51

paper, and to lay down the projections of the intersecting

cylinders upon a larger scale, as at Fig. 5. Here, the arc

h v2t, representing that marked h t in Fig. 1, but on a

scale of twice that to which that figure is drawn, is the ver-

tical projection of part of a cylinder situated horizontally,

while the semicircle h't s, corresponding with that marked

h t s in Fig. 2, is the base of a vertical cylinder. This

being premised, the construction of the curve of intersec-

tion of these two solids will manifestly resolve itself into

an application of the method already laid down, and the

lines in the drawing ‘will be found sufficiently explicit with-

out the aid of farther explanations. When the curve has

been drawn upon Fig. 5, its dimensions must be reduced

by one-half in order to reproduce it in its proper propor-

tions upon Fig. 3 ; this may be accomplished by setting

off half the distance t u (Fig. 5) from t to u (Fig. 3); then

through the point u drawing a horizontal line, and from u

setting off, on each side of the centre line, half the distance

uv (Fig. 5); which will give two points in the reduced

curve, &c.

Figs. 6, 7, and 8 are the projections of a hanging bracket

or gallows, as it is usually termed; such pieces are used

for the support of light horizontal shafts situated near to

the roof of a building or flat, and are usually bolted, as in

the present instance, to the wooden beams which support

the roof or flooring.

In executing the drawing of such an object as that now

before us, we must first prolong the vertical side B F of

the beam indefinitely, and from the point F set off the

distance (18 in.) from the end or side of the beam to

the axis C of the shaft. Then through the point thus

obtained draw a horizontal line, and lay off upon it the

distance (6| inches) from the centre to the plane of the

side B F ; the position of the centre C is thus defini -

tively fixed. After having described from this point the

various circles which represent the size of the bearing

and outlines of the brasses, as also that of the cover, fixed

to the bracket by a single bolt E, we proceed to find upon

the horizontal line C b, the centre o of the arc g h j, which

is terminated towards the outer extremity of the bracket

by a smaller arc g i. Then, with a radius of about

2 feet 10| inches, determine the centre of the arc e db;

which should be a tangent to the circle iab and to the

perpendicular passing through e; and from the same centre,

with a radius increased by f of an inch, describe another

arc, which will represent the thickness of that part of the

bracket. At the axis of the bolt E, which tends towards

the centre above named, set off from the first arc the dis-

tance 31 in., and upon the horizontal line m l determine

the point l through which the outline of the back of the

bracket is to pass ; then with a radius of 3 feet 8 inches

draw the arc k l, join Jc j by a short tangential arc and

finish the part towards l as shown in the figure. Concen-

tric with the circles k l and b e respectively, the arcs p n

and p r are to be drawn, and rounded off into each other

and the horizontal r n, as shown.

The construction of Fig. 7, which is a partial front view

of the bracket, presents no difficulty; and by the help of

Figs. 6 and 7, the plan Fig. 8 may be easily drawn, follow-

ing the same method for the determination of the curve

b c d e as that pointed out in reference to the standard.

The lines and letters of reference on the figures will suffi-

ciently explain the application to the present example.

The cover, which in Fig. 8 is supposed to be removed, is

shown in front elevation at Fig. 9.

Projections of a Rail and Chair.—Plate XIV.

Figs. 1 and 2 are an elevation and plan of a railway

chair, with a piece of rail keyed into it, in use on the

Paris and Strasburg Railway. Fig. 3 is a vertical section

by the centre line a- a in plan, with the position reversed

as respects Fig. 1, to show the inclination of the two rails

forming the way. Fig. 4 is an end view of the chair, and

Fig. 5 of the rail.

The sole, A, of the chair is flat on the under side to rest

on the sleepers, with the jaws B, B', and stiffening flanges

C, C'. The spike-holes a, a, are formed in the sole. D is

the rail, and E the wood key for fixing it in the chair.

The rail, Fig. 5, is symmetrical on both axes be, d e ; the

key is also symmetrical on its diagonals shown in Fig. 1,

so that it may be driven in with either side uppermost.

The configuration of the rail is reduced to its elements

in Fig. 5, where the elementary curves and straight lines

are indicated; the construction may obviously be com-

pleted by the aid of the elementary problems already

treated.

The configuration of the chair is also reduced to its

elements in Figs. 1 and 3; they are given with great cir-

cumstantiality, more as exercises, than as examples for

execution; as the outlines of chairs are usually sketched

in without a close adherence to circular outlines.

Projections of Ratchet Wheels and Fluted Columns.

Plate XV.

Figs. 1 and 2 are a plan and elevation of a cylinder

grooved on its circumference; of which one half is in-

dented with triangular grooves, and the other half with

rectangular grooves.

To construct the horizontal projection, Fig. 1, describe

two circles on the centre 0, the outer one, F F, to define

the extreme limits of the cylinder, and the inner one, E E,

the bottoms of the grooves. Describe a third circle, A B,

of a large diameter, and divide it into twice as many equal

parts as there are grooves; or, in the case before us, 48

parts, for 24 grooves. To do so, draw two diameters A B,

C D, at right angles, and subdivide the intervals between

them, as indicated in the drawing, and according to means

already described for dividing circles. From the points of

division draw radii cutting the given circles into the same

number of equal parts. Then, to complete the triangular

grooves, join the alternate points of intersection a, b, c, d;

the rectangular grooves require no additional lines, as they

are formed by the radial and circular lines already drawn.

Figs. 4, 5, 6, represent ratchet wheels with two different

forms of teeth. The preliminary steps are the same as in

the first case. To facilitate the drawing of the curved