

The spiral head spindle passes through the head, and is held in place by means of a nut at the small end. The front end is threaded and has a taper hole corresponding to that of the machine spindle.

Fig. 4

It is rotated by means of the worm wheel B, which is driven by the hardened worm A that is located on the shaft to which the index crank is fastened. In order to insure accuracy the worm threads are ground after hardening. Through gearing, the index plate and worm A can be driven together from the table feed screw when the index pin is in position in any hole of the pla. When worm A is turned by means of the index crank, indexing may be accomplished, and when it is geared to the table feed screw, spiral milling, in addition to indexing, is made possible. The cutting of the spiral is due to the turning of the table feed screw, which through the interposition of change gears between this screw and the gears that drive the shaft carrying worm A, causes the spindle of the spiral head to rotate as the table advances, so that the cutter produces a spiral cut in the work. For rapid indexing, when cutting flutes in taps, reamers, etc., the worm A, is disengaged and the spindle turned by hand, the divisions being made by means of the index plate c, which is fastened to the

Fig. 5

nose of the spindle, and may be locked by the pin D.

The spindle may be revolved continuously as when cutting spirals, or may be securely locked after being revolved a desired amount, as in indexing for cutters, the teeth of gears, clutches, ratchets, etc.

It is possible to swing the head in its bearings so that the front end of the spindle can be set to any desired angle from 10&deg below the horizontal up to 5&deg beyond the perpendicular without throwing the driving members out of mesh. Graduations on the front exge of the head indicate the angle of elevation to half degrees.

The design of the head is such that it permits unusually long and wide bearings. Furthermore, it isets very low and can be so firmly clamped to the base that the whole mechanism practically becomes on solid casting. Hence, it provides a particularly rigid support for the work, which is a factor of much importance in the calss of work that is done upon this mechanism.

Index Plates and Change Gears. Three index plates are furnished with the spiral head, and contain circles with the following numbers of holes:—

Plate 1—	15,	16,	17,	18,	19,	20.
Plate 2—	21,	23,	27,	29,	31,	33.
Plate 3—	37,	39,	41,	43,	47,	49.

The change gears that are furnished have the following numbers of teeth: 24 (2 gears), 28, 32, 44, 48, 56, 64, 72, 86 and 100.

Graduated Index Sector. Without the graduated index sector, much care must be exercised in counting the holes in an index plate when indexing to obtain any given number of divisions. Such a sector enables the correct number of holes to bo obtained at each indexing with little chance for error. It is shown in Fig. 5 and consists of two arms which may be spread apart when the screw A, is loosened slightly. The correct number of holes may be counted and the sector arms set to include them; or better, the graduations on the dial may be used in connection with the tables given on pages 210 to 218. To set the sector arms by this last method, follow down the column headed "Graduation" in the tables referred to, until opposite the number of divisions that is desired. Take the number that is found here and set the arms by bringing the left one against the index pin, which should be inserted in any convenient hole in the required circle, and moving the right one until the graduation corresponding to the number obtained from the table coincides with the zero on the left arm. The correct number of holes will then be contained between the two arms, and counting is unnecessary.

When setting the arms by counting the holes, the left arm should be brought against the index pin as directed above, and then the required number of holes for each division should be counted from the hole that the pin is in, considering this hole as zero.

Fig. 6

Adjustable Index Crank. The index crank of the spiral head is adjustable circumferentially. This is shown in Fig. 6. Many times it is desired to make a delicate ajustment of the work, or to bring the index pin to the nearest hole without disturbing the setting of the work. To adjust the index crank after the work has been placed in position, turn thumb screws A-A, Fig. 6, until the pin enters the nearest hole in the index plate. To rotate the work relative to the index plate, both the stop pin at the back of the plate and the index crank pin should be engaged, the adjustment being made by means of the thumb screws as before.

Throwing Worm Out of Mesh. When it is desired to turn the spindle by hand and index work by means of the plate on the front end of the spindle, it is necessary to disengage the driving worm A, Fig. 4. To do this, turn the knob E, by means of a pin wrench furnished, about one-quarter of a revolution in the reverse direction to that indicated by an arrow stamped on the knob. This will loosen nut G, that clamps eccentric bushing H; then with the fingers turn both knobs E and F, at the same time, and the bushing H, will revolve, disengaging the worm from the wheel. To re-engage the worm, reverse the above operation.

Effect of Change in Angle of Elevation on Spindle. If the angle of the spiral head spindle is changed during operation, the spindle must be rotated slightly to bring the work back to the proper position, for when the spindle is elevated or depressed, the worm wheel is rotated about the worm, and the effect is the same as if the worm were turned.

Foot-stock. The foot-stock shown in Fig. 7 is for supporting pieces of work that are milled on centres or the outer ends of arbors, and pieces that are clamped in a chuck. The centre is adjustable longitudinally, and can be elevated or depressed by means of a rack V, and pinion actuated by hex U. It can also be set at an angle out of parallel with the base when it is desired to mill drills, taper reamers, etc., so that it can be kept in perfect alignment with the spiral head

Fig. 7

centre. The advantage of this is readily appreciated from the fact that by the use of centres that cannot be adjusted, work is aprt to become cramped at certain pisitions during its revolution, and, as a result, even spacing cannot be obtained.

When set in any position, the centre is firmly held by means of the nuts W, X and Y. Set screw S prevents endwise movement of the elevating pinion.

Two taper pins, one of which is shown at Z, are used to quickly and accurately locate the foot-stock centre in line with the spiral head centre, when the centres are parallel to the top of the table. They may be loosened by twisting a little with a wrench.

Fig. 8 shows a gauge that is very handy to use for quickly adjusting the foot-stock centre in line with the spiral head centre when setting for taper work. It consists of a bushing that fits over the centre in the spiral head and a blade, the bottom edge of which is the same distance above the centre as the top of the foot-stock centre.

Fig. 8

INDEXING

The first office of the spiral head is to index or divide the periphery of a piece of work into a number of definite or given parts. This is accomplished by means of the index crank and the index plates furnished with the head; or, in the case of some of the more common coarse division, by means of the rapid index plate fastened to the nose of the spindle.

There are two practical and accurate methods of indexing, known as Plain and Differential. A third method, known as the Compound, was used extensively in the past, and is still employed by some shops having machines that are not fitted for Differential indexing. The chances for errors in making the complicated indexing moves, adn the fact that even when the moves are made correctly, exact results cannot be obtained, causes the Compound method to be of little practical value where accurate spacing is required. It has, as a aresult, been largely superseded by the Differential method, by which the same numbers can be indexed accurately, and with little liability of errors in making the indexing moves.

Most spiral heads that are not fitted for Differential indexing can be at a nominal cost, and the unusual simplicity and convenience of this method in themselves are sufficient to warrant doing this.

By the Plain method of indexing, which includes rapid indexing, using the plate on the spindle nose, all divisions up to 50, even numbers up to 100, except 96, and many numbers that are multiples of 5 up to 380, besides many others, can be indexed with the three index plates furnished. With the addition of the change gears furnished, divisions obtained by Plain indexing, together with those that cannot be obtained by that method, from 1 to 382, and many others beyond, can be indexed by the Differential method. Plain and Direct Indexing. Plain indexing on the spiral head is very similar to indexing with ordinary index centres. It depends entirely upon how many times the index crank must be turned to cause the work to make one revolution. When this ratio is known, it is an easy matter to calculate the number of turns or fractions of a turn of the index crank to produce a given number of spaces on the periphery of the work.

The worm wheel on the spindle contains 40 teeth and the worm is single threaded, hence for every turn of the index crank, theworm wheel is advanced one tooth, or the spindle makes 1/40 part of a revolution. This should be remembered, for it is used in all indexing calculations on the spiral head. If the crank is turned 40 times, the spindle and work will make one complete revolution. To find how many turns of the crank are necessary for a certain division of the work, 40 is divided by the number of the divisions which are desired. The quotient will be the number of turns, or the part of a turn of the crank, which will give each desired division. Applying this rule, 40 divisions would be made by turning the crank completely around once for each division, or 20 divisions would be obtained by turning around twice. When the quotient contains a fraction, or is a fraction, it will be necessary to give the crank a part revolution in indexing. The numerator of the fraction represents the number of holes that should be indexed for each division. If the fraction is so small that none of the plates contains the number of holes represented by the denominator, both numerator and denominator should be multiplied by a common multiplier that will give a fraction, the denominator of which represents a number of holes that is available. On the other hand, if the fraction is of large terms, it should be reduced so that its denominator will represent a number of holes that is available. For example, seven divisions are desired. 40 divided by 7, equals 5 5/7 turns of the index crank to each division. There is no plate containing so few holes as 7, so this should be raised. Multiplying by the common multiplier 3, we have 5/7x3/3=15/21.Hence, for one division of the work, the index crank pin is placed in the 21 hole circle, and the crank is given 5 complete revolutions and then is moved ahead 15 additional holes. 35 holes in the 49 hole circle mightalso be used in place of 15 in the 21 hole circle, as 35/49 is a multiple of the original fraction 5/7.

The tables on pages 210 to 218 give the correct circles of holes and numbers to index for each division of all numbers that are obtainable by plain indexing, as well as those obtainable by the differential method, and when these are used figuring, such as that above, is uneccesary.

Indexing in Degrees and Parts of Degrees. When it is desired to divide the circumference of a piece in this manner, it can often be done by plain indexing. One complete turn of the index crank produces 1/40 of a turn of the work, or 360°/40 = 9 degrees. Following this method:

2 holes in the 18-hole circle = 1 degree.

2 holes in the 27-hole circle = 2/3 degree.

1 hole in the 18-hole circle = 1/2 degree.

1 hole in the 27-hole circle = 1/3 degree.

Other odd fractional parts of a degree can be easily found by dividing the number of holes in any given circle into 9 degrees. It will be noticed that 1/4 degree spacing cannot be obtained in this way; but with differential indexing, as explained on page 57, it is easy to get 1/4 degree and other fractional spacings.

Spiral Head Geared for Differential Indexing Differential Indexing. Differential indexing enables a wide range of divisions to be indexed, which cannot be obtained by plain indexing. With the change gears and three index plates furnashed with the spiral head, it is possible to index all numbers, not obtainable by plain indexing, from 1 to 382; in addition, many other divisions beyond 382 can be indexed.

By this method, the index crank is moved in the same circle of holes, and the operation is like that of plain indexing. The spiral head spindle and index plate are connected by a train of gearing, as shown above, and teh stop pin at the back of the plate is thrown out. As the index crank is turned, the spindle is rotated through the worm and wheel, and the plate moves either in the same or opposite direction to that of the crank. The total movement of the crank at every indexing is, therefore, equal to its movement relative to the plate, plus the movement of the plate, when the plate revolves in the same direction as the crank, or minus the movement of the plate, when the plate revolves in the opposite direction to the crank. The spiral head cannot be used for cutting spirals, when it is geared for differential indexing, for when cutting spirals the head is geared to the table feed screw.

To obviate the necessity of figuring out the change gears every time a certain number of divisions is required, tables on pages 210 to 225 have been compiled. By use of these tables, all numbers obtainable by differential indexing, together with those that can be had by the plain method can be easily indexed. The tables also give the correct circle and number of holes to be indexed, graduations for setting of the index sector, and the proper change gears to use.

In order to select the proper change gears, it is first necessary to find the ratio of the required gearing between the spindle and plate. After this has been done, the correct gears can be found. The following formulae show the manner in which this gearing is calculated.

N = number of divisions required.

H = number of holes in index plate.

n = number of holes taken at each indexing.

V = ratio of gearing between index crank and spindle.

x = ratio of the train of gearing between the spindle and the index plate.

S = gear on spindle.	} Drivers.
G1 = first gear on stud.

G2 = second gear on stud.	} Driven.
W = gear on worm.

x = HV-Nn/H if HV is greater than Nn.

x = Nn-HV/H if HV is less than Nn.

x = S/W (for simple gearing).

x = SG₁/G₂ W (for compound gearing).

V is equal to 40 on the B. & S. spiral head, and the index plates furnished have the following numbers of holes: 15, 16, 17, 18, 19, 20, 21, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 49.

The gears furnished have the following numbers of teeth: 24 (2 gears), 28, 32, 40, 44, 48, 56, 64, 72, 86, 100.

In selecting the index circle to be used, it is best to select one with a number having factors that are contained in the change gears on hand, for if H contains a factor not found in the gears, x cannot usually be obtained, unless the factor is canceled by the difference between HV and Nn, or unless N contains the factor.

When HV is greater than Nn and gearing is simple, use 1 idler.

When HV is greater than Nn and gearing is compound, use no idlers.

When HV is less than Nn and gearing is simple, use 2 idlers.

When HV is less than Nn and gearing is compound, use 1 idler.

Select "n" so that the ratio of gearing will not exceed 6:1 on account of the excessive stress upon the gears.

A few examples are given herewith to illustrate the application of the above formulae:

Example 1:

N = 59. Required H, n and x.

Assume H=33, n = 22.

Then $x={\frac {\left(33\times 40\right)-\left(59\times 22\right)}{33}}={\tfrac {22}{33}}={\tfrac {2}{3}}$.

We now select gears giving this ratio, as 32 and 48, the 32 being the gear on spindle and the 48 the gear on worm. HV is greater than Nn, and the gearing is simple, requiring 1 idler.

Example 2:

N = 319. Required H, n and x.

Assume H= 29, n = 4.

Then $x={\frac {\left(319\times 4\right)-\left(29\times 40\right)}{29}}={\tfrac {116}{29}}={\tfrac {4}{1}}$.

When the ratio is not obtainable with simple gearing, try compound gearing.

4/1 can be expressed as follows:

Fig. 9 ${\frac {3\times 4}{1\times 3}}$ or ${\frac {72\times 64}{24\times 48}}$ for which there are available gears.

HV is less than Nn and the gearing is compound, requiring 1 idler.

Head Geared for 271 Divisions

Fig. 9 shows the spiral head geared, simple gearing, for 271 divisions. Referring to the table on page 216, the gears called for are: C, 56 teeth, and E, 72 teeth, with
Fig. 10 one idler D. The idler D serves to rotate the index plate in the same direction as the crank, thus in making 280 turns of the crank, nine divisions are lost, giving the correct number of divisions, 271. The sector should be set to indicate 1/7 turns, or 3 holes in the 21 hole circle, and the head is ready for 271 divisions, the indexing being made the same as for plain indexing.

Head Geared for 319 Divisions.

Fig. 10 shows the spiral head geared, compound gearing, for 319 divisions. Referring to the table on page 217, the gears called for are: C, 48 teeth; F, 64 teeth; G, 24 teeth; E, 72 teeth and one idler D, 24 teeth. The sector should be set to 4/29 turns, or 4 holes in the 29 circle; the head is then ready for 319 divisions.

Spacing for Quarter Degrees.

Example 3.

Required H, n and x for spacing 1/4 degrees, or 1440 divisions.

Assume H = 33, n = l.

Then ${\frac {\left(1440\times 1\right)-\left(33\times 40\right)}{33}}={\frac {120}{33}}{\text{or}}{\frac {64\times 100}{40\times 44}}$

One idler is required.

The following table gives data required for spacing 1/4° and 1/6°. For fractional parts of degrees obtainable by plain indexing see page 54.

Divisions	Index Circle	No. of Turns of Index	Gradua- tion	Gear on Worm	No. 1 Hole		Gear on Spindle	Idlers
					1st Gear on Stud	2d Gear on Stud		No. 1 Hole	No. 2 Hole
1/6° 1/4°	49 33	1/49 1/33		28 44	64 64	56 40	100 100		24 24

Aliquant or Fractional Spacing.

Example 4:

Required: A Vernier to read to 1/12 degree or five minutes, the scale being divided to degrees.

Each Vernier space can equal 11/12 degree.

${\frac {11\times 1}{12\times 360}}={\frac {11}{4320}}{\text{or}}{\frac {4320}{11}}$ spaces in whole circle = 392 8/11 spaces.

Assume H = 18, n = 2.

Then ${\frac {\left(392{\tfrac {8}{11}}\times 2\right)-\left(18\times 40\right)}{18}}={\frac {720/11}{18}}={\frac {720}{11}}\times {\frac {1}{18}}={\frac {40}{11}}={\frac {64\times 100}{40\times 44}}$

One idler is required.

CUTTING SPIRALS.

Spirals that are most commonly cut on milling machines embrace spiral gears, spiral mills, counterbores, and twist drills. Worms are also cut with the aid of a vertical spindle or universal milling attachment. Examples of some of these classes of work are shown in this chapter, and in operations in chapters VIII and IX.

The method of producing the spiral movement of the work has been described before, and the manner in which the head is geared is shown in Figs. 11 and 12. The four change gears are known as: gear on screw; first gear on stud (as it is the first to be put on); second gear on stud; and gear on worm. The screw gear and first gear on stud are the drivers, and the others are the driven gears. By using different combinations of the change gears furnished, the ratio of the longitudinal movement of the table to the rotary movement of the work can be varied ; in other words, the leads of the spirals it is possible to cut are governed directly by these gears. Usually they are of such ratio that the work is advanced more than an inch while making one turn, and thus the spirals cut on milling machines are designated in terms of inches to one turn, rather than turns, or threads per inch; for instance, a spiral is said to be of 8 inches lead, not that its pitch is 1-8 turn per inch.

The feed screw of the table has four threads to the inch, and forty turns of the worm make one turn of the spiral head spindle; accordingly, if change gears of equal diameter are used, the work will make a complete turn while it is moved lengthwise 10 inches; that is, the spiral will have a lead of 10 inches. This is the lead of the machine, and it is the resultant of the action of the parts of the machine that are always employed in this work, and is so regarded in making the calculations used in cutting spirals.

Fig. 11
Showing Gearing When No Idler is Required

Fig. 12

Showing Gearing With Idler in Use

Principle same as for Change Gears of a Lathe. In principle,

these calculations are the same as for change gears of a screw cutting lathe. The compound ratio of the driven to the driving gears equals in all cases, the ratio of the lead of the required spiral to the lead of the machine. This can be readily demonstrated by changing the diameters of the gears.

Gears of the same diameter produce, as explained above, a spiral with a lead of 10 inches, which is the same lead as the lead of the machine. Three gears of equal diameter and a driven gear double this diameter, produce a spiral with a lead of 20 inches, or twice the lead of the machine; and with both driven gears, twice the diameters of the drivers, the ratio being compound, a spiral is produced with a lead of 40 inches, or four times the machine's lead. Conversely, driving gears twice the diameter of the driven produce a spiral with a lead equal to ¹⁄₄ the lead of the machine, or 2+¹⁄₂ inches.

Expressing the ratios as fractions, the ${{\text{Driven Gears}} \over {\text{Driving Gears}}}={{\text{Lead of Required Spiral}} \over {\text{Lead of Machine}}}$ or, as the product of each class of gears determines the ratio, the head being compound geared, and as the lead of the machine is ten inches, the ${{\text{Product of Driven Gears}} \over {\text{Product of Driving Gears}}}={{\text{Lead of Required Spiral}} \over 10}$ That is, the compound ratio of the driven to the driving gears may always be represented by a fraction whose numerator is the lead to be cut and whose denominator is 10. In other words, the ratio is as the required lead is to 10; for example, if the required lead is 20, the ratio is 20:10. To express this in units instead of tens, the ratio is always the same as one-tenth of the required lead is to 1. And frequently this is a very convenient way to think of the ratio; for example, if the lead is 40, the ratio of the gears is 4:1. If the lead is 25, the gears are 2.5:1, etc.

To illustrate the usual calculations, assume that a spiral of 12 inch lead is to be cut. The compound ratio of the driven to the driving gears equals the desired lead divided by 10, or it may be represented by the fraction 12/10. Resolving this into two factors to represent the two pairs of change gears, 12/10 = 3/2 X 4/5. Both terms of the first factor are multiplied by such a number (24 in this instance) that the resulting numerator and denominator will correspond with the number of teeth of two of the change gears furnished with the machine (such multiplications not affecting the value of a fraction) 3/2 X 24/24 = 72/45. The second factor is similarly treated: 4/5 X 8/8 = 32/40, and the gears with 72, 32, 48 and 40 teeth are selected. ${12 \over 10}=\left({72\times 32 \over 48\times 40}\right)$ The first two are the driven, and the last two the drivers, the numerators of the fractions representing the driven gears. The 72 is the worm gear, 40 the first on stud, 32 the second on stud and 48 the screw gear. The two driving gears might be transposed, and the two driven gears might also be transposed without changing the spiral. That is, the 72 could be used as the second on stud and the 32 as the worm gear, if such an arrangement were more convenient. The following rules express in abridged form the methods of figuring change gears to cut given spirals, and of ascertaining what spirals can be cut with change gears.

Rules for Obtaining Ratio of the Gears Necessary to Cut a Given Spiral. Note the ratio of the required lead to 10. This ratio is the compound ratio of the driven to the driving gears. Example: If the lead of required spiral is 12 inches, 12 to 10 will be the ratio of the gears.

Or, divide the required lead by 10 and note the ratio between the quotient and 1. This ratio is usually the most simple form of the compound ratio of the driven to the driving gears. Example: If the required lead is 40 inches, the quotient is 40÷10 and the ratio 4 to 1.

Rule for Determining Number of Teeth of Gears Required to Cut a Given Spiral. Having obtained the ratio between the required lead and 10 by one of the preceding rules, express the ratio in the form of a fraction; resolve this fraction into two factors, raise these factors to higher terms that correspond with the teeth of gears that can be conveniently used. The numerators will represent the driven and the denominators the driving gears that produce the required spiral. For example: What gears shall be used to cut a lead of 27 inches?

${\tfrac {27}{10}}={\tfrac {3}{2}}\times {\tfrac {9}{5}}=\left({\tfrac {3}{2}}\times {\tfrac {16}{16}}\right)\times \left({\tfrac {9}{5}}\times {\tfrac {8}{8}}\right)={\frac {48\times 72}{32\times 40}}$

From the fact that the product of the driven gears divided by the product of the drivers equals the lead divided by 10, or one-tenth of the lead, it is evident that ten times the product of the driven gears divided by the product of the drivers, will equal the lead of the spiral. Hence the rule:

Rule for Ascertaining what Spiral May be Cut by Any Given Change Gears. Divide ten times the product of the driven gears by the product of the drivers, and the quotient is the lead of the resulting spiral in inches to one turn. For example: What spiral will be cut by gears, with 48, 72, 32 and 40 teeth, the first two being used as driven gears? Spiral to be cut equals ${\frac {10\times 48\times 72}{32\times 40}}=27$ inches to one turn.

This rule is often of service in determining what spirals may be cut with the gears the workman chances to have at hand.

The tables on pages 226 to 228 give the leads and approximate angles of some spirals produced by the gears furnished with our machines, and the combination of gears given in each case is such that they will properly mesh with one another. The tables on pages 229 to 247 contain all the leads that can be obtained with any possible combination of the change gears furnished, even though some of the leads are not available for use on account of the gears interfering or not reaching. Combinations of gears that are too small in diameter to reach for right-hand spirals, can generally be used for left-hand spirals, as the reverse gear is then required and will enable the gears to reach.

As we have already mentioned, the two driving gears, or the two driven gears of any combination can be transposed, but a driver must not be substituted for a driven or vice versa. Four different arrange- ments of the gears of any combination are thus possible, without changing the ratio, and when one arrangement interferes, or will not reach, the others should be tried. Thus, the gears to give a lead of 3.60" are: drivers, 100 teeth and 32 teeth; driven, 24 teeth and 48 teeth. By transposing the gears, the following four arrangements may be obtained.

Drivers.

	1st	2nd	3rd	4th
Gear on Screw	100	32	100	32
1st Gear on Stud	32	100	32	100

Driven.

2nd Gear on Stud	24	24	48	48
Gear on Worm	48	48	24	24

The first arrangement, however, is found by actual test to be the only one available, owing to the interference of the gears in the other combinations preventing their meshing properly.

When very short leads are required, it is preferable to disengage the worm wheel and connect the gearing directly to the spiral head spindle (using the differential indexing centre). This method gives leads one-fortieth of the leads given in the table for the same combinations of gears. Thus, for a lead of 6.160", the table calls for gear on worm, 56 teeth; 1st gear on stud, 40 teeth; 2nd gear on stud, 44 teeth; and gear on screw, 100 teeth. Putting the 56 tooth gear on the spindle instead of on the worm, gives a lead of 6.160/40 = .154".

By this method, very short leads may be obtained without excessively straining the mechanism, but the regular means of indexing the work cannot be employed. A method that can be used for indexing when using the differential centre is to have the number of teeth in the gear on the spindle some multiple of the number required to

Fig. 13

be indexed. Swing the gears out of mesh and advance the gear on spindle the number of teeth required to index the work one division at each indexing. Thus, if 9 divisions are required with a lead of .261 ", we select a lead from the table equal to about .261" X 40 = 10.440", when the gear on worm (which will now be the gear on spindle) is some multiple of 9, as 72. The nearest lead is 10.467", which gives 10.467/10 = .2617" lead, giving an error of .0007". To index the work, the gear on spindle is advanced 72/9 = 8 teeth at each indexing.

The short lead attachment described in the next chapter can also be used to cut short leads, an index plate being provided for use in cutting multiple threads. Position of the Table in Cutting Spirals. The change gears having been selected, the next step in cutting spirals is to determine the position at which the table must be placed to bring the spiral in line with the cutter as the work is being milled.

The correct position of the table is indicated by the angle shown at A, Fig. 13, and this angle, as may be noticed from that figure, has the same number of degrees as the angle B, which is temred the angle of the spiral, and is formed by the intersection of teh spiral and a line parallel with the axis of the piece being milled. The reason the angles A and B are alike, is that their corresponding sides are perpendicular to each other.

Fig. 14

The angle of the spiral depends upon the lead of the spiral and the diameter of the piece to be milled. The greater the lead of a spiral of any given diameter, the smaller the angle, and the greater the diameter of any spiral with a given lead, the greater the spiral angle.

If the angle wanted is not found in the tables on pages 226 to 228, it can be ascertained in two ways, graphically or more conveniently, by a simple calculation and reference to the tables on pages 309 to 317. In determining it graphically, a right-angle triangle is drawn to scale. One of the sides which forms the right angle represents the circumference of the piece in inches, and the hypothenuse represents the line of the spiral. The angle between the lines representing the path of the spiral and the lead of the spiral is the angle of the spiral. This angle can be transferred from the drawing to the work by a bevel protractor, or even by cutting a paper templet and winding it about the work as shown in Fig. 14. The machine is then set
Fig. 15 so that the spiral or groove as it touches the cutter will be in line with the cutter. Or the angle may be measured and the saddle set to a corresponding number of degrees by the graduations on the base.

The natural tangent of the angle of the spiral is the quotient of the circumference of the piece, divided by the lead of the spiral. Accordingly, the second method of obtaining the angle of the spiral is to divide the circumference of the piece by the lead, and note the number of degrees opposite the figures that correspond with the quotients in the tables of natural tangents, pages 309 to 317. The angle having been thus obtained, the saddle is set by the graduations on the base.

This second method is more satisfactory, as it is Fig. 15 more accurate, and there is less liability of error than with the first. The saddle can be set to the proper angle, but before cutting into the blank, it is well to let the mill just touch the work, then run the work along by hand and make a slight spiral mark, and by this mark see whether the qjiange gears give the right lead.

Special care should be taken in cutting spirals that the work does not slip, and when a cut is made it is well to drop the work away from the mill while coming back for another cut, or the mill may be stopped and turned to such a position that the teeth will not touch the work while the table is brought back preparatory to another cut.

Setting Cutter Central. In making such cuts as are alike on both sides, for instance, the threads of worms or the teeth of spiral gears, care must be taken to set the work centrally perpendicular with the centre line of the cutter before swinging the saddle to the angle of the spiral.

Cuts that have one face radial, especially those that are spiral, are best made with an angular cutter of the form shown in Fig. 15, as cutters of this form readily clear the radial face of the cut, keep sharp for some time and produce a smooth surface.

Fig. 16

Fig. 17

Twist Drills. The operation of milling a twist drill is shown in

Fig. 16. The drill is held in a collet, or chuck, and, if very long, is allowed to pass through the spindle of the spiral head. The cutter is brought directly over the centre of the drill, and the table is set at the angle of spiral.

The depth of groove in a twist drill diminishes as it approaches the shank, in order to obtain increased strength at the place where the drill generally breaks. The variation in depth is conditional, depending mainly on the strength it is desirable to obtain, or the usage the drill is subject to. To secure this variation in the depth of the groove, the spiral head spindle is elevated slightly, depending on the length of the flute and diameter of the drill.

The outer end of the drill is supported by the centre rest, and when quite small, should be pressed down firmly, until the cutter has passed over the end.

The elevating screw of this rest is hollow, and contains a small centre piece with a V groove cut therein to aid in holding the work central. This piece may be made in other shapes to adapt it to special work.

Another, and very important operation on the twist drill, is that of "backing off" the rear of the lip, so as to give it the necessary clearance, to prevent excessive friction during drilling. In the illustration, Fig. 17, the saddle is turned about one-half degree as for cutting a right-hand spiral, but as the angle depends on several conditions, it will be necessary to determine what the effect will be under different circumstances. A slight study of the figure will be sufficient for this, by assuming the effect of different angles, mills and the pitches of spirals. The object of placing the saddle at an angle is to cause the mill E to cut into the lip at c', and have it just touch the surface at e'. The line r being parallel with the face of the mill, the angular deviation of the saddle is shown at a, in comparison with the side of the drill.

From a little consideration it will be seen that while the drill has a positive traversing and rotative movement, the edge of the mill at e' must always touch the lip at a given distance from the front edge; this being the vanishing point, if such we may call it. The other surface forming the real diameter of the drill is beyond reach of the cutter, and is so left to guide and steady it while in use. The point e, shown in the enlarged section, shows where the cutter commences, and its increase until it reaches a maximum depth at c, where it may be increased or diminished according to the angle employed in the operation, the line of cutter action being represented by ii.

Before backing off, the surface of the smaller drills in particular should be colored with a solution of sulphate of copper, water and sulphuric acid. This solution can be applied with a piece of waste, and will give the piece a distinct copper color. The object of this is to clearly show the action of the mill on the lip of the drill, for, when satisfactory, a uniform streak of coppered surface the full length of the lip from the front edge g back to e, is left untouched by the mill.

The above-mentioned coloring solution can be made by the following formula:

Sulphate of copper (saturated solution) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 oz.

Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8 oz.

Sulphuric acid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 oz.

It is sometimes preferred to begin the cut at the shank end. By starting the cut in at this end, the tendency to lift the drill blank from the rest is lessened.

The table given on page 326 is useful for determining the cutters, pitches, gears and angles for twist drills.

Cutting Left-Hand Spirals. When giving directions for cutting spirals in any of the foregoing pages, right-hand spirals are at all times referred to. For the production of left-hand spirals, the only changes necessary are the swinging of the saddle to the opposite side of the centre line, and the introduction of an intermediate gear upon the stud, Fig. 12, to engage with either pair of change gears for changing the direction of rotation of the spiral head spindle.

Cutting Spirals with an End Mill. When spirals cannot be conveniently cut with side or angular milling cutters, as previously described, it is sometimes convenient to use end mills, as for example, when the diameter of the piece is very large, or the spiral is of such a lead that the table cannot be set at the requisite angle, the work is so held that its centre and that of the mill will be in the same plane and the saddle is set at zero.