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MATHEMATICS: J. P. LASALLE 573under X is equivalent to 7r. If 7r occurs at least once we say ir is a discrete component of X. Moreover we say that Xr occurs only a finite number of times if there exists an integer m > 1 such that it is impossible to choose Is (1 < i < m) with the above properties. The following result is an immediate consequence of Lemma 1 and Theorem 1. THEOREM 2. Let r be a discrete subgroup of G of type III and r' a subgroup of finite index in r. Put ro = yrF'y-I and suppose that N n r/N n ro is finite.,,f r Then every discrete irreducible component of the representation X of G on L2(G/r') occurs only a finite number of times in X.

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