20 KANSAS ACADEMY OF SCIENCE. 



tangent circles to a lemniscate meet the tangents at the origin are in directum 

 with the "central" circle joining the points of contact of the two "central" 

 tangent circles. 



1. The asymptotes of an hyperbola coincide with the diagonals of the rectan- 

 gle contained by the transverse and conjugate axes. 



2. The tangents to a lemniscate through the origin coincide with the diago- 

 nals of the rectangle formed by the intersections of four " central " circles whose 

 d ameters are the four semi-axes of the curve respectively. 



And in general, the inverse, with respect to the center, of a system of conies 

 given by the equation, 



a-'x^ + if " =j, (22) 



1 — e^ 



is the system of pedal curves, whose equation is, 



a' x^ + 2/2 -=( x^ + 2/2 )^ (3) 



i — e^ 



These curves belong in the general class of curves designated by the name of 



bicircular quartics. 



1. The condition that the line whose equation is y =■ nix + c shall touch the 



m^ 1 — e^ 

 conic given by equation (22) is e^ = — — + ;; — 



2. The condition that a "central" circle whose equation is y = mx + c (a;^ -f 



m} -\- 1 — e^ 

 y"^) shall touch the pedal curve given by equation (3), is c^ = -^ 



1, The equation of a tangent line at any point of the conic is ci? xx' + &^ yy' 



2. The equation of the " central" tangent circle at any point of the pedal is 



a^ xx' -j- b^ yy' = {x^ -(- y^). 



In this manner many more sets of corresponding equations in the two systems 

 of curves might be given; but the above examples are sufficient to show their 

 general relations. 



Following is a list of theorems relating to confocal conies and confocal pedal 

 curves: 



1. The equation of a system of confocal conies is 



3-2 y2 



2. The equation of a system of confocal pedal curves is 



a'j^fi h^ + a 



1. Two conies of a confocal system pass through a point. One of these conies 

 is an ellipse and the other an hyperbola. 



2. Two pedal curves of a confocal system pass through a point. One of these 

 curves is an oval and the other is a lemniscate. 



1. One conic of a confocal system and only one will touch a given straight 

 line. 



2. One pedal curve of a confocal system and only one will touch a given " cen- 

 tral " circle. 



1. Two confocal conies cut one another at right angles at all their common 

 points. 



