378 TRACING OF POLAR CURVES. . 



Hence one of the limiting values of - is zero. 



In like manner, if y* + ay*x x* = 0, 



/w\ 3 /v\ 2 



we have y - +ah- x = 0. 



y 



x 



i/ 

 Hence - has zero for one of its limiting values. Also from 



y 



the given equation we may deduce 



y+ a -- a; [ - 1 = 0- 



y \yJ 



S /ft 



Hence - has zero for one of its limiting values. Thus - 



y * 



may be zero or infinity when x and y are indefinitely dimi- 

 nished, and therefore the axes of x and y are tangents to the 

 branches through the origin. 



In connexion with the subject of tracing curves from equa- 

 tions of the form <f> (x, y) = the student may with advan- 

 tage consult Chapter xxiu. of the treatise on the Theory of 

 Equations. 



348. We shall now give some examples of polar curves. 



e 



/\ T sin - 



o *v c dr a 



Suppose r = a sec - . therefore -j- a = - , 



o at) o v 



cos - 



O 



JA & 



tan <f> = r ^- = 3 cot - . (Art. 279.) 

 dr 3 



ja a 



The polar subtangent = r 2 -j- = 3a cosec - . 



ctr o 



_. 

 "When - = , r is infinite, and the polar subtangent = 3a ; 



^ 3?r 



hence we have an asymptote. As increases from to , 



dr 



-JQ is positive, and r is positive and increases with, 0. As 



