1. Summary. In a preceding paper [7] there was presented a Galois theory, for a certaini kind of differenitial field extension called strongly normal. The Galois group of a strongly normal extension is enidowed with a structure very much like that of a group variety, as studied by Weil [14]. In the present paper the study of such Galois groups is renewed for the purpose of clarifyinig their conniection with group varieties on the onie hand and with strongly normal extensiolis on the other. Consider a differential field S of characteristic 0 with algebraically closed field of constants W, and a strongly normal extension A of 5; suppose given a universal extension 5* of X, and denote the field of constants of 5* by AV. For reasons of convenience we use the term "Galois group of & over 7" for the group of all (ipso facto strong) isomorphisms of & over 5, and not for its subgroup conisisting of all automorphisms of & over J. The field R*, being algebraically closed anid of finite transeendence degree over A, may be used as a uniiversal domain for algebraic geometry (Weil [13], ch. I, ? 1); it is the group varieties of this algebraic geometry, defilned over E and slightly generalized to permit group varieties which are reducible (i. e. which have more thani one comiponent), whieh we consider. By an "algebraic group" we miean either a Galois group or a group variety as above. Chapter I is primarily a study of " rational " homomorphisms of algebraic groups into algebraic groups; these seenm to be the homomorphismls appropriate to the consideratioin of algebraic groups. Ratioinal homomorphisms of group varities inlto group varieties, without restrictioin to fields of eharacteristic 0, were considered by Weil [14] (who omitted the adjective "rational"). Specializing the coneept of rational homomorplhism we obtaini that of birational isomorphisin. In Chapter II it is showni that every Galois group is birationally isomorphic to a group variety, and conversely that every irreducible group variety is birationallv isomorphic to a Galois group; this