ENNEPER TYPE IMMERSIONS 5

treated in Darboux [4] (see also Eisenhart [7]). In this case one

wants to solve the hyperbolic sine-Gordon equation. The treatment

in both cases is similar.

Finally, H. Dobriner [5] studied the case of Enneper type

surfaces of constant negative Gauss curvature K = -1. His

treatment is based on the book of Enneper [8] and expresses the

immersions using theta functions. His work is discussed in

Darboux [4]. The analysis in the present work follows the

approach in Darboux's treatise.

If a minimal surface in R has one family of planar curvature

lines then the same must be true of the other family. Besides the

catenoid and Enneper's minimal surface there is a one-parameter

family of such surfaces called Bonnet's surfaces. These are

described in Eisenhart [7]. In some sense these surfaces serve to

connect the catenoid to Enneper*s minimal surface. Finally, in

Section V we shall find that surfaces of revolution are the only

Joachimsthal type minimal surface in hyperbolic space, a somewhat

surprising result.