1832

1833

1834

Apply the two resulting equations to find the shortest line that can be drawn from one given curve to another given

curve.

6. Shew that the cycloid is the curve of quickest descent from one given curve to another, the motion being supposed to commence from the first, and that at the points where it meets the curves their tangents are parallel.

7. If V be a function of x, y, and the differential coefficients of y with respect to x, find the variation SV.

8. Find the shortest line that can be drawn on a surface of revolution from a given point to a given curve.

1836 9. Assuming the expression for &/Vdx, shew that the condition of /Vdx, between given limits, being a maximum or minimum, gives rise to two independent equations. Prove that the shortest line between two curves of double curvature is a straight line perpendicular to both.

SECTION XIV.

QUESTIONS IN STATICS.

1. Given a and b the arms of a straight lever which turns on 1821 an axis, the radius of which is r. P would maintain the equilibrium acting perpendicularly at the distance (a), if there were no friction but a weight p must be added to it in order to overcome the friction. Find the proportion of the friction to the pressure.

2. A beam 30 feet long, balances itself upon a point at rd of its length from the thicker end. But when a weight of 10 lbs. is suspended at the other end, the prop must be moved two feet towards it to maintain the equilibrium.

weight of the beam?

What is the

3. Explain how wheels assist the motion of a carriage.

4. Explain Archimedes' screw.

5. Investigate expressions for determining the position of the centre of gravity of a plane surface, bounded by a curve whose equation is given.

6. If a body be balanced upon a horizontal plane, and a slight motion be given to it, its centre of gravity will move. horizontally; prove this, and shew in what cases the equilibrium is stable.

7. Shew, when P sustains W upon a screw, if a slight motion be given to the machine, that P's velocity : W's velocity :: W: P.

1822

1823

8. If a triangular prismatic beam is supported at both ends, shew that it is twice as strong when the edge is uppermost, as when the base is.

of

9. Find the position of the centre of gravity of the quadrant a circular area.

10. When the same string passes over any number of pullies, and the parts of the string supporting any pulley at the lower block are not parallel to one another, find the proportion between P and W in equilibrio.

11. Two weights sustain each other on two inclined planes, having a common altitude, by means of a string parallel to the planes; compare the pressures.

12. A parallelogram and a triangle upon the same base and between the same parallels revolve round the base as an axis; prove that the solid generated by the triangle equals one third of that generated by the parallelogram.

13. State generally the principle of virtual velocities; and from it deduce the position of equilibrium of a straight rod of uniform density placed on two inclined planes.

14. The distance of the centre of gravity of a cycloid from the vertex = ths of the axis; compare, from this, the contents of the solids generated by its revolution round the base and a tangent at the vertex.

15. P supports W upon an axle, by means of a perpetual screw acting upon the circumference of the wheel. Required their proportion.

16. A plane of given form and area is supported in the air as a kite, the wind acting in a direction parallel to the horizon; the weight of the string and materials being (w), and the horizontal pressure of the wind equivalent to a weight (p) upon each square foot; required the angle made by the plane with the horizon, and the greatest weight it can support.

17. A chain of uniform density is suspended at its extremities by means of two tacks in the same horizontal line at a given distance from each other; find the length of the chain so that the stress upon either tack may be equal to the chain's weight.

18. A beam of given length and weight is placed with one end on a vertical, and the other on a horizontal plane: find the force necessary to keep it at rest, and the pressures on the two planes.

19. On a lever of uniform density, every inch weighing w oz. a weight of W oz. is suspended at a given distance from the fulcrum which is placed at one extremity. What must be the length of the lever, so that the whole may be supported by the least possible power acting in an opposite direction at the other extremity?

20. The beam of a false balance being of uniform density and thickness, it is required to shew that the lengths of the arms are respectively proportional to the differences between the true and apparent weights.

21. Determine the length of a straight line drawn through the centre of gravity of a given isosceles triangle, making a given angle with the base, and terminated by the sides.

22. Determine the conditions of equilibrium of a material point situated in a canal of indefinitely small dimensions and acted upon by any number of forces.

23. If a pole rests with one end on the ground against a 1824 wall, and the other attached to a string fixed in the wall, find the tension of the string.

24. If a, ß, y be the angles which the resultant of three forces acting at right angles to one another makes with each of them respectively, then will

where a

tension at the lowest point; and prove that its radius of curvature is equal to its normal.

26. If the sides of a triangle ABC be bisected in the points D, E, F ; then the centre of the circle inscribed in the triangle DEF is the centre of gravity of the perimeter of the triangle ABC.

1825

27. A and B are two given points in a horizontal line, to which are fastened two strings of given lengths; the string BC passes through a ring at C, and is fastened to a given weight W; find the position in which the weight will rest.

28. If any number of forces p, q, r.... in different planes, acting on a point, make the angles a, ß, y.... with the resultant, then will

29. Find the position of the centre of gravity of the area of a semi-parabola.

30. Determine the position of equilibrium of a uniform rod, one end of which rests against a plane perpendicular to the horizon, and the other on the interior surface of a given hemisphere.

31. An arch, where the equilibrium is preserved by the weights of the voussoirs, is so constructed that the centres of gravity of the voussoirs are in a catenary curve and the joints perpendicular to the curve. Required the equation from which the length of the voussoirs may be obtained.

32. When a chain fixed at two points is acted upon by a central attractive or repulsive force, the tension at any point is inversely as the perpendicular let fall from the centre of force on the tangent at that point. Required proof.

33. Determine the point in the curve surface on which a semi-paraboloid will rest on a horizontal plane.

34. A given weight is to be supported at a given point upon a straight lever of uniform density by a power acting at its extremity on the same side of the fulcrum. Required the least power which will support the system, and the corresponding length of the lever.

35. A circular hoop is supported in a horizontal position, and three weights of 4, 5, and 6 pounds respectively are suspended over its circumference by three strings meeting in the centre; what must be their positions so that they may sustain one another?