23 KANSAS ACADEMY OF SCIENCE. 



central tangent spheres in directum to two given confocal pedal surfaces is con- 

 stant. 



1. The locus of the point of intersection of three planes mutually at right 

 angles, each of which touches one of three given confocal conicoids, is a sphere. 



2. The locus of the point of intersection of three central spheres cutting one 

 another orthogonally, each of which touches one of three given confocal peda^ 

 surfaces, is a sphere. 



1. The locus of the umbilici of a system of confocal ellipsoids is the focal hy- 

 perbola. 



2. The locus of the umbilici of a system of confocal surfaces of the "firft 

 kind " is the focal lemniscate. 



1. If two concentric and co-axial conicoids cut one another everywhere at 

 right angles they must be confocal. 



2. If two concentric and co-axial pedal surfaces cut one another everywhere 

 at right angles they must be confocal. 



1. Three confocal conicoids meet in a point, and a central plane of each is 

 drawn parallel to its tangent plane at that point, then, one of the three sections 

 will be an ellipse, one an hyperbola, and one imaginary. 



2. Three confocal pedal surfaces meet in a point, and a central plane of each 

 is drawn tangent at the origin to the central sphere tangent at that point, then, 

 one of the three sections will be an oval, one a lemniscate, and one imaginary. 



1. If three lines at right angles to one another touch a conicoid, the plane 

 through the points of contact envelop a confocal conicoid. 



2. If three central circles at right angles to one another touch a pedal sur- 

 face, the central sphere through the points of contact will always touch a con- 

 focal pedal surface. 



Many other theorems relating to the curves and surfaces under discussion 

 might be obtained; but those already given show their principal properties, and 

 more would merely add comparatively uninteresting details. 



