10 10 x 4 40

By multiplying both numerator and denominator by 4, we obtain two
available gears having 24 and 40 teeth, respectively. The 24-tooth gear
goes on the spindle stud and, the 40-tooth gear on the lead-screw. The
number of teeth in the intermediate or "idler" gear _b_, which connects
the stud and lead-screw gears, is not considered as it does not affect
the ratios between gears _a_ and _c_, but is used simply to transmit
motion from one gear to the other.

We have assumed in the foregoing that the spindle stud (on which gear
_a_ is mounted) and the main spindle of the lathe are geared in the
ratio of one to one and make the same number of revolutions. In some
lathes, however, these two members do not rotate at the same speed, so
that if equal gears were placed on the lead-screw and spindle stud, the
spindle would not make the same number of revolutions as the lead-screw.
In that case if the actual number of threads per inch in the lead-screw
were used when calculating the change gears, the result would be
incorrect; hence, to avoid mistakes, the following general rule should
be used as it gives the correct result, regardless of the ratios of the
gears which connect the spindle and spindle stud:

_Rule.--First find the number of threads per inch that is cut when gears
of the same size are placed on the lead-screw and spindle, either by
actual trial or by referring to the index plate. Then place this number
as the numerator of a fraction and the number of threads per inch to be
cut, as the denominator; multiply both numerator and denominator by some
trial number, until numbers are obtained which correspond to numbers of
teeth in gears that are available._ The product of the trial number and
the numerator (or "lathe screw constant") represents the gear _a_ for
the spindle stud, and the product of the trial number and the
denominator, the gear for the lead-screw.

=Lathes with Compound Gearing.=--When gearing is arranged as shown at
_A_, Fig. 29, it is referred to as simple gearing, but sometimes it is
necessary to introduce two gears between the stud and screw as at _B_,
which is termed compound gearing. The method of figuring compound
gearing is practically the same as that for simple gearing. To find the
change gears used in compound gearing, place the "screw constant"
obtained by the foregoing rule, as the numerator, and the number of
threads per inch to be cut as the denominator of a fraction; resol