ENGINEER AND MACHINIST’S DRAWING-BOOK.
tical arcs should be traced at the same time, and by the
Figs. 17 and 18.—Required to draw the lines of pene-
tration of a cylinder and a sphere, the centre of the
sphere being without the axis of the cylinder.
Let the circle D' E' L' be the projection of the base of
the given cylinder, the elevation of which is shown at
Fig. 17, and let A B be the diameter of the given sphere ;
the curves formed by their intersection will present no
difficulty to the attentive student of our previous ex-
amples. For if a plane, as c d, be drawn parallel to the
vertical plane, it will evidently cut the cylinder in two
straight lines G G', H H', parallel to the axis, and pro-
jected vertically from the points G' and hi'. This plane
will also cut the sphere in a circle whose diameter is equal
to c'd, and which is to be described from the centre C
with a radius of half that line; its intersection with the
lines G G' and H H' will evidently give so many points
in the curves sought.
The planes a b' and e' f, which are tangents to the
cylinder, furnish only two points respectively in the
curves; of these points E and F alone are visible, the
other two, L and M, being concealed by the solid ; there-
fore, the planes drawn for the construction of the curves
must be all taken between c d' and e' f. The plane
which passes through the axis of the cylinder cuts the
sphere in a circle whose projection upon the vertical plane
will meet at the points D, N, and g, h, the outlines of the
cylinder, to which the curves of penetration are tangents.
Figs. 19 and 20.—To find the lines of penetration of
a truncated cone and a prism.
The straight line C D is the axis of a truncated cone,
which is represented in the plan by two circles described
from the centre O'; and the horizontal lines M N and
Mi N' are the projections of the axis of a prism of which
the base is square, and the faces respectively parallel and
perpendicular to the planes of projection.
In laying down the plan of this solid we have pur-
posely supposed it to be inverted, in order that the smaller
end of the cone, and the lines of intersection of the lower
surface F G of the prism may be exhibited. According
to this arrangement, the letters A' and B', Fig. 20, ought,
strictly speaking, to be marked at the points T and IF,
and conversely; but as it is quite obvious that the part
above M' N' is exactly symmetrical with that below it,
the distribution of the letters of reference adopted in our
figures can lead to no confusion.
The intersection of the plane F G with the cone is pro-
jected horizontally in a circle described from the centre O',
with the diameter F' G'. The arcs F F' A' and FT G' B'
are the only parts of this circle which require to be drawn.
To find the lines of intersection of the two solids in the
vertical projection, we have only to remark that, as the
perpendicular side of the prism is parallel to the axis, it
will cut the cone in a hyperbolic curve, which may be
drawn according to the method already described; the
lines K A and L B are the only parts of the curve which
are visible in the present example.
Figs. 21 and 22.—To describe the curves formed by the
intersection of a cylinder with the frustum of a cone ;
the axes of the tivo solids cutting each other at right
The axes of the solids and their projections are laid
down in the figures, precisely as in the preceding exam-
ple ; the plan being inverted, as before, the same expla-
natory remark should be kept in mind.
The intersections of the outlines of the cone in the ele-
vation, with those of the cylinder furnish, obviously, four
points in the curves of penetration; these points are all
projected horizontally upon the line A' B'. Now, suppose
a plane, as a b, Fig. 21, to pass horizontally through both
solids; its intersection with the cone will be a circle of
the diameter c d, while the cylinder will be cut in two
parallel straight lines, represented in the elevation by
a b, and whose horizontal projection may be determined
in the following manner:—Conceive a vertical plane/g,
cutting the cylinder at right angles to its axis, and let the
circle g ef thereby formed, be described from the intersec-
tion of the axes of the two solids; the line j h will now
represent, in this position of the section, the distance of
one of the lines sought from the axis of the cylinder.
Now set off this distance on both sides of the point A',
and through the points k and a, thus obtained, draw
straight lines parallel to A' B'; the intersections of these
lines with the circle drawn from the centre C' of the dia-
meter c d will give four points to, p, n and o, which being
projected vertically upon a b, determine two points to and
p in the curves required.
In order to obtain the vertices or adjacent limiting
points of the curves, draw from the vertex of the cone, a
straight line t e, touching the circle g ef and let a hori-
zontal plane be supposed to pass through the point of
contact e. If now, we proceed, according to the method
given above, to determine the intersections of this plane
with each of the solids in question, we shall find four
points i', r, q and s, which being projected vertically
upon the line e r determine the vertices i and r required.
Of the Helix.—Plate YII.
The Helix, or, as it is sometimes, though improperly
termed, the Spiral, is the curve described upon the sur-
face of a cylinder by a point revolving round it, and, at
the same time, moving parallel to its axis by a certain
invariable distance during each revolution. This distance
is called the pitch of the screw.
Figs. 1 and 2.—Required to construct the helical curve
described by the point A upon a cylinder projected hori-
zontally in the circle A' O' F', the pitch being represented
by the line A A3.
Divide the pitch A A3 into any number of equal parts,
say eight; and, through each point of division 1, 2, 3, &c.,
draw straight lines parallel to the ground line. Then
divide the circumference A' G F' into the same number of
parts ; the points of division B', C', E', F', &c., will be the
horizontal projections of the different positions of the
given point during its motion round the cylinder. Thus,