The comparative motion of two points at a given instant is capable of being completely expressed by one of Sir William Hamilton’s Quaternions,—the “tensor” expressing the velocity ratio, and the “versor” the directional relation.
Graphical methods of analysis founded on this way of representing velocity and acceleration were developed by R. H. Smith in a paper communicated to the Royal Society of Edinburgh in 1885, and illustrations of the method will be found below.
Division 2. Motion of the Surface of a Fluid Mass.
§ 24. General Principle.—A mass of fluid is used in mechanism to transmit motion and force between two or more movable portions (called pistons or plungers) of the solid envelope or vessel in which the fluid is contained; and, when such transmission is the sole action, or the only appreciable action of the fluid mass, its volume is either absolutely constant, by reason of its temperature and pressure being maintained constant, or not sensibly varied.
Let a represent the area of the section of a piston made by a plane perpendicular to its direction of motion, and v its velocity, which is to be considered as positive when outward, and negative when inward. Then the variation of the cubic contents of the vessel in a unit of time by reason of the motion of one piston is va. The condition that the volume of the fluid mass shall remain unchanged requires that there shall be more than one piston, and that the velocities and areas of the pistons shall be connected by the equation—
Σ · va = 0.
(1)
§ 25. Comparative Motion of Two Pistons.—If there be but two pistons, whose areas are a1 and a2, and their velocities v1 and v2, their comparative motion is expressed by the equation—
v2/v1 = −a1/a2;
(2)