(9)
The whole of the foregoing reasonings are applicable, not merely when A and B are actual regular cones, but also when they are the osculating regular cones of a pair of irregular conical surfaces, having a common apex at O.
§ 32. Screw-like or Helical Motion.—Since any displacement in a plane can be represented in general by a rotation, it follows that the only combination of translation and rotation, in which a complex movement which is not a mere rotation is produced, occurs when there is a translation perpendicular to the plane and parallel to the axis of rotation.
| Fig. 94. |
Such a complex motion is called screw-like or helical motion; for each point in the body describes a helix or screw round the axis of rotation, fixed or instantaneous as the case may be. To cause a body to move in this manner it is usually made of a helical or screw-like figure, and moves in a guide of a corresponding figure. Helical motion and screws adapted to it are said to be right- or left-handed according to the appearance presented by the rotation to an observer looking towards the direction of the translation. Thus the screw G in fig. 94 is right-handed.
The translation of a body in helical motion is called its advance. Let vx denote the velocity of advance at a given instant, which of course is common to all the particles of the body; α the angular velocity of the rotation at the same instant; 2π = 6.2832 nearly, the circumference of a circle of the radius unity. Then
T = 2π/α
(10)
is the time of one turn at the rate α; and
p = vxT = 2πvx/α