Sholes and Glidden typewriter length. The redesigned platen also introduced the modern spacing functionality (horizontal and axial movement providing letter and line spacing, respectively). The cylindrical platen became "an indispensable part of every standard [typewriter]". By the end of 1872, the appearance and function of the typewriter had assumed the form that would become standard in the industry and remain largely unchanged for the next century. Although the machine possessed a cylindrical platen and what was essentially a QWERTY keyboard, two design elements that would later become essential were lacking: the ability to write in upper and lowercase letters and "visible" print. Although the
Cylindrical harmonics Cylindrical harmonics In mathematics, the cylindrical harmonics are a set of linearly independent solutions to Laplace's differential equation, formula_1, expressed in cylindrical coordinates, "ρ" (radial coordinate), "φ" (polar angle), and "z" (height). Each function "V"("k") is the product of three terms, each depending on one coordinate alone. The "ρ"-dependent term is given by Bessel functions (which occasionally are also called cylindrical harmonics). Each function formula_2 of this basis consists of the product of three functions: where formula_4 are the cylindrical coordinates, and "n" and "k" are constants which distinguish the members of the set from each other. As a result