squeezed into hexagonal prisms, while in the latter they retain their cylindrical form. Where the polypes are comparatively few, and so have room to spread, the mutual pressure ceases to work or only tends to push them asunder, letting them remain circular in outline (e.g. Thecosmilia). Where they vary gradually in size, as for instance in _Cyathophyllum hexagonum_, they are more or less hexagonal but are not regular hexagons; and where there is greater and more irregular variation in size, the cells will be _on the average_ hexagonal, but some will have fewer and some more sides than six, as in the annexed figure of Arachnophyllum (Fig. 129). {326} Where larger and smaller cells, corresponding to two different kinds of zooids, are mixed together, we may get various results. If the larger cells are numerous enough to be more or less in contact with one another (e.g. various Monticuliporae) they will be irregular hexagons, while the smaller cells between them will be crushed into all manner of irregular angular forms. If on the other hand the large cells are comparatively few and are large and strong-walled compared with their smaller neighbours, then the latter alone will be squeezed into hexagons, while the larger ones will tend to retain their circular outline undisturbed (e.g. Heliopora, Heliolites, etc.). [Illustration: Fig. 129. _Arachnophyllum pentagonum._ (After Nicholson.)] [Illustration: Fig. 130. _Heliolites._ (After Woods.)] When, as happens in certain corals, the peripheral walls or “thecae” of the individual polypes remain undeveloped but the radiating septa are formed and calcified, then we obtain new and beautiful mathematical configurations (Fig. 131). For the radiating septa are no longer confined to the circular or hexagonal bounds of a polypite, but tend to meet and become confluent with their neighbours on every side; and, tending to assume positions of equilibrium, or of minimal area, under the restraints to which they are subject, they fall into congruent curves; and these correspond, in a striking manner, to the lines of force running, in a common field of force, between a number of secondary centres. Similar patterns may be produced in various ways, by the play of osmotic or magnetic forces; and a particular and very curious case is to be found in those complicated forms of nuclear division {327} known as triasters, polyasters, etc., whose relation to a field of force Hartog has explained[367]. It is obvious that, in our corals, these curving septa are all orthogonal to the non-existent hexagonal boundaries. As the phenomenon is wholly due to the imperfect development or non-existence of a thecal wall, it is not surprising that we find identical configurations among various corals, or families of corals, not otherwise related to one another; we find the same or very similar patterns displayed, for instance, in Synhelia (_Oculinidae_), in Phillipsastraea (_Rugosa_), in Thamnastraea (_Fungida_), and in many more.