RIGGS: ON pascal's LIMACON AND THE CARDIOID. 



91 



If two conies have a common 

 focus, two of their common chords 

 will pass through the point of in- 

 tersection of their directrices. 



Two conies have a common fo- 

 cus about which one of them is 

 turned; two of their common 

 chords will touch conies having 

 the fixed focus for focus. 



Two conies are described hav- 

 ing the same focus, and the dis- 

 tance of this focus from the cor- 

 responding directrix of each is 

 the same; if the conies touch one 

 another, then twice the sine of half 

 the angle between the transverse 

 axes is equal to the difference of 

 the reciprocals of the eccentrici- 

 ties. 



If a circle of a given radius pass 

 through the focus (S) of a given 

 conic and cut the conic in the 

 points A, B, C, and D; then 

 SA. SB. SC. SD is constant. 



A circle passes through the fo- 

 cus of a conic whose latus rectum 

 is 2 1 and meets the conic in four 

 points whose distance from the 

 focus are r^, r.-,, r^, r^, then 



r, ^ r, ^ r3 ^ r, 1 



Two points P and Q are taken, 

 one on each of two conies which 

 have a common focus and their 

 axes in the same direction, such 

 that P S and Q S are at right 



If two limacons have a common 

 node, two nodal circles passing 

 each through two points of inter- 

 section of the limacons, will pass 

 through the point of intersection 

 of their base circles. 



Two limacons have a common 

 node about which one of them is 

 turned; two of the nodal circles 

 through two of their points of in- 

 tersection will envelope limagons 

 having fixed node for node. 



If two limacons are described 

 having the same node and base 

 circles of the same diameter, and 

 if the limacons touch each other, 

 then twice the sine of half the an- 

 gle between the axes of the lima- 

 cons is equal to the difference of 

 the eccentricities. 



If a circle of a given radius 

 pass through the node (S) of a 

 given limacon and cut it in A, 

 B, C, and D; then 

 I 



SA. SB. SC. SD 

 is constant. 



A circle passes through the node 

 of a limacon whose latus rectum 

 is 2 1, meeting the curve in four 

 points whose distances from the 

 node are r^, r„, r.^, r^, then 



'■l+'"2+'"3+l'4^2l 



Two points P and Q are taken 

 one on each of two limacons which 

 have a common node and their 

 axes in the same direction, such 

 that P S and Q S are at right an- 



