the first equation to be treated and solved as an integral equation, has an extensive literature, dealing on the one hand with properties of the functions involved, and on the other hand with the solution, and conditions for solubility of the equation. In the first category one might cite, in the modern spirit, the memoirs of Hardy [1, pp. 145150] and Hardy and Littlewood [2, pp. 565-606]; and in the second category, Abel's original work [3, pp. 97-101], and the work of Tonelli [4, pp. 183-192], Tamarkin [5, pp. 219-228], Doetsch [6, pp. 192-207] and Rothe [7, pp. 375-380]. In [3], [4] and [5], the operation performed on the right hand side of (1.1) is recognized to be essentially an integration of fractional order 1 -a and the solution is obtained by making an integration of appropriate order. In Abel's memoir [3], no assumptions other than those implicitly involved in the integrations are stated about the given function F(X) and the unknown function b(T), while the Lebesgue integral is the basis of [4] and [5]. Doetsch [6] uses the Laplace transform, and assumes T>(T) to be continuous for T> 0, and differentiable. In [7] the theory of the Beta function is used, and strong differentiability conditions are imposed on F(X). In the present note, equation (1.1) is treated from the point of view of the convolution transform, and an inversion operator of integro-differential type obtained for it. The Lebesgue integral is the basis of the work, but on account of the infinite integrals which occur, an additional, but relatively mild, condition is imposed on the behaviour of (D(T) for large positive T. We assume throughout that