24 KANSAS ACADEMY OF SCIENCE. 



2. Any tangent plane to the cone tangent at the origin to a pedal surface 

 meets the surface in two equal circles in directum. 



x'^ y'^ z'- 



1. The area of any central plane section of an ellipsoid given by — ■ + — -f — 



a^ h- c- 



^ ab c , . , , . . . ^ , 



= 1 is • , , the cutting plane bemg given by the equation Ix 



(a^ P -\-b'' m^ -\- c^ n^ ) ' 



4- my -\- nz =:0. 



2. The area of any central plane section of a pedal surface of the " first kind " 



given by ^ + ^ + ^ = (a:^ + Z/' + ^'Y; is-Jabc)-'' [{b' + e'^) a^ P + (a' + c' ) 



a^ 0^ c- 2 



^2 ^ _j_ (^(2 _|_ 52^ g2 ^2 j^ ^YiQ cutting plane being given by the equation Ix -\- my 

 + nz = 0. 



1. If central plane sections of an ellipsoid be of constant area, their planes 

 touch a cone of the second degree. 



2. If central plane sections of a pedal surface of the "first kind " be of con- 

 stant area, their planes touch a cone of the second degree. 



The six central planes cutting circular sections from the conicoid given by 

 equation (23) are given by the three pairs of equations 

 1 





■ -0 I —^1 .->. 52 J P ^g2 52 J (28) 



1^ a^ e^ J yh 



Taking the case of the ellipsoid, the two systems of real circular sections are 

 given by the equations 



±_±y+Ji '^^ -T^ n.i 



b' a-] — yc 



where p is the perpendicular distance from the origin upon the cutting plane. 

 Since a circle in space may always be considered as the intersection of a sphere 

 by a plane, it follows that the inverse of a circle from any point is also a circle. 

 Hence, the inverse of a system of circular sections of a conicoid is a system of 

 circular sections of a pedal surface. And it is seen from equations (27) and (28) 

 above, that there are two such systems of real circular sections in each of the 

 three pedal surfaces. In the case of a pedal surface of the "first kind," these 

 sections all lie on the system of central spheres which are bisected by the xz 

 plane in the circles. 



^ 6^ f^2 J (^ (j2 ^2 j 1^ q2 ^f2 ^ 



When the sphere given by equation ( 29 ) is tangent to the surface, the point 

 of tangency may properly be called an umbilicus, corresponding to an umbilicus 

 in the conicoid. Hence the pedal surface has four umbilici, that is, points at 

 which tangent planes will cut out infinitely small circles. 



Following are a few additional theorems on circular sections: 



1. In a conicoid, any two circular sections of opposite systems are on a sphere. 



2. In the pedal surface, any two circular sections of opposite systems are on 

 a sphere. 



1.. If the squares of the axes of an ellipsoid are in arithmetical progression 

 the umbilici lie on the central circular sections: if they are in harmonic progres- 



