The object of this paper is to prove a theorem relating "configurationspaces" to iterated loop-spaces. The idea of the connection between them seems to be due to Boardman and Vogt [2]. Part of the theorem has been proved by May [6]; the general case has been announced by Giffen [4], whose method is to deduce it from the work of Milgram [7]. Let C. be the space of finite subsets of ~n. It is topologized as the disjoint union LI C~,k, where C~, k is the space of subsets of cardinal k, k->0 regarded as the orbit-space of the action of the symmetric group 2: k on the space ~n,k of ordered subsets of cardinal k, which is an open subset of R "k. There is a map from C. to O"S ~, the space of base-point preserving maps S"--,S ~, where S" is the n-sphere. One description of it (at least when n> 1) is as follows. Think of a finite subset c of ~ as a set of electrically charged particles, each of charge + 1, and associate to it the electric field E c it generates. This is a map Ec: R ~ c-.n~ ~ which can be extended to a continuous map Ec: RnUOO--,~nU~ by defining Ec(~)=oo if ~ec, and Ec(oo)=0. Then E c can be regarded as a base-point-preserving map S"--,S", where the base-point is oo on the left and 0 on the right. Notice that the map c~-,Ec takes Cn, k into ~"Sntk), the space of maps of degree k. Our object is to prove that C. is an approximation to Q~ S ~, in the sense that the two spaces have composition-laws which are respected by the map C~-, f l 'S ~, and the induced map of classifying-spaces is a homotopy-equivalence. In view of the "group-completion" theorem of Barratt-Priddy-Quillen [1, 8] one can say equivalently that Cn, k--~nsn(k ) induces an isomorphism of integral homology up to a dimension tending to oo with k. But to make precise statements it is convenient to introduce a modification of the space C.. If u<=v in R, let R".,~ denote the open set )u, v( x R "-~ in R ~. Then Cn is homotopy-equivalent to the space