by the walls of this fourth bubble will alter the curvature of the surfaces of the others, so far as it encloses them; and, if all four bubbles be identical in size, these surfaces which formerly we called external and which have now come to be internal partitions, will, like the others, be flattened by equal and opposite pressure, into planes. We are now dealing, in short, {317} with six planes, meeting symmetrically in a point, and constituting there four equal solid angles. [Illustration: Fig. 120.] If we make a wire cage, in the form of a regular tetrahedron, and dip it into soap-solution, then when we withdraw it we see that to each one of the six edges of the tetrahedron, i.e. to each one of the six wires which constitute the little cage, a film has attached itself; and these six films meet internally at a point, and constitute in every respect the symmetrical figure which we have just been describing. In short, the system of films we have hereby automatically produced is precisely the system of partition-walls which exist in our tetrahedral aggregation of four spherical bubbles:—precisely the same, that is to say, in the neighbourhood of the meeting-point, and only differing in that we have made the wires of our tetrahedron straight, instead of imitating the circular arcs which actually form the intersections of our bubbles. This detail we can easily introduce in our wire model if we please. Let us look for a moment at the geometry of our figure. Let _o_ (Fig. 120) be the centre of the tetrahedron, i.e. the centre of symmetry where our films meet; and let _oa_, _ob_, _oc_, _od_, be lines drawn to the four corners of the tetrahedron. Produce _ao_ to meet the base in _p_; then _apd_ is a right-angled triangle. It is not difficult to prove that in such a figure, _o_ (the centre of gravity of the system) {318} lies just three-quarters of the way between an apex, _a_, and a point, _p_, which is the centre of gravity of the opposite base. Therefore _op_ = _oa_/3 = _od_/3. Therefore cos _dop_ = 1/3 and cos _aod_ = − 1/3. That is to say, the angle _aod_ is just, as nearly as possible, 109° 28′ 16″. This angle, then, of 109° 28′ 16″, or very nearly 109 degrees and a half, is the angle at which, in this and _every other solid system_ of liquid films, the edges of the partition-walls meet one another at a point. It is the fundamental angle in the solid geometry of our systems, just as 120° was the fundamental angle of symmetry so long as we considered only the plane projection, or plane section, of three films meeting in an edge. ―――――――――― Out of these two angles, we may construct a great variety of figures, plane and solid, which become all the more varied and complex when, by considering the case of unequal as well as equal cells, we admit curved (e.g. spherical) as well as plane boundary surfaces. Let us consider some examples and illustrations of these, beginning with those which we need only consider in reference to a plane.