The principle of the Triangle of Forces is expanded, as follows, by an old seventeenth-century theorem, called Lami’s Theorem: “If three forces acting at a point be in equilibrium, each force is proportional to the sine of the angle contained between the directions of the other two.” That is to say
P : Q : R : = sin QOR : sin POR : sin POQ.
or
P ⁄ sin QOR = Q ⁄ sin ROP = R ⁄ sin POQ.
And from this, in turn, we derive the equivalent formulae, by which each force is expressed in terms of the other two, and of the angle between them:
P2 = Q2 + R2 + 2 QR cos(QOR), etc.
From this and the foregoing, we learn the following important and useful deductions:
- (1) The three forces can only be in equilibrium when any one of them is less than the sum of the other two: for otherwise, the triangle is impossible. Now in the case of a drop of olive-oil upon a clean water surface, the relative magnitudes of the three tensions (at 15° C.) have been determined as follows:
- Water-air surface
- 75
- Oil-air surface
- 32
- Oil-water surface
- 21
- No triangle having sides of these relative magnitudes is possible; and no such drop therefore can remain in equilibrium. {297}
- (2) The three surfaces may be all alike: as when a soap-bubble floats upon soapy water, or when two soap-bubbles are joined together, on either side of a partition-film. In this case, the three tensions are all equal, and therefore the three angles are all equal; that is to say, when three similar liquid surfaces meet together, they always do so at an angle of 120°. Whether our two conjoined soap-bubbles be equal or unequal, this is still the invariable rule; because the specific tension of a particular surface is unaffected by any changes of magnitude or form.
- (3) If two only of the surfaces be alike, then two of the angles will be alike, and the other will be unlike; and this last will be the difference between 360° and the sum of the other two. A particular case is when a film is stretched between solid and parallel walls, like a soap-film within a cylindrical tube. Here, so long as there is no external pressure applied to either side, so long as both ends of the tube are open or closed, the angles on either side of the film will be equal, that is to say the film will set itself at right angles to the sides.
- Many years ago Sachs laid it down as a principle, which has become celebrated in botany under the name of Sachs’s Rule, that one cell-wall always tends to set itself at right angles to another cell-wall. This rule applies to the case which we have just illustrated; and such validity as the rule possesses is due to the fact that among plant-tissues it very frequently happens that one cell-wall has become solid and rigid before another and later partition-wall is developed in connection with it.
- (4) There is another important principle which arises not out of our equations but out of the general considerations by which we were led to them. We have seen that, at and near the point of contact between our several surfaces, there is a continued balance of forces, carried, so to speak, across the interval; in other words, there is physical continuity between one surface and another. It follows necessarily from this that the surfaces merge one into another by a continuous curve. Whatever be the form of our surfaces and whatever the angle between them, this small intervening surface, approximately spherical, is always there to bridge over the line of contact[342]; and this little fillet, or “bourrelet,” {298} as Plateau called it, is large enough to be a common and conspicuous feature in the microscopy of tissues (Fig. [102]). For instance, the so-called “splitting” of the cell-wall, which is conspicuous at the angles of the large “parenchymatous” cells in the succulent tissues of all higher plants (Fig. [103]), is nothing more than a manifestation of Plateau’s “bourrelet,” or surface of continuity[343].
| Water-air surface | 75 |
| Oil-air surface | 32 |
| Oil-water surface | 21 |