J : I H t 





iuation [M] show. .> if the equation 



tents an ellipse, there b no value of $ for which p becomes infinite. 



Therefore there b NO direction in which a line may be drawn to meet an 



If * I, i *., if the equation represents a parabola, 



.' value of 0. vi/.. o, for whHi p Incomes infinite. There- 



here b OM direction , a line may be drawn to meet a perab- 



If r . - 1, ir-., if the equation represents an hyperbola, 



there are two values of $, vuu, $ - cos" 1 r which p becomes 



Therefore there are fvo directions itt which a line may be drawn 



to meet an hyperbola at 



I threo species of conic sections may therefore be dbtingubhed 



each other by the number of direction-, m >\hi< h lines may be drawn 



e focus to meet the curve at sine* parallel lines 



Met ut infinity. .11. the plane may be used instead of the focus. 



132. From the polar equation of a conic to trace the curve. Suppose 



oppose equation [.V*J represents an hyperbola. When $ = 0^ 



p as . hence p b negative; as $ increases, cos $ decreases, and co*f 



beoomes numerically more and more nearly equal to 1 ; therefore p re- 

 negative and be- 

 comes larger an< 

 , co when 



l> n when 



say; as $ increases 

 through thb value, p 

 beoomes + oo and then 

 decrease*, but remains . , 



equal to / h.-n $ = W; as $ increases through 90 to 180 s , p remains 

 positive, l>ut continues to decrease, reaching its smallest value, viz. 

 when $ = 180; as $ increase* from ISO 9 to 270", p remains 

 and increases from - to /; a* $ increases from l.'7u j to 



- a, p increase* from /to + oo ; a* $ increase* through 300* - a, p 

 ao ; and finally, as $ increases from 300* - a to 300*, p r*- 



iains negative, but decreases numerically, reaching the value 



whan I 



