But before we go further in this enquiry, it will be necessary to consider, to some small extent at least, the _curvatures_ of the six different surfaces, that is to say, to determine what modification {221} is required, in each case, of the general equation which applies to them all. We shall find that with this question is closely connected the question of the _pressures_ exercised by, or impinging on the film, and also the very important question of the limitations which, from the nature of the case, exist to prevent the extension of certain of the figures beyond certain bounds. The whole subject is mathematical, and we shall only deal with it in the most elementary way. We have seen that, in our general formula, the expression 1/_R_ + 1/_R′_ = _C_, a constant; and that this is, in all cases, the condition of our surface being one of minimal area. In other words, it is always true for one and all of the six surfaces which we have to consider. But the constant _C_ may have any value, positive, negative, or nil. In the case of the plane, where _R_ and _R′_ are both infinite, it is obvious that 1/_R_ + 1/_R′_ = 0. The expression therefore vanishes, and our dynamical equation of equilibrium becomes _P_ = _p_. In short, we can only have a plane film, or we shall only find a plane surface in our cell, when on either side thereof we have equal pressures or no pressure at all. A simple case is the plane partition between two equal and similar cells, as in a filament of spirogyra. In the case of the sphere, the radii are all equal, _R_ = _R′_; they are also positive, and _T_ (1/_R_ + 1/_R′_), or 2 _T_/_R_, is a positive quantity, involving a positive pressure _P_, on the other side of the equation. In the cylinder, one radius of curvature has the finite and positive value _R_; but the other is infinite. Our formula becomes _T_/_R_, to which corresponds a positive pressure _P_, supplied by the surface-tension as in the case of the sphere, but evidently of just half the magnitude developed in the latter case for a given value of the radius _R_. The catenoid has the remarkable property that its curvature in one direction is precisely equal and opposite to its curvature in the other, this property holding good for all points of the surface. That is to say, _R_ = −_R′_; and the expression becomes (1/_R_ + 1/_R′_) = (1/_R_ − 1/_R_) = 0; in other words, the surface, as in the case of the plane, has _no {222} curvature_, and exercises no pressure. There are no other surfaces, save these two, which share this remarkable property; and it follows, as a simple corollary, that we may expect at times to have the catenoid and the plane coexisting, as parts of one and the same boundary system; just as, in a cylindrical drop or cell, the cylinder is capped by portions of spheres, such that the cylindrical and spherical portions of the wall exert equal positive pressures.