The following work is, with some modifications, the same as that of which an abstract was presented at the summer meeting of the American Mathematical Society in 1899. With regard to the imprimitive groups of degree fifteen, which form the subject matter of the first part of this paper, it should be stated that I have added two new groups to the list as originally presented, namely, the groups with five systems of imprimitivity simply isomorphic to the alternating and symmetric groups of degree 5, and that Dr. Kuhn reported at the February meeting of the Society, 1900, that he had carried the investigation further, adding 28 to the 70 groups that I succeeded in finding.
In the second part of this paper the determination of the primitive groups of degree 18 depends to a great extent upon the lists of transitive groups of lower degrees already determined. Any new discovery of groups of degree less than 18 would necessitate an examination of such groups to determine whether they can be combined with others in such a way as to generate a primitive group of degree 18. This list, therefore, cannot claim to be absolutely complete, since omissions are always possible.