[Ill.u.s.tration: Fig. 1237.]

The term "compounded" as applied to the change gears of a lathe, means that there exists in it a shaft or some equivalent means by which the velocity of the wheels may be changed. Such a shaft is shown at R in Fig. 1236, and it affords a means of compounding by placing on its outer end, as at D, a wheel that has a different number of teeth to that in wheel C. In Fig. 1237 this change is made, wheel D having 40 teeth instead of the 20 it had before. As in the former case, however, it will make one revolution to one of C or one of A, but having 40 teeth it will move 40 of the teeth in I past the line of centres, and this will cause the lead screw wheel S to make two revolutions, because it has 20 teeth only. Thus, the compounding of C and D on shaft R has caused S to make two revolutions to one of A, or, what is the same thing, one revolution of A will in this case cause S to make two revolutions, and the thread cut would be twice as coa.r.s.e as the lead-screw thread. In the case of a lathe geared as in either Fig. 1235 or 1236, all the wheels that we require to consider in calculating the change wheels are D and S. Now, the shaft R is called the "mandrel," the "stud," or the "spindle," all three terms being used, and the wheel D is the wheel on the stud, mandrel, or spindle, while in every case S is that on the lead screw, and the revolutions of this wheel D and those of the lead screw will be in the same proportion as exists between their numbers of teeth. In considering their revolutions it is to be borne in mind that when D has more teeth than S the speed of the lead screw is increased, and the lathe will cut a thread coa.r.s.er than that of its lead screw, or when D has less teeth than S the speed of the lead screw is diminished, and the pitch of thread cut will be finer than that of the lead screw.

[Ill.u.s.tration: Fig. 1238.]

Another method of compounding is shown in Fig. 1238, the compounded pair C D being on a stud carried in the swing frame F. Now, suppose A has 32, C 64, D 32, and S 64 teeth, the revolution being in the same proportion as the numbers of teeth, C will make one-half a revolution to one revolution of A, and D, being fast to the same stud as C, will also make one-half revolution to one revolution of A. This one-half revolution of D will cause S to make one-quarter of a revolution; hence the thread cut will be four times as fine as the pitch of the thread on the lead screw, because while the lathe makes one turn the lead screw makes one-quarter of a turn. In this arrangement we are enabled to change wheel C as well as wheel D (which could not be done in the arrangement shown in Fig.

1236), and for this reason more changes can be made with the same number of wheels. When the wheel C makes either more or less revolutions than the driver A, it must be taken into account in calculating the change wheels. As arranged in Fig. 1236, it makes the same number as A, which is a very common, arrangement, but in Fig. 1238 it is shown to have twice as many teeth as A; hence it makes half as many revolutions. In the latter case we have two pairs of wheels, in each of which the driven wheel is twice the size of the driver; hence the revolutions are reduced four times.

Suppose it is required to cut a thread of eight to an inch on a lathe such as shown in Fig. 1235, the lead screw pitch being four per inch, and for such simple trains of gearing we have a very simple rule, as follows:--

_Rule._--Put down the pitch of the lead screw as the numerator, and the pitch of thread you want to cut as the denominator of a vulgar fraction, and multiply both by the pitch of the lead screw, thus:

Pitch of lead screw. {the number of teeth for the Pitch of lead screw 4 4 16 = {wheel on the spindle.

- - = -- Pitch to be cut 8 4 32 = {the number of teeth for the {wheel on the lead screw.

There are three things to be noted in this rule; and the first is, that when the pitch of the lead screw and the pitch of thread you want to cut is put down as a fraction, the numerator at once represents the wheel to go on the stud, and the denominator represents the wheel to go on the lead screw, and no figuring would require to be done providing there were gear-wheels having as few teeth as there are threads per inch in the lead screw, and that there was a gear-wheel having as many teeth as the threads per inch required to be cut. For example, suppose the lathe in Fig. 1236 to have a lead screw of 20 per inch, and that the change wheels are required to cut a pitch 40, then we have 20/40, the 20 to go on at D in Fig. 1236 and the 40 to go on the lead screw. But since lead screws are not made of such fine pitch, but vary from two threads to about six per inch, we simply multiply the fraction by any number we choose that will give us numbers corresponding to the teeth in the change wheels. Suppose, for example, the pitch of lead screw is 2, and we wish to cut 6, then we have 2/6, and as the smallest change wheel has, say, 12 teeth we multiply the fraction by 6, thus: 2/6 6/6 = 12/36. If we have not a 12 and a 36 wheel, we may multiply the fraction by any other number, as, say, 8; thus: 2/6 8/8 = 16/48 giving us a 16 wheel for D, Fig. 1236, and a 48 wheel for the lead screw.

The second notable feature in this rule is that it applies just the same whether the pitch to be cut is coa.r.s.er or finer than the lead screw; thus: Suppose the pitch of the lead screw is 4, and we want to cut 2. We put these figures down as before 4/2, and proceed to multiply, say, by 8; thus: 4/2 8/8 = 32/16, giving a 32 and a 16 as the necessary wheels.

The third feature is, that no matter whether the pitch to be cut is coa.r.s.er or finer than the lead screw, the wheels go on the lathe just as they stand in the fraction; the top figure goes on top in the lathe, as, for example, on the driving stud, and the bottom figures of the fraction are for the teeth in the wheel that goes on the bottom of the lathe or on the lead screw. No rule can possibly be simpler than this. Suppose now that the pitch of the lead screw is 4 per inch and we want to cut 1-1/2 per inch. As the required pitch is expressed in half inches, we express the pitch of the lead in half inches, and employ the rule precisely as before. Thus, in four there are eight halves; hence, we put down 8 as the numerator, and in 1-1/2 there are three halves, so we put down 3 and get the fraction 8/3. This will multiply by any number, as, say, 6; thus: (8/3) (6/6) = (48/18), giving us 48 teeth for the wheel D in Fig. 1236, and 18 for the lead screw wheel S.

In a lathe geared as in Fig. 1235 the top wheel D could not be readily changed, and it would be more convenient to change the lead screw wheel S only. Suppose, then, that the lead screw pitch is 2 per inch, and we want to cut 8. Putting down the fraction as before, we have 2/8, and to get the wheel S for the lead screw we may multiply the number of teeth in D by 8 and divide it by 2; thus: 32 8 = 256, and 256 2 = 128; hence all we have to do is to put on the lead screw a wheel having 128 teeth. But suppose the pitch to be cut is 4-1/4, the pitch of the lead screw being 2. Then we put both numbers into quarters, thus: In 2 there are 8 quarters, and in 4-1/4 there are 17 quarters; hence the fraction is 8/17. If now we multiply both terms of this 8/17 by 4 we get 32/68, and all we have to do is to put on the lead screw a wheel having 68 teeth.

When we have to deal with a lathe compounded as in Fig. 1238, in which the combination can be altered in two places--that is, between A and C and between D and S--the wheel A remaining fixed, and the pitch of the lead screw is 2 per inch, and it is required to cut 8 per inch--this gives us the fraction 2/8, which is at once the proportion that must exist between the revolutions of the wheel A and the wheel S. But in this case the fraction gives us the number of revolutions that wheel S must make while the wheel A is making two revolutions, and it is more convenient to obtain the number that S requires to make while A is making one revolution, which we may do by simply dividing the pitch required to be cut by the pitch of the lead screw, as follows: Pitch of thread required, 8; pitch of lead screw, 2; 8 2 = 4 = the revolutions S must make while A makes one. We have then to reduce the revolutions four times, which we may do by putting on at C a wheel with twice as many teeth in it as there are in A, and as A has 32, therefore C must have 64 teeth. When we come to the second pair of wheels, D and S, we may put any wheel we like in place of D, providing we put on S a wheel having twice as many.

But suppose we require to cut a fractional pitch, as, say, 4-1/8 per inch, the pitch of lead screw being 2, all we have to do is to put the pitch of the lead screw into eighths, and also put the number of teeth in A into eighths; thus: In two there are 16 eighths, and in the pitch required there are 33 eighths; hence for the pitch of the lead screw we use the 16, and for the thread required we use the 33, and proceed as before; thus:

Pitch of thread Pitch of lead required. screw.

33 16 = 2-1/16 = the revolution which A must make while wheel B makes one revolution.

The simplest method of doing this would be to put on at C a wheel having 2-1/16 times as many teeth as there are in A. Suppose then that A has 32 teeth, and one sixteenth of 32 = 2, because 32 16 = 2. Then twice 32 is 64, and if we add the 2 to this we get 66; hence, if we give wheel C 66 teeth, we have reduced the motion the 2-1/16 times, and we may put on D and S wheels having an equal number of teeth. Or we may put on a wheel at C having the same number as A has, and then put on any two wheels at D and C, so long as that at S has 2-1/16 times as many teeth as that at D.

Again, suppose that the pitch of a lead screw is 4 threads per inch, and that it be required to find what wheels to use to cut a thread of 11/16 inch pitch, that is to say, a thread that measures 11/16 inch from one thread to the other, and not a pitch of 11/16 threads per inch: First we must bring the pitch of the lead screw and the pitch to be cut to the same terms, and as the pitch to be cut is expressed in sixteenths we must bring the lead screw pitch to sixteenths also. Thus, in an inch of the length of the lead screw there are 16 sixteenths, and in this inch there are 4 threads; hence each thread is 4/16 pitch, because 16 4 = 4. Our pitch of lead screw expressed in sixteenths is, therefore, 4, and as the pitch to be cut is 11/16 it is expressed in sixteenths by 11; hence we have the fraction 4/11, which is the proportion that must exist between the wheels, or in other words, while the lathe spindle (or what is the same thing, the work) makes 4 revolutions the lead screw must make 11.

Suppose the lathe to be single geared, and not compounded, and we multiply this fraction and get--

4 4 16 = wheel to go on lead screw.

-- - = -- 11 4 44 = " " stud or mandrel.

4 5 20 = wheel to go on lead screw.

Or, -- - = -- 11 5 55 = " " stud or mandrel.

4 6 24 = wheel to go on lead screw.

Or, -- - = -- 11 6 66 = " " stud or mandrel.

But suppose the lathe to be compounded as in Fig. 1235, and we may arrange the wheels in several ways, and in order to make the problem more practical, we may suppose the lathe to have wheels with the following numbers of teeth, 18, 24, 36, 36, 48, 60, 66, 72, 84, 90, 96, 102, 108, and 132.

[Ill.u.s.tration: Fig. 1239.]

Here we have two wheels having each 36 teeth; hence we may place one of them on the lathe spindle and one on the lead screw, as in Fig. 1239; and putting down the pitch of the lead screw, expressed in sixteenths as before, and beneath it the thread to cut also in sixteenths, we have:

4 6 24 = wheel to be driven by lathe spindle, -- - = -- 11 6 66 = " to drive lead screw wheel;

the arrangement of the wheels being shown in Fig. 1239.

We may prove the correctness of this arrangement as follows: The 36 teeth on the lathe spindle will in a revolution cause the 24 wheel to make 1-1/2 revolutions, because there are one and a half times as many teeth in the one wheel as there are in the other; thus: 36 24 = 1-1/2.

Now, while the 24 wheel makes 1-1/2, the 66 will also make 1-1/2, because they are both on the same sleeve and revolve together. In revolving 1-1/2 times the 66 will cause the 36 on the lead screw to make 2-3/4 turns, because 99 36 = 2-3/4 (or expressed in decimals 2.75), and it thus appears that while the lathe spindle makes one turn, the lead screw will make 2-3/4 turns.

Now, the proportion between 1 and 2-3/4 is the same as that existing between the pitch of the lead screw and the pitch of the thread we want to cut, both being expressed in sixteenths; thus:

Pitch of lead screw in sixteenths 4 } }, and 11 4 = 2-3/4; " to be cut in sixteenths 11 }

that is to say, 11 is 2-3/4 times 4.

Suppose it is required, however, to find what thread a set of gears already on the lathe will cut, and we have the following rule:

_Rule._--Take either of the driven wheels and divide its number of teeth by the number of teeth in the wheel that drives it, then multiply by the number of teeth in the other driving wheel, and divide by the teeth in the last driven wheel. Then multiply by the pitch of the lead screw.

[Ill.u.s.tration: Fig. 1240.]

_Example._--In Fig. 1240 are a set of change wheels, the first pair of which has a driving wheel having 36 teeth, and a driven wheel having 18 teeth. The second pair has a driving wheel of 66 teeth, and a driven wheel of 48.

Let us begin with the first pair and we have 36 18 = 2, and this multiplied by 66 is 132. Then 132 48 = 2.75, and 2.75 multiplied by 4 is 11, which is the pitch of thread that will be cut. Now, whether this 11 will be eleven threads per inch, or as in our previous examples a pitch of 11/16 inch from one thread to another or to the next one, depends upon what the pitch of the lead screw was measured in.

[Ill.u.s.tration: Fig. 1241.]

If it is a pitch of 4 threads per inch, the wheels will cut a thread of 11 per inch, while if it were a thread of 4/16 pitch, the thread cut will be 11/16 pitch.

Let us now work out the same gears beginning from the lead screw pair, and we have as follows:

Number of teeth in driver is 66, which divided by the number in the driven, 48, gives 1.375. This multiplied by the number of teeth in the driver of the other pair = 36 gives 49.5, which divided by the number of teeth in the driven wheel of the first pair gives 2.75, which multiplied by the pitch of the lead screw 4 gives 11 as before.

Taking now the second example as in Fig. 1240, and beginning from the first pair of gears, we have, according to the rule, 36 48 66 18 4 = 11 = pitch the gears will cut; or proceeding from the second pair of gears, we have by the rule, 66 18 36 48 4 = 11 = the pitch the gears will cut. It is not often, however, that it is required to determine what threads the wheels already on a lathe will cut, the problem usually being to find the wheels to cut some required pitch. But it may be pointed out that when the problem is to find the result produced by a given set of wheels, it is simpler to begin the calculation from the wheel already on the lathe spindle, rather than beginning with that on the lead screw, because in that case we begin at the first wheel and calculate the successive ones in the same order in which we find them on the lathe, instead of having to take the last pair in their reverse order, as has been done in the examples, when we began at the wheel on the lead screw, which we have termed the second pair.

The wheels necessary to cut a left-hand thread are obviously the same as those for a right-hand one having an equal pitch; all the alteration that is necessary is to employ an additional intermediate wheel, as at I in Fig. 1241, which will reverse the direction of motion of the lead screw. For a lathe such as shown in Fig. 1235, this intermediate wheel may be interposed between wheels D and I or between I and S. In Fig.

1236, it may be placed between D and I or between I and S, and in Fig.

1238 it may be placed between A and C or between D and S.

[Ill.u.s.tration: Fig. 1242.]

Here it may be well to add instructions as to how to arrange the change wheels to cut threads in terms of the French centimetre. Thus, an inch equals 254/100 of a centimetre, or, in other words, 1 inch bears the same proportion to a centimetre as 254 does to 100, and we may take the fraction 254/100 and reduce it by any number that will divide both terms of the fraction without leaving a remainder; thus, 254/100 2 = 127/50.

If, then, we take a pair of wheels having respectively 127 and 50 teeth, they will form a compound pair that if placed as in Fig. 1242 will enable the cutting of threads in terms of the centimetre instead of in terms of the inch.

Thus, for example, to cut 6 threads to the centimetre, we use the same change wheels on the stud and on the lead screw that would be used to cut 6 threads to the inch, and so on throughout all other pitches.

CUTTING DOUBLE OR OTHER MULTIPLE THREADS IN THE LATHE.--In cutting a double thread the change wheels are obviously arranged for the pitch of the thread, and one thread, as A in Fig. 251 is cut first, and the other, B, afterwards. In order to insure that B shall be exactly midway between A, the following method is pursued. Suppose the pitch of the lead screw is 4 threads per inch, and that we require to cut a double thread, whose actual pitch is 8 per inch, and apparent pitch 16 per inch, then the lead screw requires to make half a turn to one turn of the lathe spindle; or what is the same thing, the lathe spindle must make two turns to one of the lead screw, hence the gears will be two to one, and in a single-geared lathe we may put on a 36 and a 72, as in Fig. 1243, in which the intermediate wheels are omitted, as they do not affect the case. With these wheels we cut a thread of 8 per inch and then, leaving the lead screw nut still engaged with the lead screw and the tool still in position to cut the thread already formed, we make on the change wheels a mark as at S T, and after taking off the driving gear we make a mark at s.p.a.ce _u_, which is 18 teeth distant from S, or half-way around the wheel. We then pull the lathe around half a turn and put the driving gear on again with the s.p.a.ce _u_ engaged with the tooth T, and the lathe will cut the second thread exactly intermediate to the first one. If it were three threads that we require to cut, we should after the driving gear was taken off give the lathe one-third a revolution, and put it back again, engaging the twelfth s.p.a.ce from S with tooth T, because one-third of 36 is 12.

[Ill.u.s.tration: Fig. 1243.]