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KANSAS UNIVERSITY QUARTERLY. 



angles, S being the common focus. gles, S being the common node. 



Then the tangents at P and Q meet Then the nodal tangent circles at 



on a conic the square of whose P and Q intersect on a limacon 



eccentricity is equal to the sum of the square of whose eccentricity 



the squares of the eccentricities of is equal to the sum of the squares 



the original conies. of the eccentricities of the origi- 

 nal limacons. 



A series of conies are described 

 with a common latus rectum; the 

 locus of points upon them at 

 which the perpendicular from the 

 focus on the tangent is equal to 

 the semi-latus rectum is given by 

 the equation 



p^ — r cos 2X 



If POPj be a chord of a conic 

 through a fixed point O, then will 

 tan 1/2 Pj SO tan >^PSO be a 

 constant, S being the focus of the 

 conic. 



Conies are described with equal 

 latera recta and a common focus. 

 Also the corresponding directrices 

 envelop a fixed confocal conic. 

 Then these conies all touch two 

 fixed conies, the reciprocals of 

 whose latera recta are the sum and 

 difference respectively of those of 

 the variable conic and their fixed 

 confocal, and which have the same 

 directrix as the fixed confocal. 



Every focal chord of a conic 

 is cut harmonically by the curve, 

 the focus, and the directrix. 



The envelope of circles on the 

 focal radii of a conic as diameters 

 is the auxiliary circle. 



If a series of limacons are de- 

 scribed with the same latus rectum, 

 the locus of points upon them at 

 which the diameter of the nodal 

 tangent circle is equal to the 

 semi-latus rectum, is given by the 

 equation 



pr = — cos 2JC 



If P O Pj be a nodal circle of a 

 limacon passing through a fixed 

 point O, then will tan. J/2 P^ S O 

 tan. }4 P S O be a constant, S be- 

 ing the node. 



Limacons are described with 

 equal latera recta and a common 

 node. Also the director circles 

 envelop a fixed limacon having a 

 common node. Then these lima- 

 cons all touch two fixed limacons 

 whose latera recta are the sum and 

 difference respectively of the re- 

 ciprocals of the variable limacon 

 and of the fixed limacon, and 

 which have the same base circle 

 as the fixed limacon. 



Every nodal chord of a lima- 

 con is bisected by the base circle. 



The envelope of the perpendic- 

 culars at the extremities of the 

 nodal radii of a limacon is a circle 

 having for the diameter the axis of 

 the limacon. 



