in approaching the line of centres, while the teeth of the wheel conduct the staves of the trundle in their progress from that line*. II. To find the figures of the teeth of the wheel, when the staves of the trundle are cylinders of a finite diameter. Consider the trundle at first as having infinitely small staves, represented by the centres of the staves, A, E, H, &c. and trace, as above mentioned, the teeth CL P, AQN, &c. of the wheel, as if it had to conduct a trundle with infinitely small staves: observing to leave a small space, such as A L, between all the teeth, in order that they may act freely. Describe, with the radius of the staves, upon the plane of each tooth, as many See Article 6 of this chaper. small arcs as may be convenient, having all their centres in the two epicycloids which form the teeth. Trace, by means of these little arcs, two curves, such as RO, SO, parallel to the epicycloids, and then you will have inclosed the space, R, O, S, which is the figure all the teeth of the wheel ought to have beyond its proportional circle. For if we suppose, that the centre, E, of a stave, is conducted by the teeth C, P, L: the curve, RO, which is parallel to the epicycloid, CP, and which is placed at the distance of the radius of the stave E, shall always touch the circumference of that stave. Thus the curve, RO, shall conduct the cylindric stave, as if the tooth, CPL, conducted the centre of that stave, and consequently the tooth ROS, shall be a proper figure to conduct the trundle, with cylindric staves, The curved parts of the teeth of the wheel, being determined as above, the spaces, TS, Z &, &c. should be cut out, in order to admit that part of the staves which extends beyond the proportional circle of the trundle. Having considered the case of a wheel and trundle, with cylindric staves acting together, I am now to explain that of a wheel and pinion, two sides of the figure of whose teeth are straight lines directed to its centre. Of the Wheel and Pinion. 4. To find the figure of the teeth and leaves of a wheel and pinion, when that part of the teeth and leaves, which lies within their respective proportional circles are straight lines directed to the centres of these circles. Having set off upon the proportional circles, the points, G, Q, L, and, O,o, H, &c. according to the thickness of the |