Degeneration of a compact Riemann surface of genus 2

A problem in the degeneration of a compact, two-sheeted, Riemann surface of genus 2 is studied, using theta function techniques. The three moveable branch points coalesce to the fourth branch point on the first limiting surface while the triangles formed by these points are all essentially similar. Applying a conformal map, we see that these points represent the finite branch points on the second of the limiting surfaces.