Infinite discontinuous groups of birational transformations which leave certain surfaces invariant

It was shown by SCHWARZ t that no algehraic curve of genus greater than unity can remain invariatlt under a continuous group of birational transforma-tions. Later HURWITZ t showed that no such curve could belong to any birational group of infinite order. The corresponding theory for surfaces is by no means complete. WhileA those belonging to continuous groups have been determined, only a few isolated examples are known of surfaces having an infinite discontinuous group. § All the groups which have been discussed are illustrations of two principles, the first of which refers to quartic surfaces and will be considered in two parts, the latter including the former as a particular case; the second principle is appli-cable to a much wider category. I propose to discuss these principles and apply the second to the determination of an extended family of new surfaces having an infinite group.