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for a certain c. In fact I shall prove that if n is highly composite, then the next highly composite number is less than n+n(log y&)-C ; and the result just stated follows immediately from this. At, present I cannot, decide whether the number of highly composite numbers not exceeding z is greater than (logx)” for every k. The principal tool in the proof will be Ingham’s improvement,$ on Hoheisel’s theorem. This asserts that if x is sufficiently large, then the number of primes in the int,erval (x, x+&) is asymptot.ic to cxg(logz)-1. First we state three lemmas, which will be proved at the end of the paper. They are contained subst’antially in the paper of Ramanujan, but we prove Ohem here for completeness. Let n = 22 3~ . . . p+ be a sufficiently large highly composite number. Plainly

M. D. ATKINSONIn a series of papers [3, 4 and 5] on insoluble (transitive) permutation groupsof degree p = 2q +1, where p and q are primes, N. Ito has shown that, apart from asmall number of exceptions, such a group must be at least quadruply transitive.One of the results which he uses is that an insoluble 2q grou +1 p of degree p =which is not doubly primitive must be isomorphi (3, 2)c wit to PSh p =L 7. Thisresult is due to H. Wielandt, and ltd gives a proof in [3]. It is quite easy to extendthis proof to give the following result: a doubly transitive group of degree 2q + l,where q is prime, which is not doubly primitive, is either sharply doubly transitiveor a group of automorphisms of a bloc A = 1 ank desigd k = 3n wit. Ouh rnotation for the parameters of a block design, v, b, X, k, i r,s standard; see [9].In this paper we shall prove two results about doubly transitive but not doublyprimitive groups which resemble the two results mentioned above.

where the integral is in the appropriate sense; let CL(a, b), D*(a,b) and D(a, b) be the associated separated spaces. Miss W. L. C. Sargent [3] has established several theorems of Banach-Steinhaus type for these spaces; from a modern point of view it is evident that these results may be interpreted as proving that each of the above spaces is barrelled. In this paper we present a direct proof of this observation. In particular these spaces provide the earliest examples of barrelled spaces which are meager and normable; the standard example of such a space [1; p. 157, Ex. 10] is perhaps less natural and less accessible as it is obtained essentially from the fact that the tensor product of two Banach spaces equipped with its projective tensor product topology is barrelled and normable but not, in general, a Baire space. 2. The main theorems