TWEXTY-SrXTH ANXUAL MEETIXG. 23 



tral properties"; that is, are dependent vipon the center of the conicoid. The 

 term "central sphere," analogous to "central ch-cle," will be used to denote the 

 inverse of a plane; it is a sphere passing through the origin or center of inversion. 



1. All plane sections of a conicoid are conies. 



2. All central plane sections of a pedal surface are plane pedal curves. 



1. The equation of the tangent plane at any point of the conicoid, 



«.r^ + bi/^ + cz' -{- d = 0, is 

 a.rx' + bi/y' + czz' -{- d= 0. 



2. The equation of a central tangent sphere at any point of the pedal surface, 



ax- + btj- + cz'' + d {x- + y^ + z^)- = 0, is 

 axx' + byy' + czz' + d {x- + y^ -\- z'^) = 0. 



1. The condition that the plane Ix + niy ■-\- nz -\- 2^ = shall touch the above 



. . ^ . P ni- n- p^ 



conicoid IS — -f -[- — + — = 0. 



a b c d 



2. The condition that the central tangent sphere Ix + my -{- nz -\- 2^ {x- -{- y"^ 

 -f- 0- ) = shall touch the above pedal surface is, 



P m' n^ »^ 

 a b o d 



1. The asymptotic cone to the conicoid given by equation (23) is 



x"^ y- z- 



— ± — ± — = 0. 

 a^ b- c^ 



2. The tangent cone at the origin to the pedal surface given by equation (24) is 



.T- y'^ z'^ 

 o. b' c- 



1. The condition that the conicoid shall be one of revolution is (6 — a) (c — a) 



2. The condition that the pedal surface shall be one of revolution is {b — a) 



(e — a) = 0. 



1. The sum of the squares of the reciprocals of any three diameters of an 

 ellipsoid which are mutually at right angles is constant. 



2. The sum of the squares of any three diameters of a pedal surface of the 

 "first kind" which are mutually at right angles is constant. 



1. The locus of the point of intersection of three tangent planes to the conicoid 

 given by equation (23) which are mutually at right angles is a;^ + ^/^ + ^^ = o,'^ ± 

 b- + C-. This is the director sphere of the conicoid, and is real in the case of an 

 ellipsoid ; in the other cases it depends upon the values of a, b, and c. 



2. The locus of the point of intersection of three central tangent spheres to 

 the pedal surface given by equation (2i) which cut one another orthogonally is 



3-2 _j_ 2^2 _|_ 2.2 _ _ This may be called the director sphere of the pedal 



a'^ + 6^ + e- 

 surface, and is always real in the case of a pedal surface of the "first kind." 



1. The locus of the foot of a perpendicular from the center upon any tangent 

 plane to a conicoid is a pedal surface. 



2. The locus of the extremity of the "central" diameter of any "central" 

 sphere tangent to a pedal surface is a conicoid. [By "central diameter" is 

 meant the diameter which passes through the origin.] 



1. Any tangent plane to the asymptotic cone of a conicoid meets the coni- 

 coid in two parallel straight lines, equidistant from the center. 



