threads per inch and
subtract the quotient from the outside diameter. Expressing this rule as
a formula,

/1.299\
_r_ = _D_ - ( ----- )
\ _N_ /

in which _D_ equals outside diameter; _N_, the number of threads per
inch; and _r_, the root diameter. The number 1.299 is a constant that is
always used.

[Illustration: Fig. 10. End View of Lathe Headstock]

=Cutting a Left-hand Thread.=--The only difference between cutting
left-hand and right-hand threads in the lathe is in the movement of the
tool with relation to the work. When cutting a right-hand thread, the
tool moves from right to left, but this movement is reversed for
left-hand threads because the thread winds around in the opposite
direction. To make the carriage travel from left to right, the
lead-screw is rotated backwards by means of reversing gears _a_ and _b_
(Fig. 10) located in the headstock. Either of these gears can be
engaged with the spindle gear by changing the position of lever _R_.
When gear _a_ is in engagement, as shown, the drive from the spindle to
gear _c_ is through gears _a_ and _b_, but when lever _R_ is raised thus
shifting _b_ into mesh, the drive is direct and the direction of
rotation is reversed. The thread is cut by starting the tool at _a_,
Fig. 8, instead of at the end.

[Illustration: Fig. 11. End of Square Thread Tool, and Graphic Method of
Determining Helix Angle of Thread]

=Cutting a Square Thread.=--The form of tool used for cutting a square
thread is shown in Fig. 11. The width _w_ is made equal to one-half the
pitch of the thread to be cut and the end _E_ is at an angle with the
shank, which corresponds to the inclination _x--y_ of the threads. This
angle _A_ depends upon the diameter of the screw and the lead of the
thread; it can be determined graphically by drawing a line _a--b_ equal
in length to the circumference of the screw to be cut, and a line
_b--c_, at right angles, equal in length to the lead of the thread. The
angle [alpha] between lines _a--b_ and _a--c_ will be the required angle
_A_. (See end view of thread tool). It is not necessary to have this
angle accurate, ordinarily, as it is simply to prevent the tool from
binding against the sides of the thread. The end of a square thread tool
is shown in section to the right, to illustrate its position with
relation to the threads. The sides _e_ and _e_{1}_ are ground to slope
inward, as shown, to provide additional clearance.