On non-existence of periodic solutions of an important differential equation

summary:The equations of variation with respect to the straight-lineequilibrium points $L_1,L_2,L_3$ of the elliptic three-dimensional restricted problem of three bodies are equivalent to a system of two differential equations of the second order and one Hill's equation. In the paper presented here, this Hill's equation is studied and a proof is given that this differential equation has no nontrivial periodic solution