380 ASYMPTOTES TO POLAR CURVES. 



measured in the ordinary way round from OA be -- + 

 the corresponding value of r is 



or or - a \/2 (\/3 + 1) ; 



1 /STT TT\ TT 



C S 3U + lJ Sln T2 



hence we take OP = a \/2 (\/3 + 1) measuring it along $0 

 produced. In this way, as 6 changes from - - to STT, we 

 obtain the portion ECFA of the curve. 



If we suppose 6 negative, or positive and greater than 

 STT, we shall only obtain repetitions of the branches already 

 found. 



349. A very common mistake in drawing polar curves is 

 made with respect to the asymptotes. For example, if r is 

 infinite when = 0, it is assumed that the initial line is an 

 asymptote. This involves a double error, for in the first 

 place it does not follow that because r is infinite there is an 

 asymptote ; and secondly, if there be an asymptote it may be 

 parallel to the initial line instead of coinciding with it. 



For example, the polar equation to the parabola from the 

 vertex is 



_ 4a cos 9 

 sin*0 * 



Here when & = 0, r is oo , but the curve has no asymptote. 

 In the curve 



a 



r= e> 

 sin l 



when 6 = 0, r. is infinite ; there is an asymptote, but it does 

 not coincide with the initial line ; it will be found to be 

 parallel to it and at a distance 3a from it. 



350. Trace the curve 



a sin 



