We may now illustrate some of the foregoing principles, before we proceed to the more complex cases in which more bodies than three are in mutual contact. But in doing so, we must constantly bear in mind the principles set forth in our chapter on the forms of cells, and especially those relating to the pressure exercised by a curved film.
| Fig. 102. (After Berthold.) | Fig. 103. Parenchyma of Maize. |
Let us look for a moment at the case presented by the partition-wall in a double soap-bubble. As we have just seen, the three films in contact (viz. the outer walls of the two bubbles and the partition-wall between) being all composed of the same substance {299} and all alike in contact with air, the three surface tensions must be equal; and the three films must therefore, in all cases, meet at an angle of 120°. But, unless the two bubbles be of precisely equal size (and therefore of equal curvature) it is obvious that the tangents to the spheres will not meet the plane of their circle of contact at equal angles, and therefore that the partition-wall must be a curved surface: it is only plane when it divides two equal and symmetrical cells. It is also obvious, from the symmetry of the figure, that the centres of the spheres, the centre of the partition, and the centres of the two spherical surfaces are all on one and the same straight line.
Fig. 104.
Now the surfaces of the two bubbles exert a pressure inwards which is inversely proportional to their radii: that is to say p : p′ :: 1 ⁄ r′ : 1 ⁄ r; and the partition wall must, for equilibrium, exert a pressure (P) which is equal to the difference between these two pressures, that is to say, P = 1 ⁄ R = 1 ⁄ r′ − 1 ⁄ r = (r − r′) ⁄ r r′. It follows that the curvature of the partition wall must be just such a curvature as is capable of exerting this pressure, that is to say, R = r r′ ⁄ (r − r′). The partition wall, then, is always a portion of a spherical surface, whose radius is equal to the product, divided by the difference, of the radii of the two vesicles. It follows at once from this that if the two bubbles be equal, the radius of curvature of the partition is infinitely great, that is to say the partition is (as we have already seen) a plane surface.
The geometrical construction by which we obtain the position of the centres of the two spheres and also of the partition surface is very simple, always provided that the surface tensions are uniform throughout the system. If p be a point of contact between the two spheres, and cp be a radius of one of them, then make the angle cpm = 60°, and mark off on pm, pc′ equal to the {300} radius of the other sphere; in like manner, make the angle c′pn = 60°, cutting the line cc′ in c″; then c′ will be the centre of the second sphere, and c″ that of the spherical partition.
