Using Tangential Force Method to Detect the Singular Points and to Discriminate Their Stability Conditions of a Movable Mass Point on any Guide Curve Rotating about a Vertical Axis without Friction

Ever since one has used generally the state plane method to search the singular points and to decide their eq uilibrium state for a mass points sliding on guide ail rotating about a vertical axis with friction disregarded. For the same purpose,this paper presents another method which wight be briefly named "The Tangential Force Method". In contrast with the state plane method,the new method is much simpler both in argumentation and calculation,especially when one resorts to the five criteria in section XIII.Throughout the paper the function for defining the guide rail was introduced,with great endeavor,in the equations newly set up,in order to avoid deducing them each time,i.e.,the useful equations are set up somewhat once for ever.Moreover,the condition of letting the tangential force vanish yields two solutions,the parabolic and the exponential curves of the shape of the guide rails;they are two additional orthogonal curve families although not conjugate harmonics.In the last part of the paper,we present nine examples to show the superiority of this method against the state plane and the potential function methods;seven of the nine examples might be considered as newly introduced in this paper.