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lines and dimensions necessary to be given or assumed in

laying down the projections of any bevil wheel, namely:

the axis 0 S of the primitive cone, the diameter A B of

its base, the angle A S O which the side of the cone makes

with the axis, and the straight lines A 0, D O', perpen-

dicular to AS, and representing the sides of two cones

between which the breadth of the wheel (or length of the

teeth) is comprised. These lines having been laid down

according to the required conditions, the first operation

is to divide the primitive circle, described with the radius

A C, into a number of equal parts corresponding to the

number of teeth, or pitch of the wheel. Then upon the

section, Fig. 3, draw with the radius O A or O B sup-

posed to move parallel to itself outside the figure a small

portion of a circle, upon which construct the outlines of

a tooth M, and of the rim of the wheel, with the same

proportions and after the same manner as we have

announced and explained in reference to spur wheels;

set off from A and B the points a, d, and /,—denoting

respectively the distances from the pitch line to the

points and roots of the teeth, and to the inside of the

rim, and join these points to the vertex S of the primi-

tive cone, terminating the lines of junction at the lines

D O', E O'; the figure abed will represent the lateral

form of a tooth, and the figure c df e a section of the rim

of the wheel, by the aid of which the face view (Fig. I)

may easily be constructed.

The points a, b, c, d, and e, having been projected upon

the vertical diameter A' B', describe from the centre O' a

series of circles passing through the points thus obtained,

and draw any radius, as C' L, passing through the centre

of a tooth. On either side of the point L set off the dis-

tances L 1c, L l making up the thickness of the tooth M at

the point, and indicate, in like manner, upon the circles

passing through the points B' and d', its thickness at the

pitch-line and root; then draw radii through the points

i, l, k, g, m, &c., terminating them respectively at the

circles forming the projections of the corresponding parts

at the inner extremity of the teeth; these radial lines

will represent the rectilinear edges of all the teeth. The

curvilinear outlines may be delineated by arcs of circles,

tangents to the radii g C and i C, and passing through the

points obtained by the intersections of the radii and the

various concentric circles. The radii of these circular arcs

may in general, as in the case of spur wheels, be taken

equal to the pitch, and their centres upon the interior and

exterior pitch-circles; thus the points g and i, n and o,

for example, are the centres for the arcs passing through

the corresponding points in the next adjacent teeth, and

vice versa.

The drawing of the teeth in the edge view (Fig. 2), and of

such portions of them as are visible in the section (Fig. 3)

will present no difficulty to the attentive student, and is

sufficiently explained by inspection of the lines of projec-

tion which we have partially introduced into the plate for

this purpose. We have only to remark, that in the construc-

tion of these views every point in the principal figure from

which they are derived is situated upon the projection of

the circle drawn from the centre C' and passing through

that point. Thus the points g and i, for example, situated

upon the exterior pitch-circle will be determined in Fig. 2

by the intersection of their lines of projection with the

base A B of the primitive cone; and the points k and l

will be upon the straight line passing through a a (Fig. 3),

and so on. Farther, as the lateral edges of all the teeth

in Fig. 1 are radii of circles drawn from the centre C, so,

in Fig. 2, they are represented by lines drawn through

the various points found as above for the outer extremi-

ties of the teeth, and converging towards the common

apex S; while the centre-lines of the exterior and interior

extremities themselves all tend to the points O and O'

respectively. This circumstance will suggest a mode of

materially simplifying the operation of drawing the edge

view of the teeth when the wheels are small or executed to

a small scale; and in all cases it affords a means of testing

the accuracy of the operations, if the method of projecting

numerous points be adopted. Finally, having represented

the projection of the rim, arms, and eye, according to the

dimensions indicated, the drawing of this subject will be

complete.

Projection of a Bevil Wheel and Pinion in Gear.—

Plate XXYI.

This plate contains two views of a pair of bevil

wheels working into each other, the number of their

teeth, and consequently the ratio of their velocities, being

as 11 to 7.

Let A S and S B be the axes of the two wheels, and

D E and E F the projections of their pitch-circles. Then

regarding Fig. 1, in the first instance, as a section made

by a plane in which the lines A S and B S are situated,

proceed to represent the form of a tooth X and Y in each

wheel. Then construct upon Fig. 2 the face view of the

wheel, substituting appropriate circular arcs for the spheri-

cal epicycloids which give the exact curvature of the teeth;

the details of this operation are laid down with sufficient

minuteness in the Plate which last occupied our attention,

as also the mode of deriving from Fig. 2 the edge view of

the wheel in Fig. 1. And, since the face view of the

pinion is not given in either of the figures, it is necessary

to lay down a portion of it on any convenient part of the

paper, taking the centre, of course, upon the line A S;

this we have shown in dotted lines on the Plate, together

with the lines of projection; the teeth being disposed sym-

metrically on each side of the axis, a quadrant of the

pinion is all that is required to be thus represented. The

position of the teeth of the pinion in Fig. 2 with respect

to the axis A S being precisely the same as in Fig. 1, it is

unnecessary to draw a second plan in order to obtain the

edge view, but simply to copy that already constructed in

Fig. 1.

The forms and proportions of the teeth in the wheels

here represented differ slightly from those embodied in

the preceding example; but they are such as are ap-

proved and adopted in practice by many mill-wrights.

I