Dupre has described an arrangement by which the surface tension of a liquid film may be illustrated. A piece of sheet metal is cut out in the form AA' (fig. 2). A very fine slip of metal is laid on it in the position BB' and the whole is dipped into a solution of soap. When it is taken out the rectangle AA'CC' is filled by a liquid film. This film tends to contract on itself and the loose strip of metal will, if let go, be drawn towards AA'. If S is the area of one face of the film the potential energy is oS. If AA' =b and AC= a it is equal to oab. Hence if F is the force by which the slip is pulled towards A A', F = (a ab) = a so that db the force per unit length for one face is a. Hence, a is either the force per unit length or the potential energy per unit area.
It must be added that we have only considered one of the two faces and the force due to it. There is an equal force on the strip due to the second surface so that the total force is twice as great as the value taken. But the total area is twice as great also so that the conclusion remains unchanged.
There are other ways of illustrating the existence of the tension. Form a film as before but in a fixed frame. Tie a short length of spider line or fine unspun silk so as to form a flexible ring. If it is placed carefully on the film it usually takes an irregular shape. If the film inside the ring is destroyed by piercing it with a hot wire the spider line opens out into a circular form, this being the form which makes the surrounding surface least for a given length of line. Again, a small drop of mercury or a falling rain drop is practically spherical—the sphere being the form which has the smallest area for a given volume. A large drop fails to be spherical because gravitational forces are then strong enough to compete successfully with surface forces. Sometimes a difficulty is felt in connection with the assertion that the surface tension is constant. Take a film such as we considered and place it vertically. The upward force on AA'=2aa for the two faces, the downward force at the level CC' is also 2aa. The remaining force on the portion of the film between these lines is its weight. Hence the resultant force on the mass of the film is its weight and its acceleration should there fore be that of a freely falling body. Yet whatever may be found true for a pure substance, it cannot be true in general that the tension is the same at all parts of a large area. In the case of a soap film some adjustment must automatically take place (either by alteration of concen tration or otherwise) which makes the upper tension a little greater than the lower. It is probably the power of such an adjustment being made in the case of a soap solution which is the leading requisite permitting a durable film to be made. Other
wise the film would literally fall to pieces.
plained by assuming the existence of forces between elements (i.e., infinitesimal volumes) of matter can be illustrated by supposing the substance of a body to be distributed continuously instead of partitioned into molecules. Such was the mode in which Laplace, more than a century ago, regarded it ; and indeed it is the assump tion that has always been made until recent times regarding the distribution of the substance of a body.
Let the force between two elementary volumes be supposed proportional to the product of the volumes and inversely as the nth power of their distances apart. The force required is that across unit area drawn anywhere well inside the body arising from the volumes on each side of it.
The calculation is made in suc cessive stages.
(i.) Take a lamina of thick ness dy. The force on an elementary volume v due to a zone of 27rxdx a radius x and width dx is v dy since the tangential comr" r ponents will cancel each other. Hence the whole lamina at tracts the element with a force (iii.) The force (per unit area) on such a semi-infinite body due to a semi-infinite body in contact with it is obtained by in tegrating with respect to y from y = o to y = cp.. This force is the cohesion per unit area and is given by The unit of force used is that between two unit volumes separated by unit distance.
Lord Rayleigh remarks : "The pressure will therefore be in finite whatever n may be. If n-4 be negative the attraction of infinitely distant parts contributes to the result; while if n-4 be positive the parts in immediate contiguity act with infinite power. For the transition case, discussed by W. Sutherland (Phil. Mag. xxiv. p. 113, 1887) of 15.= 4, K is also infinite." He adds "It seems therefore that nothing satisfactory can be arrived at under this head." In more recent times, however, it has been realized (par ticularly by the Dutch school of physicists) that to treat the body as a continuum is bound to give erroneous results. Every sub stance is known to be built up of molecules. If we assume the existence of attractions depending on the distance between the centres of the molecules it is at the same time necessary to recog nize that the two halves of the body can never approach to absolute contact. The lower limit in the last integration should be the least distance of separation between the lines of centres of contiguous layers. This will be a magnitude comparable with the molecular diameter, say s. The value we seek then becomes There is now no objection on the above grounds to the power taw provided that n> 4.