22 



KANSAS ACADEMY OF SCIENCE. 



(23) 



(24) 



Let the general equation of a system of central conicoids be 



a^ b^ g'- 

 where a^h^c, then that of the system of inverse surfaces will be 



a'' V c- 



This is the first pedal surface from the center of the conicoid given by the equa- 

 tion cC- x^ + &^ 2/- + g'' z- = 1, (25) 

 which in turn is the polar reciprocal with respect to the center, of the original 

 conicoid. Hence, in general, the inverse of a conicoid from the center is the 

 pedal surface of its polar reciprocal with respect to the center. 



Resuming equation (24), it is evident that the three semi-axes of the surface 



are — , — ^^— and — ^-; and that the surface is symmetrical about any coordinate 

 ah G 



plane. The eccentricities of the sections made by the xy, xz, yz planes respect- 

 ively are 



a^ + If' a? + c^ + h- + c^ 



ez = : — , ey^ = ; — , e^ = 



a- 



(26) 



d- a' + b'- 



When the signs in equation (24) are taken all positive, the surface is the pedal 

 of an ellipsoid, and so far as this paper is concerned, may conveniently be called 

 a pedal surface of the "first kind." It varies from a smooth to an indented 

 surface, the transition occurring when either ez or By or each becomes equal to 

 l/y. The two limiting forms are, a sphere when Cz = Cy = e^ = 0, and two 

 spheres tangent at the origin to the xy plane when ey = 6^;.= 1. 



Z' 



When the sign of — in 



equation (24) is taken nega- 

 tive, the surface is the first 

 pedal from the center of an 

 hyperboloid of one sheet: 

 and may be called a pedal 

 surface of the "second kind." 

 The general form of the 

 surface is represented by 

 fig. 4. 



When the signs of 



Z- 7/^ 



— and '—, in equation (24) are taken both negative, the sur- 

 c- b- 



face is the pedal from the cen- 

 ter of an hyperboloid of two 

 sheets; and may be called a 

 pedal surface of the "third 

 kind." The general form of 

 the surface is shown in fig. 5; 

 it is much the shape of two 

 tops placed point to point. 



Following is a list of some 

 of the more important theo- 

 rems on the above pedal sur- 

 faces. They were obtained by inverting the properties of conicoids, the center 

 of the conicoid being taken as the center of inversion. As in the case of plane 

 pedal curves, only those properties were used which may be denominated "cen- 



