care being taken to attend to the sign of α, so that when that is negative the arithmetical values of γ and α are to be added in order to give that of β.
The whole of the foregoing reasonings are applicable, not merely when aaa and bbb are actual cylinders, but also when they are the osculating cylinders of a pair of cylindroidal surfaces of varying curvature, A and B being the axes of curvature of the parts of those surfaces which are in contact for the instant under consideration.
| Fig. 93. |
§ 31. Instantaneous Axis of a Cone rolling on a Cone.—Let Oaa (fig. 93) be a fixed cone, OA its axis, Obb a cone rolling on it, OB the axis of the rolling cone, OT the line of contact of the two cones at the instant under consideration. By reasoning similar to that of § 30, it appears that OT is the instantaneous axis of rotation of the rolling cone.
Let γ denote the total angular velocity of the rotation of the cone B about the instantaneous axis, β its angular velocity about the axis OB relatively to the plane AOB, and α the angular velocity with which the plane AOB turns round the axis OA. It is required to find the ratios of those angular velocities.
Solution.—In OT take any point E, from which draw EC parallel to OA, and ED parallel to OB, so as to construct the parallelogram OCED. Then
OD : OC : OE :: α : β : γ.
(8)
Or because of the proportionality of the sides of triangles to the sines of the opposite angles,
sin TOB : sin TOA : sin AOB :: α : β : γ,