Thus, the form of the tooth can be obtained when that of the pitch-line is known.
Now, when two disks, turning on fixed centers, touch each other at any point out of the straight line joining these centers, there is a slippiiig of the one surface over the other; and therefore, in order that the pitch.lines may roll together, they must be so shaped as that the point of contact may be in the line of centers. It can be shown that, for any assumed contour of the wheel A, another contour having its center at 13, and rolling upon A, is possible. But, except in one or two special cases, the working out of the problem has not been accomplished. It will be enough here to mention the single case of elliptic wheels. The action of these is founded on the well-known prop erty of the ellipse, that the sum of the distances of any point in it from the two foci is constant, and that the curve makes equal angles with these two lines. Hence two equal ellipses turning on their foci, when their centers are at a distance equal to the major 'axis of the ellipse, will roll upon each other; and teeth formed upon these as pitch-lines will work perfectly. When the ellipses have their major and minor axes in the propor tion of 5 to 4, the focus is at one-fifth part of the major axis from one end; and there fore one focus, at one part of the revolution, moves four fast—at another part, four times as slowly—as the other focus.
Sometimes one of the wheels has to be quite at rest during part of the motion of the other wheel. This is accomplished by causing some part of the wheel that is to be sta tionary, to bear upon a part of the circumference of the moving-wheel which is concentric with its axis. This is exemplified in the arrangement for counting wheels shown in fig. 5. The object of this apparatus is to count and record the revolutions of the wheel' B. As this wheel turns round, a pin 13 attached to it enters into the slit GH, and thus carries the wheel A round as long as the pin remains in the slit, that is, until the slit GH be brought into the position IK. As
soon as E leaves the slit at I, there would be no further con nection between the two wheels, and A could he moved any how, altogether independently of B. In order to prevent this, the disk B is made nearly five-sixths entire, and parts of A are scooped out between the slits so as to receive and to fit B. By this means A is prevented from being turned either backward or forward until the pin E again comes into one of the slits. When this happens, the projecting. part at G finds room in the recess F. If there be seven slits, GFI, round the wheel A, and if B turn once in twenty-fou• hours, an index attached to A would show the days of the week; and the index might be made to be stationary all day, the change being effected during the night. Another example of this kind of interrupted motion is seen in the ordinary dead-beat clock escapement, in which the detaining surface of the pallet is concentric with the axis of, the crutch.
When the axes are inclined to each other, beveled wheels are used. Just as common wheels may he regarded as fluted cylinders, beveled wheels may be described as fluted cones having a common apex. principles which regulate the formation of the teeth of these are the same as for plane wheels, but the application of these principles is con siderably more intricate. bince both the teeth and the spaces between them are tapered, it is impossible to notch-out the intervals by means of a revolving cutter. Attempts have been made to construct machinery for r ming the teeth by means of a cutter moving in a line toward the apex of the cone, but he complexity of the apparatus, and the slow ness of the process, have prevented its introduction; and thus the accurate formation of beveled wheels has still to be accomplished by hand.