As is known, Lagrange, in his "Analytical Mechanics", when presenting the theory of small oscillations of a system of points, made the incorrect assertion that when the roots of the characteristic equation are multiple, the solution always contains terms with integer powers of time outside the sine and cosine signs. The fallacy of this assertion was pointed out in 1858 by Weierstrass, who, however, considered only the special case of a system of equations containing terms with second-order derivatives and terms linear in the unknown functions. Routh, in his treatise on the stability of motion, considered the general case of equations also containing terms with first derivatives. This note is devoted to the same question. While containing no essentially new results, it aims to provide a comprehensive study of the conditions under which the so-called secular terms are absent from the solution of a system of linear homogeneous second-order equations of the most general type. The approach is based on Cauchy's method of integrating linear equations—that is, the very method that Weierstrass applied in studying the aforementioned special case. The advantage of this method lies in its naturalness: we seek expressions for the coefficients in the general integral of the system and, by setting them to zero, obtain the desired conditions.