size of the spheres, and which will be plane if the latter are all of the same dimensions; but whether plane or curved, the three partitions will meet one another at an angle of 120°, in an axial line. Various pretty geometrical corollaries accompany this arrangement. For instance, if Fig. [114] represent the three associated bubbles in a plane drawn through their centres, c, c′, c″ (or what is the same thing, if it represent the base of three bubbles resting on a plane), then the lines uc, uc″, or sc, sc′, etc., drawn to the {308} centres from the points of intersection of the circular arcs, will always enclose an angle of 60°. Again (Fig. [115]), if we make the angle c″uf equal to 60°, and produce uf to meet cc″ in f, f will be the centre of the circular arc which constitutes the partition Ou; and further, the three points f, g, h, successively determined in this
Fig. 115.
manner, will lie on one and the same straight line. In the case of coequal bubbles or cells (as in Fig. [114], B), it is obvious that the lines joining their centres form an equilateral triangle; and consequently, that the centre of each circle (or sphere) lies on the circumference of the other two; it is also obvious that uf is now {309} parallel to cc″, and accordingly that the centre of curvature of the partition is now infinitely distant, or (as we have already said), that the partition itself is plane.
When we have four bubbles in conjunction, they would seem to be capable of arrangement in two symmetrical ways: either, as in Fig. [116] (A), with the four partition-walls meeting at right angles, or, as in (B), with five partitions meeting, three and three, at angles of 120°. This latter arrangement is strictly analogous to the arrangement of three bubbles in Fig. [114]. Now, though both of these figures, from their symmetry, are apparently figures of equilibrium, yet, physically, the former turns out to be of unstable
Fig. 116.
and the latter of stable equilibrium. If we try to bring our four bubbles into the form of Fig. [116], A, such an arrangement endures only for an instant; the partitions glide upon each other, a median wall springs into existence, and the system at once assumes the form of our second figure (B). This is a direct consequence of the law of minimal areas: for it can be shewn, by somewhat difficult mathematics (as was first done by Lamarle), that, in dividing a closed space into a given number of chambers by means of partition-walls, the least possible area of these partition-walls, taken together, can only be attained when they meet together in groups of three, at equal angles, that is to say at angles of 120°. {310}
Wherever we have a true cellular complex, an arrangement of cells in actual physical contact by means of a boundary film, we find this general principle in force; we must only bear in mind that, for its perfect recognition, we must be able to view the object in a plane at right angles to the boundary walls. For instance, in any ordinary section of a vegetable parenchyma, we recognise the appearance of a “froth,” precisely resembling that which we can construct by imprisoning a mass of soap-bubbles in a narrow vessel with flat sides of glass; in both cases we see the cell-walls everywhere meeting, by threes, at angles of 120°, irrespective of the size of the individual cells: whose relative size, on the other hand, determines the curvature of the partition-walls. On the surface of a honey-comb we have precisely the same conjunction, between cell and cell, of three boundary walls, meeting at 120°. In embryology, when we examine a segmenting egg, of four (or more) segments, we find in like manner, in the great majority of cases, if not in all, that the same principle is still exemplified; the four segments do not meet in a common centre, but each cell is in contact with two others, and the three, and only three, common boundary walls meet at the normal angle of 120°. A so-called polar furrow[358], the visible edge of a vertical partition-wall, joins (or separates) the two triple contacts, precisely as in Fig. [116], B.
In the four-celled stage of the frog’s egg, Rauber (an exceptionally careful observer) shews us three alternative modes in which the four cells may be found to be conjoined (Fig. [117]). In (A) we have the commonest arrangement, which is that which we have just studied and found to be the simplest theoretical one; that namely where a straight “polar furrow” intervenes, and where, at its extremities, the partition-walls are conjoined three by three. In (B), we have again a polar furrow, which is now seen to be a portion of the first “segmentation-furrow” (cf. Fig. [155] etc.) by which the egg was originally divided into two; the four-celled stage being reached by the appearance of the transverse furrows {311} and their corresponding partitions. In this case, the polar furrow is seen to be sinuously curved, and Rauber tells us that its curvature gradually alters: as a matter of fact, it (or rather the partition-wall corresponding to it) is gradually setting itself into a position of equilibrium, that is to say of equiangular contact with its neighbours, which position of equilibrium is already attained or nearly so in Fig. [117], A. In Fig. [117], C, we have a very different condition, with which we shall deal in a moment.