45

the torus, and let the circle IT' /' L', and the rectangle

HIML be the analogous projections of the cylinder,

which passes perpendicularly through it. If now we con-

ceive, as before, a plane a b, Fig 13, to pass horizontally

through both solids, it will obviously cut the cylinder in

a circle which will be projected in the base H' /' L' itself,

and the ring in two other circles of which one only, part

of which is represented by the arc /' 63 b', will intersect

the cylinder at the points/' and A, which being projected

vertically to Fig 13, will give two points / and b2 in the

upper curve of penetration.

Another horizontal plane, taken at the same distance

below the centre line A B, as that marked a b is above it,

will evidently cut the ring in circles coinciding with those

already obtained ; consequently the points/' and 63 indi-

cate points in the lower, as well as in the upper curves

of penetration, and are projected vertically at d and e.

Thus we find that by laying down two planes at equal

distances on each side of A B, we obtain, by one operation,

four points in the curves required.

In order to determine the vertices to and n, we must

again have recourse to the method explained in the pre-

ceding problem; that is, to draw a plane O' n, passing

through the axis of the cylinder and the centre of the

ring, and to conceive this plane to be moved round the

point O', as on a hinge, until it has assumed the position

O' B', parallel to the vertical plane ; the point li, repre-

senting the extreme outline of the cylinder in plan, will

now be at r', and being projected vertically, that outline

will cut the ring in two points p and r, which would be

the limits of the curves of penetration in the supposed

relative position of the two solids ; and by drawing the

two horizontal lines r n and p to, and projecting the point

n' vertically, we obtain, by the intersections of these lines,

the two points to and n, which are the vertices of the

curves in the actual position of the penetrating bodies.

The points at which the curves are tangents to the out-

lines H I and L M of the cylinder, may readily be found

by describing arcs of circles from the centre O' through

the points H and L', which represent these lines in the

plan, and then proceeding, as above, to project the points

thus obtained upon the elevation. Lastly, to determine

the points, as j, z, &c., where the curves are tangents to

the horizontal outlines of the ring, draw a circle P' s' /

with a radius equal to that of the centre line of the ring,

namely P D ; the points of intersection z and j’ are the

horizontal projections of the points sought.

Required to represent the sections which would be

made in the ring now before us, by two planes, one of

which, N' T', is parcdlel to the vertical plane, while the

other T' E' is perpendicular to both planes of projection.

The section made by the last-named plane must obvi-

ously have its vertical projection in the line C D, which

indicates the position of the plane ;* but the former will

* We may here observe, that in drawing the details of machinery,

particularly if on a large scale or of the actual size, a mode of repre-

sentation similar to that exemplified in Fig. 13, is frequently resorted

to, and is attended with considerable economy of time and space.

be represented in its actual form and dimensions in the

elevation. To determine its outlines, let two horizontal

planes g q and i h, equidistant from the centre line A B,

be supposed to cut the ring; their lines of intersection

with it will have their horizontal projections in the two

circles g o' and li q which cut the given plane N T in o'

and f. These points being projected vertically to o, q, k,

&c., give four points in the curve required. The line

N' T' cutting the circle A' E' B' at N', the projection N

of this point is the extreme limit of the curve.

The circle P s' j', the centre line of the rim of the torus,

is cut by the planes N' T' at the point s', which being pro-

jected vertically upon the lines D P and G l, determines s

and l, the points of contact of the curve with the horizontal

outlines of the ring. Finally, the points, t and u are ob-

tained by drawing from the centre 0, a circle T' v tangent

to the given plane, and projecting the point of intersection

v to the points v and x, which are then to be replaced

upon C D by drawing the horizontals v t and x u.

Penetrations of Cylinders, Prisms, Spheres,

and Cones.—Plate VI.

Figs. 15 and 16.—Required to delineate the lines of

penetration of a sphere and a regular hexagonal prism

ivhose axis qicisses through the centre of the sphere.

The centres of the circles forming the two projections

of the sphere are, according to the terms of the problem,

upon the axis C C of the upright prism, which is projected

horizontally in the regular hexagon D' E' F' G' H' I'.

Hence it follows, that as all the lateral faces of the prism

are equidistant from the centre of the sphere, their lines

of intersection with it will necessarily be circles of equal

diameters. Now, the perpendicular face represented by

the line E' F' in the plan, will meet the surface of the

sphere in two circular arcs E F and L M, Fig. 15, described

from the centre C, with a radius equal to c b' or a' c.

And the intersections of the two oblique faces D' E' and

F G' will obviously be each projected in two arcs of an

ellipse whose major axis clg is equal to the diameter of

the circle a cb, and the minor axis is the vertical projec-

tion of that diameter, as represented at e' f, Fig. 16. But

as it is necessary to draw small portions only of these

curves, the following method may be employed.

It is sufficiently evident that the horizontal line D G

will pass through all the points where the edges of the

visible faces of the prism intersect the surface of the

sphere. Now, if we divide the portions E F and F G

respectively into the same number of equal parts, and,

drawing perpendiculars through the points of division,

set off from F G the distances from the corresponding

points in E F to the circular arc E' c F, we shall have as

many points in the elliptical arc required as we have

taken divisions upon the chords. The remaining ellip-

When the form of a piece is exactly symmetrical on either side of a

centre line, it is sufficient for all practical purposes to represent on

one side of the centre line, half the external elevation of the object,

and on the other, half the section or plan, according as its nature or

form may require.