The article by Lieut.-Col. A. G. M'Kendrick explores the application of mathematical methods to various medical problems, particularly focusing on the dynamics of disease transmission and population dynamics. The author uses vector diagrams to illustrate the movement of individuals through different stages of infection, treating each individual as a system of cells that interact and evolve over time. The study is divided into one-dimensional and two-dimensional cases, with reversible and irreversible transitions. In the one-dimensional case, the author models the progression of an infection, such as the common cold, where individuals move through compartments representing the number of attacks they have experienced. The equations derived from this model help in understanding the probability of transitions between compartments and the distribution of cases over time. The author also discusses the effects of correlation between variables, such as the relationship between external and internal infections in a house. The two-dimensional case extends the analysis to two variables, such as the number of males and females in a community, and considers both forward and backward movements. The author introduces the concept of "oblique" and "shear" correlation, which describes the relationship between different types of events and their impact on the distribution of cases. The article includes several examples, such as the distribution of house infections, the behavior of epidemics, and the age distribution of populations. These examples illustrate how the mathematical models can be applied to real-world scenarios, providing insights into the dynamics of disease spread and population changes. Finally, the author generalizes the models to continuous variables, leading to equations that describe diffusion and hydrodynamics. The article concludes with a discussion on the application of these models to vital statistics and the course of epidemics, emphasizing the importance of understanding the underlying mathematical principles in medical research.The article by Lieut.-Col. A. G. M'Kendrick explores the application of mathematical methods to various medical problems, particularly focusing on the dynamics of disease transmission and population dynamics. The author uses vector diagrams to illustrate the movement of individuals through different stages of infection, treating each individual as a system of cells that interact and evolve over time. The study is divided into one-dimensional and two-dimensional cases, with reversible and irreversible transitions. In the one-dimensional case, the author models the progression of an infection, such as the common cold, where individuals move through compartments representing the number of attacks they have experienced. The equations derived from this model help in understanding the probability of transitions between compartments and the distribution of cases over time. The author also discusses the effects of correlation between variables, such as the relationship between external and internal infections in a house. The two-dimensional case extends the analysis to two variables, such as the number of males and females in a community, and considers both forward and backward movements. The author introduces the concept of "oblique" and "shear" correlation, which describes the relationship between different types of events and their impact on the distribution of cases. The article includes several examples, such as the distribution of house infections, the behavior of epidemics, and the age distribution of populations. These examples illustrate how the mathematical models can be applied to real-world scenarios, providing insights into the dynamics of disease spread and population changes. Finally, the author generalizes the models to continuous variables, leading to equations that describe diffusion and hydrodynamics. The article concludes with a discussion on the application of these models to vital statistics and the course of epidemics, emphasizing the importance of understanding the underlying mathematical principles in medical research.