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.
Pitch of lead screw 4 × 4 = 16 = {the number of teeth for the 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 coarser 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 coarser 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 11⁄2 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 11⁄2 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 41⁄4, the pitch of the lead screw being 2. Then we put both numbers into quarters, thus: In 2 there are 8 quarters, and in 41⁄4 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.