The laplace condition in the adjustment of a chain of triangles

Summary

The purpose of this treatise is to study in detail the influence of the approximations usually applied in the Laplace condition, on the adjusted elements of a chain of triangles. In § 1 the Laplace condition is derived once again from the well-known Laplace equation, because in our opinion the usual deriviations are not quite correct. In §§ 2–6 the Laplace condition is converted to a form—terms containing first powers of s: R inclusive—suitable for an application in the adjustment of a chain of triangles. The usual approximate form is derived from it by omitting the terms mentioned. In § 7 the influence of these approximations is examined by adjusting a chain of triangles twice over: applying the exact resp. the approximate Laplace condition and comparing the results. From this it is concluded that in all normal cases the influence of applying the approximations is negligible and that the validity of this conclusion is independent of a possible future increase of the accuracy of observation.