This paper, authored by S. Pancharatnam, explores the generalized theory of interference and its applications, particularly focusing on coherent pencils in anisotropic media. The introduction highlights the complexity of studying optical properties in anisotropic media, such as transparent optically active crystals like quartz, where waves are elliptically polarized rather than linearly polarized. The author discusses the interference phenomena observed in absorbing biaxial crystals, such as iolite, and how these phenomena can be studied using polarizers and analyzers. The main content of the paper is divided into several sections: 1. **Introduction**: Outlines the background and motivation for the study, emphasizing the complexity of interference phenomena in anisotropic media. 2. **The Poincaré Sphere and the Stokes Parameters**: Introduces two methods for specifying the state of polarization: the analytical method using Stokes parameters and the geometrical method using the Poincaré sphere. The paper primarily uses the Poincaré representation, providing a self-contained derivation of its properties. The paper aims to address specific problems in crystal optics, including the interference of two coherent beams in different states of elliptic polarization, the resolution of a polarized beam into two beams in given states of polarization, and the composition of two coherent beams of different polarization. The introduction also hints at the broader implications of these problems, which will be explored further in Part II, particularly in the context of partial coherence of polarized beams.This paper, authored by S. Pancharatnam, explores the generalized theory of interference and its applications, particularly focusing on coherent pencils in anisotropic media. The introduction highlights the complexity of studying optical properties in anisotropic media, such as transparent optically active crystals like quartz, where waves are elliptically polarized rather than linearly polarized. The author discusses the interference phenomena observed in absorbing biaxial crystals, such as iolite, and how these phenomena can be studied using polarizers and analyzers. The main content of the paper is divided into several sections: 1. **Introduction**: Outlines the background and motivation for the study, emphasizing the complexity of interference phenomena in anisotropic media. 2. **The Poincaré Sphere and the Stokes Parameters**: Introduces two methods for specifying the state of polarization: the analytical method using Stokes parameters and the geometrical method using the Poincaré sphere. The paper primarily uses the Poincaré representation, providing a self-contained derivation of its properties. The paper aims to address specific problems in crystal optics, including the interference of two coherent beams in different states of elliptic polarization, the resolution of a polarized beam into two beams in given states of polarization, and the composition of two coherent beams of different polarization. The introduction also hints at the broader implications of these problems, which will be explored further in Part II, particularly in the context of partial coherence of polarized beams.