In this paper,by use of a type of auxiliary functions,the uniqueness of limit cycles surron-ding multiple singular points is investigated for the system,\dot{x}=p(y),\dot{y}=-f(x,y)h(y)-g(x)\varphi(y),and a simple sufficient condition for the uniqueness is obtained.Moreover,the relations betweensome usual methods for proving the uniqueness of limit cycles is presented,and the methodfor structuring Dulac function for the system is studied.