In this paper, a general function Y and its variational principles are applied to solve the problem of gravity-influenced free surface plane flows. First of all, the physical plane is transformed into a new image plane. On the new image plane, the general function Y is introduced and the variational principle concerning the function Y is written. The solution of the gravity-influenced free surface flows by means of variational principles and finite element method overcome the difficulty of the unknown free surface boundary. Thus, a new way for solving gravity-influenced free surface flows is found. The solution of the dam out flow problem is cited as an example. The result shows that this method is very effective.