43

Let Z' Z', Fig. 6, denote the position of the cutting

plane, as seen in horizontal projection: then the section

of the cone exhibited in the elevation will be seen in its

actual dimensions. Now it is obvious that the very same

method of construction which we have already explained

In reference to the two preceding examples is also appli-

cable to the present; therefore we shall only indicate the

application in a single instance.

Having drawn any horizontal line E F at pleasure,

bisect it, and from the centre S', with half of that line as

radius describe a circle, which will be intersected by the

line Z' Z' at the points G' and Hj these being projected

to G and H, on the line E F, determine two points in the

curve. The vertex I of the curve, which in this case is a

hyperbola, will be obtained by describing from the centre

S' with the distance S' I', a circle intersecting A B in

K', projecting this latter point to K, and drawing the

horizontal line K L. Should the cutting plane pass through

the axis of the cone, the section obviously will resolve

itself into an isosceles triangle.

SECTION Y.

Penetrations or Intersections of Solids.

Having thus minutely explained the mode of projecting

the conic sections, it will be quite unnecessary to enter

into any detail with regard to the sections of the cylinder.

Supposing it to stand in an upright position, a plane cut-

ting it obliquely will obviously form an ellipse, of which

the minor axis is the diameter of the cylinder, and the

major axis may easily be determined by the methods

already laid down in treating of the cone.

On examining the minor details of most machines, we

shall find numerous examples of cylindrical, and other

forms, fitted to, and even appearing to pass through each

other in a great variety of ways. We shall, therefore, pro-

ceed to examine attentively the outlines formed by the

penetrations of various solid bodies; our examples are

grouped in Plate YI11, and are selected with the view of

exhibiting those cases which are of most frequent occur-

rence, and at the same time, of elucidating the general

principles which are applicable in every case.

Penetrations of Cylinders.—Plate IV.

#

Figs. 1 and 2 represent the projections of two cylinders

of unequal diameters, meeting each other at right angles ;

one of which is denoted by the rectangle A B E D in the

vertical, and by the circle A' H' B' in the horizontal pro-

jections ; while the other which is supposed to be hori-

zontal, is indicated in the former by the circle L P I N,

and in the latter by the figure L' I' K' M'. From the

position of these two solids, it is evident that the curves

formed by their junction will be projected in the circles

A' FI' B' and LPIN; and further, that such would also

be the case even although their axes did not intersect

each other.

But if we suppose the position of these bodies to be

changed into that represented at Figs. 3 and 4, the lines

of their intersection will assume in the vertical projection,

a totally different aspect, and may be accurately deter-

mined by the following construction.

Through any point taken upon the plan, Fig. 4, draw

a horizontal line a b', which is to be considered as indi-

cating a plane cutting both cylinders, parallel to their

axes ; this plane would cut the vertical cylinder in lines

drawn perpendicularly through the points c and d'. To

find the vertical projection of its intersection with the

other cylinder, conceive its base I' L', after being trans-

ferred to I2 L2, to be turned over parallel to the horizontal

plane; this is expressed by simply drawing a circle of the

diameter I2 L2; and producing the line a' b' to a? ; then

set off the distance a2 e, on each side of the axis I K, and

draw straight lines through these points parallel to it.

These lines ab, g h denote the intersection of the plane

a b' with the horizontal cylinder, and therefore the points

c, d, m, o, where they cut the perpendiculars c c', d d' are

points in the curve required. By laying down other

planes, similar to a b', and operating as before, any number

of points may be obtained. The vertices i and k, of the

curves are obviously projected directly ; and their extreme

points are determined by the intersections of the outlines

of both cylinders. When the cylinders are of unequal

diameters, as in the present case, the curves of penetration

are hyperbolas.

Figs. 5 and 6.—When the diameters of the cylinders

are equal, and when they cut each other at right angles,

the curves of penetration are projected vertically in straight

lines perpendicular to each other. For, if we proceed to

apply the method given above, we shall soon discover that

the various points in these curves are situated in two planes

at right angles to each other, and to the vertical plane,

the sections formed by them being, in fact, ellipses equal

and similar to each other. We need not enter into any

details in illustration of this case, further than to call at-

tention to the figures, where the projections of some of the

points are indicated, in elevation and plan, by the same

letters of reference.

Figs. 7 and 8.—To delineate the intersections of tivo

cylinders of equal diameters at right angles, when one

of the cylinders is inclined to the vertical plane.

Supposing the two preceding figures to have been drawn,

we may easily ascertain the projection c, of any point such

as c', by observing that it must be situated in the perpen-

dicular c c, and that, since the distance of this point, (pro-

jected at c in Fig. 5), from the horizontal plane remains

unaltered, it must also be in the horizontal line c c. TJpon

these principles all the points indicated by literal refer-

ences in Fig. 7 are determined; the curves of penetration

resulting therefrom intersecting each other at two points

projected upon the axial line L K, of which that marked

q, alone is seen. The ends of the horizontal cylinder are

represented by ellipses, the construction of which will also

be obvious on referring to the figures ; and they do not

require further consideration here.