Abstract : The present paper may be considered as a sequel to our previous paper in the Proceedings of the Cambridge Philosophical Society, Theorie generale de noyaux reproduisants-Premiere partie (vol. 39 (1944)) which was written in 1942-1943. In the introduction to this paper we outlined the plan of papers which were to follow. In the meantime, however, the general theory has been developed in many directions, and our original plans have had to be changed. Due to wartime conditions we were not able, at the time of writing the first paper, to take into account all the earlier investigations which, although sometimes of quite a different character, were, nevertheless, related to our subject. Our investigation is concerned with kernels of a special type which have been used under different names and in different ways in many domains of mathematical research. We shall therefore begin our present paper with a short historical introduction in which we shall attempt to indicate the different manners in which these kernels have been used by various investigators, and to clarify the terminology. We shall also discuss the more important trends of the application of these kernels without attempting, however, a complete bibliography of the subject matter. (KAR) P. 2