TWENTY-SIXTH ANNUAL MEETING. 21 



2. Two confocal pedal curves cut one another at right angles at all then- com- 

 mon points. 



1. The difference of the squares of the perpendiculars drawn from the center 

 on any two parallel tangents to two given confocal conies is constant. 



2. The difference of the squares of the reciprocals of the diameters of any two 

 "central" tangent circles in directum with one another, to two given confocal 

 pedal curves is constant. 



1. If a tangent to one of two confocal conies be perpendicular to a tangent to 

 the other, the locus of their point of intersection is a circle. 



2. If a " central " circle tangent to one of two confocal pedal curves cut ortho- 

 gonally a "central" tangent circle to the other, the locus of their point of inter- 

 section is a circle. 



1. From any point :Z'the two tangents TP, TP' are drawn to one conic, and 

 the two tangents TQ, TQ to a confocal conic ; then will the straight lines (^P, 

 Q^P make equal angles with the tangent at P. 



2. From any point I' the two "central" tangent circles TP, TP' are drawn 

 to one pedal curve, and the two "central" tangent circles TQ, TQ' to a confocal 

 pedal curve; then will the "central" circles QP, Q'P cut the tangent at Pat 

 equal angles. 



1. TP, TQ are tangents one to each of two fixed confocal conies; then, if the 

 tangents are at right angles to one another the line PQ will always touch a third 

 confocal conic. 



2. TP, TQ are two "central" tangent circles one to each of two confocal 

 pedal curves; then, if these circles cut one another orthogonally, the " central " 

 circle PQ will always be tangent to a third confocal pedal curve. 



1. If an ellipse have double contact with each of two confocal conies, the tan- 

 gents at the points of contact will form a rectangle. 



2. If an oval (concentric) have double contact with each of two confocal ovals,. 

 the "central" tangent circles at the points of contact will cut one another orthog- 

 onally. 



1. A triangle circumscribes an ellipse and two of its angular points lie on a 

 confocal ellipse; then will the third vertex lie on another confocal ellipse. 



2. Three " central " circles are tangent internally to an oval, and two of their 

 points of intersection lie on a confocal oval; then will the third point of intersec- 

 tion lie on another confocal oval. 



, In figure 3 is shown the general appear- 



• ance of a system of confocal pedal curves, 



I consisting of ovals and lemniscates. 



/ 



, If the foregoing method of inversion be 



applied to a system of central conicoids, it 

 y gives rise to a system of surfaces of the 



fourth degree. These surfaces bear to the 

 ■plane pedal curves just considered many of 

 ' f^ ' ' 5C the relations that conicoids bear to plane 

 conies They have three general forms ac- 

 cJ^'l^. 3- cording as the sections made by the coordi- 



nate planes are ovals or lemniscates, just as the central conicoids have the three 

 forms of ellipsoid, hyperboloid of one sheet and hyperboloid of two sheets. 



X 1 ^y 



