This paper deals with an on-line signature verification system. It is assumed taht the system requires a person approaching it to declare his name and to write his signature. The system compares the written signature with reference signatures registered in advance and admits his access if the dissimilarity is below a threshold. New ideas in this paper on the design of such a system are to construct an effective dissimilarity measure by including the direction of pen movement, to select a few representative signatures from the reference set by using a clustering procedure, and to decide the threshold as the maximum of dissimilarity measures among reference signatures multiplied by a constant. The effectiveness of the designed system is examined experimentally using a database provided by CADIX Co., Ltd. It consists of 2203 writings for 28 signatures which contain 10 in Latin letters (English alphabet) by foreigners, 14 in Japanese letters by Japanese, and 4 in Latin letters by Japanese. It comprises genuine signatures and forgeries. As a result of the experiment, the system turned out to work highly satisfactory: it achieved about 99% of an average correct verification rate. The inclusion of the direction of pen movement into the dessimilarity measure leads to about 3% of an average increase in the correct verification rate.