[Illustration: Fig. 29. (A) Lathe with Simple Gearing for Thread Cutting. (B) Compound Geared Lathe]
Suppose that nine threads to the inch are to be cut and the lead-screw has six threads per inch. In this case the work must make nine revolutions while the lead-screw makes six and causes the carriage and thread tool to move one inch, or in other words, one revolution of the lead-screw corresponds to one and one-half revolution of the spindle; therefore, if the lead-screw gear _c_ has 36 teeth, the gear _a_ on the spindle stud should have 24 teeth. The spindle will then revolve one and one-half times faster than the lead-screw, provided the stud rotates at the same rate of speed as the main lathe spindle. The number of teeth in the change gears that is required for a certain pitch can be found by multiplying the number of threads per inch of the lead-screw, and the number of threads per inch to be cut, by the same trial multiplier. The formula which expresses the relation between threads per inch of lead-screw, threads per inch to be cut, and the number of teeth in the change gears, is as follows:
threads per inch of lead-screw teeth in gear on spindle stud ------------------------------ = ----------------------------- threads per inch to be cut teeth in gear on lead-screw
Applying this to the example given, we have 6/9 = 24/36. The values of 36 and 24 are obtained by multiplying 6 and 9, respectively, by 4, which, of course, does not change the proportion. Any other number could be used as a multiplier, and if gears having 24 and 36 teeth were not available, this might be necessary. For example, if there were no gears of this size, some other multiplier as 5 or 6 might be used.
Suppose the number of teeth in the change gears supplied with the lathe are 24, 28, 32, 36, etc., increasing by four teeth up to 100, and assume that the lead-screw has 6 threads per inch and that 10 threads per inch are to be cut. Then,
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.