SOS POLAE SUBTANGENT. 



283. The polar equations in Arts. 281 and 282, may also 

 be derived from the rectangular equations to the tangent 

 and normal of Arts. 257 and 258, by transforming these to 



polar co-ordinates, using the value of ~ given in Art. 278. 



284. From S draw 8Y perpendicular to the tangent PT\ 

 then 



c<v er>T rttm'SPT 

 SY=rsm SPT = 



Hence, if SY=p, we have 



11 



, , fdu\- . 1 



= U+ (d6) lu = r- 



285. From S draw 8T at right angles to the radius vector 

 SP, then ST is called the polar subtangent ; its value is 



jn 



rianSPT, that is r 2 ^-. 

 dr 



286. Since an asymptote is a tangent which remains at 

 a finite distance from the origin when the point of contact 

 moves off to an infinite distance, if a polar curve has an 

 asymptote, SP or r must be infinite while ST remains finite. 

 Hence to determine the asymptotes to a polar curve, we must 

 first find those values of 6, if any, which mate r infinite. 



