Recently, it has been widely recognized that control and mechanical systems cannot be designed separately due to their tight interdependency. However, there still leaves much to be understood about how well-balanced coupling between control and mechanical systems can be achieved. Therefore, as an initial step toward this goal, this study intensively discusses the effect of the intrinsic dynamics of a robot's body on the resulting behavior, in the hope that mechanical systems appropriately designed will allow us to significantly reduce the complexity of control algorithm required. More precisely, we focus on the property of leg elasticity of a passive dynamic running biped, and investigate how this influences the stability of running. As a result, we have found that a certain type of nonlinearity in the leg elasticity plays a crucial role to enhance the stability of passive dynamic running. To the best of our knowledge, this has never been explicitly considered so far.