of the thread; describe also the curves k and b, which terminates the returning thread, and completes the delineation as required.

Fig. 3, as to mode of construction, is exactly similar to that of fig. 2, but intended, by displacement of the cylinder, to delineate a continuous vein of the spiral in its proper form around the whole circumference: hence, being deemed by the preceding figure already sufficiently described, further elucidation must be considered unnecessary.

43. To determine the proper forms for a pair of bevel wheels, the required angle of the shafts and diameters of the wheels being given.

Draw at the given angles lines representing the shafts on which the wheels are to be fixed, as A B, A C; make the lines

a b, a c, parallel with and at a distance from A B, A C, equal to the radius of each respective wheel at the greatest pitch circle; draw the line A a through the intersection at d; then from a at right angles

with A B, A C, draw the lines a e, a f, making each in length equal to the wheel's diameter; draw the lines A e, A f, and from a, with the intended breadth of the wheels on the face, cut the line A a in d; draw the lines d i, dg, parallel to a e, a f, (hence the proper conical forms of the wheels and the pitch circles;) draw at right angles with A a, and through the intersection of the lines a e, a f, the line c B, also the lines в e, cf, d ki, d lg; from в and c, with

the radius в a, c a, describe portions of circles, as a G, a H, on which describe the greatest dimensions of, and proper form of the teeth; then from d, and parallel with A B, A C, draw the lines a v, a x, cutting the line с в in v and x; from в and c, with the distances c v, в x, describe the portions of circles, which determines the dimensions of the teeth on the interior pitch circle, and completes the proper forms of the wheels as required.

Proportions for the construction of toothed wheels. Length of the teeth

Thickness

وو

of the pitch.

=

do.

Breadth on face-23 times the pitch.

Edge of the rim

Projecting rib inside do. each of the pitch.

Thickness of flat arms

$

Breadth of arms at rim=2 teeth and the pitch, increasing in breadth towards the centre of the wheel, in the proportion of an inch for every foot in length. Thickness of the ribs or feathers on the arms=1 of the pitch.

Thickness of metal around the eye, or centre,=Z of the pitch.

Wheels and other circular bodies are very conveniently transferred from plan to that of a projected perspective by means of a peculiar appropriation of straight lines, commonly called orthographic projection, the principle of which will be readily understood by reference to the diagrams and illustrations given for the purpose in Plate C.

Fig. 1 is a circle divided into equal parts, and its form in projection is required, v n being supposed the line of intersection: parallel with the diameter of the

circle bf, draw at right angles, and through the centre, the line c e; draw also the line or chord fg, cutting the line v n in g; then, with the distance v g, describe the quadrant ir s; bisect the arc in r, from which and parallel with v n, draw the line rt; draw also the lines ft, g t, which determines the breadth of the ellipsis or projected form of the circle, and which may be drawn as described (page 5, fig. 12): hence all the other lines being so distinctly described, by mere inspection of the diagrams, further explanation is un

necessary.

Fig. 2 is a projected representation of an undershot water-wheel, of which a is the plan, v n the line of intersection, and в the diameter and breadth, laid at a proper angle or inclination, as determined by the principles of the diagrams: the breadth of the ellipsis is found and formed as there described, and the lines in the illustration, or fig. 2, whereby to obtain the proper angles for projection, render sufficiently explicit the proper mode by which the representation is effected.