First-order analysis of a three-lens zoom system with the last lens fixed

A general analysis for the first-order solutions of three-lens zoom system with the last lens fixed is presented. The reasonable solution areas in the focal length diagrams are derived and shown graphically. The relation between the separation of the two front lenses and the image distance of middle lens in zooming is found to be a hyperbola. According to the different locations of centre of hyperbola, four cases are analysed. From the four hyperbolic graphs, we get six different types of zoom system. For each zoom type, we find the maximum range of focal length and the position where the maximum or minimum system length occurs during zooming.