This book delves into partial differential equations, with a specific focus on those related to surfaces. The author builds upon the work of Professor Craig, examining equations connected to the theory of surfaces. By considering linear partial differential equations of the second order, the book provides a thorough understanding of Laplace's method for integration. It explores the concept of invariants, which enjoy a significant role in the study of the proposed equations, and demonstrates how these functions can be used to derive general formulas for calculating invariants. The author also investigates the relationship between the equations and the curvature of surfaces, showing that under certain conditions, the lines of curvature form an isothermal system. The book's insights on partial differential equations connected to surfaces offer valuable knowledge for mathematicians and researchers working in differential geometry and related fields.

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Excerpt from Certain Partial Differential Equations Connected With the Theory of Surfaces: Dissertation Submitted to the Board of University Studies of the Johns Hopkins University, for the Degree of Doctor of Philosophy
In this paper the work done there has been continued. The first section is introductory, and gives a brief resume of Laplace's method for the integration of linear partial differential equations of the second order.
In the next two sections Professor Craig's equations (e01) and. Are deduced together with their equivalent equations and the corresponding Laplace's series, and the general formulae for calculating their invariants are derived. The form of substitution required to pass from an equation of one series to one of the other series has also been found.
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