47

when the point is at E' in the plan, its vertical projection

will be the point of intersection B of the perpendicular

drawn through B' and the horizontal drawn through the

first point of division. Also when the point arrives at C',

in the plan, its vertical projection is the point C where

the perpendicular drawn from C' cuts the horizontal passing

through the second point of division, and so on for all the

remaining points. The curve A B C F A3, drawn through

all the points thus obtained is the helix required.

Figs. 1 and 2.—To draw the vertical elevation of the

solid contained betiveen two helical surfaces, and tivo

concentric cylinders.

A helical surface is generated by the revolution of a

straight line round the axis of a cylinder; its outer end

moving in a helix, and the line itself forming with the

axis a constant and invariable angle.

Let A' C' F' and K' M' O' represent the concentric bases

of the cylinders, whose common axis S T is vertical; the

curve of the exterior helix A C F A3 is first to be drawn

according to the method pointed out above. Then having

set off from A to A2 the thickness of the required solid,

draw through A2 another helix equal and similar to the

former. Now construct, according to the method given

above, another helix K C 0 of the same pitch as the last,

but on the interior cylinder; as also another K2 C2 O2,

equal and parallel to the former. The lines A' K', B' L',

C' M', &c., represent the horizontal projections of the vari-

ous positions of the generating straight line, which, in the

present example has been supposed to be horizontal; and

these lines are projected vertically at A K, B L, &c.

It will be observed that in the position A K, the gene-

rating line is projected in its actual length, and that, at

the position C' M', its vertical projection is the point C.

The same remark applies to the generatrix of the second

helix. The parts of both curves which are visible in the

elevation may easily be determined by inspection.

Figs. 3 and 4.—To determine the vertical projection of

the solid formed by a sphere moving in a helical curve.

Let A' C' E' be the base of a cylinder upon which the

centre point, C, of a sphere whose radius is a C, describes

a helix, which is projected on the vertical plane in the

curve A C E F. After determining, as above, the various

points A, B, C, D, &c., in this curve, draw from each of

these points as centres, circles of the radius a C; the cir-

cumferences of these circles will denote the various posi-

tions of the sphere during its motion round the cylinder;

and if lines be drawn touching these circles, the curves

thereby formed will constitute the figure required. One

of these curves will disappear at 0, which is its point of

contact with the circle described from the point E, the

intersection of the helix with the perpendicular E E'. It

will again re-appear at the point I, where it becomes a

tangent to the circle described from the point J, in the

prolongation of the line A A'. The exterior and interior

circles in Fig. 4, represent the horizontal projection of the

solid in question.

T ART

DRAWING OF

Our design is, in the first place, to show by simple ex-

amples, how different objects, by their various necessities,

require various treatment for their complete illustration;

and to lead the way to the more complicated treatment

and illustration of machinery in larger masses.

SECTION I.

DRAWING OF SCREWS.

The screw is a cylindrical piece of wood or metal, in the

surface of which one or more helical grooves are formed.

The thread of the screw is the solid portion left between

the grooves; and the pitch of the screw is the distance,

measured on a line parallel to the axis of the cjdinder,

between the centres of any two contiguous threads.

A screw is said to have a triangidar thread when the

thread is triangular in section.

FOURTH.

MACHINERY.

The screw is usually accompanied by a Nut, which is

a detached piece, formed of suitable material, and perfo-

rated by a cylindrical hole having grooves formed on its

periphery corresponding in all respects to those on the

screw ; so that the threads of the screw shall fit exactly

into the grooves of the nut, and all the corresponding

points in the two surfaces shall coincide.

Projections of a Triangular-threaded Screw and

Nut.—Plate IX.

The ground line A M having been drawn throughout

the entire length of the paper, as also the centre lines C C' of

the figures, as before directed, the next step is to lay

down the circles which are to represent the exterior and

interior cylinders. For this purpose, observing that the

diameter of the exterior cylinder is 4-}-g- inches, the half of

this length is, by the aid of the foot-rule, to be taken up

in the compasses ; and with this radius, at a suitable dis-

tance C' from the ground line, a semicircle A' G' B' is to be