Excerpt from Lectures on the Differential Geometry of Curves and Surfaces

In the second section, consisting of chapters II - VI, the subject-matter is the properties of curves upon any general surface in space. Some classes of these curves (e.g. lines of curvature) are organically connected with the surface; they are completely determined by the elements of the surface to which they belong. Other curves, such as geodesics, have an equally organic relation with the surface; but they are not determined solely by the elements of the surface, for they can satisfy some arbitrarily assigned condition or conditions. Again, quite arbitrary curves and families of curves can be assumed upon a surface; not a little attention has been devoted to methods for constructing differential invariants which, being in value independent of parameters of reference, express the geometrical magnitudes of the curves, subject, of course, to the dominance of the intrinsic magnitudes of the surface containing the curve or curves.

In the third section, consisting of chapters VII - XI, the subject-matter is surfaces in general, rather than particular configurations on surfaces. The most ordinary methods of point-to-point correspondence and comparison of surfaces are explained. Surfaces, which are defined (wholly or partially) by intrinsic properties, are considered, special attention being paid to minimal surfaces. Families of surfaces are discussed, according to the respective definitions that ultimately establish the families; the most obvious instance relates to those surfaces which have plane or spherical sets of lines of curvature. Lastly, a brief sketch of the simplest fundamental properties of triply orthogonal systems is given.

The book concludes with a single chapter that contains an introduction to the elementary theory of congruences of curves, specially of straight lines and of circles.

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