We may obtain a whole series of wheels, A',A",A", etc., from the original B; and from A as an original, we may obtain another series, B ,B , etc., having various numbers of teeth. Aud it has been shown that any wheel of the series A will work accurately along with any one of the series B. So far well; but then the wheel A of 20 teeth may not be like the wheel B of the same number of teeth. It becomes, Therefore, a desideratum to choose the form of the teeth of B in such a manner that its conjugate of the same number of teeth may have the same form; by such an arrangement, we shall obtain a series of wheels, any One of which will work with any other.
If the number of the teeth of B be augmented indefinitely, the outline of the pitch-line will become nearly straight; and so draw ing through C (fig. 2) a straight line to touch the pitch-line of A, we shall have the pitch-line of the straight rack, as it is called, which could be worked by any wheel of the series A. The reverse of this rack would work with 'any one of the series B, and there fore, if the series A and B be identic with each other, the rack must be its own reverse. Thus we obtain a very important gen eral result—viz., that if we mark off along a straight line dis tances, CD equal to the desired interval between the teeth, and then draw any line CKLAID, consisting of 4 equal parts, CK, KL, LM, MD, symmetrically arranged, all the wheels obtained from this as the original, will work into each other; and moreover the forms thus obtained answer for internal as well as external teeth.
Being then at liberty to choose any line whatever, subject to the above condition of symmetry, for the figure of the straight rack, we may inquire whether it may not be arranged so as to bring about other desiderata. This line, it may be noted, is not necessarily curved; it may be composed of straight lines, or partly of straight and partly of curved lines.
The general appearance of this wavy line recalls that curve known as the curve of sines, which. indeed. is the simplest known curve, consisting of equal and symmetric undulations, and unlimited in extent. By changing the ordinates in any ratio, say in the ratio of PQ to PR, the waves of the curve may be made shallower or deeper; and on studying the effects of such a change, we discover some new and very important laws concerning the contacts of the teeth of wheels.
Beginning with the curve of sines proper, in which the greatest ordinate, SK, is equal to the radius of a circle of which CD is the length of the circumference, it is found that wheels traced from it can only touch each other at one point: of course such wheels can not work, because the solitary contact is now on the back and now on the front of the tooth. In this case the contour of the tooth crosses the pitch line at an angle of On deepening the teeth, still keeping to the same kind of curve, it is found that the wheels begin to touch at more points than one; and when they are made so deep as that the contour crosses the pitch-line at an angle of 65°, there are always three contacts, neither more nor less. If the teeth be still further deepened, the contacts become more
numerous; they appear and disappear in pairs, so that with an inclination of, say, there would be sometimes three, and sometimes five contacts. When it becomes 70° 17', there are always five; and with an inclination or 73° 11', there are always seven points in contact at once.
Of these points of contact, some are on the sides of the teeth, and others are near the top and bottom; the latter, on account of the obliquity of their action, are of no ps• in driving; they may be called supplementary, and their number is always one less than the number of useful or working contacts. In the system of seven contacts, four are useful, two of them being forward, and two backward, so that two teeth are always in action at once; an arrangement by which a gradual improvement in the equality of the teeth is secured by their wearing.
When two properly formed wheels are put in motion, the points of contact move also, and describe a peculiarly shaped line, the nature of Which depends on the charac ter of the primary form adopted for the tooth of the straight rack. Conversely, if this path of the points of contact be first assumed, and the law of motion in it be observed, the form of the tooth of any wheel may thence be obtained; and this leads us to the most convenient way of making the delineation.
In fig. 3, the form of the straight rack and the corresponding shape of the teeth of a wheel of 20 are shown in contact, the depth of the tooth being such as to give five con tacts, which in the drawing arc at the five points marked 0. If we suppose the rack to be slid upward, carrying the wheel along with it, the of contact will change; and when the motion has been one-eighth part of the interval between two teeth,these points will occupy the positions marked 1. When a motion of another eighth is made, the two upper contacts on the left hand merge into one, and are about to disappear; at the instant, nstant, two new contacts begin at the lower point, marked 2; and thus the motion 'j continues in the order of the numbers marked along the peculiarly shaped path of the points of contact. Those contacts which occur along the crossing lines of the curve are working contacts; those which happen along the two external arcs, are supple mentary. When the form of this path, and the posi tions of the successive points in it have been obtained by calculation, the outline of any wheel is easily traced geometrically.