In setting out a wheel, the basis of the operation is the pitch circle, or the working circumference ; the term pitch itself means the distance apart of the centres of the teeth upon that circle. The only condition which regulates the pitch is, that the material should be strong enough to bear the effort to which it is to be exposed ; and in practice it is found that cast-iron wheels work satisfactorily with pitches varying between 1 and 3 inches in large wheels, and between a quarter of an inch and three quarters of an inch in very small ones. The number of teeth depends upon the circumference, and upon the pitch ; or calling the number of teeth n, the circumference c, and the pitch p; then n= all the dimensions being in inches ; or the diameter d will be found by making P n d = The velocities at which the various wheels of a piece of machinery are required to revolve determines their respective diameters, and they am calculated upon the principles to be described in the sequel ; but it is to be observed that the true radii aro alwaya rather larger than the prim/lire radii, which latter serve to define the pitch circle ; the true radii, on the contrary, define the extremities of the teeth. Another general law is that the number of teeth in the spur (or driving-) wheel is to the number of teeth in the pinion (or follower) in the ratio of their respective radii ; or calling the diameter of the spur-wheel A n; the diameter of the pinion e D; and the number of teeth on the former sr ; then the number of teeth on the pinion will be found thus :—AB:CD::11: a. The outlines of the teeth are sneer s In small wheels with few teeth the depth of the addendum must be increased above the proportion above-mentioned.
Bcvilled gearing consists of frustra of cones, which are supposed to roll upon one another, and whose apices are supposed to meet in one point ; and in this case the form of the addenda ought to be a portion of a spherical epicycloid, according to strict theory, but in practice a much more simple form, devised by Telford, and described in Buchanan's Treatise on Mill-work,' p. 58 (1841), is used. Indeed, it is very rarely that the faces of wheels are made with cyeloidal forms of any descrip tion ; and practical men adopt, instead of them, simpler circnlar forms. Professor Willis, in his Principles of Mechanism,' has described an instrument of his own invention, for drawing the teeth of wheels by arcs of circles; and has accompanied the description by tables illus trating its use. The reader is referred to the various books above mentioned, and to Warr's Construction of Machinery,' Camus ' On the Teeth of Wheels,' Ilachette's Trait6 des Machines,' Armengaud's Dessin Industriel; and W. Johnston's Practical Mechanic,' for more specific details with respect to the principles upon which teethed wheels of the various kinds enumerated are designed. Professor Willie's Principles of Mechanism' may, perhaps, he cited as containing the most valuable information ou the subject.
It is necessary here to add that in some cases it is desirable to pro duce in machinery an intermittence in its action, or a certain alterna tion of motion and of rest, in the wheels gearing into one another.
This is effected by leaving a portion of the circumference of the driving-wheel without teeth for a length corresponding with the desired period of repose ; precautions, however, must be taken to insure that the teeth should fall into their proper places when contact is restored, for which purpose pins and guides ou the face of the wheel are commonly introduced. Ratchet-wheels, or those which only revolve in one direction, and have a species of alternate reciprocating action on the driver (in this sense, that they are raised gradually for a certain portion of the revolution, and aro then suddenly released), are introduced for the purposes either of preventing the wheel-work from revolving in more than in the direction originally selected, or for ob taining alternate vertical or horizontal motion in the shafts of a piece of machinery. The pin which prevents the alteration of the rotation of a ratchet-wheel, is usually placed on a pivot, and is able to be thrown out of gear at will : it is commonly known by the name of a paull. Mill work of all kinds may either be kept permanently in gear, or it may be constructed so as to allow any of its parts to work, or to rest, independently of the general combination, by means of couplings, clutch-boxes, friction, or reversing-gear. In the construction of ma chinery, wheels play very important parts by the action of drivers upon the various combinations for producing change of motion, such as racks, endless screws, cams, eccentrics, teethed arcs, &c. ; but the consideration of these functions of wheels, as also of those by which they are made to regulate the motion of the machinery to which they are attached, by reason of their power of retaining momentum, belongs especially to the province of applied mechanics. Before closing this article, however, it is toselitial to allude cursorily to some of the mechanical conditions involved iu the application of wheels.
The simplest manner iu which those contrivances for the trans• mission of power are used, is the one known by the term of the wheel anti axle. [WIIEEL•AND-AXLE.] In the windlass the power is apPlied to the axle by means of a cranked lever revolving in a circular path, instead of by a wheel ; but the principle of the action of this form of lever is precisely the same as that of the wheel, with only the difference that the moments of inertia of the moving machinery are slightly changed. (WINDLAss.) In tread-wheels the power is obtained by causing the men or animals who act upon the machinery to exercise their effect, by the application of their weight, upon the periphery of the wheel. When equable motion is required to be produced by a wheel animated by a variable power, it is obtained by the interposition of a cone, or of some such contrivance for increasing the leverage of the 'bower in proportion as the power itself diminishes; as, for instance, in the spiral springs and fasces of watch-work.