106 3. T h e e r r o r -control function 108 4. C onnection coefficien ts; identification of solutions 109 4.1. The identification of solutions 110 5. E stimation of t h e variation of t h e e r r o r -control function 111 6. M ajorant functions 113 6.1. Application 114 R eferences 114 An account is given of the Liouville-Green method for the approximate solution, with error estimates, of linear second-order differential equations, together with certain extensions of the method. The purpose is

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... o make readily available a range of tech niques for use in the two final parts of the present series. The topics treated include: (a) the construction of approximations in terms of both elementary and higher transcendental functions, (b) the relations between approximations of the same solution in terms of different functions, (c) the identification of solutions and the estimation of connection coefficients, (d) uniform estimation of the error-control function in problems with more than one widely ranging parameter, (e) the construction of majorants for approximating functions, the last two being required for the derivation of satisfactory error estimates. There is little in this part that is new, though a method of constructing approxi mations in terms of Bessel functions is developed specifically for application to the Mathieu equation. Apart from this, some aspects of the presentation are thought to be novel.