30 Similarly we have SSy dx dy = 36ba. zogpab sin (a + B) {a sin a + b sin ß}. 7. To determine the pressure on a loop of the Lemniscata of James Bernoulli, the axis of the loop being vertical, and its vertex just touching the surface of the fluid. The polar equation to the curve is gee = a* cos 20, the axis of the loop being the prime radius vector, and the vertex being the pole. Hence, the depth of an elementary polar area rdo dr below the surface of the fluid being r cos 0, the pressure on the loop will be equal to gefsr do dr.r cos 0 Sta cos Ꮎ dᎾ dr = sel 22 - 29/8 . 39ρα 1 1 - 0 0 8. A hemisphere, with a flat lid, as nearly as possible filled with fluid, is held with a point in its edge uppermost; to find its position when the sum of the pressures on the curve and the plane surfaces is the greatest possible. r or = Let O be the inclination of the axis of the hemisphere to the vertical, and let r denote the radius. Then the area of the base is equal to ar, and that of the curve surface to 27r?. Also the depth of the centre of gravity of the base below the highest point of the rim is equal to r sin 0, and the depth of the centre of gravity of the curve surface, which is at the middle point of the axis of the hemisphere, below the same point, is equal to r sin 0 + ir cos 0. Hence the whole pressure on the two surfaces of the hemisphere is equal to gp {arré. r sin 0 + 2ar?. (r sin 6 + įr cos 6)} nagpms (3 sin @ + cos 6). When this expression is a maximum, 3 cos 0 - sin 0 = 0, tan 0 = 3, which determines the required position. 9. A rectangular board is immersed vertically in a fluid, one side of the board being coincident with the surface; to find the pressure on the board. If a denote the length of a horizontal and b of a vertical side of the board, and p the density of the fluid, the required pressure will be equal to žgpab?. Bossut: Traité d'Hydrodynamique, tom 1. p. 33. 10. A triangle of any form is immersed vertically in a fluid with one side in the surface; to determine the whole pressure on the triangle. Let c represent the length of the side in the surface of the fluid, and h the distance of the opposite angle from this side ; then the required pressure will be equal to égpch. 11. A board in the shape of an isosceles right-angled triangle with the squares on its sides, is placed with the upper side of the square opposite to the right angle in the surface of the fluid; ; a a a -1 -1 tan tan to compare the sum of the pressures on the squares containing the right angle with the pressure on the square opposite to the right angle. If P = the pressure on the square opposite to the right angle, and Q = the sum of the pressures on the squares containing the right angle, Q = 3P. 12. A rectangle is immersed vertically in a fluid with one angle in the surface of the fluid; to find the pressure upon it, and to determine the inclination of its sides to the surface when the pressure is a maximum. If 2a, 2b, be two of its unequal sides, and the inclination to the horizon of the diagonal through the angle which is in the surface of the fluid, the pressure will be equal to 4gpab (a? + ). sin , which will evidently be the greatest possible when 0 = žit, that is, when its sides 2a, 2b, are inclined to the surface at angles 6 õ' respectively. 13. ABCD is a parallelogram, the diagonals AC, BD, of which intersect in E, AB being in the surface of the fluid: to compare the pressures on the three triangles AEB, BEC, CED. If P, Q, R, represent the three pressures, P:Q:R:: 1:3 : 5. 14. A triangle, of which the area is A, is immersed in a fluid, the angular points being at depths h, l', h", below the surface of the fluid ; to find the pressure on the triangle. The pressure is equal to 3gp A (h + k + h"). 15. A rod, inclined at any angle to the horizon, is just immersed in a fluid ; to divide it into four parts, which shall be l equally pressed. Let a represent the length of the rod, and x, x', x", x'", the lengths of the four parts taken in order, beginning with the highest. Then x = ža, a' = }a (V2 - 1), a" = ja (V3 - V2), X'" = }a (2 - V3). a : 16. A cubical vessel, filled with fluid, is held so that one of its diagonals is vertical; to compare the pressures on one of its higher and one of its lower faces. If P be the pressure on a higher and Q on a lower face, then Q = P (2 V3 - 1). 17. A symmetrical pyramid is immersed in fluid with the surface of which its vertex just coincides; the axis of the pyramid is vertical: to find the pressure on the surface of the pyramid, exclusively of the base, the length of each of the n sides of its polygonal base being a, and the height of each of its triangular faces being h. The required pressure is equal to 3gpnah (ha- ja coť? ’ 18. An ellipse is placed with its axis vertical and the extremity of its axis major in the surface of a fluid; to compare the pressure on the ellipse with the pressure on the circle of curvature at its highest vertex. If P denote the former and Q the latter of these pressures, then, 2a, 2b, being the major and minor axes of the ellipse, P:Q:: a : . a 19. A circular area is immersed vertically in a fluid, and has a point of its circumference in the surface; the circular area being divided into two parts by a straight line drawn from its highest point at an angle of 45° to the horizon, to determine the ratio of the pressure on the larger to the pressure on the smaller portion. The required ratio is equal to 97 + 8 a 20. A quadrant of a circle ACB is divided into two sectors ACP, PCB, and is immersed in a fluid so that AC is coincident with the surface; to compare the areas of the two sectors when they both experience the same pressure. The area of the sector ACP must be double of that of the sector PCB. 21. A semicircular area is placed in a fluid with its vertex downwards, and its diameter coincident with the surface; to divide the area into n portions by horizontal ordinates, such that the pressures upon all the portions may be equal. If a be the radius of the circle, and y, the lower ordinate of the æth portion, the diameter of the semicircle being the higher ordinate of the first portion, then or the values of Yı, Y2, Yz ,. ... will be respectively 22. An elliptic area is immersed vertically in fluid, the major axis coinciding with the surface; to find the pressure on the area included between the arc and the line joining the extremities of two conjugate diameters. If a, b, denote the semi-axes of the ellipse, and a, b, the depths of the extremities of the two conjugate diameters, the required pressure will be equal to έπρα (α + β). 23. A parabola is immersed in a fluid with its axis vertical, its vertex A being at the surface; S being its focus and P a point in its arc, to determine the pressure on the area included between the straight lines SA, SP, and the parabolic arc AP. If SA = m and SP = r, the required pressure will be equal to i'sgp (rm – mo).(2r2 + rm + 2m%). If r = 2m, or if SP be the semi-latus-rectum, the pressure will be equal to gpm. 24. A cycloidal area is immersed in a fluid, so that the tangent at its vertex lies in the surface; to compare the pressure |