A computationally useful initial-value theory for determining the intensity of radiation emerging normal to the surface of the atmosphere for comparison with observed profiles. In this theory the emergent intensity E is the solution of an initial-value problem in which the independent variable is the interval length, or x, the optical thickness. The solution is determined as the thickness is varied from x equals zero when E equals zero, to x equals the desired thickness value. The computational procedure is based on the ability of modern computers to effectively solve large systems of ordinary differential equations subject to a complete set of initial conditions. The differential-integral equations of the exact theory are replaced by a system of ordinary differential equations in which the definite integrals are approximated by sums according to a quadrature formula. A suitably chosen quadrature formula can yield a very good approximation. 24 pp. Refs.