96. A hollow sphere formed of a rigid inelastic substance, and filled with inelastic fluid, is let fall on a horizontal plane; find the whole impulse on its curved surface, and on each half of its surface above and below the horizontal plane through its

centro.

Also determine the resultant impulses on each of these surfaces.

97. A flexible and elastic cylindrical tube is placed within a rigid hollow prism, in the form of an equilateral triangle, which it just fits when unstretched; if there be no air between the tube and the prism, and if air at a given pressure be forced into the tube, find the extension and the portion in contact with the sides of the prism.

98. A conical bag, which is filled with liquid, has its rim fastened to a horizontal plate, and is then inverted; prove that the tension at any point, in the direction of a generating line, varies as the square of the distance from the vertex.

99. A bag, in the form of a paraboloid, formed of thin flexible substance, is supported by its rim, and is filled with water; find the tension at any point in direction of the tangent to the generating parabola at the point.

Hence prove that the tension in every direction at the vertex =gpah, if h be the depth of the bag, and 4a the latus rectum. Also obtain this last result independently by aid of Art. (167).

100. If the same bag, when filled, be closed and inverted, prove that the tension at any point P, in direction of the tangent to the generating parabola, varies as AN. SP, A being the vertex of the bag, S the focus, and AN the depth of P below the vertex.

SPECIFIC GRAVITIES.

Ratios of the Specific Gravities of different substances to that of water at €0o.

Ratios of the densities of gases and vapours of different substances to that of atmospheric air at the same temperature and under the same pressure.

2. 1st. 4329 lbs. on a square inch. 2nd. about 58 lbs.

4. 7311 lbs. on a square inch, neglecting atmospheric pressure.

11. If h be the vertical side, the depth of the horizontal

3. The line divides the opposite side in the ratio of 3: 1.

10. The point lies in the line from the vertex bisecting the

(the depth of the vertex).

base and at a depth

11. 1: √√2-1.

12. (1+√10) inches.

13. 14: 9.

14. If a, π-a, be the angular spaces occupied by the liquids p, p', the inclination to the vertical of the bounding dia

p2+p).

sin a p

16. The increase=14 (the weight of the fluid).

18. 1-(1-1).

radius.

20. Produce the rectangle to the surface; then, knowing the centres of pressure of the whole and of the upper part, and the pressures on these parts, the position of the centre of pressure of the lower part can be inferred.

15. Weight of wood + weight of water it displaces.

16. If σ, o' be the specific gravities, V, V' the volumes, and p the specific gravity of water, the condition is

3. Surface divides altitude in ratio 1 : √2-1.

15. If w, w be the weight of the cone and of the fluid displaced, the force=w-w', and its line of action must be at a distance from the centre of gravity of the solid cone equal to