Relation between the length of the Catenary and the depth (D) of its lowest point below the points of attachment. I. Dynamics.--That branch of mechanics which investigates the different kinds of motion produced by forces. II. Velocity or Rate of Motion.-The number of units of space described by a body in a unit of time. III. Rate of Acceleration. The velocity generated in a unit of time by the continued action of a force. IV. Units of Time and Space.-1 second and 1 foot. V. Mass.-The quantity of matter in a body as indicated by its weight. VI. Unit of Mass.-A body weighing g lbs. g being the rate of acceleration due to gravity. VII. Momentum or Dynamical Effect.-The effect of a force with reference to its time of action. G VIII. Centripetal Force.-A force which tends to draw a body towards the centre of a circle in which it is constrained to move. IX. Centrifugal Force.-The outward reaction tending from the centre of a circle in which a body is constrained to move. X. Vis viva of a particle.-The product of the mass of a particle into the square of its velocity. XI. Centre of Oscillation.-That point in a pendulum at which the whole mass might be collected without altering the time of oscillation. XII. Centre of Percussion.-That point where an impulsive force may strike a body, having a fixed axis, without producing any pressure upon the axis. DYNAMICAL PRINCIPLES. 1. Parallelogram of Velocities.-If two adjacent sides of a parallelogram represent the magnitude and direction of two velocities with which a particle is simultaneously animated, then the intermediate diagonal will represent the magnitude and direction of the resultant velocity. 2. First Law of Motion.-If a body is not acted upon by any force it will remain at rest; and if in motion, it will continue to move uniformly in the same direction. 3. Second Law of Motion.—If any number of forces act simultaneously upon a body at rest or in motion, each force produces the same effect in the direction of its action, as if it acted singly on the body at rest. 4. Third Law of Motion.-The velocity generated in any time by a force continually acting upon a body, is propor tional to the force. 5. Motion along an inclined plane or curve.-The velocity acquired by a body acted upon by gravity whilst ascending or descending a smooth straight or curved line in a vertical plane, will be the same as if it ascended or descended freely through the same vertical space. 6. Isochronous Chords.-The time of descent down any chord of a circle drawn from either extremity of a vertical diameter, is equal to the time down that diameter. 7. Conservation of the Motion of the Centre of Gravity.-The common centre of gravity of any number of free particles acted upon by any forces, moves in the same manner as if the particles were concentrated in one body and acted upon by the same forces, in directions parallel to their actual directions. 8. D'Alembert's Principle.-When a system of material particles, rigidly or otherwise connected, is in motion under the action of any finite forces, the externally impressed moving forces will together with the effective moving forces on the same particles, applied in directions contrary to that in which they act, satisfy the conditions of statical equilibrium. 9. Vis Viva.-The vis viva of a connected system is, at any time, the same as if each particle, being free, had been acted upon by the same impressed forces through the same spaces. 10. Laws of Gravitation.-(1) The force of gravity on a body exterior to the earth's surface, is inversely proportional to the square of its distance from the earth's centre. (2) The force of gravity on a body within the earth's surface, is directly proportional to its distance from the earth's centre. W = DYNAMICAL FORMULE. weight of a body. M = mass of a body. General Notation. velocity of a body moving uniformly ; velocity acquired by a body moving with an accelerated s = space described by a body in time t. f = rate of acceleration due to a pressure (P) continually applied to a body. 9 rate of acceleration due to gravity; = = 32.2 nearly, at Greenwich. 1. Composition and Resolution of Velocities.-Employ the statical formulæ, substituting velocity for force. For falling bodies, f = g = 32.2. 9. For bodies projected vertically upwards, attaining the greatest |