( 319 ) 



CHAPTER XXII. 



SINGULAR POINTS. 



296. UNDER the common title of " Singular Points " are 

 included all those points on a curve which offer any sin- 

 gularity depending on the curve itself and independent of 

 the position of the co-ordinate axes. We proceed to define 

 the different singular points and to investigate the conditions 

 of their existence. 



Points of Inflexion. 



297. These points have been considered in Arts. 288... 29 5 ; 

 the condition for their existence is that -J^ should change 

 sign. 



Multiple Points. 



298. DEFINITION. A multiple point is a point through 

 which two or more branches of a curve pass. 



Let < (x, y} = be an equation in a rational form ; by 

 Art. 177 



dx dy dx 



Now since two or more branches of a curve pass 

 through a multiple point, it will be possible to draw more 



than one tangent to the curve at that point ; hence -j- must 

 admit of more than one value. But since the equation 

 <f> (> y)= is supposed rational, -r- and -3- will each have 

 but one value for the given values of x and y. Hence from 



