S40 Mr. Stubbs on a netio Method in Geometty. 



A sin 2 fl B sin j cos 9 Ccos 2 9 D sin 9 E cos 9 



+ F = 0, 

 or multiplying by r 4 , 



A ? J2 sin 2 9 + B r' 2 sin 9 cos 9 + C i n cos 2 9 + D r' 3 sin 9 

 + E; J3 cos9 -f F/ J4 = 

 is the polar equation of the inverse conic section, its equation 

 in rectangular coordinates being 



A y* + Bx y + C x* + D^ (.r 2 + y 2 ) + E x (.r 2 + f) 



+ (x*+f)' 2 = 0. 

 1. If the focus be the pole, the distance from any point P to 

 the focus is to its distance from the directrix in a constant ratio 

 as e to 1. 



Fig. 3. Fig. 4. 



Now if we invert the line D O into a circle and the curve into 

 the inverse focal ellipse whose equation is r = k(l — ecos w), 

 we can construct the focal inverse ellipse by a circular direc- 

 trix; (in fig. 4) let S be the pole (which is the focus), S O any 

 circle passing through S ; (in fig. 3) produce S P to T so that 



S T = -J^ , and S D to R so that RS= ^-^ (which is the 



o P o J J 



same as to invert the curve and directrix) from the similar tri- 

 angles R T S, DPS RT: RS::DP:PS::l:e 



v TRS = DPS = PSV. 

 Hence the circle circumscribing R S T is a tangent to S V at 

 S ; from this may be constructed the inverse focal conic section ; 

 for (in fig. 4) draw any chord S R to meet the circular direc- 

 trix S R O, through S and R describe a circle S R T tangent 

 to S V (the axis) at S, and inflect R T in a given ratio to R S, 

 T is a point in the curve. As the cardioide is only a particu- 

 lar case of the focal conic section, this construction applies to 

 it, making the ratio that of equality. 



From the focal properties of conic sections we may deduce 

 by inversion the following properties of the curve whose equa- 

 tion is r = k (1— e cos «.). 



In the parabola the perpendicular from the focus on the tan- 

 gent meets it in the vertical tangent. Hence in the cardioide, 

 if a polar circle be drawn tangent to the curve, the locus of 



