Abstract: In this paper，the analytical method of Lagrange gyroscope motion equation is established．The condition for regular precession of Lagrange gyroscope is derived，and the analytical solution for nutation with small angle and precession near regular precession is obtained by series expansion technique．A concise picture of the gyroscope rotation axis motion is given，that is，the superposition of the regular precession around the vertical axis caused by gravity and the free precession without gravity. According to the analytical solution，the mystery of the gyroscope not falling down is explained.

Key words: Lagrange gyroscope, analytic method, regular prece, nutation