The reciprocity, , has been shown to hold when g(x) is in B* [Proc. Edin. Math. Soc., 12 (1960), 85]. The purpose of the present note is to show that it holds more generally and in particular when g(x) is in a subclass B″, of functions of bounded variation B, such that B* ⊂ B″ ⊂ B′ ⊂ B; it is assumed that f(x) is any function in D1 and that f(x) and g(x) are both continuous on the left and possibly with simultaneous points of discontinuity.

Dual integral equations of the form

where f(x) and g(x) are given functions, ψ(x) is unknown, k≧0, μ, v and α are real constants, have applications to diffraction theory and also to dynamical problems in elasticity. The special cases v = −μ, α = 0 and v = μ = 0, 0<α2<1 were treated by Ahiezer (1). More recently, equations equivalent to the above were solved by Peters (2) who adapted a method used earlier by Gordon (3) for treating the (extensively studied) case μ = v, k = 0.

Dr. A. D. Sands has pointed out that the following sentence, occurring in (2, p. 60, lines 2 to 4), should be deleted: “Hence Sinkov's group of order 1344 … is the holomorph of the Abelian group {pi}, of order 8 (1, pp. 111–117).” In fact, Sinkov's group and the holomorph of C2×C2×C2 are not isomorphic. For, the holomorph is representable on 8 letters by definition (1, p. 87), whereas Sinkov's group is not representable on 8 letters. To see this, we recall that Sinkov's group (3, p. 584) is generated by two elements of periods 2 and 3 (namely, QP3 and QP2) whose commutator is of period 8. If these two generators could be represented as permutations of 8 letters, their commutator would be an even permutation and thus could not be of period 8.

In the solution of boundary value problems in mathematical physics by means of integral transforms, we often find that the solution of a particular problem can be expressed in terms of integrals of the type

where r and z are positive, and v and n are integers satisfying the convergencecondition v+n>−1.