MISCELLANEOUS PROPOSITIONS. 413 



382 . The remarks which we shall now give will illustrate an, 

 instructive mode of considering the singular points of curves. 

 It will be seen that in effect we transfer the origin to the 

 point to be examined, and then employ polar co-ordinates. 



383. Suppose that from any point of a curve as centre a 

 circle is described with an infinitesimal radius ; then by the 

 aid of diagrams the following statements become obvious : 



If the point is an ordinary point the circle cuts the curve at 

 two points, and the radii of the circle drawn to the two points 

 include an angle which differs infinitesimally from two right 

 angles. 



If the point is a singular point we have ether results which 

 depend on the nature of the singularity. 



If the point is a conjugate point the circle does not cut the 

 curve. 



If the point be a point darrfa the circle cuts the curve at 

 only one point. 



If the point is a cusp the circle cuts the curve at two 

 points ; but the radii of the circle drawn to the two points 

 include an infinitesimal angle. 



If the point is a point saillant the circle cuts the curve at 

 two points ; but the radii of the circle drawn to the two 

 points include an angle which is neither infinitesimal nor 

 infinitesimally different from two right angles. 



If the point is a multiple point the circle cuts the curve 

 at more than two points. 



384. Now suppose that < (x, y] = is the equation to the 

 curve in a rational form. Let x and y be the co-ordinates of 

 a point on the curve ; and let x + h and y + k be the co-ordi- 

 nates of any adjacent point. 



Since </> (x, y) = 0, we have, by Chapter xiv., 



1 

 2 



<j> (x + h, y + Tc} = Ah + Bk + \ (Ch* + 2Dkk + Etf) + R ; 



