which gives 0, and therefore the complement of 0, or the inclination

sought.

684.

on the base is

Let r be the radius of the sphere; then the pressure

and the depth of its centre of gravity (Vince's Flux. p. 119,)

685. face is

Let be the radius of the interior; then the sur

specific gravity of the fluid, the internal pressure of

P = 4πr xr x s = 4πr3s.

Also the weight of the fluid is

686. Let be the radius of the inner sphere; then if y be the radius of any section parallel to the base, and a -x (a being the altitude of the fluid,) its depth in the fluid, the pressure upon its circumference surface is measured by (Vince, Prop. VII.)

and resolving this, the pressure vertically upon this section is y Xs= (›13 —x2) (a−x)

2xу (a−x) × 1

r

Hence the whole vertical pressure is

P =

2π

2π

· ƒ (r2 — x2) (a — x) ds = 2% [√] (r2 —x2) (a—x) dx

2π

.. P = awx √ (x2 − x2) + 2TM (r2 — a2)3+rar sin.~~ + C

3

Hence when x = 0; the whole vertical pressure of the fluid whose altitude is a (S being the specific gravity of the fluid) is

Again, if S' denote the specific gravity of the vessel, its solid part being in magnitude

which, by the question, being equated to Q, will give

In the problem a =

2π

:: Q = 1 × (8−6√√3—*) ×S= 2= { (r+t)—r3} xS

'12

3

687. The surface pressed is the same in both cases; . the pressures will be as the depths of the centres of gravity. Now the distance of the centre of gravity of the surface from the vertex

688.

Let a be the axis of the paraboloid, and 8 the radius of its base, s its specific gravity, and s' that of the fluid; then the part immersed is (see Vince, Prop. XVI.)

Hence, if a denote the depth of the axis immersed, and p the parameter of the generating parabola, we have

Again, the distance of the centre of gravity of the whole solid, and of the part immersed from the vertex, are respectively (Vince's Fluxions, p. 117,)

is

Again, the area of the section made by the surface of the water

πy = πρι = πρα

x

and if du denote an element of this area, and a denote the distance of this particle from the diameter about which the solid will revolve (if at all); then by Poisson's Mech. vol. II. p. 416, the body will be stable, unstable, or in a state of indifference, according as

fx2 du d x Q

is positive, negative, or zero, fx du being taken through the whole extent of the area ра

S

Now if generally R be the radius of this section, it is easily found that

Hence the equilibrium is stable, unstable, or indifferent, accord

which will shew the state of the equilibrium when p, a and

are given in numbers.

When the paraboloid tends to fall, a must be increased to a', so that

689.

Let ABC (Fig. 104) be a vertical section, parallel to the base of the horizin, and the B being immersed, let DE be the surface of the water, and suppose E the given point which meets the water; also if AC, DE, be bisected in F, H, and BF, BH be trisected in G, g; G, g, will be the centres of gravity of the whole section ABC, and of the part immersed DBE. Now the condition of equilibrium of the prism is evidently the same for the prism as for this section; it is therefore that Gg be vertical or ↓ DE.

Since in the same fluid the volume of the part immersed is constant, the area of DBE is constant, and the locus of the points H, G is an hyperbola (see Problem 44, vol. II.); to find whose equation, let