GEOMETRY 41 



If of the first degree in x, this represents a rectilinear asymptote; if of a higher 

 degree, a curvilinear asymptote. 



2.250 Singular Points. If the equation of the curve is F (x, y) = o, singular 



points are those for which 



dF dF 



-T- = -r- = O. 



dx dy 

 Put, 



= &F d*F I d*F V 



" dx 2 dy 2 (dx dy) 



If A<o the singular point is a double point with two distinct tangents. 

 A>o the singular point is an isolated point with no real branch of the curve 



through it. 



A = o the singular point is an osculating point, or a cusp. The curve has two 

 branches, with a common tangent, which meet at the singular point. 



/J 77 /1 77 r$T? rJ277 r)2/7 



If t > -> -j - simultaneously vanish at a point the singular 

 dx dy dx 2 dy 2 dx dy 



point is one of higher order. 



PLANE CURVES, POLAR COORDINATES 



2.270 The equation of the curve is given in the form, 



In figure 2, OP = r, angle XOP = 6, angle XTP = T, angle pPt = <. 



2.271 6 is measured in the counter-clockwise direction from the initial line, 

 OX, and s, the arc, is so chosen as to increase with 6. The angle <j> is measured 

 in the counter-clockwise direction from the positive radius vector to the positive 

 tangent. Then, 



T = + <f>. 



rd6 

 2.272 tan = - 



rdd 



sin <p = j 

 ds 



dr 



cos 9 = 3- 

 ds 



