SINGULAR POINTS. 



325 



an ordinary multiple point, at a cusp, and at a 'conjugate 

 point, 



= . 

 dy 



Hence, whenever we have found values of x and y which 

 satisfy these three equations, we must, by examining the 

 particular curve, and tracing it in the vicinity of the point 

 in question, determine what species of singular point exists. 



We now pass to some other singular points which occur 

 but rarely, and, as the student will find by experience, never 

 present themselves in curves the equations to which are of an 

 algebraical form. See Art. 6. 



Points d'arret. 



804. A point d'arret is a point at which a single branch 

 of a curve suddenly stops. 



Example. Let y x log x. 



Here when x = we have y = ; but if x be negative, y 

 becomes impossible. Hence the origin is a point d'arret. 

 _i 



Again, suppose y = e x . 



Here if x be made indefinitely small and positive, we have y 

 approaching the limit zero ; but if x be negative and indefi- 

 nitely small, y is indefinitely great. 



Hence the curve has the form represented in the figure, the 

 origin being a, point d'arret; the dotted line is aa asymptote 

 having for its equation y= 1. 



