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In order to accomplish the transition from avascular to vascular growth, solid tumours secrete a diffusible substance known as tumour angiogenesis factor (TAF) into the surrounding tissue. Neighbouring endothelial cells respond to this chemotactic stimulus in a well-ordered sequence of events comprising, at minimum, of a degradation of their basement membrane, migration and proliferation. A mathematical model is presented which takes into account two of the most important events associated with the endothelial cells as they form capillary sprouts and make their way towards the tumour i.e. cell migration and proliferation. The numerical simulations of the model compare very well with the actual experimental observations. We subsequently investigate the model analytically by making some relevant biological simplifications. The mathematical analysis helps to clarify the particular contributions to the model of the two independent processes of endothelial cell migration and proliferation.

In this paper we use a method due to Williams (l) to discuss the scattering of surface waves of small amplitude on water of infinite depth by a fixed vertical plane barrier extending indefinitely downwards from a finite depth.1. Introduction. In 1966 Williams (l) presented a new approach to the problem of the scattering of two-dimensional surface waves of small amplitude on water of infinite depth by a fixed vertical plane barrier extending from above the free surface to a finite depth. This problem had previously been considered by Ursell (2), with whose results Williams agreed. Subsequently Faulkner (3,4) extended Wnliams's method, in conjunction with the Wiener-Hopf technique, to discuss the scattering of three-dimensional waves. However, the second of Faulkner's papers, which considers the case when the barrier extends downwards indefinitely from a finite depth, contains an error of formulation which vitiates his results.In this paper we first extend Williams's method to the two-dimensional case where the barrier extends indefinitely downwards from a finite depth and then give a correct formulation of the problem considered by Faulkner.2. Statement of the problem. We consider the scattering of surface waves of small amplitude on water of infinite depth by a fixed vertical plane barrier extending indefinitely downwards from a finite depth I. We take a non-dimensional Cartesian coordinate system with the axis of y vertically downwards, the mean free surface of the water being the plane y = 0, and such that the barrier is given by x = 0, y ^ 1, -oo < z < oo. Assuming the motion to be irrotational and simple harmonic in time, the velocity potential may be taken as the real part of x(z, y, z) e~i ai: , where (6) % is a single-valued harmonic function in the region occupied by the water satisfying the boundary conditions

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