POLAK SUBTANGENT. ASYMPTOTES. 309 



Suppose a such a value of 6 ; if for this value of 6 the polar 

 subtangent r 2 -y- is infinite, there is no corresponding asymp- 



G/i 



tote. If r 2 -j- be finite there is an asymptote which may be 



constructed thus : conceive a straight line drawn from 8 at 

 an angle a to the initial line ; draw from 8 a second straight 



7/1 



line at right angles to the first, to the right of it, if r 2 -j- be 



JQ 



positive, and to the left of it, if r z -j- be negative, and equal 



in length to r 2 -,- ; through the end of this second straight 



line draw a straight line parallel to the first, and it will be 

 the required asymptote. 



The terms right and left in the above rule are to be under- 

 stood with respect to the straight line first drawn, the eye 

 being supposed to look along that line from S. The reason 

 of the rule must be collected from the figure of Art. 284 and 

 the general principle of the interpretation of signs ; that 



figure makes r increase with 6, and therefore r 2 -j- is positive. 



If we draw a figure in which r diminishes when 6 increases, 



dr 



so that -jx and the polar subtangent are negative, we shall 

 do 



find that ST falls to the left of 8P. 



287. Example. r = -. a . 



sm0 



Here r is infinite when 6 is any multiple of TT. 

 dr _ a (sin 6 6 cos 0) 



-t\.lSO 7/1 o x 



dd BUT 6 



therefore r 2 -r- = -^g ~ - a . 



dr sin00cos0 



Hence, when sin = 0, the value of the polar subtangent 



a0 

 is 5. 



COSC7 



When = TT, the polar subtangent = air. 



