same plane lies upon it; given the coefficient of friction between the beam and sphere; also that the sphere cannot slide but only roll along the plane: find the inclination of the beam to the horizon when the sphere is just about to roll away.

(45). A heavy beam lies over a peg with one extremity against a vertical wall; given the coefficient of friction between the beam and peg, and between the beam and wall; find the limiting positions of equilibrium.

(46). One end of a beam, whose weight is W, is placed on a smooth horizontal plane; the other end, to which a string is fastened, rests against another smooth plane inclined at an angle a to the horizon; the string passing over a pulley at the top of the inclined plane hangs vertically supporting a weight P. Shew that the beam will rest in all positions if a certain relation hold between P, W, and a.

(47). A beam whose length is 2a has two beams each equal, a jointed one at each of its extremities; the three beams are then placed on the top of a sphere whose radius is 2a, in such a way that the middle point of the middle beam rests on the highest point C of the sphere: shew that the pressure on the sphere at C is equal to of the whole weight of the three beams.

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(48). Two equal smooth spheres connected by a string are laid upon the surface of a cylinder, the string being so short as not to touch the cylinder: determine the position of rest and the tension of the string.

(49). A weight Q is placed inside a smooth hemispherical cup, a string attached to it passes over the smooth edge of the cup and supports a weight P hanging vertically; the hemisphere rests with its curved surface in contact with a smooth horizontal plane: find the position of equilibrium. If the cup be without weight, how must the enunciation of the problem be altered in order that equilibrium may be possible?

(50). A series of equal weights are fixed at equal distances along a string which is suspended from two fixed points; shew that the successive portions of the string are inclined to

the horizon at angles whose tangents are in arithmetical progression.

(51). A light cord with one end attached to a fixed point passes over a pulley in the same horizontal line with the fixed point, and supports a weight hanging freely from the other extremity. A heavy weight being fixed upon the cord at different places successively between the fixed point and pulley, it is required to find the locus of its positions of equilibrium.

(52). An elliptical plane, in which A and B are the extremities of the major and minor axes and S the focus respectively, is acted upon by two forces applied at the point B, which are in directions of, and proportional to BA and BS: what point in the major axis must be fixed in order that motion may not ensue ?

(53). A triangular board has one right angle, and is placed in a vertical plane with one side horizontal; a heavy string covers the hypothenuse and hangs down at the lower end; the upper end of the string is held by a force parallel to its length: find a point in the board which must be fixed in order that the base may remain horizontal.

(54). Three equal spheres lying in contact on a horizontal plane are held together by a string. A cube whose weight is 3 W is placed with one of its diagonals vertical so that its lower sides touch the spheres: shew that the tension of the string is W√3.

(55). The piston rod of a locomotive engine is attached by a free joint to a point in one of the radii of the driving wheel; shew that the pressure upon the axis of the wheel tending to force it forward is the same for symmetrical positions of the piston rod as the wheel revolves, whether the rod be pushing or pulling.

(56). A hemisphere of given radius rests upon the top of a rough fixed sphere; find the radius of this sphere such that the equilibrium may be neutral.

(57). If a cylinder has its base united concentrically to the base of a hemisphere of equal radius, find the height of the cylinder in order that the solid may stand upon a smooth horizontal plane on any point of its spherical surface.

(58). A sphere of radius r rests upon the concave surface of a sphere of radius R; if the first sphere be loaded so that the height of its centre of gravity from the point of contact be r, find R such that the equilibrium may be neutral.

(59). Out of a solid right cone is cut another cone having coincident base and axis; the remaining solid is placed like an inverted wine-glass upon a pointed rod: find the relation between the lengths of the axis of the cones, according as the equilibrium is stable, neutral, or unstable.

(60). When the solid cut out of a cylinder by two planes passing through the axis is in equilibrium with its cylindrical surface placed upon an horizontal plane and edge uppermost, prove that the equilibrium is stable however small be the inclination of the planes.

DYNAMICS.

SECTION I.

PRELIMINARY.

(1). It has been defined (Art. 3, Statics) that when the constituent particles of a body occupy different positions in space at successive instants of time, the body is in motion. Its particles are said to be animated with velocity.

If during its motion a material particle always describes any given space in the same time, wherever in its course the space be taken it is said to be moving uniformly, or to have a uniform velocity; i.e. its velocity at every instant is the same throughout the motion.

On the contrary, if a given space, wherever taken, be not passed over in the same time, the motion is not uniform; or the velocity of the particle varies from instant to instant.

Supposing the motion to be such that the given space is passed over more quickly in the latter part of the particle's course than in the first, the particle's velocity would be said to have increased, and vice versa.

It is not difficult, from such considerations as the preceding, to form a distinct idea of velocity as being a property of a moving particle, differing in degree from instant to instant; its magnitude at any one moment being perfectly independent of the subsequent movement of the particle.