The famous example of the double-Watt mechanism given by Connelly and Servatius raises some problems concerning the classical definitions of higher-order flexibility and rigidity. Recently, the author was able to give a proper redefinition of the flexion/rigidity order for bar-joint frameworks, but the question for the flexes associated with higher-order flexible structures remained open. In this paper we properly define these flexes based on the theory of algebraic curves and demonstrate their computation by means of Puiseux series. The presented algebraic approach also allows reality issues to be taken into account.