*KINEMATICS OF MACHINERY.*

Fig. 119.

become points only, the paths they describe in rolling become the centroids themselves. These may be used as profiles if the cylindric axoids bearing them be so pressed together that sliding at the point of contact is prevented by friction, so that they are compelled to roll on one another. This is the only case where the profiles of elements have a pure rolling motion. Circular centroids give us cylindric wheels, like those known as friction wheels (Fig. 119), The applications of this method are not few, the most important and most familiar being perhaps the wheels of railway carriages. We shall in the next chapter consider in detail the subject of axoids pressed together, or restrained, by a force.

**§ 38**
**Generalisation of the foregoing Methods.**

In the foregoing paragraphs we have throughout limited ourselves to axoids for general cylindric rolling, but the methods employed are equally applicable to the case of non-cylindric axoids. For conic axoids this is easily seen, but not so easily with the higher rolling and twisting axoids (see § 13). Considerable difficulties appear here in the theoretic examination even of motions occurring according to simple laws, and still greater difficulties in their practical presentation. It is a part of Applied Kinematics to consider so far as may be necessary the more important cases. It must be admitted that in general the actual forming of profiles for the higher axoids, even those for instance for the teeth of hyperboloidal wheels, presents as yet considerable

upon a circle larger or smaller than that by a distance equal to two-thirds of the intended depth of the face or the flank of the tooth. The first of these points is assumed and the second found from it by a very accurate approximation. A normal to the roulette at the second point is drawn, and the intersection of this line with the normal bisector of a line joining the two points gives the centre for the circular arc to be substituted for the roulette. This arc passes through both the points, and has a common tangent with the roulette at one of them.