  The structure equations for a surface are introduced and two required results
based on the Codazzi equations are obtained from them. Important theorems
pertaining to isometric surfaces are stated and a theorem of Bonnet is
obtained. A tranformation formula for the connection forms is developed. It is
proved that the angle of deformation must be harmonic. It is shown that the
differentials of many of the important variables generate a closed differential
ideal. This implies that a coordinate system exists in which many of the
variables satisfy particular ordinary differential equations and these results
can be used to characterize Bonnet surfaces.
