Eight examples are given to indicate the desirability of having a general theory treating sets of curves which can be transformed into parallel curves. With this theory it is shown that a set of curves can be rendered parallel if and only if every pair of functions in the completion of a particular group associated with the curves is uncrossed. Under general conditions any two transformations rendering a set of curves parallel are related by a linear transformation. Methods for calculating transformations are proposed. Several structural properties of sets of curves which can be rendered parallel are proven.
