A class of statistics, large enough to comprise those used in all the known distribution-free tests of fit for continuous distribution functions, is characterized by a structure called "structure ($d$)." A number of statistics of this class may be constructed and used for tests of fit. To make a reasonable choice among all these statistics, it appears desirable to introduce in the space of continuous distribution functions a distance which would reflect the type of discrepancy the proposed test is intended to detect. By studying the power of various statistics with regard to this distance one may then be able to choose those with optimal properties.