///' 



poo*0 and jr p in 0, and equation (4) at oiuw rsdneas to 



% lii.-li i-i tberefoTC the required polar equation of the lemui 



) shows that: when 0-0, p a; when *<44*.p has 



I. i.t . | |Miu> values, each of which U smaller than ; when 



= *:. . p - (), ,.r. f the angle which the cunre makes with the initial line 



0<135*. pis imaginary; when 1CT<*< lW f p has 



qual but opposite values, each of which b smaller than a ; and when 



$ a 180*. p a o. Th< M-refore, consists of two otsis meeting to 



( each lyiiiK in the same angle between the asymptote* of the hyperbola 



as does the corresponding branch of thai cunre, and these asymptotes are 



Htnl to the lemniaoate at the point 



lf the two poinU F, and P be so located that 



(r t y) be any point on the lemnJMHSjn, 





Hence the lemniscate may be defined as the locos of a point which 

 mores so that the product of its distances from two fixed points b con- 

 stant, and equal to the square of half the distance between the find 



foot- not* . 



Thin <). tu.m..u *.f the curve easily leads to the equation already 

 <> enables one to readily construct the cunre thus: 



>. describe an arc; then, with 



center, an. I ;i tlnr.l as radius, 6>> 



anotlii-r .1- intersection 5 is a point on the 



1 as many points as desired may be constructed in the same 

 way 



