Abstract
The symbol $ indicates that the principal value of the integral is to be taken and the coefficients cn are given constants. The special case of all cn = 0 has been dealt with extensively, for instance by Glauert [ l ] , Fuchs [2], Hamel [3], Schroeder [4] and Sohngen [S]. The values of the coefficients cn might be determined by the condition that a given kernel i£(£—#), for instance i£ = l /sinh (£—#), is approximated as nearly as possible by the kernel of equation (1). The purpose of the present note is to derive the solution of (1) for a finite number of non vanishing cn. The method of solution is an extension of the method applicable when all cn = 0. Equation (1) is first transformed by the substitutions (2) x = cos #, £ = cos 0, (3) g(*)=G( ), f(Q=F(fi) into
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