This paper by G. Floquet, published in the *Annales scientifiques de l'É.N.S.* in 1883, focuses on linear differential equations with periodic coefficients. The author studies the analytical form of the solutions to the equation \( P(y) = \frac{d^m y}{dx^m} + p_1 \frac{d^{m-1} y}{dx^{m-1}} + p_2 \frac{d^{m-2} y}{dx^{m-2}} + \ldots + p_m y = 0 \), where the coefficients are uniform and periodic with period \(\omega\). Floquet introduces a fundamental system of solutions \( f_1(x), f_2(x), \ldots, f_m(x) \) and shows that these solutions can be expressed in terms of a fundamental equation \(\Delta = 0\), which is an \(m\)-degree polynomial in \(z\). The roots of this equation correspond to the multipliers of the periodic solutions. He proves that the roots of this fundamental equation are independent of the choice of the fundamental system and that the solutions can be written as linear combinations of uniform functions periodic of the second kind. The paper also discusses the behavior of the elements of the fundamental system when \(x\) is replaced by \(x + \omega\), leading to the conclusion that the solutions can be expressed in a specific form involving uniform functions periodic of the second kind. Finally, Floquet provides a detailed analysis of the structure of these solutions, showing that they are combinations of functions with specific multipliers and periods.This paper by G. Floquet, published in the *Annales scientifiques de l'É.N.S.* in 1883, focuses on linear differential equations with periodic coefficients. The author studies the analytical form of the solutions to the equation \( P(y) = \frac{d^m y}{dx^m} + p_1 \frac{d^{m-1} y}{dx^{m-1}} + p_2 \frac{d^{m-2} y}{dx^{m-2}} + \ldots + p_m y = 0 \), where the coefficients are uniform and periodic with period \(\omega\). Floquet introduces a fundamental system of solutions \( f_1(x), f_2(x), \ldots, f_m(x) \) and shows that these solutions can be expressed in terms of a fundamental equation \(\Delta = 0\), which is an \(m\)-degree polynomial in \(z\). The roots of this equation correspond to the multipliers of the periodic solutions. He proves that the roots of this fundamental equation are independent of the choice of the fundamental system and that the solutions can be written as linear combinations of uniform functions periodic of the second kind. The paper also discusses the behavior of the elements of the fundamental system when \(x\) is replaced by \(x + \omega\), leading to the conclusion that the solutions can be expressed in a specific form involving uniform functions periodic of the second kind. Finally, Floquet provides a detailed analysis of the structure of these solutions, showing that they are combinations of functions with specific multipliers and periods.