McCallum and Smith (1) have developed a logical computer which solves problems which are essentially of the following kind.

The object of this note is to prove the following generalization (from Riemann to Lebesgue integrable functions) of a theorem already proved by the authors (1). The notation and terminology of the earlier paper are used.

Let G denote a group of order a power of the prime p, and let G′ be the derived group of G. The lower central series of G will be written

For any subgroup H of G we denote by P(H) the subgroup of H generated by all elements xp as x runs through H, and by Φ(H) the Frattini subgroup of H. We write (H:Φ(H)) = pd(H); thus d(H) is the minimal number of generators of H.

Let R be a ring, not necessarily associative. A derivation of R is a mapping D:R → R, not identically zero, such that

for all x, y ε R. If R is a linear algebra over a field F, then D also satisfies

for all x ε R and α ε F.

We shall be concerned with the differentiation of an arbitrary set function ψ from certain classes of set functions with respect to another fixed set function φ. In (1), § 2·5, a halo evanescence condition is shown to be necessary and sufficient for the differentiation of a fixed measure ψ with respect to φ. Here we formulate a type of halo condition which ensures the differentiability of entire classes of set functions. Although the necessity of this condition is not shown, we demonstrate by means of an example that the results cannot be improved upon in certain respects.

We shall understand the norm, h(A) of a regular matrix A = (amn) to be

The method has a norm, ‖ ‖, ‖ ‖ ≥ 1, given by ‖ ‖ = inf h(A) where the inf is taken over all the matrix methods equivalent to for bounded sequences (b-equivalent). These definitions are due essentially to Brudno(1) though his definition of

A number of writers have noticed Fourier sine or cosine transform relations between certain pairs of series. For example, Duffin(4) has shown that in a variety of circumstances

are a pair of sine transforms, and Guinand(7) that

are cosine transforms, using L2 theory. Other examples are given in (2) and (5); and in (6) power series and general transforms are considered.

If φ belongs to L2(0, ∞), the Watson transform W φ of φ is defined by

where k(u)/u belongs to L2(0, ∞) and

The operator W is unitary in L2(0, ∞), and its inverse is given by

In the last few years it has been found useful to apply known theorems in the theory of functions of several complex variables to solve problems arising in the quantum theory of fields (11). In particular, in order to derive the dispersion relations of quantum field theory from the general postulates of that theory it appears useful to apply known theorems on holomorphic continuation for functions of several complex variables ((2), (10)). The most important theorems are those which enable a determination to be made of the largest domain to which every function which is holomorphic in a domain D may be continued. This domain is called the envelope of holomorphy of D, and denoted by E(D). If D = E(D) then D is termed a domain of holomorphy. E(D) may be defined as the smallest domain of holomorphy containing D. Only in the special cases that D is a tube, semi-tube, Hartogs, or circular domain has it been possible to determine the envelope of holomorphy E(D) ((3), (7)). An iterative method for the computation of envelopes of holomorphy has recently been given by Bremmerman(4). It is also possible to use the continuity theorem (1) in a direct manner, though in most cases this is exceedingly difficult.

The methods introduced into the theory of atomic spectra by Racah are applied to the calculation of electric quadrupole line strengths for transitions involving d and s electrons. The transitions considered are of the types dn−dn−1s, dn−1s−dn−1s and dn−1s−dn−2s2. Sum rules and rules for almost closed shells are obtained. Detailed numerical results are given for various transitions in and between the configurations d9 and d8s, and d7, d6s and d5s2.

In an earlier paper (Helliwell and Mackie(3)) it was shown that steady two-dimensional flow patterns of a compressible inviscid fluid at high subsonic speed past a finite wedge could be determined quite simply when sonic velocity is attained at the shoulder of the wedge and thereafter the flow breaks away from the shoulder with a free streamline. In a subsequent paper (Helliwell (4)) a similar method of analysis has been applied to determine a flow pattern of the same general type past a finite wedge symmetrically placed in a channel, from which the case of the wedge in the free stream may be deduced as a special case. However, in a general investigation into transonic flow past wedges (Mackie and Pack (5)) it was argued that when the wedge angle or the free stream (subsonic) velocity is too small no supersonic region would develop on the wedge side, and the flow would break away from the wedge shoulder with some higher subsonic velocity, giving a free stream line. The present note examines the flow pattern which develops under these conditions for a wedge symmetrically placed in a channel with parallel walls.