ve
both numerator and denominator into two factors each, and multiply each
"pair" of factors by the same number, until values are obtained
representing numbers of teeth in available change gears. (One factor in
the numerator and one in the denominator make a "pair" of factors.)

Suppose the lathe cuts 6 threads per inch when gears of equal size are
used, and that the number of teeth in the gears available are 30, 35, 40
and so on, increasing by 5 up to 100. If 24 threads per inch are to be
cut, the screw constant 6 is placed in the numerator and 24 in the
denominator. The numerator and denominator are then divided into factors
and each pair of factors is multiplied by the same number to find the
gears, thus:

6 2 x 3 (2 x 20) x (3 x 10) 40 x 30
-- = ----- = ------------------- = -------
24 4 x 6 (4 x 20) x (6 x 10) 80 x 60

The last four numbers indicate the gears which should be used. The upper
two having 40 and 30 teeth are the _driving_ gears and the lower two
having 80 and 60 teeth are the _driven_ gears. The driving gears are
gear _a_ on the spindle stud and gear _c_ on the intermediate stud,
meshing with the lead-screw gear, and the driven gears are gears _b_ and
_d_. It makes no difference which of the driving gears is placed on the
spindle stud, or which of the driven is placed on the lead-screw.

=Fractional Threads.=--Sometimes the lead of a thread is given as a
fraction of an inch instead of stating the number of threads per inch.
For example, a thread may be required to be cut, having 3/8-inch lead.
The expression "3/8-inch lead" should first be transformed to "number of
threads per inch." The number of threads per inch (the thread being
single) equals:

1 3 8
--- = 1 / - = - = 2-2/3
3/8 8 3

To find the change gears to cut 2-2/3 threads per inch in a lathe having
a screw constant of 8 and change gears varying from 24 to 100 teeth,
increasing by 4, proceed as follows:

8 2 x 4 (2 x 36) x (4 x 24) 72 x 96
----- = --------- = ----------------------- = -------
2-2/3 1 x 2-2/3 (1 x 36) x (2-2/3 x 24) 36 x 64

As another illustration, suppose we are to cut 1-3/4 thread per inch on
a lathe having a screw constant of 8, and that the gears have 24, 28,
32, 36, 40 teeth, etc., increasing by four up to one hundred. Following
the rule:

8 2 x 4 (2 x 36) x (4 x 16) 72 x 64
----- = --------- = --------------------