"Selecting the z-axis as an axis of revolution, a point on the surface generated by rotating the curve r = f(z) is defined by two coordinates... z and \theta. ...Now ds^2 = ds_1^2 + ds_2^2 where ds_1 is the displacement along the meridian and ds_2 the displacement along the parallel of latitude. ...since ds_1^2 = dz^2 + dr^2 ...The [arbitrary] line element ds is... defined by the relation {{center|1=ds_1 = dz\sqrt{1 + (\frac{dr}{dz})^2}}}and The line element ds is thus defined by the relation:{{center|1=ds^2 = dz^2[1 + (\frac{dr}{dz})^2] + r^2d\theta^2 = A^2dz^2 + B^2d\theta^2 \qquad (1.1)}}where{{center|1=A = \sqrt{1 + (\frac{dr}{dz})^2} \quad and \; B = r \qquad \qquad (1.2)}}This is the first of the generalized forms of equations in curved surface theory in which A and B are parameters. ... For a generalized curved surface with an arbitrarily selected orthoganal coordinate system defined by the coordinates \alpha and \beta, eq. (1.1) assumes the generalized form...the coefficients will now be functions of \alpha and \beta. We may again write:{{center|1=ds_1 = Ad\alpha \quad \text{for} \quad \beta = c_1 ds_2 = Bd\beta \quad \text{for} \quad \alpha = c_2}}Equations (1.1) and (1.3) are of great importance in the theory of curved surfaces and hence in comprehending shell theory. By means of these equations the geometry of the surface is described as a two-dimensional configuration similar to the method used to define a point on a flat surface, i.e. ...by two normalized orthogonal coordinates. ...If a set of orthogonal coordinates can be selected such that A and B are independent of \alpha and \beta, the geometry in the neighborhood of a point on the curved surface does not differ from that of a flat plate. Then the cartesian-coordinate relationship:is still valid. This classification includes the s such as the cone and the cylinder. ...the distance between two points on the surface does not change in the development. For that reason, when a curved surface defined by the generalized equation, eq. (1.3), can be reduced by using a suitable set of coordinates \alpha and \beta to the form of eq. (1.4) with A and B constant, the so-called conditions of euclidean geometry will be satisfied. ...When it becomes impossible to select \alpha and \beta coordinates for which A and B are constant, the geometry of the curved surface becomes different from that of a flat surface... eq. (1.4), is no longer valid and a non-euclidean geometry must be applied. Such surfaces are not developable, i.e. they cannot be folded out into a flat surface under the condition that any line element ds remains invariant. This class of surfaces includes the , the , the and the hyperboloid."