of the pinion; the diameter of the generating circle for describing the cycloidal teeth should be half the proportional diameter of the pinion. See Buchanan's Practical Essays on Mill-work. Tredgold's edition.

Bevel geer.-We have already stated, that when the axes of wheels are angular to each other, they are called bevel geer, in order to distinguish them from spur geer, whose axes are parallel; it therefore now remains for us to describe in what manner the teeth of bevel geer differ from the teeth of spur geer.

Bevel geer is represented by the two cones at fig. 50, where A B and B C are the axes, and DE and EF their proportional diameters or pitch lines.

If these two cones are placed in close contact, and motion is communicated to the one, that motion will, as is already stated, be communicated to the other, and the motion of both, as we have shown, when speaking of spur geer, will be equal.

The epicycloid for forming the teeth of bevel geer, is generated by one cone rolling upon the surface of another, while their summits coincide: for example, if a cone C, fig. 51, having a point a, move upon the surface of the cone D, the point a will, in its revolutions, describe the line A EF, A being the place from where it starts, E its greatest height, and F its lowest depth; therefore a curved line drawn from A to E, and continued from E to F, gives what is called a spherical epicycloid; and the base of the cone C is the generating circle of the spherical epicycloid. The method of using the spherical epicycloid for forming the teeth of bevel geer is, in every respect, similar to the method of using the exterior and interior epicycloid for forming the teeth of spur geer, consequently it will be needless to repeat it.

Fig. 52. To construct bevel geer we must calculate the proportional diameters or pitch lines of the wheel and pinion that are to act upon each other, and then draw their axes A B and B C. Draw parallel to the axis AB of the wheel the line DE, and the line FD parallel to the axis of the pinion, and from the point D, where these two lines intersect, draw the line D G perpendicular to A B, and DH perpendicular to B C, and make IG equal to DI, and KH equal to DK; then DG gives, what is called the principal diameter, or diameter of the pitch line of the wheel, and DH that of the pinion.

Proceed to draw the teeth of the wheel, by fixing one

foot of the compasses in the point at A, and, having extended the other foot to the distance G, sweep the small arc Ga, then set off the length of the tooth from G to b, draw the line bc, tending to a, and sweep the arc ce, concentric to ba. Set off from G to f part of the required length of the tooth, from the principal diameter to the root; and draw the line fg tending to A, which gives the root of the tooth. Parallel to fg, draw ae, and afge will represent a section of the solid ring of the wheel.

In an excellent article on mill-work, in Dr. Rees's Cyclopedia, the author states, "that the manner of setting out the teeth of cog-wheels, in such a form that they shall act in the most equable manner upon each other, and with the least friction, has been a subject of much investigation among mathematicians and theoretic mechanics; but the practice and observation of the mill-wrights have produced a method of forming cog-wheels, which answers nearly, if not fully, as well in practice, as the geometrical curves which theory has pointed out to be the most proper. This they have effected by making the teeth of the modern wheels extremely small and numerous. In this case, the time of action in each pair of teeth is so small, that the form of them becomes comparatively of slight importance; and the practical methods of the mill-wrights (using arcs of circles for the curves) approximates so nearly to the truth, that the difference is of no consequence: and this method is the best, because it so easily gives the means of forming all the cogs exactly alike, and precisely the same distance asunder, which, by the application of any other curve than the circle, is not so easy. The method, which is extremely simple, is explained in fig. 53. The wheel being made, and the cogs fixed in much larger than they are intended to be, a circle, a a, is described round the face of the rough cogs upon its pitch diameter, that is, the geometrical diameter, or acting line of the cogs; so that when the two wheels are at work together, the pitch circles, a a, of the two are in contact. Another circle, bb, is described within the pitch circle for the bottom of the teeth, and a third, dd, without it, for the extremities. After these preparations, the pitch circle is accurately divided into the number which the wheel is intended to have: a pair of compasses are then opened out to the extent of one and a quarter of these divisions, and with this radius arcs are struck on each side of every division, from the pitch line a, to the outer circle dd. Thus the point of the compasses being set in the division e, the curve ƒ g, on one side of the

cog, and no on one side of the other, are described; then the point of the compasses being set on the adjacent division k, the curve Im is described. This completes the curved portion of the cogs e, and this being done all round completes every tooth; the remaining portion of the cog within the circle a, is bounded by two straight lines drawn from the points g and m towards the centre; this being done to the cogs all round, the wheel is set out, and the cogs, being dressed or cut down to the lines, will be formed ready for work, every cog being of the same breadth; and the space between every one and its neighbour is exactly equal to the breadth, provided the compasses are opened to the extent of one division and a quarter as first described."

COUPLINGS.

Coupling boxes are used to connect the shafts of wheels; they are either round or square, and with single or double bearings. The square coupling with double bearings, is represented in fig. 54, where B, between the bridges CD, is a square shaft with the coupling box resting upon it, ready to be thrust, when occasion requires it, upon the shaft A, which is out of geer, and to which it can be fastened by means of a pin, as shown at F, where the shafts are in geer. The round coupling, represented in fig. 55, is, when fastened on the shafts, engaged by two bolts A B, and C, which pass through the box at right angles to each other, and one of them through each of the shafts. As it is almost impracticable to form the axes of two shafts with such accuracy that they shall present one truly straight line; and as the shafts will, though made never so accurate, wear unequally, both these couplings have been found to be somewhat disadvantageous in millwork. The square coupling with one bearing, is decidedly superior to either of the above-mentioned, as it possesses, to a certain degree, the property of being flexible in all directions. In conveying motion through a great length of shafts, where there is but little lateral pressure, it can be used to great advantage; but where there is much lateral pressure the sockets are found to wear away and get loose, which occasions a hobbling and inaccurate motion. A longitudinal section of this coupling is represented in fig. 56, where A is the square of one shaft, B the square of the other, CC the coupling box, and DD two pins, one of which passes through each square of the shafts, in order to support the square B in a line with the square A. Sometimes the square B is held in a line with the square A by means of a round projection F, from the centre