310 POLAR FORMULAE. EXAMPLES. 



When 6 2-Tr, the polar subtangent = 2a7r, 

 and generally when 6 = mr, the polar subtangent = ( l)*~Via7r. 



To draw the first asymptote, for which = TT, the eye must 

 be supposed to look from S along the direction opposite to 

 Sx, and then measure from S at right angles to Sx and 

 towards the right, a straight line in length a-rr; a straight 

 line drawn parallel to the initial line and at a distance air 

 from it is the required asymptote. 



To draw the second asymptote, for which = 2?r, the eye 

 must be supposed to look along Sx, and then measure to the 

 left (since the subtangent is negative) a length 2a?r. Hence 

 the asymptote is parallel to the initial line at a distance 2a?r 

 from it, and above the initial line. 



Proceeding in this way we find an infinite number of 

 asymptotes parallel and equidistant, and all above Sx. 



If we ascribe to 6 negative values, we shall in like manner 

 obtain a series of asymptotes all parallel to Sx, and equi- 

 distant, lying below Sx. 



EXAMPLES. 



1. In the curve r = a sin 6, shew that < = 0. 



2. Determine the points in the curve r = a (1 4- cos 6} at 



which the tangent is parallel to the initial line. 



3. Shew that in the curve rd = a the polar subtangent is 



of constant length. 



4. In the curve r (ae 6 + be" 9 } = ab, the length of the polar 



subtangent is a - 



ae e be- 6 



5. In any conic section, the focus being the pole, the locus 



of the extremities of the polar subtangents is a straight 

 line at right angles to the axis major. 



6. Find the angle between the radius vector and tangent 



at any point of an ellipse, (1) the focus being the pole, 

 (2) the centre being the pole. Determine in each case 

 when the angle is a maximum. 



