If the film be removed from the reservoir, and if o denote subsequently the mass of unit of area, prove that

7. Any number of soap-bubbles are blown from the same liquid and then allowed to combine with one another. Find the radius of the resulting bubble, and prove that the decrease of surface bears a constant ratio to the increase of volume.

8. The surface tension of water exposed to air is such that the stress across an inch is equal to the weight of about 3.3 grains. If 1,000,000,000 spherical drops combine to form a single spherical rain-drop inch in diameter, shew that the work done by the surface tensions is equal to about 0001277 foot-pounds.

9. If a film under unequal and external pressure form a surface of revolution, prove that the inclination of the tangent plane at P to the axis is given by the equation

x being the perpendicular from P on the axis and a, b constants.

10. A drop of liquid with uniform surface-tension is made to revolve about an axis. Prove that the meridian curve of the surface will be the roulette of the pole of the

11. Two soap-bubbles are in contact; if r,, r,, be the radii of the outer surfaces, and r the radius of the circle in which the three surfaces intersect,

12. If a frame of fine straight wire in the form of a tetrahedron be lowered into a solution of soap and water and drawn up again, there are found in certain cases plane films starting from the edges and meeting in a point. Shew that this is not a possible form of equilibrium for every tetrahedron, and that it is so if one face be an equilateral triangle and the others isosceles triangles, whose vertical angles are each less than sec(-3).

13. If water be introduced between two parallel plates of glass, at a very small distance d from each other, prove that the plates are pulled together with a force equal to

A being the area of the film and B its periphery.

14. A hollow right circular cone of glass is placed with its axis vertical and vertex upwards in homogeneous liquid. Find the height to which the liquid will be raised in the cone, and write down the differential equation of the surface inside. Deduce results for a cylinder.

15. A needle floats on water with its axis in the natural level of the surface; if σ be the specific gravity of steel referred to water, ẞ the angle of capillarity, and 2a the angle subtended at the axis by the arc of a cross-section in contact with the water, prove that

16. A capillary tube in the form of a surface of revolution is partly immersed in a liquid with its axis vertical. Find the equation of the generating curve if the liquid is in equilibrium at whatever height it stands in the tube.

17. A soap-bubble is filled with a mass m of a gas whose pressure is kx (its density) at the temperature considered. The radius of the bubble is a, when it is first placed in air. The barometer then rises, the temperature remaining unaltered. Shew that the radius of the bubble increases or

B. H.

15

diminishes according as the tension of the film is greater or

represents a possible form of a liquid film, the pressure on both sides being the same. (Catalan.)

19. If two needles floating on water be placed symmetrically parallel to each other, shew that they will be apparently attracted to each other, and that this is due to the surface tension.

20. A small cube floats with its upper face horizontal, in a liquid such that its angle of contact with the surface of the cube is obtuse and equal to π

α.

If p is the density of the liquid, and σ of the cube, and if gpc2 is the surface tension, prove that the cube will float if

21. Two equal circular discs of radius a are placed with their planes perpendicular to the line which joins their centres, and their edges are connected by a soap film which encloses a mass of air that would be just sufficient in the same atmosphere to fill a spherical soap-bubble of radius c. If the film be cylindrical when the distance between the discs is b, prove that in order that it may become spherical the distance between the discs must be lessened to 2z where

≈ (3a2 + 22o) {8c2

3ab +

6a2b-8c3)

√a+z2

=

6abc2 (2a — c).

22. A framework of wires forms a prism of height b, the bases being equilateral triangles of side a. If the framework is dipped into soapy water, describe the arrangement of plane films in the state of equilibrium. Prove that for equilibrium to be possible with plane films b must be greater than a/√6.

23. A film of fluid adheres to two wires each of which forms one turn of a helix, the axes of the two helices being coincident, and their steps equal. Shew that the condition of equilibrium of the film will be satisfied if the differential equation to any section of the film through the axis is of the form

when 2πα = step of either helix: (i.e. distance between consecutive threads).

24. To the extremities of the axis of a wire helix of pitch b, whose length is very great compared with its diameter, an elastic string (modulus of elasticity E) is fastened, the wire being bent over radially at each end so as to meet the axis. The string when straight is tight but unstretched. If the helix and string be dipped into a solution of soap and then removed with a film adhering to the wire and string, shew that, except near the ends, the string will be drawn into a helix of radius r where r is given by the equation

(16π1h3T2 – 64π°E2) r2 + 32π1h2TEr3 + 8π2h*T2p2

+8m2 TEr+ hoT2 = 0,

T representing the whole tension per unit of length (of both surfaces) of a soap-film.

CHAPTER XI.

THE EQUILIBRIUM OF REVOLVING LIQUID, THE PARTICLES OF WHICH ARE MUTUALLY ATTRACTIVE.

171. IF a liquid mass, the particles of which attract each other according to a definite law, revolve uniformly about a fixed axis, it is conceivable that, for a certain form of the free surface, the liquid particles may be in a state of relative equilibrium; since, however, the resultant attraction of the mass upon any particle depends in general upon its form, which is unknown, a complete solution of the problem cannot be obtained.

For any arbitrarily assigned law of attraction, the question is one of purely abstract interest, and it is only when the law is that of gravitation that it becomes of importance, from its relation to one of the problems of physical astronomy.

We shall consider the fluid homogeneous, and confine our attention to two cases; in the first of these the attractive forces are supposed to vary directly as the distance, and, in the second, to follow the Newtonian law.

172. A homogeneous liquid mass, the particles of which attract each other with a force varying directly as the distance, rotates uniformly about an axis through its centre of gravity; required to determine the form of the free surface.

The resultant attraction on any particle is in the direction of, and proportional to, the distance of the particle from the