E. Buckingham's paper, "On Physically Similar Systems; Illustrations of the Use of Dimensional Equations," published in the Physical Review in 1914, explores the principles of dimensional analysis and their application to physical equations. The paper begins by defining the most general form of physical equations involving \( n \) different kinds of quantities, which can be expressed as \( f(Q_1, Q_2, \cdots Q_n, r', r'', \cdots) = 0 \). Under the assumption that certain ratios \( r \) remain constant, the equation simplifies to \( F(Q_1, Q_2, \cdots Q_n) = 0 \). Buckingham then introduces the concept of dimensional conditions, stating that all terms in a physical equation must have the same dimensions. This leads to the requirement that the exponents of the dimensions of the quantities must satisfy dimensional equations. He demonstrates how to find the number of independent dimensionless products \( \Pi \) and their values by solving these dimensional equations. The paper provides several examples to illustrate the application of these principles. For instance, Buckingham shows how to derive the general form of an equation describing a physical relation among seven quantities, and how to solve for specific quantities using dimensional analysis. He also discusses the concept of physically similar systems, where two systems are considered similar if they satisfy the same physical laws under certain transformations. Buckingham critiques Richard C. Tolman's "Principle of Similitude," arguing that it is a consequence of dimensional homogeneity and not a new principle. He provides detailed examples, such as the energy density of an electromagnetic field, the relation between mass and radius of an electron, and radiation from an accelerated electron, to demonstrate the practical application of dimensional analysis. Finally, Buckingham applies the method to a thermal problem, specifically the transmission of heat between a metal pipe and a fluid, to illustrate its utility in planning experiments and interpreting results.E. Buckingham's paper, "On Physically Similar Systems; Illustrations of the Use of Dimensional Equations," published in the Physical Review in 1914, explores the principles of dimensional analysis and their application to physical equations. The paper begins by defining the most general form of physical equations involving \( n \) different kinds of quantities, which can be expressed as \( f(Q_1, Q_2, \cdots Q_n, r', r'', \cdots) = 0 \). Under the assumption that certain ratios \( r \) remain constant, the equation simplifies to \( F(Q_1, Q_2, \cdots Q_n) = 0 \). Buckingham then introduces the concept of dimensional conditions, stating that all terms in a physical equation must have the same dimensions. This leads to the requirement that the exponents of the dimensions of the quantities must satisfy dimensional equations. He demonstrates how to find the number of independent dimensionless products \( \Pi \) and their values by solving these dimensional equations. The paper provides several examples to illustrate the application of these principles. For instance, Buckingham shows how to derive the general form of an equation describing a physical relation among seven quantities, and how to solve for specific quantities using dimensional analysis. He also discusses the concept of physically similar systems, where two systems are considered similar if they satisfy the same physical laws under certain transformations. Buckingham critiques Richard C. Tolman's "Principle of Similitude," arguing that it is a consequence of dimensional homogeneity and not a new principle. He provides detailed examples, such as the energy density of an electromagnetic field, the relation between mass and radius of an electron, and radiation from an accelerated electron, to demonstrate the practical application of dimensional analysis. Finally, Buckingham applies the method to a thermal problem, specifically the transmission of heat between a metal pipe and a fluid, to illustrate its utility in planning experiments and interpreting results.