Chap. XI.

MECHANICS.

941

dividing the teeth accurately, the tracer will, by the rolling of the two circles, describe thecurves sought. The teeth of wheels are usually set out by arcs of circles after their centreand radius are accurately determined.

Iron was first cast into wheels and used in machinery about the year 3 782, at the worksat Colebrook Dale, though wrought-iron had been partially employed long before on thecontinent.

When two wheels engage each other the smallest is called the pinion, and the numberof their teeth being determined, the line uniting their centres is termed the line of theircentres, and if this line be divided in the ratio of the number of teeth in each wheel, thispoint limits the circle, or what practically is called the pitch line. If from each centre acircle is described, passing through this common point, the diameters of these circles arenamed primitive or proportional, and the circles themselves the pitch lines; the pitch ofthe teeth being the distance from each other, as measured across this line as a chord, andnot along its circumference. Suppose C D the lineof centres, E E the pitch line of the smaller wheel,

F F that of the larger, and the centre A to bein the line of centres C D. Let A be the stave ofa trundle, from the centre of which describe thearc bc t which shows the form of the tooth, whichwithin the pitch line may be drawn circular. Theradius for describing the teeth is equal to thepitch, minus half the diameter of the stave of thetrundle, and the centres of these arcs will be in theproportional circle, which we have called the pitchline.

Having set off upon the proportional circles thepoints G, 2 L, and OoH, according to the thick-ness of the teeth and leaves, draw lines from thesepoints towards the centre of their respective circles,to serve as the sides of the spaces between theteeth and between the leaves; the depth of whichspaces must be such as to give room for the actionof the curved part of the teeth and leaves. Thendescribe upon the extremity of the sides of eachtooth epicycloids, such as Q D, LD, with the generating circle Y, the diameter of whichis equal to the proportional radius of the pinion, upon the circumference of the propor-tional circle of the wheel as a base.

The mode of forming the teeth being shown, that of the leaves will be obvious, V V beingthe generating circle of the epicycloid upon the circumference of the circle of the pro-portional pinion as a base. When the teeth are small, and do not begin to act till theyarrive at the line of the centres, the teeth of the wheel when the wheel drives the pinion,or the leaves of the pinion when the pinion drives the wheel, may be described by acircular arc, of which the radius is equal to the pitch, and of which the centre is in thepitch line of the wheel or pinion.

Another method is shown, which may be considered more simple : —

Let A A represent the pitch ine, the circleB B the bottom of the teeth, and D D thetop.

After the pitch line is described, it is dividedinto the number of teeth the wheel is to have;with a radius equal to one and a quarter of thesedivisions, strike circles upon the pitch line A ;thus the curved part of each side of the cogis formed. The remaining portion of the toothwithin the pitch line A is bounded by twostraight lines, drawn from the point gmn towardsthe centre of the wheel; the spaces between, aswell as the teeth, are all equal.

Another method may be shown, which re-quires, however, to be executed with attention.

Let the tooth GCH have the form of a curve traced by the extremity of a thread FC,unlapped from the circumference, and let the acting face of the tooth b be formed in asimilar manner: then tfie line FCE, drawn perpendicularly to the touching point C, isin the direction of the threads, by which the two acting faces are formed; this line, beingthe common tangent to the circumferences of the two wheels, will always cut the line A Bin the point D : thus the teeth can act on each other at the same time, and the pressure isspread over a number, instead of being confined to one.

Fig.1288

I'ig. 1236.