Folded rocks having bedding surfaces which are approximately parallel are said to lie in parallel folds. Utilizing the principle of evolutes and involutes, the author offers a more precise definition of parallel folds and points out inconsistencies in other concepts. With the idea of classifying parallel folds and possibly of deducing the mechanics of their formation, methods are presented for obtaining the differential equations of the families of involutes which, in certain cross sections, represent the traces of stratigraphic surfaces; and for obtaining the equations of the corresponding evolutes. Geometric methods are also given. These equations and geometric constructions may be deduced either from assumed structural postulates or from actual field data. Another part of the paper deals with the application of mean trigonometric functions to the measurement of thickness of strata, depth and distance to a stratum, and other stratigraphic dimensions in sections oblique to the strike of the rocks. This topic is considered under two headings: (1) where such measurements can be made from data collected at several stations along a line of traverse, and (2) where they must be made from a series of structural observations, considered in pairs. In the first case, no assumption is made regarding the curvature of the strata, but instead the mean values of the required functions are derived by mechanical integration. In the second case, the usual assumption of circular curvature is made, and the necessary functions are obtained by the use of definite integrals. Tables of the logarithms of these mean functions, with an increment of 5 degrees for the argument, are also presented. © 1940 Geological Society of America.