The teeth of wheels for watches and clocks need particular care in shaping, and it may be of interest if I describe briefly the principles upon which these wheels are made. What is required is that the motion shall not be communicated by jerks as the teeth successively engage one another, but that the motion shall be perfectly smooth. The problem therefore becomes this: How are we to arrange the teeth of the wheels so that as one of them turns and drives the other round the leverage or turning power exercised by the driving wheel on the driven wheel shall always be uniform? Now if the teeth were simple spikes one can easily see that this would not be the case. For instance, as the arm a c turned round, driving before it the arm b d, the point c would scrape along, and the leverage between the two teeth would constantly alter. Evidently some other construction must be adopted. Before we can determine what it is to be, we must inquire what the leverage would be between two rods, a c and d b, mounted on pivots at a and d. The answer to this question is, that when a lever such as a c presses with its end against another, d b, the power is exercised in a direction c e at right angles to d b. Hence the leverage between the two arms is in the ratio of a e to d c. The system is just as if we had a lever a e united to a lever d c by a rigid rod e c at right angles to both of them.
Fig. 75.
Whence then the ratio of the power is as a e is to d c.
Fig. 76.
But since the triangles a e f, d c f, are similar, a e is to d c as a f to f d. Whence then we get this general proposition: If one body mounted on an axis is pressing upon another body mounted on an axis, the pressure exerted between them is always exercised in a direction, shown by the dotted line, at right angles to the two surfaces in contact; and the ratio of the leverage is found by drawing a line from one axis to the other, so as to cut the line of direction of pressure in f. The leverage of one on the other is then as a f to f d. Our problem has now become the following: Given a rod b d, suppose that it is pressed upon by a curved surface mounted on an axis at a. Then the direction of the pressure that the curved surface (called in engineering language a cam) will exercise on the rod b d is shown by the dotted line; and the ratio of the driving power to the driven power is as d f to a f. Now how can we shape the cam so that as it moves round, and different parts of its surface come successively into contact with b c, the ratio of the leverage is always the same; that is to say, the ratio of a f to f d shall always be constant; that is to say, the line drawn through the point of contact perpendicular to the curve at that point, shall always pass through the point f?
Fig. 77.
