Simultaneous Velocities of a Point and the Increments of its Velocities.
Ratio of the elementary displacement and the velocity of a point to the displacement, and velocity of its projection upon a straight line or plane. Use of infinitesimals to determine these ratios. Example:—Oscillatory motion of the projection upon a fixed axis of a point moving uniformly upon the circumference of a circle.
Analogous considerations for polar co-ordinates. Relations of the velocity of a point, of its velocity of revolution and its angular velocity about a fixed pole; of its velocity in the direction of the radius vector; of the velocity of increase of the area which this radius describes.
Simple Motions of Solids, or Rigid Systems.
1. Motion of rectilinear or curvilinear translation; simultaneous displacements, and velocities of its different points.
2. Motion of rotation about a fixed axis; relation of the velocities of different points to the angular velocity.
Geometrical notions and theorems relative to the instantaneous center of rotation of a body of invariable figure and movable in one plane, or to the instantaneous axis of rotation of a rigid system situated in space, and movable parallel to a fixed plane. Relation of the velocities of different points to their common angular velocity. Use of the instantaneous center of rotation for tracing tangents; examples—and amongst others—that of the plane curve described by a point in a straight line of given length, whose extremities slide upon two fixed lines. Rolling of a curve upon another fixed curve in a plane. Descartes’ theorems upon the inter of the normals at the successive points of contact: cycloids, epicycloids, involutes, and evolutes. Extension of the preceding motions to the instantaneous axis of rotation of a rigid system movable about a fixed point.
COMPOSITION OF MOTIONS.
Lessons 3–6. Composition of the Velocities of a Point.
Polygon of velocities. Example of movements observed relatively to the earth. Particular cases; composition of velocities taken along three axes; composition of the velocity of a point round a fixed pole, and its velocity along the radius vector. Method of Roberval for tracing tangents.