Let c and d represent two circles about the respective cones, being equidistant from e, and therefore of equal diameters and circumferences, and it is obvious that at every point in the length of each cone the velocity will be equal to a point upon the other so long as both points are equidistant from the points of intersection of the axes of the two shafts; hence if one cone drive the other by frictional contact of surfaces, both shafts will be rotated at an equal speed of rotation, or if one cone be fixed and the other moved around it, the contact of the surfaces will be a rolling contact throughout. The line of contact between the two cones will be a straight line, radiating at all times from the point e. If such, however, is not the case, then the contact will no longer be a rolling one. Thus, in [Fig. 57] the diameters or circumferences at a and b being equal, the surfaces would roll upon each other, but on account of the line of contact not radiating from e (which is the common centre of motion for the two shafts) the circumference c is less than that of d, rendering a rolling contact impossible.
Fig. 58.
We have supposed that the diameters of the cones be equal, but the conditions will remain the same when their diameters are unequal; thus, in [Fig. 58] the circumference of a is twice that of b, hence the latter will make two rotations to one of the former, and the contact will still be a rolling one. Similarly the circumference of d is one half that of c, hence d will also make two rotations to one of c, and the contact will also be a rolling one; a condition which will always exist independent of the diameters of the wheels so long as the angles of the faces, or wheels, or (what is the same thing, the line of contact between the two,) radiates from the point e, which is located where the axes of the shafts would meet.
Fig. 59.
The principles governing the forms of the cones on which the teeth are to be located thus being explained, we may now consider the curves of the teeth. Suppose that in [Fig. 59] the cone a is fixed, and that the cone whose axis is f be rotated upon it in the direction of the arrow. Then let a point be fixed in any part of the circumference of b (say at d), and it is evident that the path of this point will be as b rolls around the axis f, and at the same time around a from the centre of motion, e. The curve so generated or described by the point d will be a spherical epicycloid. In this case the exterior of one cone has rolled upon the coned surface of the other; but suppose it rolls upon the interior, as around the walls of a conical recess in a solid body; then a point in its circumference would describe a curve known as the spherical hypocycloid; both curves agreeing (except in their spherical property) to the epicycloid and hypocycloid of the spur-wheel. But this spherical property renders it very difficult indeed to practically delineate or mark the curves by rolling contact, and on account of this difficulty Tredgold devised a method of construction whereby the curves may be produced sufficiently accurate for all practical purposes, as follows:—