Overhead power transmission lines rely on cables to carry electricity across large distances. To ensure the stability and functionality of these lines, it is crucial to understand the shape and tension experienced by the cables under the influence of their weight.

A generalized loading function is employed to analyze a cable subjected to its own weight. This function considers the force acting along the cable's arc length rather than its projected length, providing a more accurate representation of the cable's behavior. To further analyze the cable, a small segment of the cable is considered, and a free-body diagram is drawn. The diagram helps to visualize the forces acting on the cable segment and serves as a basis for applying the equilibrium equations.

A set of three equations can be obtained by applying the equilibrium equations to the cable segment. The first and second equations represent the horizontal and vertical components of the tensile force acting on the cable, respectively.

Using Pythagoras' theorem, a relationship can be established between the change in vertical distance (dy) and the arc length of the cable (ds). The relationship is then substituted into the third equation obtained from the equations of equilibrium. Finally, by rearranging the terms in the equation and integrating the obtained equation, an expression for the shape of the cable can be determined.

This expression allows engineers to calculate the sag and tension in the cable, ensuring the stability and efficiency of overhead power transmission lines.