tool-head, is determined as follows: sin_b_ = (sin_a_ x _h_) / _v_, in
which _h_ represents the rate of horizontal feed and _v_ the rate of
vertical feed. The angle corresponding to sine _b_ is next found in a
table of sines. We now have angles _b_ and _a_, and by subtracting the
sum of these angles from 90 degrees, the desired angle _x_ is obtained.
To illustrate: The sine of 30 degrees is 0.5; then sin _b_ = (0.5 x 1/4)
/ 3/16 = 0.6666; hence angle _b_ = 41 degrees 49 minutes, and _x_ =
90 deg.-(30 deg. + 41 deg. 49') = 18 degrees 11 minutes. Hence to turn
the casting to angle _a_ in a boring mill having the horizontal and
vertical feeds given, the tool-head would be set over from the vertical
18 degrees and 11 minutes which is equivalent to about 18-1/6 degrees.

If the required angle _a_ were greater than angle _y_ obtained from the
combined feeds with the tool-bar in a vertical position, it would then
be necessary to swing the lower end of the bar to the left rather than
to the right of a vertical plane. When the required angle _a_ exceeds
angle _y_, the sum of angles _a_ and _b_ is greater than 90 degrees so
that angle _x_ for the tool-head = (_a_ + _b_) - 90 degrees.

=Turret-lathe Type of Vertical Boring Mill.=--The machine illustrated in
Fig. 16 was designed to combine the advantages of the horizontal turret
lathe and the vertical boring mill. It is known as a "vertical turret
lathe," but resembles, in many respects, a vertical boring mill. This
machine has a turret on the cross-rail the same as many vertical boring
mills, and, in addition, a side-head _S_. The side-head has a vertical
feeding movement, and the tool-bar _T_ can be fed horizontally. The
tool-bar is also equipped with a four-sided turret for holding turning
tools. This arrangement of the tool-heads makes it possible to use two
tools simultaneously upon comparatively small work. When both heads are
mounted on the cross-rail, as with a double-head boring mill, it is
often impossible to machine certain parts to advantage, because one head
interferes with the other.

The drive to the table (for the particular machine illustrated) is from
a belt pulley at the rear, and fifteen speed changes are available. Five
changes are obtained by turning the pilot-wheel _A_ and this series of
five speeds is compounded three times by turning lever _B_. Each spoke
of pilot-wheel _A_ indicates a speed which is engaged only when the
spoke is in a vertical position,