TWEXTY-^siXrH AXXUAL MEETING. 27 



as ft increases: so that the limiting form of the surface when /i = cc is a point 

 sphere at the origin. 



If /? is negative and less than e', the surface is a pedal of the "first kind," 

 which recedes more and more from the spherical form, like an hour-glass, until, 

 when /i = -c'\ it folds over on itself, as it were, and embraces all of the xi/ plane 

 exterior to the oval whose equations are 



~V--^ + T^-T = (•^•' + Pr, 2 = 0. (31) 



When /i is between -e^ and -6^, the surface is a pedal of the "second kind " 

 which approaches as a limit, when /? is very nearly equal to -&' that portion of> 

 the xy plane inclosed by the oval given by equation (31); and when /? is very 

 nearly equal to -b^, the surface is very nearly coincident with that part of the 

 xz plane which is exterior to the lemniscate whose equations are 



a'-b- c^-b' 

 If H is between -6- and -a', the surface is a pedal of the "third kind." When 

 fi is very nearly equal to -6-, the surface is very nearly coincident with that por- 

 tion of the X z plane which is inclosed by the lemniscate given by equation (32). 

 When y =-a-, the surface becomes imaginary; but analytically it is all that por- 

 tion of the y z plane exterior to the imaginary curve 



77^— + T^ = (y^ + ^^)'> X = 0. (33) 



And when /i is between — a'- and — oc the surface continues imaginary. The two 

 pedal curves given by equations (31) and (32) may be called the focal pedal curves 

 of the system, corresponding in their properties to the focal conies of a system of 

 confocal conicoids. Following is a list of corresponding theorems on confocal 

 conicoids and confocal pedal surfaces : 



1. Three conicoids of a confocal system pass through any given point : one of 

 them is an ellipsoid, one an hyperboloid of one sheet, and one an hyperboloid of 

 two sheets. 



2. Three pedal surfaces of a confocal system pass through any given point : 

 one of them is a pedal surface of the "first kind," one is of the "second kind," 

 and one is of the "third kind." 



1. One conicoid of a given confocal system will touch any plane. 



2. One pedal surface of a confocal system will touch any central sphere. 



1. Two conicoids of a confocal system will touch any straight line. 



2. Two pedal surfaces of a confocal system will touch any central circle. 



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

 points. 



2. Two confocal pedal surfaces cut one another orthogonally at all their com- 

 mon points. 



1. If a straight line touch two confocal conicoids, the tangent planes at the 

 points of contact will be at right angles. 



2. If a central circle touch two confocal pedal surfaces, the central tangent 

 spheres at the points of contact will cut one another orthogonally. 



1. The difference of the squares of the perpendiculars from the center on any 

 two parallel tangent planes to two given confocal conicoids is constant. 



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



