1. _Definition of the Radiolaria._--RADIOLARIA are marine Rhizopoda, whose unicellular body always consists of two main portions, separated by a membrane; an inner _Central capsule_ (with one or more nuclei) and an _Extracapsulum_ (the external calymma, which has no nucleus, and the pseudopodia); the endoplasm of the former and the exoplasm of the latter are connected by openings in the capsule-membrane. The central capsule is partly the general central organ of the Radiolarian cell, partly the special organ of reproduction, since its intracapsular protoplasm, along with the nuclei embedded in it, serves for the formation of flagellate spores. The extracapsulum is partly the general organ for intercourse with the outer world (by means of the pseudopodia), partly the special organ of protection (calymma) and nutrition (sarcomatrix). The majority of Radiolaria develop also a skeleton for support and protection, which presents the utmost variety of form, and is generally composed of silica, sometimes of an organic substance (acanthin). The Radiolarian cell usually leads an isolated existence (Monozoa _vel_ Monocyttaria); only in a small minority (of one legion) are the unicellular organisms united in colonies or coenobia (Polyzoa _vel_ Polycyttaria). The extent of the Radiolaria, as limited by the above definition, which I have made as compact as possible, differs in several important respects from that allowed to the group by all previous diagnoses. The shortest expression of its scope might perhaps be:--Rhizopoda with central capsule and calymma; for the most important character of the Radiolaria, and that by which they are distinguished from all other Rhizopoda, is the differentiation of the unicellular body into two principal parts of equal importance and their separation by a constant capsule-membrane. 2. _The Two Subclasses of the Radiolaria._--The systematic catalogue of the Radiolaria, which forms the second part of this Report, and is brought up to the year {ii}1884, contains 20 orders, 85 families, 739 genera, and 4318 species. The consideration that but a small proportion of the ocean his yet been investigated renders it likely, however, that even this large number does not include the half of the recent species. The great progress which our knowledge of the organisation of the Radiolaria has made, by means of comparative study, renders it possible to arrange this enormous mass of forms in four main divisions or legions, and these are again related in pairs, so that two divisions of the highest rank or subclasses are constituted, the _Porulosa_ (or _Holotrypasta_) and _Osculosa_ (or _Merotrypasta_).