In the last paragraph but one, I have spoken of the difficulty of so arranging the meeting-places of a number of cells that at each junction only three cell-walls shall meet in a line, and all three shall meet it at equal angles of 120°. As a matter of fact, the problem is soluble in a number of ways; that is to say, when we have a number of cells, say eight as in the case considered, enclosed in a common boundary, there are various ways in which their walls can be made to meet internally, three by three, at equal angles; and these differences will entail differences also in the curvature of the walls, and consequently in the shape of the cells. The question is somewhat complex; it has been dealt with by Plateau, and treated mathematically by M. Van Rees[392]. [Illustration: Fig. 158. Various possible arrangements of intermediate partitions, in groups of 4, 5, 6, 7 or 8 cells.] If within our boundary we have three cells all meeting {375} internally, they must meet in a point; furthermore, they tend to do so at equal angles of 120°, and there is an end of the matter. If we have four cells, then, as we have already seen, the conditions are satisfied by interposing a little intermediate wall, the two extremities of which constitute the meeting-points of three cells each, and the upper edge of which marks the “polar furrow.” Similarly, in the case of five cells, we require _two_ little intermediate walls, and two polar furrows; and we soon arrive at the rule that, for _n_ cells, we require _n_ − 3 little longitudinal partitions (and corresponding polar furrows), connecting the triple junctions of the cells; and these little walls, like all the rest within the system, must be inclined to one another at angles of 120°. Where we have only one such wall (as in the case of four cells), or only two (as in the case of five cells), there is no room for ambiguity. But where we have three little connecting-walls, as in the case of six cells, it is obvious that we can arrange them in three different ways, as in the annexed Fig. 159. In the system of seven cells, the four partitions can be arranged in four ways; and the five partitions required in the case of eight cells can be arranged in no less than thirteen different ways, of which Fig. 158 shews some half-dozen only. It does not follow that, so to speak, these various {376} arrangements are all equally good; some are known to be much more stable than others, and some have never yet been realised in actual experiment. The conditions which lead to the presence of any one of them, in preference to another, are as yet, so far as I am aware, undetermined, but to this point we shall return. ――――――――――