The finite element method is described briefly and a systematic approach to the construction of finite elements is presented. A particular element, which is suitable for problems requiring a trial function of class C1, is developed in detail and some applications given. The natural choice of generalized co-ordinates is shown to be over restrictive; the necessary continuity in the trial function can in fact be obtained with a more flexible choice of generalized co-ordinates. This is shown to be particularly useful in dealing with a problem involving singularities.