From the theory of resistance of profiles we know that the resistance of a line to exterior normal forces is directly proportional to its degree of curvature, and consequently inversely proportional to the radius of the curve. Hence the sectional profile of a tunnel excavated through hard rock, where there are no lateral pressures owing to the great cohesion of the material, and having to resist only the vertical pressure, should be designed to offer the greatest resistance at its highest point, and the curve must, therefore, be sharper there, and may decrease toward the base. In quicksand, mud, or other material practically without cohesion, the pressures will all be normal to the line of the profile, and a circular section is the one best suited to resist them. These theoretical considerations have been proved correct by actual experience, and they may be employed to determine in a general way the form of section to be adopted. Applying them to very hard rock, they give us a section with an arched roof and vertical side walls. In softer materials they give us an elliptical section with its major axis vertical, and in very soft quicksands and mud they give us the circular section. These three forms of cross-section and their modifications are the ones commonly employed for tunnels. An important exception to this general practice, however, is met with in some of the city underground rapid-transit railways built of late years, where a rectangular or box section is employed. These tunnels are usually of small depth, so that the vertical pressures are comparatively light, and the bending strains, which they exert upon the flat roof, are provided for by employing steel girders to form the roof lining. [Illustration: FIG. 7.--Diagram of Polycentric Sectional Profile.]