This will alter the formula given above'to this extent. The apparent pitch of the lead-screw will be in-increased, and before calculating its gear, multiply the number of threads per inch by 2 if the stud revolves one-half as fast as the spindle, by 3 if the ratio is 1 :3, or by 1.5 if the ratio is 3 : 2. Then use the apparent pitch of the lead-screw for Lp.

There is one fractional thread that is very often cut, and that is the thread used in pipe work, 11 1/2threads per inch. It may be interestingto use this example as an illustration in assisting the amateur to understand the foregoing directions. Let us assume that the lead screw has 6 threads per inch and that the number of threads to be cut is 11 1/2perinch. The gear on the spindle is driven direct and at the same speed as the spindle, and the constant number between the gears is 4. We wish to find the gears for the spindle and lead-screw.

The pitch of the lead-screw is 1-6, and the pitch of the thread to be cut is 1-11 1/2, or 2-23. Reducing these two fractions to a common denominator, we have 12-138 and 23-138, respectively. These two screws, therefore, will be in the proportion of 12 to 23, the denominators of the fractions, and we can assume that the threads to be cut are 23 per inch and that the lead-screw has 12 threads per inch. Multiplying each by the constant 4, we have 4x12=48, the number of teeth in the spindle gear, and 4x23=92, the number of teeth for the lead-screw. These two gears, therefore, cut 11 1/8 threads per inch, or, since this is the ratio that is to be preserved and not the number of teeth, two gears of 24 and 46 teeth would accomplish the same result, as would 12 and 23 teeth. Should it happen that the stud travels one-half as fast as the spindle, multiply 12 by 2, and this product by 4, which would give 96 as the number of teeth required in the gear on the stud, the gear on the lead-screw remaining the same.

But in a train of simple gearing, such as has been described above, the number of threads possible with a limited number of gears is small. We have, how ever, recourse to another method which greatly enlarges the number of possible threads and renders it possible to cut threads of an almost infinite number of pitches with a comparatively small set of gears, because the limit to the number is only reached after every possible combination with each gear has been made. This method consists of interposing between the gear on the stud and that on the lead-screw, two other gears mounted on a stud carried by an adjustable spider, the combination of which may either reduce or increase the ratio existing between the numbers of teeth on the spindle and lead-screw gears. This is known as compound gearing, and consists of the substitution of two gears that run together on a sleeve, for the immediate idle gear. One of these two gears is driven by the gear on the stud, while the other drives the gear on the lead-screw.

The ratios given in the formula above no longer hold when compound gearing is used. Suppose that we placed on this intermediate stud two gears whose ratios were as 1 :2, driving the large gear from the spin-dle and allowing the smaller gear to drive the lead-screw. We would then have multiplied the number of threads to be cut by 2, because the lead-screw will revolve just one-half as fast. Thus, if we use two gears with this particular ratio on the idler stud, we could multiply the number of threads on the lead-screw by 2 and proceed as in simple gearing; just so if the ratio was 1 : 3, multiply by 3 and proceed.

But another rule is often given. The spindle gear is a driving gear, as is also the smaller gear that drives the lead-screw gear; the larger gear of the pair revolving on the same sleeve (to which they are immovably fixed by a key) is a driven gear, and so is the gear on the lead-screw. Therefore, to cut a given number of threads when the lathe is compounded, select the three gears except the one for the lead-screw, at random. Place them in position and multiply together the number of teeth in the driving gears and the number of threads to be cut. Then multiply together the number of threads in the lead-screw and the number of teeth in the driven gear on the sleeve; divide the first product by this product and the quotient will be the number of teeth required in the lead-screw gear, which is the remaining driven gear. Where a ratio other than 1 :1 exists between two revolutions of the spindle and the first stud, proceed exactly as in simple gearing, multiplying the threads of the lead-screw by the ratio, and using this figure in the calculations.

Signifying the two intermediate gears used in compounding in the following manner, the gear meshing with and being driven by the gear on the spindle of stud being known as Cd and the other erear mounted beside it on the sleeve and driving the lead-screw gear as Ca, we have the formula:

No. of threads to be cut x number of teeth in spindle gear x No. of teeth in Ca= No. teeth in Cd x threads per inch in lead screw gear, or Lg = P x Sg x Ca / Cd x Lp

Using this formula, a table may be computed, showing all the possible threads, fractional and whole, that may be cut with the set of change gears accompanying the lathe.

Owing to the fact that the metric system is used in several works in this country, this article would fall short of its purpose if the cutting of metric threads was not described. In the metric system, screw threads are usually expressed in threads per centimeter, and by proper compounding any lathe may be made to cut metric threads. A centimeter is equal to 50-127 or 50 divided by 127 equals .3937.

Let it be required to cut a screw having 8 threads per centimeter, look in the thread table for the simple gears to cut 8 threads per inch and put them in position. Then fill the intermediate space by two gears mounted immovably on a sleeve which may revove on a stud, having 50 and 127 teeth respectively, the gear with 127 teeth meshing with the gear on the spindle, while the gear with 500 teeth drives the lead-screw gear. The threads cut will be 8 per centimeter, and furthermore, the thread table accompanying the lathe may be used for cutting any other number of threads in the metric system, because the number of threads per inch and per centimeter remain the same throughout.

The usual forms of thread used in this country are the sharp V thread, the United States Standard thread or the Sellers system, the square thread and the acme thread. Each type has its particular advantage or disadvantage. The sharp V thread, owing to the bottom of the thread terminating in a very acute angle, weakens to a great extent the strength of the piece upon which it is cut. There seems to be a tendency for the sharp point of the tool to injure the metal beyond, perhaps starting an incipient crack which reduces the effective cross-sectional area of the piece. It is a difficult thread to cut owing to the fact that the fine point of the tool enters the metal first and at all times takes the heaviest load; being unsupported by the metal at the sides, a hard spot may crack off the point entirely. Then the sharp edge of the threads is very easily injured by coming in contact with other objects, and, after once being battered out of shape, it is almost impossible to get a close-fitting nut to go on. It is also very easily ruined if a small portion of the edge happens to become torn away and rolls up inside the nut.

The other threads will be taken up in the next chapter, together with the tools required to cut them ; multiple pitch will also be spoken of in connection with screw cutting.