POINTS OF INFLEXION. . EXAMPLES. 317 



294. A curve referred to polar co-ordinates is said to. be 

 concave or convex to the pole at any point, according as the 

 curve in the neighbourhood of that point does, or does not, lie 

 on the same side of the tangent as the pole. 



If p be the perpendicular from the pole on the tangent at 

 a point whose co-ordinates are r, 0, it will be seen from a 

 figure, that if the curve be concave to the pole, p increases if 



r increases, and decreases if r decreases ; hence -? must be 



dr 



positive. Similarly if the curve be convex to the pole HP- must 



dp dr 

 be negative. Thus at a point of inflexion -~ must change 



sign. 



295. Since \ = U* + (-} , Art. 284, 



p \duj 





I dp ( d 2 u\ du 



f dp / d*u\ 



therefore -f- = p (u + -^ 9 . 



du, r V dv/ 



dp dp du 1 dp 

 But -j 5- = -f j- = -- 2 j 



dr du dr r du 



p 3 f d\( 



4__ I / J __ 



Hence, at a point of inflexion we must have generally 



d*u , . . . 

 u + -jTp changing its sign. 



EXAMPLES. 



cc 5 



1. lfy = -z -, there is a point of inflexion at the origin, 



Q/ ~T~ & 



and also when x = a A/3. 



2. If y = } ^- . there is a point of inflexion when 



a(xd) 



x = -a(y$- 1). 



