A computational burden has always been associated with the stability analysis of a time delay system over its entire parameter ranges. Indeed, this has led many previous investigators to develop control schemes under the assumption of small or negligible time delay. In this paper we have attempted to explore an alternative approach, which has been previously called as Critical or Single Points, Hale＇s Method, Projection Series and Centre Manifold. Here we have deliberately omitted the relevant theorems, lemmas or corollaries of this approach and rather focus mainly on a calculation spirit to make the underlying ideas accessible to investigators seeking more insight into the wealth of time delay dynamics. The nonlinear delay equation around which we present this approach is a variational form of equations studied earlier by Bentsman et al. (1991), Lehman et al. (1992) and Lehman and Bentsman (1992, 1994). Furthermore, using the integral averaging method explicit bifurcation equations of the form g(a, mu) = 0 are derived and from the signs of their characteristic exponents, the stability of the trivial and nontrivial solutions is obtained.