This paper presents a generalized theory of interference and its applications, focusing on coherent pencils. The author, S. Pancharatnam, discusses the interference phenomena in anisotropic media, particularly in absorbing biaxial crystals. He notes that the interference patterns observed in such crystals can be studied by examining an extended source through a plate cut normal to an optic axis, using materials like iodite. The interference rings are visible even without an analyser, and the patterns change when an analyser is introduced. The paper also explores the analysis of interference phenomena involving polarised beams, leading to the study of elliptically polarised waves and their interactions. The author formulates problems related to the interference of two coherent beams in different states of elliptic polarisation, the resolution of polarised beams into given states of polarisation, and the composition of beams at the second face of the crystal plate. The introduction of an analyser leads to the study of interference of beams brought to the same state of elliptic vibration. The paper also touches on the partial coherence of polarised beams, which is discussed in Part II. The Poincaré sphere and Stokes parameters are introduced as methods for specifying the state of polarisation of a beam. The Poincaré representation is used in Part I, while the Stokes representation is introduced in Part II. The paper provides a self-contained derivation of the properties of the Poincaré sphere using a new procedure. The study of partial polarisation naturally leads to the use of the Stokes representation in Part II.This paper presents a generalized theory of interference and its applications, focusing on coherent pencils. The author, S. Pancharatnam, discusses the interference phenomena in anisotropic media, particularly in absorbing biaxial crystals. He notes that the interference patterns observed in such crystals can be studied by examining an extended source through a plate cut normal to an optic axis, using materials like iodite. The interference rings are visible even without an analyser, and the patterns change when an analyser is introduced. The paper also explores the analysis of interference phenomena involving polarised beams, leading to the study of elliptically polarised waves and their interactions. The author formulates problems related to the interference of two coherent beams in different states of elliptic polarisation, the resolution of polarised beams into given states of polarisation, and the composition of beams at the second face of the crystal plate. The introduction of an analyser leads to the study of interference of beams brought to the same state of elliptic vibration. The paper also touches on the partial coherence of polarised beams, which is discussed in Part II. The Poincaré sphere and Stokes parameters are introduced as methods for specifying the state of polarisation of a beam. The Poincaré representation is used in Part I, while the Stokes representation is introduced in Part II. The paper provides a self-contained derivation of the properties of the Poincaré sphere using a new procedure. The study of partial polarisation naturally leads to the use of the Stokes representation in Part II.