The equations that determine the point characteristic V of a refracting plane are obtained. They are solved in closed form for a special choice of the position of the posterior base point. The position of the point of incidence, as a function of the coordinates of the end points, is found in the form of infinite series. An ordinary nonlinear differential equation for V is determined, the required solution being its singular integral. By these means, the problem of obtaining step by step the coefficients of successive terms of the power series for V becomes much more tractable. An alternative method gives the coefficients in terms of those of the Taylor expansion of a certain elementary algebraic function; the general coefficient is exhibited explicitly as a finite sum over Legendre polynomials. Finally, the displacement is briefly considered.
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