The well-known method of differences for the analysis of plates of constant thickness, which until a few years ago was very laborious, deserves renewed attention because the advent of the electronic computer has removed this drawback. It is now also rif interest to extend the method to plates whose thickness changes abruptly at regular distances in two directions. Plates with apertures are a special case. It is shown that the difference equation for a plate of abruptly varying thickness, as derived in this paper, can also be used for establishing the conditions for arbitrary boundaries. This equation gives directly, i.e., without having to introduce "external" points as is usually done in the existing literature, a difference equation for the points at the edge and the points located at a distance of one mesh width therifrom. The theory is further elucidated with the aid of a worked example, and the results are verified by means of tests peiformed on a model by employing the moire method. Attention is called to the possibility of programming the entire procedure.